A multi-scale model of virus pandemic: Heterogeneous interactive entities in a globally connected world
Nicola Bellomo, Richard Bingham, Mark A.J. Chaplain, Giovanni Dosi, Guido Forni, Damian A. Knopoff, John Lowengrub, Reidun Twarock, Maria Enrica Virgillito
aa r X i v : . [ q - b i o . P E ] J un A multi-scale model of virus pandemic:Heterogeneous interactive entities in a globallyconnected world
Nicola BellomoUniversity of Granada, Departamento de Matemática Aplicada, SpainIMATI CNR, Pavia, Italy, and Politecnico of Torino, ItalyRichard BinghamDepartments of Mathematics and BiologyYork Cross-disciplinary Centre for Systems AnalysisUniversity of York, U.K.Mark A.J. ChaplainMathematical Institute, School of Mathematics and StatisticsUniversity of St Andrews, Scotland, UKGiovanni DosiInstitute of EconomicsEMbeDS, Scuola Superiore Sant’Anna, Pisa, ItalyGuido ForniAccademia Nazionale dei Lincei, Roma, ItalyDamian A. KnopoffCentro de Investigación y Estudios de Matemática (CONICET)Famaf (UNC), Córdoba, ArgentinaJohn LowengrubDepartment of Mathematics, University California Irvine, USA eidun TwarockDepartments of Mathematics and BiologyYork Cross-disciplinary Centre for Systems AnalysisUniversity of York, U.K.Maria Enrica VirgillitoInstitute of EconomicsEMbeDS, Scuola Superiore Sant’Anna, Pisa, ItalyJune 9, 2020 Abstract
This paper is devoted to the multidisciplinary modelling of a pandemicinitiated by an aggressive virus, specifically the so-called
SARS–CoV–2 Se-vere Acute Respiratory Syndrome, corona virus n.2 . The study is developedwithin a multiscale framework accounting for the interaction of different spa-tial scales, from the small scale of the virus itself and cells, to the large scaleof individuals and further up to the collective behaviour of populations. Aninterdisciplinary vision is developed thanks to the contributions of epidemi-ologists, immunologists and economists as well as those of mathematicalmodellers. The first part of the contents is devoted to understanding thecomplex features of the system and to the design of a modelling rationale.The modelling approach is treated in the second part of the paper by showingboth how the virus propagates into infected individuals, successfully and notsuccessfully recovered, and also the spatial patterns, which are subsequentlystudied by kinetic and lattice models. The third part reports the contributionof research in the fields of virology, epidemiology, immune competition, andeconomy focused also on social behaviours. Finally, a critical analysis isproposed looking ahead to research perspectives.
Key words:
COVID-19, living systems, immune competition, complexity,multiscale problems, spatial patterns, networks
AMS Subject Classification:
Motivations and plan of the essay
The onset of the
SARS–CoV–2 virus responsible for the initial
COVID-19 outbreakand the subsequent pandemic, has brought to almost all countries across the globehuge problems affecting health, safety/security, economics, and practically allexpressions of collective behaviour in our societies. Data are constantly updatedand presented at various dedicated websites [44, 69, 79, 121]. A significantpercentage of societies and governments believed this to be a so-called blackswan [123] event for our society, including a number of scientists. However, thisevent is definitely not a black swan, although it has already had a great impact allover the world. Indeed, this event should have been predictable (and indeed waspredicted by a few [8, 63]) but many of our societies appear to be unprepared totackle this problem.If we look at COVID-19 from the view point of applied mathematicians, somepreliminary remarks concerning how best to approach modelling such a pandemic,which include an interdisciplinary vision and the role of mathematical models inscience and our society, are as follows:• The modelling approach should go far beyond deterministic population dy-namics, since individual reactions to the infection and pandemic events areheterogeneously distributed throughout the population. Spatial dynamicsand interactions are an important feature to be included in the modellingapproach, since the dynamics are generated by nonlocal interactions andtransportation devices.• The modelling ought to be developed within a multiscale vision, as thedynamics of individuals depend on the dynamics at smaller scales insideeach individual by the competition between virus particles and the immunesystem. It is clear that applied mathematicians cannot tackle the modellingproblem by a stand-alone approach – an interdisciplinary vision is necessarythrough mutually enriching and beneficial interactions with scientists inother fields including virology, epidemiology, immunology and biology ingeneral.• The approach adopted and described in this paper looks firstly for a modellocal-in-space accounting for the infection dynamics and, subsequently forthe competition inside each individual, between the proliferating virus andthe immune-system specific to the individual. Subsequently the approachfocuses on collective behaviour. Finally, the mathematical description of3patial propagation is studied, since we are aware that this is not a problemof diffusion, but that of directed propagation with finite speed (which can andshould be related to endogenous communication networks where individualsmove).• The scope of such a research project should not be confined only to “bio-logical and medical sciences”, but also be addressed to wider aspects of andother communities in our society. This requirement is motivated not onlyby the strong impact that a virus pandemic can have on a society, but alsoto the need that science should look forward with predictive aims. On theirown, mathematical models do not and cannot solve the problems of biology,medicine and economics. However, once refined and informed by empiricaldata, they can produce insightful provisional simulations which can evenuncover dynamics which were not previously observed (cf. emergent be-haviour). Hence mathematical models can and should also be viewed as atool to generate dialogue and wider communication between the hard andapplied sciences. This dialogue can in turn lead to a perspective on andinsight into possible future events.• Since the modelling approach aims and intends to capture the complex fea-tures of living systems, this effort often requires new mathematical methodsand techniques, even new mathematical theories. This requirement was al-ready posed by Hartwell [70], accounting for multiscale problems as hadalready appeared in the celebrated essay by Schrödinger [118], but the gen-eral premise and vision goes back over a century to Hilbert [72, 73, 74].Indeed, the words of Hilbert opening his lecture “
Mathematische Probleme ”delivered before the International Congress of Mathematicians in Paris,1900, are worth recalling here since they are prophetically apposite:
Who of us would not be glad to lift the veil behind which the futurelies hidden; to cast a glance at the next advances of our scienceand at the secrets of its development during future centuries?What particular goals will there be toward which the leadingmathematical spirits of coming generations will strive? What newmethods and new facts in the wide and rich field of mathematicalthought will the new centuries disclose?
As noted above, the contribution of mathematics to model complex naturalphenomena can be effective only if sufficiently well integrated with the work,4nowledge and insight of scientists active in virology, epidemiology, immunologyand biology. Once integrated in this fruitful interdisciplinary manner, the bestmathematical models (e.g. through predictive computational simulations) arecapable of presenting a broad vision of a range of possible system outcomes whichcan be used towards both therapeutic actions and, in the case of a pandemic,confinement strategies to control the dynamics of an infection. The recent eventsrelated to COVID-19 have shown that the pandemic has exerted a huge impacton societies concerning both their social behaviour and economic strategies. Thispaper accounts for some specific, but highly relevant, features of the pandemicand aims to construct a framework which can hopefully contribute to developingfuture strategies by which governments can fight to reduce and possibly controlthe impact of such an event. Bearing in mind all of the above, the contents of ourpaper, which can be viewed in three parts, can be presented as follows.Part 1 is devoted to understanding the complex features of the system underconsideration whose biological dynamics are described in Section 2. A first steptowards the design of a modelling rationale is proposed in the same section tocontribute to defining the general mathematical framework deemed to provide theconceptual basis for the derivation of the models.Part 2 focuses on modelling topics. Firstly the case of spatial homogeneity isstudied in Section 3, where a model is derived accounting for the heterogeneity ofthe immune defence of individuals and describing how the virus propagates intoinfected individuals. Simulations enlighten the main features of the dynamics aswell as the role of possible therapeutic actions. Subsequently, the more realisticcase of the spatial development of the disease is developed in Section 4, accountingfor recent studies on the modelling of crowd dynamics [2, 9]. Different featuresof spatial dynamics and spread are studied to show how the infection propagatesthrough space.Part 3 reports on the contribution of research in the fields of virology (inSection 4), immune competition (in Section 5), and economics, focused also onsocial behaviour (in Section 6). These sections propose a critical analysis whichaccounts for the modelling achievements and indicates key modelling targets.Section 7 looks ahead to research perspectives suitable for deeper consideration ofthe various initiatives proposed in the preceding sections. This section providesan answer to the general key objective described below by tracing the guidelinesof a forward look to a future vision of the systems approach adopted here.Summary statement:
The key target of this paper consists in designing a multiscale mod- lling approach and framework suitable for designing and producinga simulation device with the capacity to explore crucial aspects of thespread of a pandemic. More specifically, this should be capable of ex-ploring the different scenarios corresponding to possible strategies tocontrol a virus pandemic, so that crisis managers can select the mostappropriate or most effective choices towards effective control withinthe shortest timeframe. At a later stage, once reliable data are avail-able, these data will be properly selected and stored in a database,and then the simulation device can become predictive, while the modelcan be further refined and developed to include additional details ofthe virus such as mutations, selection and evolution. This section provides a general introduction to the technical and formal contents ofour paper developed in Sections 3–7. The presentation is in three parts: first, a briefhistorical introduction to epidemiology is delivered, subsequently a concise reporton the phenomenology of the biological system under consideration is presented,and finally a rationale towards the modelling approach is brought to the reader’sattention in view of the mathematical formalization proposed in Section 3.Firstly we provide a concise introduction to the contribution of mathematicsto epidemics, subsequently an outlook to the phenomenological behaviour of thesystem is presented enlightening some heterogeneity and multiscale features of thedynamics, lastly we focus on the key features of the rationale towards the modellingapproach developed in the next sections.The mathematical modelling of infectious diseases goes back over 250 yearsto Daniel Bernoulli who developed a model to predict the number of deaths due tosmallpox and the effect on mortality of inoculation [23]. This problem was alsoconsidered by D’Alembert who produced an alternative model to Bernoulli [48],in fact a critique of Bernoulli’s approach which led to a heated debate betweenthe two [55]. Our modern day subject of “theoretical epidemiology” goes backalmost 100 years to the seminal paper of Kermack and McKendrick in 1927 [84].Kermack and McKendrick proposed and developed the first “compartmental”model of disease dynamics and disease spread by considering a population ofindividuals sub-divided into three separate classes - susceptible, infectious andrecovered - with the transmission of the disease/infection being dependent uponthe number of interactions between individuals and the underlying rate of infection.6his is the first
SIR model of disease transmission. In formulating their model inthis way, the so-called reproduction number or R was defined and introduced.In the mid-1970s, interest in the subject was rekindled [10] with both theoreticaldevelopments of the models and systems [54] and prescient applications to realepidemic outbreaks such as the cholera epidemic in Bari, Italy, in the early 1970sby Capasso [33, 34]. Modelling the spatial spread of epidemics, although notmuch focussed on in general (using reaction-diffusion equations and the theory oftravelling waves [35]), has been applied to the spread of rabies [81] which led to anestimate of the speed of the invasive wave of infection. Applied to historical datafrom Europe, this approach has also been used to estimate the spread of the blackdeath/bubonic plague through Europe in the 14th century [104]. Subsequent workby, e.g. Diekmann and others, has formalised much of the theory behind epidemicmodels and introduced structured population models [52, 53].In the 1980s and into the 1990s, with the onset of AIDS/HIV, work by Anderson& May and others [4, 5, 6, 101, 105, 106] highlighted the role of mathematicalmodelling in predicting possible outcomes of disease spread in large populations.Building on this, ground-breaking work by Perelson and others [111, 112, 113]on viral load, virion clearance rate and the role of the immune system/responsehelped lead to breakthroughs in treatment for AIDS/HIV.Applications of predictive and insightful mathematical modelling to other dis-eases include the work of Grenfell and co-workers which has helped to quantify thedynamics of measles epidemics, in a range of settings and the impact of vaccinationstrategies against measles and other childhood infections [27, 28, 66, 60]. Prior tothe current COVID pandemic, other applications of epidemiological modelling todisease outbreaks over the past 30 years or so have included: bovine spongiformencephalopathy (BSE), Foot-and-Mouth disease (FMD), SARS, MERS, and theflu pandemic among others [71].The virus, Corona-virus disease 2019 (COVID-19) is caused by the infectionof the SARS–CoV–2 virus, a corona-virus. The so called Corona-viruses are partof a large family of viruses that cause illness ranging from the common cold tomore severe diseases including Severe Acute Respiratory Syndrome (SARS) andCOVID-19. The
SARS–CoV–2 virion is made up of four structural proteins knownas Spike, Envelope, Membrane and Nucleocapsid.The World Health Organization (WHO) upgraded the state of
SARS–CoV–2 infection from epidemic to pandemic in March 2020. Apparently, the epidemicfirst started in China, followed by South Korea and then Italy. Progressively many(almost all) countries around the world have have been invaded and have reacted byimplementing different strategies including the imposition of mobility and physical7ocks-downs and border limitations. The behaviour of all countries to counteractthe pandemic appears to be quite heterogeneous, as it is the national organizationwhich cares about infected patients. This heterogeneity is also confirmed by thediffusion of the disease inside each country and the distribution of the level of thepathology.As we shall see in the next sections, accounting for heterogeneity of the pop-ulations is not an essential requirement for the modelling approach, but it cancontribute to a deeper understanding towards the contribution that economics andsociology can provide to all decision makers, from crisis managers all the way upto policy makers.Although the knowledge of this topic is still evolving, some specific featurescan be extracted, in view of a deeper analysis to be developed in sections 4–6,with the aim of developing some reasonings on the modelling rationale towardsthe mathematical formalization developed firstly in Section 3 and subsequentlyalso in Section 7. The amount of scientific literature which has appeared in thelast few months after the appearance of Covid-19 is impressive and, in some cases,not easy to follow due to a contrast of view points which can be related to thenon-exhaustive knowledge in the field.Therefore, we need to issue a warning to the reader that we do not claimcompleteness, while we refer to the report [37], which is bimonthly updated andit is available at the website of the Italian Lincei Academy. The description isdelivered in the following, where some of the Items have been extracted from [37].•
Virus transmission: SARS–CoV–2 is mainly transmitted through the respi-ratory route 10-12 via respiratory droplets, up to 1 millimetre in diameter,that an infected person expels when she/he coughs or sneezes. As the virusmultiplies, an infected person may shed copious amounts of it.•
Binding and cell attack:
The large Spike protein forms a sort of crown onthe surface of the viral particles and acts as an anchor allowing the virus tobind to the Angiotensin-Converting Enzyme 2 (ACE2) receptors on the hostcell. After binding, the host cell transmembrane proteases (TMPRLRSS2and Furin) cut the Spike proteins, allowing the virus surface to approach thecell membrane, fuse with it and the viral RNA enter the cell.•
Virus proliferation:
Then, the virus hijacks the cell machinery and the celldies releasing millions of new viruses thus generating a virus infection.COVID-19 starts with the arrival of
SARS–CoV–2 virions to the respiratory8ucosal surfaces of the nose and throat that express high levels of ACE-2receptors on the surface .•
Immune system actions:
When the virus manages to overcome the barrierof the mechanisms and the mucus secreted by goblet cells from a firsteffective reaction, a rapid release of danger signals activates the reaction ofthe host’s innate immunity. Corona viruses are successful at suppressingvarious mechanisms in an immune response, but a protective immunity canbe however developed.This brief description induces a concept of the amount of infection roughlyrelated to the quantity of initial virus corresponding to the initial infection. There-fore, it is wise to account for the heterogeneous level of infections and of theheterogeneous ability of each individual to express his/her immune defence.In addition, the modelling approach should include some ability to predictthe trends of infected individuals towards full recovery and of those who do notsucceed in surviving. The overall modelling should also be deemed to support thestrategic choice of hospitalization of infected individuals based on the estimate ofthe progression of the pathology.Subsequently, the focus moves to the contagion in space , where it occurs bycontact thus creating a population of infected individuals who move in space andpropagate the epidemic in a globally connected world. The role of space can bestudied by treating the following sequential steps of the modelling approach:1.
Dynamics in the case of spatial homogeneity , where the virus is transmittedfrom infected to healthy individuals by short-range interactions. In generalcontagion depends on the infectivity features of each individual and on thedistances they maintain homogeneously in a crowd, namely on the localdensity.2.
Contagion in a crowd , where individuals move in a spatial domain, namelywith the loss of space homogeneity. Contagion depends on the local dis-tance between individuals which depends on time and space. Contagionprobability is somehow related to local density and infectivity.3.
Spatial dynamics in exogenous networks, where individuals move by trans-portation across nodes of a physical network, while the dynamics withineach node follows the dynamics outlined in the preceding Items.9he various reasonings given above indicate that we are studying a complexsystem which presents multiscale features, where the microscopic scale, in shortmicro-scale, corresponds to virus particles and immune cells, which induces thedynamics at the higher scale of individuals who carry an infection (meso-scale),while collective behaviours are observed at the macro-scale of all individuals. Asmentioned previously, an ability to develop an immune defence is heterogeneouslydistributed over the whole population.Heterogeneity and multiscale features address the modelling approach to theso-called kinetic theory of active particles [18] which has already been appliedto various branches of the so-called behavioural sciences [12, 94]. For instance,the immune competition [22], collective learning [30, 31], and the dynamics ofcrowds and swarms [2, 9], where understanding the dynamics of the collectivebehaviour leads to a deeper understanding of the contagion dynamics.The key features of the aforementioned mathematical approach, presented as amathematical theory in [18], can be summarized in the following sequence:1. Subdivision of the overall system into functional subsystems, in short FSs.2. Representation of the state of each FS by probability distributions over themicro-state of interacting entities.3. Deriving a general mathematical framework deemed capable of capturingthe complex features of living systems constituted by very many interactingentities.4. Modelling interactions and deriving specific models by inserting these mod-els into the said mathematical framework.5. Simulation of the dynamics at the micro-scale by computing macroscopicquantities by weighted moments of the probability distribution functionsmentioned in Item 2.An important step of the overall approach is the modelling of interactions whichneeds a multidisciplinary vision as it links knowledge in biology to advanced math-ematical methods as interactions are, generally, nonlinearly additive, nonlocal, andstochastic. This rationale, once transferred into a mathematical framework, goeswell beyond classical SIR models. In addition, the problem is multiscale as eachindividual is a carrier of an internal competition, at the micro-scale, between thevirus and the host immune system. 10igure 1: A crowd with aggregation of multiple groups.Indeed, it is a broad vision although we acknowledge that recent developments,for instance [97] have effectively contributed to a deeper understanding of theCOVID-19 pandemic.
We consider a population of a large number of individuals, where a small fractionis infected and transfers the virus to healthy individuals by short range interactions.In general, contagion depends on the frequency of contacts, on the level of theinfection within each individual, and on the level of physical protection used byindividuals aware of the risk of contagion. The frequency can be related to the dis-tance, generally called social distance , while the dynamics within each individualincludes a competition between the virus and the immune system. Figure 1 showshow, in a crowd, the social distance can differ due to local aggregations. Individu-als move according to certain walking strategies [20], while a special case (usefulnonetheless to model the dynamics of contagion) refers to a crowd homogeneouslydistributed in space.This section is devoted to developing a modelling approach suitable to describe11he dynamics over time and space of the whole population which includes healthy,infected, and recovered individuals. The model is also required to describe thegrowing number of those who could not recover and therefore die. The study isdeveloped along some sequential steps devoted to the following topics: Modellingcontagion under the assumption of spatial homogeneity; Simulations of the conta-gion dynamics linked to individual competition against the undesired host virus;Modelling the contagion dynamics in complex spatial environments. These topicsare treated in the next subsections, while the last subsection is devoted to a criticalanalysis and further developments of the modelling strategy.
We consider a population consisting of N individuals homogeneously distributedin space. A small number of them ε N is initially infected, while ( − ε ) N isconsidered healthy. The general framework supporting the modelling approach isdefined by a selection of key features somehow related to the general descriptiongiven in Section 2 with the additional aid of the technical report [37]. We supposethat modelling contagion dynamics, followed by the competition for survival withineach individual, can be developed according to the following rationale:1. Individuals are viewed as active particles , in short a-particles, namely par-ticles which are carriers of an internal state, called activity . The level ofinfection of each a-particle can progress in time due to a prevalence of thevirus aggressiveness over the immune defence or can regress due to the suc-cessful actions of the immune defence. The ability to express an individualimmune defence is heterogeneously distributed throughout the population.2. Contagion depends on the level of the infection as well as on the physicaldistance between individuals which is a constant parameter in the case ofspatial homogeneity.3. Dynamics within each individual transmission depends on the competitionbetween a proliferative virus and the immune system. Hence, the dynamicsinvolve individuals which are carriers of an internal dynamics.In general, medical actions either to weaken the virus or to activate the immunesystem can contribute to drive the competition towards full recovery. On the otherhand, the model might consider both the dynamics induced by isolation and death12f individuals whenever the action of the virus has a gain over the immune defence.Additional features of the modelling approach are:– The overall population is subdivided into four sub-populations labeled by thesubscripts i = , , ,
4. The abbreviation i -FS is used to denote the i -th populationviewed as a functional subsystem.– The micro-state of a-particles includes two variables u ∈ [ , ] and w ∈ [ , ] corresponding, respectively, to the progression of virus invasion and to the levelof activation of the immune defence. In this sense, u = u > u towards 1 correspond to more aggressive states. Similarly, w = w = u = { u = , . . . , u j = j − m − , . . . , u m = } , and w = { w = , . . . , w k = k − n − , . . . , w n = } , where u = w = i =
1: Corresponds to healthy individuals with distribution f , k ( t , u , w k ) ,where t is the time belonging to the interval [ , T ] .• i =
2: Corresponds to infected individuals with distribution f j , k ( t , u j , w k ) ,with 1 < j < m .• i =
3: Corresponds to individuals recovered from the infection with distri-bution f ( t ) , namely infected individuals that succeed in reaching back tothe state j =
1. Here, the dependence on w k is not any longer relevant withrespect to the specific dynamics under consideration.• i = f ( t ) is the number of individuals of the infected population whodo not succeed to recover, that are infected individuals who reach the state13 = m . Here again, the dependence on w k is no longer relevant with respectto the specific dynamics under consideration.Therefore, an individual with micro-state ( u j , w k ) can be either healthy ( i = j =
1, or infected ( i =
2) if 2 ≤ j < m . If an individual from 2-FS reaches thestates ( u , w k ) or ( u m , w k ) , then he/she recovers ( i =
3) or dies ( i = ddt f ri j = G ri j ( f ) − L ri j ( f ) = m Õ s = n Õ h , k , p , q = η pqhk ( r , s )( f )A pqhk ( hk → i j )( f ) f rhk f spq − f ri j m Õ s = n Õ p , q = η pqi j ( f ) f spq , (1)where (1) refers to the dynamics of a large number of active particles whose stateis defined by the probability f ri j to find an active particle from the r functionalsubsystem with micro-state i j . The subscripts h , k and p , q denote the micro-statescorresponding to the r , s FSs which by interactions lead to the dynamics of f r .In addition, η pqhk , η pqi j , denote the interaction rates, and A pqhk the transition rate intothe micro-state i , j of the r-FS. The time dynamics are then ruled by a gain termof particles which at time t gain the state ( i , j ) and a loss term related to particleswhich lose such a state.The modelling of interactions can be developed according to the followingassumptions:1. i =
1: Active particles belonging to 1-FS interact with a-particles from2-FS and can become, with some probability, infected. The rate of infectiondepends on the physical interaction rate η , supposed to be constant, and tothe level of progression u j of the infected individuals as the probability ofinfection grows with u j . 14. i = ,
2: The interaction rate depends on the social distance. Interactions donot modify the levels of the immune defence, while particles which movefrom 1-FS to 2-FS take the value u and start their competition to survivethe attack from the immune system. The model allows the study of differentscenarios corresponding to various possible situations.3. i =
2: The dynamics within 2-FS are induced by the competition betweenprogressing particles of the virus and the immune defence. Viral particlesprogress (proliferate) thanks to foraging of the surrounding tissues, whilethe immune defence counteracts the progression by inducing a regression.Interactions occur with a constant interaction rate µ , η . In addition, thedynamics should account for the inflow from 1-FS and outflows from 2-FSto 3-FS and 4-FS.4. i = , ,
4: A-particles from 2-FS move to 3-FS if the immune defencesucceeds to obtain a regression down to u . A-particles from 2-FS move to4-FS if the immune defence does not succeed to obtain a regression and thevirus progression reaches the highest value u m . A-particles recovered fromthe infection are not subject to a new infection.The dynamics described in the Items above are shown in Figure 2, whileFigure 3 depicts the progression and competition dynamics within each individual,corresponding to the block i = N for every time t .The model, which is presented in the following, takes into account only inter-actions which have an influence on the dynamics of the overall system, namelytransitions from 1-FS to 2-FS, within 2-FS, and transitions from 2-FS into 3-FSor 4-FS. Then, simulations should put in evidence different scenarios dependingon different distributions corresponding to the levels of the defence ability of theimmune system. Possible developments of the model are discussed in the lastsubsection to account for additional features which are not included in this model.15nfected i = i = i = i = u Recovered u Healthy u j + u j − u j u j + u m − u m Dead β u j γ w k Figure 3: Dynamics within infected population.16earing in mind all the above, let us consider, in sequence, the dynamics relatedto each FS.
Infection dynamics:
A healthy individual from 1-FS with state u interacts, withrate η , with an infected individual from 2-FS with state u j , j >
1, and becomesinfected with a probability which depends on a parameter α ∗ and on the levelof infection of the individual from 2-FS. The dynamics refers to f , k ( t ) and it isgoverned only by the loss term: ∂ t f , k ( t ) = − L k ( t ) = − n Õ s = m − Õ j = α u j f , k ( t ) f j , s ( t ) , (2)for k = , . . . , n and α = η α ∗ . The interaction rate η depends on the localdensity of individuals. It grows with increasing value of the density thus reachingthe maximal value in the case of maximal packing density (a technical value is6-7 individuals per square metre). The use of the subscript 0 refers to the spatiallyhomogeneous case, where the density is the same throughout the spatial domain.Steady cases, but with different crowding levels, as shown in Figure 1, can betaken into account by a space-dependent interaction rate so that α = α ( x ) , where x denotes the position. Further developments are discussed in the last subsection. Dynamics of infected individuals:
Each infected individual is the carrier of astruggle between virus particles and immune system. The virus takes advantagefrom the foraging of surrounding tissues and increases its micro-state from each j -level to the higher j + β ∗ and on the j -th level.The immune system acts to decrease the j -level to the lower j − γ ∗ and on the k -th level. Individuals, whose virus progression levelsreach the values u and u m move, respectively, to 3-FS and 4-FS. The dynamicsrefers to f j , k ( t ) and it is governed by both gain and loss terms: ∂ t f j , k ( t ) = G j , k ( t ) − L j , k ( t ) , (3)Detailed calculations yield: G j , k ( t ) = δ j L k ( t ) + β u j − f j − , k ( t ) + γ w k f j + , k ( t ) , (4)for j = , . . . , m − k = , . . . , n , β = µ β ∗ , γ = µ γ ∗ , δ is the Kroneckerdelta function, and where the three terms correspond, respectively, to inflow from1-FS, progression of the virus and regression of the virus due to the immune action.In addition L j , k ( t ) = β u j f j , k ( t ) + γ w k f j , k ( t ) (5)17or j = , . . . , m − k = , . . . , n , while the two terms correspond to theoutflow of recovered persons who move into 3-FS and dead individuals who moveinto 4-FS. Dynamics of recovered individuals:
The dynamics are caused by the inflow from2-FS into 3-FS: ∂ t f ( t ) = γ n Õ k = w k f , k ( t ) . (6) Dynamics of dead individuals:
The dynamics are caused by the inflow from 2-FSinto 4-FS: ∂ t f ( t ) = β u m − n Õ k = f m − , k ( t ) . (7)Collecting all equations of the differential system yields ∂ t f , k ( t ) = − α n Õ s = m − Õ j = u j f , k ( t ) f j , s ( t ) ,∂ t f j , k ( t ) = α n Õ s = m − Õ j = u j f , k ( t ) f j , s ( t ) δ j + β u j − f j − , k ( t ) + γ w k f j + , k ( t ) − β u j f j , k ( t ) − γ w k f j , k ( t ) ,∂ t f ( t ) = γ n Õ k = w k f , k ( t ) ,∂ t f ( t ) = β u m − n Õ k = f m − , k ( t ) , (8)where for ( ) one has k = , . . . , n , while for ( ) one has j = , . . . , m − k = , . . . , n .It can be rapidly verified that model (8) is a particular case of (1) correspondingto specific models of interactions. It is the same at both scales and accounts forheterogeneity at each scale. A mathematical model has been proposed in the preceding subsection, with therationale behind the model being proposed and described in Subsection 3.1, and the18echnical details given in Subsection 3.2. We do not naively claim that the complexbiological dynamics have been exhaustively captured by our model, but we remarkthat it includes some important features such as the ability of the immune systemwhich is heterogeneously distributed throughout the population.Important features still need to be considered, for instance, modelling the spatialdynamics, typical of crowds, which introduce a dynamically varying heterogeneityin the contagion dynamics. Therefore, a more detailed study of the modellingdevelopments is postponed firstly to the next subsection mainly to account for therole of spatial dynamics and in Section 7 to account for the hints of Sections 4,5, and 6 devoted to virology, immune competition and economics. Some samplesimulations are presented in this subsection simply to enlighten the role of theparameters of the model.Let us study the behaviour of the system corresponding to different choicesof the parameters. We consider, at time t =
0, only one infected individual ina population of 10 million people ( ε = − ). Simulations are developed for m = n = β = . γ = .
2, while three values of α = . , . , .
25, areconsidered.As expected, simulations show that the larger α is, the greater is the increasein the number of infected people, while reducing interactions reduces contagion,see Figures 4-6. Time I n f e c t ed popu l a t i on Time P opu l a t i on Figure 4: α = .
4. Left: Total number of active cases (blue), active cases requiringhospitalization (red) and the number of available beds (black). Right: Cumulativeinfected population (red), recovered (blue) and dead (black) vs time.An additional key factor considered in the simulations is the pressure onthe local health-care system. The red line representing active cases requiringICU admission can be compared with the black line representing the number of19
500 1000 1500
Time I n f e c t ed popu l a t i on Time P opu l a t i on Figure 5: α = .
3. Left: Total number of active cases (blue), active cases requiringhospitalization (red) and the number of available beds (black). Right: Cumulativeinfected population (red), recovered (blue) and dead (black) vs time.
Time I n f e c t ed popu l a t i on × -3 Time P opu l a t i on Figure 6: α = .
25. Left: Total number of active cases (blue), active casesrequiring hospitalization (red) and the number of available beds (black). Right:Cumulative infected population (red), recovered (blue) and dead (black) vs time.available ICU beds. For instance, recent data shows that, in Italy, about 12% of
SARS–CoV–2 positive cases required ICU admission and that, if in practice about1 per thousand of the patients are infected at the same time, then the total ICUcapacity of the country would be saturated, as reported in [37].A natural question can be posed at this stage, namely how does the epidemicpeak depend on model parameters? Moreover, we argue that the epidemic canexist in a population in some equilibrium state in which the rate of newly infectedpeople is compensated by those who recover. Figure 7 shows the magnitude of theepidemic peak in the population (namely the maximum number of people who are20nfected simultaneously) and the time at which this peak occurs, as a function of α .On the left side of Figure 7, we see that the magnitude of the peak is an increasingfunction of α . It is shown that the peak could even reach a level of almost the 30%of the whole population when α is close to 1.On the other hand, if α is small enough, then an outbreak of the epidemic isprevented. As mentioned, these dynamics correspond to equal rates of infected andrecovered. Additionally, the right hand plot of Figure 7 shows that lower valuesof α effectively delay the peak, and social distancing can thus be considered as astrategy to prepare and equip the health system, if necessary. A deeper analysis ofthis feature is further treated in Section 7. α P ea k o f a c t i v e c a s e s α T i m e o f t he pea k Figure 7: Left: Proportion of people infected at the same time when the peak ofinfection is reached as a function of α . Right: Time at which this peak occurs asfunction of α Finally, let us study the effect of implementing social distancing at a given time.Consider for instance the dynamics shown in Figure 4, where α = . N ( t ) = n Õ k = © « f , k ( t ) + m − Õ j = f j , k ( t ) ª®¬ + f ( t ) + f ( t ) , which keeps a constant value equal to N due to the conservative nature of thetransitions. 21
300 1200 2000
Time I n f e c t ed popu l a t i on Time I n f e c t ed popu l a t i on × -3 Figure 8: α = . t < α = .
25 at t = t = Time I n f e c t ed popu l a t i on Time I n f e c t ed popu l a t i on × -3 Figure 9: α = . t < α = .
25 at t = t = j , or simply a fraction of the infectedindependently of the said level.Therefore, it is interesting to show how hospitalization influences the contagion.A simple model can be developed by supposing that individuals who show someclinical evidence/symptoms larger than a critical state, say u ≥ u c are hospitalized.These patients undergo clinical treatments, which are not studied in this section,but our model shows how the contagion evolves accounting for dynamics wherehospitalized individuals do not contribute to infecting healthy people (of coursebesides those within each hospital).Let φ = φ ( t ) be the population of hospitalized individuals. Then, the modelcan be given by a technical modification of Eq. (8) as follows: ∂ t f , k ( t ) = − α n Õ s = m − Õ j = u j f , k ( t ) f j , s ( t ) ,∂ t f j , k ( t ) = α n Õ s = m − Õ j = u j f , k ( t ) f j , s ( t ) δ j + β u j − f j − , k ( t ) + γ w k f j + , k ( t ) − β u j f j , k ( t ) − γ w k f j , k ( t ) ,∂ t f ( t ) = γ n Õ k = w k f , k ( t ) ,∂ t φ ( t ) = β m Õ j = c u j n Õ k = f j , k ( t ) , (9)where the second equation is now valid for j = , . . . , c −
1. Individuals reachingthe state u c are immediately removed from 2-FS and contribute as a gain in the lastequation.Figure 10 shows how the healthy and infected population depend on the criticalstate u c at which people are isolated. 23
500 1000 1500
Time H ea l t h y popu l a t i on Time I n f e c t ed popu l a t i on Figure 10: Healthy and cumulative infected populations for different values of u c ,namely c = c = c = m = Some reasonings on the modelling of the dynamics of contagion propagationthrough space are developed in this subsection. Spatial homogeneity in and ofitself already provides useful information and insight. However, it is only anapproximation of the physical reality of how the infection is transmitted spatiallyand additional work is needed to study more fully the dynamics in space and therelated pattern formation that is created over the spatial domain. This is not an easytask, and it should focus not only on the dynamics in open areas, but also considermovement in complex venues across interconnected areas and networks. In bothcases, the venue might include restricted aggregation areas, namely zones of highcontagion diffusion. This study poses challenging problems which require newideas but at the same time opens interesting research perspectives. This subsectionpresents some of these perspective ideas towards possible research programs whichshould also include computational simulations.The literature in the field of mathematical biology, specifically the dynamicsof multicellular systems, already provides some perspective ideas which can con-tribute to a modelling strategy for the complex system under consideration. As anexample, an analytic and computational study is proposed in [21] for the transportof an epidemic population model by a reaction-diffusion-chemotaxis system. Astudy of the modelling of the movement of invasive cancer cells through the ex-tracellular matrix has been developed in [120] focusing on the multiscale featuresof the dynamics. The second part of [120] presents a model of cell dynamicsat the micro-scale for the cancer cell movement through the extracellular matrixaccounting for the different way by which cells can move, namely mesenchymal24nd ameboid. The authors specifically apply this technique to the phenomenon ofcancer invasion, but some concepts can contribute also to a deeper understandingof the problems studied in our paper. An additional problem, which cannot beneglected, is the derivation of models at the macro-scale from the underlying de-scription at the micro-scale by asymptotic methods [16, 29] somehow inspired byHilbert’s sixth problem [74].A multiscale vision requires that the approach to modelling spatial dynamicsshould take into account both the literature in the field of biology to understandhow cells and viruses move over and through human tissues, and of the literatureon crowd dynamics to understand how individuals move in venues and acrossterritory. This section focuses on some selected topics and provides, for eachof them, some perspective ideas to develop a modelling strategy to be furtherextended in an appropriate research program. Firstly, we focus on the contagiondynamics in crowds corresponding to different levels of awareness of the risk ofcontagion, and subsequently, the pattern formation of infected areas in a hetero-geneous territory/domain are studied accounting also for the role of transportationnetworks. • Contagion dynamics in crowds:
Consider a crowd moving in an open domainwhere the distance between individuals varies in space and time. A small numberindividuals in the crowd are initially infected, and our target is to study the dynamicsof infection.Applications of kinetic theory methods to crowd dynamics indicates that a keymodelling approach has been introduced in [17] for a crowd supposed to movealong a finite number of discrete velocity directions. This model has been sub-sequently developed in [19] to include continuous velocity dependence, the roleof emotional states, and a self-organization ability homogeneously shared by thewhole crowd. Derivation of a macroscopic equation from the underlying descrip-tion at the micro-scale has already been developed in [15]. Further developmentsand related computational schemes have been proposed in [86]. The spatial prop-agation of emotional states by a consensus dynamics towards a commonly sharedstatus in one space dimension has been studied in [24, 137], where the term con-tagion has been used to identify the dynamics by which individuals have a trendto share a common emotional state. Recent research activity is specifically fo-cused on congestion problems in crowds [96]. Congestion in urban areas might beinduced by traffic as shown in Figure 3.3.Focusing on contagion problems, the dynamics in more than one spatial di-mension has been studied in [20], where different types of social interactions have25igure 11: A crowd during city traffic.been modelled. The coupling between a kinetic model of crowd dynamics [86]and infection transmission has been proposed in the pioneering paper [87]. Thisis an important difference with respect to models of evacuation dynamics wherethe key social state is the level of stress which promotes aggregation of walkersrather than rarefaction which is different from the awareness of the contagion thatinduces walking strategies by which individuals try to maintain a safe distancefrom each other.The study of crowd dynamics related to evacuation problems has brought aboutthe derivation of quite sophisticated models which depict a detailed computationof the local density and the mean speed of the flow. Indeed, both quantities areimportant in the modelling approach as high densities correspond to lack of safety,while high speed and optimal search of walking trajectories contributes to rapidevacuation. However, the study of contagion dynamics needs to be developed ina technically different framework. Hence we show how a modelling approach canbe developed to account for the technical modifications induced by the awarenessof the risk of contagion.In more detail, let us consider the dynamics where individuals move in twodimensional space with direction θ and speed v : v = v ( cos θ i + sin θ j ) = v ν , (10)26here i and j are orthogonal unit vectors of a plane frame, θ denotes the velocitydirection, and ν is a unit vector which defines the velocity direction. Individuals inthe crowd are viewed as a-particles whose microscopic state is defined by position x , velocity v , and activity u .Dimensionless quantities can be used by referring the components of x to somespatial scale ℓ , while the velocity modulus is divided by the limit velocity V ℓ whichdepends on the quality of the environment, V ℓ is the speed which can be reachedby a fast pedestrian in free flow conditions, while ℓ is the diameter of the circulararea containing the domain where the crowd is initially localized.The overall system is subdivided into n functional subsystems, while theirmesoscopic (kinetic) representation is obtained from the statistical distribution attime t , over the microscopic state: f i = f i ( t , x , v , u ) = f i ( t , x , v , ν , u ) , (11)where x ∈ R , v ∈ [ , ] , θ ∈ [ , π ) , u ∈ [ , ] . If f i is locally integrablethen f i ( t , x , v , u ) d x d v du is the (expected) infinitesimal number of pedestriansof the i-th FS whose micro-state, at time t , is comprised in the elementary volume [ x , x + d x ] × [ v , v + d v ] × [ u , u + du ] of the space of the micro-states, correspondingto the variables space, velocity and activity.We refer to [18] to propose the following structure which consists in an integro-differential system suitable for describing the time dynamics of the distributionfunctions f i : ( ∂ t + v · ∇ x ) f i ( t , x , v , u ) = J i [ f ]( t , x , v , u ) = n Õ k = ∫ D η ik [ f ]( x , v ∗ , v ∗ , u ∗ , u ∗ ) A ik [ f ]( v ∗ → v , u ∗ → u | v ∗ , v ∗ , u ∗ , u ∗ )× f i ( t , x , v ∗ , u ∗ ) f k ( t , x , v ∗ , u ∗ ) d v ∗ d v ∗ du ∗ du ∗ − f i ( t , x , v , u ) n Õ k = ∫ D η ik [ f ]( x , v , v ∗ , u , u ∗ ; α ) f k ( t , x , v ∗ , u ∗ ) d v ∗ du ∗ , (12)where the integration domain is D = D v × D u , where D v and D u correspond,respectively, to the domains of v and u . In addition, the following quantities modelinteractions: η hk [ f ]( x , v ∗ , v ∗ , ν ∗ , ν ∗ , u ∗ , u ∗ ) , is the interaction rate modelling the frequency bywhich a candidate (respectively test) h -particle in x develops contacts, in Ω s , witha field k -particle. 27 hk [ f ]( ν ∗ → ν , v ∗ → v , u ∗ → u | v ∗ , v ∗ , ν ∗ , ν ∗ , u ∗ , u ∗ ) is the transition density mod-elling the probability density that a candidate h -particle in x with state { v ∗ , ν ∗ , u ∗ } shifts into the state { v , ν , u } of the i -test particle due to the interaction with a field k -particle in Ω s with state { v ∗ , ν ∗ , u ∗ } . Both η and A can depend on the localdensity.Interactions correspond to a decision process by which each a-particle modifiesits activity and decides on its mechanical dynamics depending on the micro-state and distribution function of the neighbouring particles in its interactiondomain. This process modifies the velocity direction and speed. Three types ofa-particles are involved in the interactions. The test particle , the field particle , andthe candidate particle . Their distribution functions are, respectively f i ( t , x , v , u ) , f k ( t , x , v ∗ , u ∗ ) , and f h ( t , x , v ∗ , u ∗ ) . The test particle is representative, for each FS,of the whole system, while the candidate particle can acquire, in probability, themicro-state of the test particle after interaction with the field particles. The testparticle loses its state by interaction with the field particles.The derivation of the mathematical models is obtained by inserting into (12)models of interactions specialized to account for each specific physical situationunder consideration. The strategy expressed by individuals is that they first modifythe dynamics of the emotional state, then they select the walking direction andfinally, the walking speed.The modelling approaches known in the literature, see [9] as a reference, sug-gest that each individual interacting with others in his/her sensitivity domain firstselects the walking direction by a weighted choice from the following directions:A trend towards the target θ T , attraction towards the main stream ξ , and the searchof paths with less congested local density θ V . In more detail, the selection dependson the parameter u and is weighted by the local density ρ , where increasing valuesof u correspond to a trend towards the stream ξ with respect to the trend towardsthe target, while the local density increases the trend towards vacuum or emptyzones. Therefore, the model depend, for each FS, on the micro-state quantity u which differs in each FS and on two local macro-scale quantities, namely ρ and ξ .The selection is modelled by theoretical tools of stochastic game theory, wherethe output of the interaction is govened by the local density and a parametermodelling the level of individual stress in the crowd. In detail, increasing thevalue of the said parameter increases the attraction toward the main stream againsttrajectories across less crowded areas, while decreasing the density increases theattraction towards the target.Somehow different is the case of a crowd of individuals trying to avoid conta-28ion, where the contrast is between the trend towards the target and the search forless crowded areas. These dynamics can be modelled by a parameter σ ∈ [ , ] modelling the level of awareness of the risk of contagion, where σ = σ = σ is that the directional strategy is dominatedby the search for low concentration trajectories rather than attraction towards thestream which generate high concentration. Then, modelling infective contagionin crowds, as shown in [87], can be developed by coupling a selected model ofcrowds to the contagion model presented in the preceding subsection.The modelling approach can therefore be technically developed according tothe following rationale:1. A model of crowd dynamics is selected corresponding to a binary mixtureof infected and healthy individuals.2. Interactions can even disregard the aforementioned attraction towards themain stream, while some technical assumptions can simplify the model, forinstance by constant speed and variable velocity directions.3. The contagion dynamics modelled by Eq. (3.2) should be based on an ap-propriate modelling of the contagion term α depending on the local density.4. If the domain where the crowd moves includes walls and/or obstacles, non-local boundary conditions should be implemented as shown in Section 4of [9].5. A technical generalization consists in the modelling of the crowd acrossinterconnected domains, where flow conditions depend on the geometry andquality of each domain of the venue.A simplification of the mathematical structure, mentioned in Item 2, can be ob-tained by assuming that the speed is equally shared by all walkers. This hypothesis,29ith obvious meaning of notations, yields: (cid:0) ∂ t + v · ∇ x (cid:1) ϕ i ( t , x , θ ) = η ( ρ ) n Õ h = ∫ π ∫ π A ( ϕ )( θ ∗ → θ ; θ ∗ , θ ∗ , κ ) ϕ i ( t , x , θ ∗ ) ϕ h ( t , x , θ ∗ ) d θ ∗ d θ ∗ − η ( ρ ) ϕ i ( t , x , θ ) n Õ h = ∫ π ϕ h ( t , x , θ ∗ ) d θ ∗ , (13)where the crowd has been divided into n different groups to include infectedindividuals. In this case the representation of the crowd is achieved as follows: ϕ i , i = , . . . , n , corresponding to the i-th walking group with level of contagion i = , . . . , n , where i = ϕ denotes the wholeset of ϕ i .The design of computational codes towards the simulation of the spatial dynam-ics of ϕ i can be developed by different techniques depending on the mathematicalstructure of the specific model used for the simulations. Different examples canbe found in the pertinent literature, for instance, finite differences have been usedin [17], while Monte Carlo particle methods [110] and splitting methods have beendeveloped, respectively, in [19] and in [86].If the domain where the crowd moves includes walls and/or obstacles, nonlocalboundary conditions should be implemented as shown in Section 4 of [9], wherethis topic is treated at each modelling scale, namely microscopic, mesoscopic(kinetic), and macroscopic (hydrodynamical). A technical generalization, whichconsists in the modelling of the crowd across interconnected domains is discussedbelow. • On the role of networks:
The propagation of infective states can occur incomplex venues and in transportation networks. We define a complex venue asa small-sized network of interconnected areas, where individuals walk to reach acertain target without using transportation systems. We define a globally connectedworld as a network whose nodes are connected by transportation means, say train,bus, airplanes, etc. Each node of a globally connected world is constituted by asubnetwork of complex venues, typically it is a town.The modelling of the contagion dynamics in complex venues can be developedusing in each area the approach defined above in this subsection, but accountingfor the specific physical and geometrical features of each area. The modelling30pproach to the dynamics in large networks is an open problem and here we simplypresent some perspective ideas towards a rationale to tackle the said approach.1. The globally connected world is subdivided into an exogenous networkconstituted by interconnected nodes.2. Each node is subdivided in a network of interconnected complex venues.3. The overall dynamics are modelled by coupling the various networks, wherethe input and output flows in each node are described by models of migrationdynamics across nodes in the line of the modelling approach to migrationphenomena and networks interaction proposed in [89, 90, 91].4. A possible simplification to reduce the computational complexity consistsin local averaging of the dynamics in complex venues and even in nodes.
A mathematical computational model has been proposed in this section. Focusingon the descriptive skills of the model, the main features, selected among variousones, can be summarized as follows:• It describes the dynamics of the virus transmission, from infected to healthyindividuals, depending on an encounter rate related to the confinement dis-tance for a population characterized by a heterogeneous distribution of theability of the immune system.• The model predicts the time evolution of the number of healthy, infected,recovered and dead individuals. These dynamics are related to a model ofcompetition internal to each individual between virus particles and immunecells.• Inside each individual, progression and recovery are modelled, resulting inthe evolution of the time dynamics of the number of recovered and deadindividuals.• Explorative ability of the hospitalization policy which can be related to thelevel of progression of the pathology. Simulations can show how planninghospitalization on the basis of the level of the pathology can influence theoutput of the dynamics. 31 Explorative investigation on the confinement strategy referring both to thedifferent levels of confinement and to the time interval of the application ofthis action.In addition, we have proposed a rationale for the modelling of spatial patternformation, due to the spread of the infection throughout a given spatial domain, ina small world as well as in the network of a globally connected large world.The achievements reported in the items above cannot, however, be consideredthe end of the story, as it is necessary to look ahead to additional work to bedeveloped within dedicated research programs which can take advantage of theflexibility of the mathematical framework and of the computational tools proposedin this paper. For instance, new specific features of the dynamics can be includedin the general framework also accounting for the perspective ideas presented in thenext sections related to virology, immunology and economics.Focusing on the modelling approach, we mention that it goes far beyond SIRmodels and recent developments, as the contents of this section shows how thepredictions of our model can contribute to the planning of health care. Indeed,the model is derived within a multiscale vision, where the dynamics at the scaleof particles is linked to that of populations whose dynamics depends on that atthe low scales. The description of the dynamics at the higher scale of populationsprovides information useful to the planning of hospitalization as it provides notonly numbers but also levels of the infection and the type of hospitalization to beaddressed to specific levels of the infection. These features are enlightened by thesimulations proposed in Subsection 3.3.Definitely, challenging research perspectives should look at the small-scale,namely at the dynamics inside each individual by a more detailed description ofthe dynamics of virus progression which might include also darwinian mutations,as well as by specializing the different actions that the immune system can developto counteract the virus progression. In addition, the transfer of the dynamics fromthe small to the large world requires additional work to be technically related tothe heterogeneous features of the territory.A detailed presentation of research perspectives is proposed in Section 7 ac-counting, as mentioned, not only of the contribution of this present section, butalso of that of the next Sections 4, 5, and 6. In particular, Section 7 reports arepresentation of the overall flow which is extended, with respect to Figure 2, toaccount for spatial dynamics and the role of hospitalization.The final goal consists in developing a systems approach towards pandemicdiseases suitable to lead an explorative model with descriptive ability to contribute32o depict a broad panorama of simulations useful to support the selection medicaland biological strategies as well as to the strategic indications that might bedelivered within the framework of economical sciences.
Viruses constitute one of the most abundant species on the planet, and play impor-tant roles in all kingdoms of life. Phages, viruses infecting bacteria, are essentialfor areas as diverse as the ecosystem of the oceans and gut health, and epidemicscaused by plant and animal viruses make severe impacts on agriculture and humanhealth.
SARS–CoV–2 , the causative agent of COVID-19, is a prominent example[141]. Like SARS-CoV and MERS, which caused outbreaks in 2003 and 2012,respectively, it is a betacoronavirus. However, in contrast to these viruses it hasevolved properties that make it far more dangerous, such as its ability to spreadbetween hosts with ease, and in many cases stay asymptomatic for a significanttime after infection. Mathematical modelling of individual viral particles can playa key role in understanding how changes in viral genomes due to mutation resultin dramatically different properties of the virus. • Modelling of viral geometry.
Viral genomes encode instructions for the produc-tion of the proteins required to build progeny virus (the structural proteins), aswell as proteins with a range of other functions in the viral life cycles. This in-cludes virally encoded polymerases required that catalyze genome translation andtranscription, and in larger and more complex viruses (such as HIV) also proteinsto counteract host defense mechanisms. The genetic information is stored in theform of ribonucleic acid (RNA) or deoxyribonucleic acid (DNA). Smaller viruses,with genomes ranging between about 1k to 30k nucleotides (nts) in length, mostlyhave RNA genomes, whilst larger viruses predominantly use DNA and can havegenomes up to a size of 2.5M nts. However, all viruses face the same challenge ofprotecting their genomes between rounds of infection. For this, they use proteincontainers, called viral (nucleo)capsids, and/or a lipid membrane, within which thegenetic material is packaged. In RNA viruses, genome packaging usually occursconcomitant with capsid assembly, whilst in larger and more complex viruses, ad-ditional molecular machinery (packaging motors) is required in order to packagethe genome into a preformed capsid.For RNA viruses, such as SARS-CoV-2, that package their genomes duringparticle assembly, an understanding of viral geometry is important. This is becausethe geometric shapes of the structural proteins and their assembly properties are33ntimately linked [131, 132], and also affect other aspects of the viral life cycle,such as the structural transitions that in some viruses are important for infection[76]. In particular, in order to attribute as little coding sequence as possible tothe viral capsid - a phenomenon called the principle of genetic economy - virusesencode blueprints of only a minimal number of distinct proteins (as small as onefor the simplest form of virus), that are then repeatedly synthesized from the samegenome segment, thus delivering multiple identical copies for virus assembly. Asmultiple identical proteins form the same types of interactions with each other,this results in protein assemblies with symmetry. As Crick and Watson noted in aseminal paper [46], they must either form spherical shells with the symmetries ofthe Platonic solids - as is the case in icosahedral viruses, which constitute the vastmajority of viruses - or be rod shaped. RNA viruses with large RNA genomes,such as coronaviruses, typically exhibit helical geometries. This is, perhaps, astheir long genomes would be difficult to compact inside a spherical container.Coronaviruses, by contrast, wrap their 30k nt long genomes around a helicalcore formed from the nucleocapsid (N) protein [40]. The nucleocapsid complex,formed from genomic RNA and N protein, is enveloped by a lipid membrane, theviral envelope, that is studded with three types of glycoproteins: the Spike protein(S), which is important for receptor binding and thus virus entry into a host cell,as well as the membrane (M) and envelope (E) proteins. While these componentsare common to all coronaviruses, genetic changes in new emerging variants canlead to substantially different infection dynamics. For example, the S proteinof
SARS–CoV–2 contains a novel, short sequence of amino acids which enablesthe virus to enter the cell more rapidly than previously circulating coronaviruses[139]. Understanding how changes to the protein structure affect the dynamics ofvirus assembly and virus-host cell interactions can be key to the identification ofeffective drug targets.Any repeat organisation of protein units can be modelled via lattice theory[128, 129]. The first such theory has been proposed by Caspar and Klug in 1962,where they classified the surface architectures of icosahedral viruses formed fromclusters of 5 (pentamers) and 6 (hexamers) identical protein subunits. Extensions ofthis theory have provided models also for viruses that fall out of this scheme, suchas the noncrystallographic architectures of the cancer-causing papillomaviruses[126], and tubular variants formed from the same proteins [127]. A mathematicalapproach based on Archimedean lattices, that embody the concept of identicallocal interactions for different types of proteins, has provided an overarchingframework for the modelling of icosahedral viruses that accommodates Caspar-Klug theory as a special case [130]. A dual view, from a mathematical point of34iew, is to use affine extended symmetry groups to model material boundaries inicosahedral viruses [82, 76, 83, 50]. This approach is also directly applicable tothe core architecture of coronavirus, which is given by a translation and rotationoperation that in combination define the 70 degree angle and the helical pitch of140 Angstrom for a unit of 16 N proteins [40]. • Implications for viral life cycles.
Models of viral geometry provide the opportunity to the analysis of the intra-cellular dynamics of a viral infection. Models of intracellular dynamics includereactions, or equations, for all processes inside the cell pertaining to viral replica-tion. They typically include the copying of the viral genome (transcription) andprotein production (translation), as well as reactions describing virus assembly[42]. In general, virus assembly is described by a single reaction. However, thismisses the intricate interdependence of its different functions, both as a templatefor replication and as a packaging substrate. Insights into viral geometry enablesrefinement of these reactions [57], and therefore provides a much more realisticview of intracellular dynamics, revealing aspects of viral life cycles that had pre-viously been overlooked. Model outcomes can then be included into infectionmodels at different scales, including intercellular models of within host dynamicsand between host dynamics, and thus provide a foundation for a deeper under-standing of viral infection dynamics. Models of the within-host progress of viralinfections can be used to inform pharmaceutical interventions [68, 43] or provideinsights into viral phenomenology [115]. The long genomes of coronavirusesencode at least 19 distinct proteins, which in turn results in a complex interac-tion network. Viral proteins frequently perform multiple roles in the viral lifecycle, creating degeneracy in the network. The assembly of infectious virus par-ticles requires non-uniform amounts of the constituent proteins, with many morestructural proteins required than non-structural ones [13]. Many viruses, includ-ing coronaviruses, therefore produce shorter fragments of their genomes, knownas subgenomic fragments, to control this process. Identifying these subgenomicfragments [85] is a key step for building viral life cycle models, as each fragmentshould be a node in the corresponding interaction network. Accurate modelling ofviral life cycles is built upon comprehensive knowledge of the interactions of viraland cellular proteins and genetic material, which is still incomplete at this pointand an active area of research for
SARS–CoV–2 . • Implications for viral evolution.
Viruses occur as populations of genetically related viral strain variants calledquasispecies [107]. Their distribution is important for their ability to adapt to35nvironmental conditions, as well as to different hosts. In the case of
SARS–CoV–2 , it is likely that the virus has spread from animal hosts, perhaps at a wet market inChina, as its genetic sequence exhibits high similarity with coronaviruses infectingbats and pangolins [3].The accumulation of mutations over time allows for the reconstruction of thespread of the infection through the population, an essential step in developingpublic health interventions [65]. Any virus accumulates mutations as it passesthrough a population, as replication inside each host brings the chance for novelmutations to occur. However, the majority of mutations do not significantlyaffect the phenotype (and therefore fitness) of the virus, but some mutations conferselective advantages to the virus and therefore become fixed in the population. Suchmutations are often interpreted as adaptation by the virus to environmental changes,but there are publications cautioning that this is likely an over-interpretation of thedata, particularly for widely circulating viruses [99]. • Implications for therapy.
Mutations also enable viruses to circumvent challenges from anti-viral ther-apy. Mutation rates are much higher in RNA viruses (typically 10 − substitutionsper nucleotide per cell infected), as opposed to DNA viruses (typically 10 − substitutions per nucleotide per cell infected) [117], making the development oflong-lasting vaccines and antiviral therapies difficult. The rapid evolution of theRNA genome of the influenza virus requires the development of a new vaccineevery winter [64], whereas the smallpox vaccine has effectively eradicated the dis-ease [59]. Better understanding of the mutational landscape of SARS–CoV–2 willenable predictions of the efficacy of vaccination efforts[32, 122]. Many existingantiviral therapies are developed against widely shared viral mechanisms, such aspolyprotein cleavage and genome replication, however these are often susceptibleto rapid adaptation by the virus [61]. It is therefore essential to better understandthe geometric constraints on viruses, as these can point to evolutionarily conservedfeatures that could serve as more stable and effective drug targets.Due to its large genome size compared to other RNA viruses, and the vastfrequency of infection, and thus opportunities for mutation, during a pandemic,understanding the consequences of mutations, and finding evolutionarily stabletargets, is particularly important for viruses such as
SARS–CoV–2 . Mathematicalmodelling can support the discovery of such novel therapeutic targets in many ways.Traditionally, this occurs through the analysis of individual viral components andtheir dynamic properties. For example, since the start of the COVID-19 pandemic,normal mode analysis and quantum mechanical computations have been used to36etter understand potential drug targets on viral proteins. This includes structuralproteins, most importantly S-protein, which is involved in ACE2 receptor binding[116, 1], as well as non-structural proteins, such as proteases [56]. The rapidemergence of COVID-19 has also spurred the development of novel methodologiesto support the discovery of antiviral therapies, in particular in silico screeningtechniques of multiple compounds against
SARS–CoV–2 proteins [58, 142, 62].Once effective antiviral therapies have been developed, mathematical mod-elling can provide valuable insights into effective treatment [67] and mechanismsof drug resistance [80, 103]. The modelling of viral life cycles based on viralgeometry adds a new dimension to the modelling. It provides a framework inwhich the merits of different forms of antiviral strategies can be compared andsynergies explored [25], thus revealing novel opportunities for antiviral therapy.
A systems approach to model, by a differential system, the dynamics of the
SARS–CoV–2 in a population has been developed in Section 3 within a general frameworkof a multiscale vision which includes the competition inside each individual be-tween a progressing viral infection and the immune system. This competitioninitiates when the virus is transferred from infected to healthy individuals by adynamics modeled at the higher scale. This section provides a description ofthe immune competition at the low scale and outlines the relevant features of theaforementioned dynamics according to the present state of the art. • The contagion.
The contagion by the
SARS–CoV–2 occurs mainly by air, breathing respiratorydroplets released by an infected person when coughing, sneezing or speaking. Thevirus load in the droplets increases as the infection progresses [119]. Respiratorydroplets finally land on various surfaces where the virus maintains its infectivecapacity for various times [134]. Therefore,
SARS–CoV–2 can also be transmittedby contact between a susceptible person and the infected one or touching contam-inated surfaces. In this case, the infection takes place if the susceptible persontouches the mucosa of the mouth, nose or eye after capturing the virus [138]. Theprobability of infection rests on the viral load carried by the droplets expelledthrough respiratory emissions and the persistence over time of virus infectivitydepending on the environmental conditions.With the arrival of the
SARS–CoV–2 virions on mucosal surfaces a quantitativeand time-sequence competition between the virus ability to infect and host defense37echanisms is putted in motion. The final outcome (inhibition of the viruses,minor and asymptomatic infection, a severe and fatal disease) rests on a seriesof confrontations between the virus infective ability and distinct immune defensemechanisms [100]. • First line of defense: the mucosal barrier.
The number of infecting
SARS–CoV–2 particles is a first and crucial variablein the competition between the virus and host defense mechanisms [100]. Thevirus load arriving on mucosal surfaces has to deal with the very effective barriermade by the mucus. Mucus is a complex mixture of glycoproteins continuouslyproduced by goblet cells in the mucosal membranes and by particular glands. Itcontains salts, lactoferrin , enzymes and antibodies (secretory IgA and IgM) [26].This viscoelastic gel covers the mucosa lining the nose, throat and lungs. Byharnessing and neutralizing viral particles, it prevents a direct contact of theviruses with the surface of the epithelial cell of the mucosa. Mucus productionis regulated mainly by two lymphokines (IL-13 and IL-22) secreted by sentinellymphocytes associated with the mucous membranes. IL-13 is mainly producedby Innate Lymphoid Cells (ILC), IL-22 by T-helper lymphocytes17 (Th17). Theoverproduction of mucus gives rise to phlegm [125].Mucus is continually transported by the coordinate beats of the cilia of the hairycells in the mucous membranes, then it is swallowed and destroyed in the stomach.Mucus transport is fundamental for its protective action. Under normal conditions,the ciliary beat frequency is around 700 beats per minute. The intensity of thebeats is negatively regulated by IL-13 and is lowered by environmental pollutantspresent in the breathed air, by the humidity and by the low temperature [95].The importance of the mucus barrier in hindering
SARS–CoV–2 infection itis unexplored, although it is well know the influence that environmental factors(humidity, temperature, air pollution, etc.) have on the defense against coronavirusinfections that cause winter colds. It is thus probable that in the great majorityof cases mucus and the beats of cilia clear the invaders. Nevertheless, whenthe delivered viral load is very high and mucus production and its transport aredisturbed, a few virus particles might be able to sneak through and establish adirect contact with the surface of mucosal epithelial cells.The peculiar current distribution of the
SARS–CoV–2 pandemic, which some-what has spared the warmer countries while has hit harder areas with high airpollution could depend, in addition to a different survival of the virus in a warmerand drier environment, on a more efficient barrier effect created by mucus andciliated cells. On the other hand, circumstances requiring speaking aloud in a low38emperature and humid environment favor
SARS–Cov–2 super–spread [92]. • Overcoming the barrier.
An efficient barrier defense confronted with a low viral load probably efficientlystops the infection. In all cases, however, the barrier drastically reduces the numberof viral particles that manage to reach the surface of the cells of the mucosa. Ifany
SARS–CoV–2 particles get through the mucus layer, they can reach the cellmembrane of mucosal cells. In the event that a physical contact takes place, the
SARS–CoV–2
Spike glycoprotein, that forms a sort of crown on the surface of
SARS–CoV–2 particles, anchors the virus to the cell surface.The Spike glycoprotein forms homotrimers protruding from the virus surface.Each Spike monomer is made by an external S1 domain and a S2 domain connectedwith the virus membrane. The distal S1 domain comprises the Receptor BindingDomain (RBD), a molecular subunit that engages with high affinity a region of theN-terminal domain of human Angiotensin-Converting Enzyme (ACE2). ACE2is an ectoenzyme that is normally highly expressed on the surface of the cells ofhuman respiratory and intestinal epithelia. It is also expressed on the surface ofmany other kind of cells, even if at lower density [78].Following Spike-ACE2 binding, other cell ectoenzymes (Transmembrane Pro-tease Serine 2 TMPRLRSS2 and Furin) present on the surface of human cells cutthe Spike protein separating S1 from S2 domain. As a consequence of the cut,the S2 domain, connected to the virus membrane, exposes particular sequences ofamino acids (fusion peptides) that facilitate the fusion between the viral capsid andthe membrane of the human cell [47]. Thanks to this fusion, the RNA of the virusenters the cell where it is directly translated into proteins by the human ribosomes.New virions are assembled as reported in the previous Chapter and the cell diereleasing millions of new viruses. • Second line of defense: intracellular reaction against viral RNA.
The invaded cell is endowed with several mechanisms to sense and blockviral invasions. In the cell cytoplasm, the
SARS–CoV–2
RNA are recognizedand destroyed by RNA helicases. The peculiarities of viral RNA are also isrecognized by RIG-1-like receptors and Toll like receptors (TLR) present in thecell cytosol. Upon activation, these receptors induce signaling cascades leading tothe phosphorylation of transcription factors ultimately conducting to transcriptionof Type I interferon (IFN), a cytokine that activates antiviral programs in theinvaded cell and induces the expression of families of transmembrane proteins thatinhibit virus entry in nearby cells [134].In addition, the signals transduced by RIG-1-like and Toll like receptors trigger39he assembly of the inflammasomes, cytoplasmatic multimeric complexes. Onceassembled, the inflammasome activate caspases which permit the production ofhigh amounts of important pro-inflammatory cytokines (IL1, IL18, . . . ). Thesecretion of the mature form of these cytokines promotes the release of additionalcytokines and the induction of an innate immunity inflammatory response. More-over, inflammatory caspase 1 cleaves gasdermin D to cause cytokine release andpyroptotic cell death, a kind of programmed cell death that occurs most frequentlyupon viral infection. The death of virus infected cell is an effective way to blockthe progression of the viral infection [102].Faced with this sophisticated series of receptors and intracellular reactionmechanisms,
SARS–CoV–2 has elaborated several mechanisms to avoid viral RNArecognition or to antagonize with receptor signal transduction. The first proteincoded by the
SARS–CoV–2
RNA penetrated inside the mucosal cell, ORF1ab,is a chain of 16 proteins joined together. Two portions of this long proteinmake the cuts that free the different proteins. The activity of several of them isto interfere with intracellular reaction mechanisms: NSP1 protein prevents thecell from assembling antiviral proteins; NSP3 protein alters the regulation of cellprotein, thus reducing cell ability to put in motion the antiviral mechanisms; NSP10and NSP16 proteins protect viral RNA from destruction; NSP13. Other viruscoded protein, such as ORF6 interfere with the signals activating cell reaction toviral RNA; ORF9b protein suppress intracellular signaling thus limiting antiviraldefenses nonspecifically [45]. Other viral proteins interfere with the signalingdownstream of IFN release: ORF3a, ORF6 viral proteins alter various steps of thesignal transduction pathway that bridge the IFN receptor to the STAT proteins thatactivate transcription of IFN-stimulated genes [134]. • Third line of defense: the inflammatory reaction.
If the infecting viral load is enough to overcome the mucosal barrier andsneak through intracellular defense mechanisms, the spread of
SARS–CoV–2
RNAbecomes evident in nostril, pharynx and eye mucosal surfaces. Here, viral infectionis confronted by multiple reaction mechanisms of innate immunity. The intensity,efficacy and features of this third line of defense decides the natural history ofinfection: whether the virus spread will be efficiently blocked in upper airwaysand how many viruses will reach the lungs [100].Humoral elements of innate immunity, including the complement and coagu-lation systems, soluble proteins of innate immunity such as the Mannose BindingLectin, natural antibodies and cross-reactive antibodies induced by previous in-fections by different viruses are immediately confronted with the virus spreading40nd the local cell damages. The damages associated with the viral invasion, alarmsignals and cytokines induce the dilatation of local post capillary venule with thesubsequent exit of plasma and of reactive leukocytes. The reactivity of leukocyteswill be driven by multiple alarm signals, cytokines and interferons released byinfected and dying epithelial cells.Metatranscriptomic sequencing performed to profile the inflammatory reac-tion shows a markedly elevated expression of several chemokine genes. CXCL8chemokine gene overexpression is directly connected with the recruitment of neu-trophils into the inflamed tissues, while the upregulation of CCL2 and CCL7chemokine genes plays a central role in the recruitment of monocytes/macrophages.Consonant with this pattern of gene expression, neutrophils, activated dendriticcells, monocytes are the most abundant cells present in the reactive infiltrate [14].
SARS–CoV–2 infection triggers the expression of several IFN-inducible genes.While this suggest that a robust IFN response is ongoing, the IFN gene is notupregulated. This discrepancy, probably due to the interference exerted by
SARS–CoV–2 proteins on the IFN gene expression and on the signaling cascade of IFNreceptors, has a crucial importance in shaping the efficacy of the inflammatoryreaction and the evolution of the viral infection [134, 140]. This confrontationbetween host’s innate immunity and
SARS–CoV–2 decides whether infection willbe blocked in upper airways or the virus will reach the lungs, making the patientsick or very sick [100]. Epidemiological data suggest that this reaction is extremelyeffective as the vast majority of
SARS–CoV–2 infected people are asymptomaticor minimally symptomatic. • Pathogenic fallout of innate immunity reactions.
The feature of initial local viral spread, the genetic characteristics and healthstatus of the infected person decides whether the innate immunity reactions willlead to the containment of the
SARS–CoV–2 infection or will worsen its evolution.For example, IFN is protective in the very early stages of the inflammatory re-sponse, while later it becomes pathogenic [134]. The expression of ACE2 on thecell membrane is modulated by IFN, and its upregulation in airway mucosal cellsfavors the expansion of
SARS–CoV–2 infection [143].The intracellular recognition of viral RNA and the presence of
SARS–CoV–2 coded ORF3, ORF8b, and Envelope proteins triggers the leukocyte assemblyof the inflammasome and the consequent massive production of IL-1 and IL-18.Similarly, the intracellular presence of
SARS–CoV–2 coded NSP9 and NSP10proteins trigger IL-6 production [134]. These three cytokines together induceanti-viral programs and markedly enhance the inflammatory response. However,41levated serum concentrations of IL-6, IL-1 and IL-18 may result in a systemic“cytokine storm” involving the secretion of vascular endothelial growth factors(VEGF), monocyte chemoattractant protein-1 (MCP-1), IL-8 and additional IL-6. The vascular permeability and vessel leakage are increased leading towardhypotension and acute respiratory distress syndrome (ARDS) [143]. In this way,if the
SARS–CoV–2 is able to reach the lung alveolar cells, innate immunitymechanisms driven by the cytokine storm may dramatically contribute to diseaseseverity [134]. • Fourth line of defense: The T and B cell adaptive response.
T cells play a central role in the healing of numerous infectious diseases. CD4T cells are required for the activation of the production of high affinity antibodiesby B cells. Moreover, they modulate and guide a more efficient inflammatoryresponse. CD8 T cells are able to find and kill virus infected cells before theybecome factories of millions of new viral particles. Given the central role T cellshave in the healing of viral infections, it is not surprising that an overall reduction ofT cells in peripheral blood is associated the aggravation of
SARS–CoV–2 infection.T cell reduction during
SARS–CoV–2 infection is likely due to a number of factors,including the cytokine storm [134].A persistent and robust T reactivity to the peptides of Nucleocapsid, Membraneand Spike
SARS–CoV–2 proteins is evident in patients recovering from
SARS–CoV–2 infection. Often a preferential specific reaction against Spike peptideswas found. While the induction of a robust T cell reactivity is essential for viruscontrol, a too high CD4 T cell reactivity may facilitate the development of lungimmunopathologies and exacerbate the cytokine storm [134, 102].While the reaction of CD8 T cells is directed to the killing of virus infected cellsin which millions of new viral particles are assembling, the action of antibodiesis against each viral particle. An antibody binding to the proteins exposed on theexternal surface of a virus severely limits it capacity to infect the target cells. Inmost cases the antibodies are mainly directed towards the Spike protein. In somecases, it has been possible to identify antibodies that specifically bind the RBD,the region of the Spike glycoprotein that binds to the ACE2 enzyme on humantarget cells. Often, the appearance of virus specific IgA and IgM and Ig (serumconversion) is associated with a complete virus clearance: the
SARS–CoV–2
RNAis no more detected in the swabs.The effectiveness of antibodies in neutralizing viral infectivity is also shown bythe success of therapies based on the transfer of plasma obtained from convalescentpatients. Moreover, in several cases it has been possible to obtain monoclonal42ntibodies that interact with RBD and that block
SARS–CoV–2 infectivity startingfrom B cells obtained from patients in convalescence. Antibodies play a role notonly in the healing but also in preventing
SARS–CoV–2 re-infections [93]. • The dark side of the adaptive responses.
In many cases of
SARS–CoV–2 infection, the subsequent activation of adaptiveresponses corresponds to healing. However, the situation is not always so clearlydefined. Some people recovering from
SARS–CoV–2 infection seem unable toproduce a significant antibody response. Moreover, there are various reportson high antibody titer associated with a more severe clinical outcome. Theseobservations highlight that in certain cases antibodies can facilitate viral infectivity,a phenomenon known as Antibody-Dependent Enhancement (ADE) [134, 108].When the various mechanisms of immune reactivity fail to control virus spreadand the infected person starts to produce and shed copious amount of
SARS–CoV–2 , the virus may diffuses to all the body districts infecting the cells that express theACE2 receptor enzyme. The expression of these receptor enzymes is particularlyabundant on the cell membrane of both lung alveolar epithelial cells and enterocytesof the small intestine [136]. If the virus infects and replicates in lung alveolar orcells in the cell of intestinal mucosa, the immune attack may be responsible ofmajor pathophysiological complications. The deposition of immunocomplexesmade by antibodies and
SARS–CoV–2 virions triggers Complement activation,vasodilatation, inflammatory reactions, coagulation and tissue damage. Localcytokine and chemokines contribute to the local vasodilatation, recruitment ofinnate immunity cells and serum antibodies extravasation [100, 134].The massive killing of virus infected cells leaves the lung alveoli stew offluid and dead cells. Some patients slowly recover while others develop a pooroxygenation index and a condition called acute respiratory distress syndrome(ARDS) requiring intensive care therapy and mechanical ventilation. Similarlesions occur in the intestine. The spread of
SARS–CoV–2 throughout the bodyalso promote blood clots, hearth attack, and cardiac inflammation. At this stage,almost half of the patients slowly recover whereas the other die [136]. • Additional reasonings.SARS–CoV–2 is a sophisticated single stranded RNA virus endowed with thecapacity to infect in a short time over 6 million people around the world andto kill over 370,000 of them [44], causing dramatic health, social and economicproblems. Epidemiological studies suggest that the number of people who havebeen infected with the virus is probably much higher [49]. Therefore, even theinfection by this extraordinary virus appears to be managed by the immune system43n well over 90% of cases. Paradoxically, much of the virus lethality come from a“friendly fire” due to an excessive immune reactivity [134, 136].Immune defense mechanisms that are activated to block
SARS–CoV–2 infectionare here schematically distinct in four lines. Each one is based on peculiar anddistinct reaction mechanisms and follows a different strategy. Nevertheless, thesefour lines of reaction are highly interconnected with a very integrated transitionfrom one line of defense to another.In the vast majority of cases, each of these lines of defense wins its confrontationwith the virus. Even if, as to now, it is not yet possible to know what thepercentage of complete victory is, each line of defense proves to be extremelyeffective. Probably, the percentage with which the defensive system wins againstthe infectious ability of
SARS–CoV–2 is particularly high as regards the first linesof defense. The elevated lethality of the virus in elderly people due to the naturalimmunodeficiency connected with aging (the immunosenescence [109]) highlightshow important the defense role played by the immune system is in the control of
SARS–CoV–2 infection.
In this section we intend to uncover the socio-economic implications of the spreadof COVID-19. After distinguishing between the direct impact of the pandemic andthe indirect one of the lockdown, we discuss the potential limitation of generalisedlockdown policies facing localised contagion dynamics. We emphasize the roleof heterogeneity in order to properly account for the contagion dynamics itself.Moreover we suggest the explicit account of clustered networks as a potentialmodelling extension. Additionally, we stress the importance of the timing of thelockdown and the need to introduce early warning indicators. We conclude byoutlining a series of policy actions to be put in place beyond lockdowns to dealwith the spread of collective infectious diseases.
The explosion of the COVID-19 pandemic and the ensuing policy of social dis-tancing undertaken by many countries have put the organization of the productionand the economy as a whole under an unprecedented stress. Analysis of the impactof the pandemic on the labour market are now spurring with scaring projections interms of number of jobs and related income losses. The ILO projects two hundreds44illions losses worldwide. The Internationally Monetary Fund estimates a declineof world GDP by 3%, a drop much more severe than the 2008-2009 crisis, whilethe Euro-Area estimated loss currently is 7 .
5% of GDP [77]. Both are conserva-tive forecasts. However these numbers are expected to increase. Already after onemonth since the lock-down, around 11 millions European workers have been hit bythe consequence of the pandemic, with an increase of four millions unemployedpeople and with seven millions of short-term contract workers at risk, accordingto ETUC estimates [75].The direct and indirect impacts of the pandemic invest many realms of theeconomic analysis, from the organization of production and global value chains(GVCs), to patent systems and appropriability conditions in the pharmaceuticalsector, to the provision of health as a public good, up to the study of unconventionalfiscal and monetary policies (see [11]). On top of that, implications in terms ofthe organization of the workplace and related of the work-process are going to behuge. Indeed, social distancing is expected to jeopardize business and employmentopportunities in a labour market at the outset marked by strong inequalities.To detect the economic impact of the COVID-19 pandemic it is important todistinguish the direct economic effects and the indirect ones of the policy of socialdistancing. The direct impact is and is likely to be per se quite limited. Amongdeveloped countries, in Europe, mortality has been concentrated on the elderlypopulation, with significant losses only in the range of plus 70 years. Differently,in the U.S., whose healthcare system is largely private, impeding provision ofhealth assistance for poor and more vulnerable communities, the impact of thevirus is spreading remarkably across Blacks, Coloureds, and Latinos of relativelyyoung age (less than fifty). Indeed, the existence of previous health diseasesas diabetes and obesity, more diffused among the poorest population, has beenrecognised a factor aggravating the infectiousness of the virus.Notwithstanding heterogeneity across countries, in general it is not likely toexpect the pandemic impacting on labour supply to a magnitude recalling the BlackDeath or even the Spanish Flu (see [14]). Together, this pandemic, unlike otherhistorical episodes such as the Plague of the 14th century, will not serve to alleviateincome and wealth inequalities, by increasing the wages of a scarce labour forceand reducing the value of real estates on sale for the death of their primary owner.On the contrary it is, and will be much more so, amplifying existing inequalities,ranging from access to hospitalization, possibility to work-remotely, benefitingof a stable income, risk of unemployment (see [38]). On top of that, even risksof contagion are strongly heterogeneous, much more concentrated among workercategories directly exposed, such as those in the health sectors, and relatively less45oncentrated for workers able to work remotely.Granted the unequal effects of the direct and indirect impacts of the pandemic,the modelling effort has to focus on heterogeneity to meaningfully capture also itseconomic consequences. In particular, given the concentrated direct impact onthe elderly population, one might not consider any direct economic consequencearising from the death of the elderly. This does not mean that deaths are ac-ceptable because they do not impinge directly on the economic system. It meansthat we refrain from attributing any direct value-estimation to the life of humanbeings. Clearly, deaths are an enormous social and humanitarian cost which haveto be considered as such well beyond any economic consideration grounded ondismissible cost-benefit analyses.Therefore, the economic impact of the pandemic is largely indirect :• The economic damage of the pandemic increases with the amplitude andseverity of the lock-down.• The economic damage, arising from the lock-down, unevenly hits the popu-lation, with low-income individuals more harshly affected than high-incomeones.Defining the all set of potential variables and transmission mechanisms affect-ing the economy via the lock-down is out of the scope of the paper. As mentioned,the economic impacts are quite diverse, including production, global value chains,business closures, job and income losses, fiscal burden of policy interventions,debt accumulation, financial instability and markets volatility, just to mention afew. All these mechanisms, diverse as they are, might interact via cascade andcumulative effects along propagation channels, all contributing to fuel the longestlast and most severe crisis the economy has faced since the 1929. The end re-sult will be various possible “damage functions”, which we shall present below,combining the diverse economic effects and their potential interaction.Let as define a damage function D( L ( α ) , σ ( ω i / ω max )) depending on the in-tensity of the lock-down and on a proxy of inequality of a given system. L ( α ) , thelock-down, is governed by the parameter α , discussed in Section 3, increasing withthe amplitude and duration of the policy, represented by the reduction of α , while σ ( ω i / ω max ) represents the spread between the actual and the maximum incomelevel in the system, with low-income individuals hit harder than high-income ones,where ω i defines the income level of individual i .We can graphically sketch its functional form. Figure 12 depicts a positivenon-linear behaviour of the damage function vis-a-vis the intensity of the lock-46 ( α )D( L ( α )) L max | α = L min | α = Lock-down max D Figure 12: The economic damage as a function of the lockdowndown, meaning that the higher the intensity of the lock-down, given by a reductionof α , ranging from [ , ] , the higher the economic damage, and non-linearly so.The lock-down policy is implemented by reducing the parameter α , controllingfor social distance. For low intensity of the lock-down, the economic damage israther negligible (e.g. local lock-downs). Extending the lock-down after the firstinflection point harshly hits the economy with more than proportional incrementsof the damage, while after very persistent lock-downs (second inflexion point), thecumulative damage is so high that it can only increase less than proportionally.The intersection between the curve and dashed vertical axis at α = Ω = ( ω i + , . . . , ω i + n ) , the damage will be higher, the higher the ratio σ between the minimum and the maximum income level, that is when ω i → ω min ,conversely it will be lower when ω i → ω max .Note a major implication of the foregoing argument: the relationships betweenhuman losses and economic ones not only is not linear but under some lock-down47 = ( ω , .., ω n )D( L ( α )) σ ( ) σ ( ω min / ω max ) max D Figure 13: The economic damage function across income levelsconditions might not be even monotonic. Tighter degrees of lock-down for sureimply higher economic losses but in some circumstances, as we shall see, mightalso yield higher rates of infection . The model developed in Section 3 allows to overcome the limitations of the mostincumbent compartmental modelling approaches. Indeed, it accounts for infectionrates endogenously affected by the state of the virus invasion in infected individuals ( u j ) . This is a first major advancement in that one can account for the importance ofthe viral load in determining the evolution of the disease after the individual entersthe infectious status. The second major advancement relevant to the discussionhere concerns the definition of an intra-body interaction process between the virusand the individual immunity ( w k ) . Now, considering the infection process resultingfrom the coupling of these two types of heterogeneity, this will not yield a singleparameter R but a distribution of R , t , i = ( R , i + , ..., R , i + n ) across individuals,evolving in time, according to the state of the infection.Such an analysis encompassing heterogeneity fundamentally differs from SIRcompartmental structures modelled by differential equations (as pointed out long48go by [114]), so that the level of R parameter decreases with:• The degree of individual heterogeneity in susceptibility to infection by thevirus.• the clustering of the contact structure.The foregoing model introduces individual heterogeneity and local interac-tions while considering the rate of interaction η as constant and equal across allindividuals (e.g., space homogeneity). Under that set-up, the simulation analysispoints at a general positive impact of the policy of social distancing, implementedby the reduction of the α parameter, decreasing the peak of contagion and theactual number of infected individuals.The current model specification can be extended in two directions:1. The interaction dynamics across individuals is made to evolve in structurednetworks, say lattices which can represent homes, workplaces, schools,hospitals. This would imply a modification from η ⇒ η i , j .2. The level of infection, namely u j , may be affected by the interaction processitself, u j ⇒ u j ( η i , j ) , implying that the virulence increases with repeatedcontacts.The relevance of the two extensions is well supported by the evidence both onthe infection dynamics and on its spatial heterogeneity. Indeed, the R estimatedin hospitals is by far higher than the one which might have been recorded on thestreets. Contagion has spread even during the lock-down, mainly occurring athome and in hospitals. In the following we shall outline two potential venues toaccount for the above dynamics. Extension 1: occasional versus structured contacts
Let us suppose the existence of different types of α parameters, one for occasionalcontacts, say occurring along a street, and one for structured contacts, occurring athome and in hospitals: α o and α s . Under two different social distance parameters,the policy of the lock-down has opposing effects: from the one hand it reduces α o by reducing occasional contacts, while, on the other, it increases α s by increasingstructured contacts. If this is the case, the overall contagion rate might not bereduced by lock-down policies. On the other hand, the lock-down will have astrong economic damage as discussed above.49hus, one could envisage heterogeneous probabilities of interaction and dif-ferent interaction dynamics, e.g. (i) homogenous/random contacts vs structuredcontacts in lattice, and (ii) presence or absence of the possibility of long-rangeinfection (small world). This will result in a distribution of individual social in-teractions, say ( α i + , ..., α i + n ) . Understanding the extent to which the contagiondynamics is clustered also entail alternative policy measures. So, for example, generalised lock-down policies might be ineffective under clustered dynamicsof contagion . Extension 2: coupling interaction with virulence
Actual recorded contagion in schools and workplaces has been very low. Notwith-standing the limited robustness of such evidence due to the absence of a generalisedtesting strategy, a possible higher immunity response in the youth and adult popu-lation demands a second potential extension of the model, namely the possibilityof making the virulence dependent also on the “types” of interacting agents withhigher immunity rates in youth and young adults.
Countries have reacted very differently in terms of the management of the COVID-19 crisis: some countries, including South-Korea, Taiwan, New Zealand, Japan,Germany report case-fatality rates ranging from [1.4 - 5] %, while some othercountries like Italy, France, Spain, U.K., Belgium do report far higher rates, in therange [10 - 15] % (see [79], data retrieved on the 26 th of May). Sweden, with nolock-down policy at all records a fatality rate proximate to the Italian one [11.9 vs14.3] %. These numbers are clearly biased by data collection and testing strategies.However, country heterogeneity is a robust fact.More than the amplitude of the lock-down, a variable strongly affecting theoverall dynamics of the pandemic seems to be the timing of the policy intervention.In general, timely and selective closures have been more effective than delayedgeneralised closures . This has been the case of South Korea, with massive contacttracing technologies, but also of Germany, which has undertaken a massive testingstrategy and very early selective isolation. In both cases, the lock-down was notgeneralised.Together, hospital capacity and the number of ICU beds has been the othercrucial variable. However, while ICU beds in the very short-term might beconsidered as relatively fixed, this is not the case for early detection, which mighteven prevent hospitalization. 50hen the policy measure should be put in place? Is there any thresholdindicator useful to understand the timing of the policy action? Together, when thelock-down should end? Which type of indicators should be used? These questionsare extremely important but addressing them entails the appropriate understandingof the contagion dynamics. The saturation of ICU beds indicator strongly reducesrooms for policy actions. Being a “downstream” indicator, the system will detectthe saturation only when the contagion is peaking, or at least accelerating.Early warning indicators, upstream, are of crucial importance. And they in-volve monitoring systems communicating suspect phenomena, like, in the Coronaviruses case, anomalous picks of pneumonia and persistent flue symptoms acrosspatients. Early warning indicators are also crucial to avoid unnecessary long lock-down phases, exerting strong impact on the socio-economic systems but less so onthe contagion dynamics. Extension 3: introduction of early warning indicators and the interactionwith policies
At the current stage the model includes in the dynamics of infection the socialdistance parameter as a crucial variable of policy action. The model might beextended by introducing an interaction from an early warning indication of thestate of the system to the containment, hospitalization and treatment policies alsoable to reduce the pool of susceptible individuals, isolating and treating earlydetected cases, thus avoiding the spreading of the contagion.
Extension 4: endogenizing the timing of the policy action
The time of the policy action is currently exogenous to the model. However,the timing in which the social distance parameter is changed might become en-dogenous and depend on some state of the system as well, e.g. the fraction ofinfected/hospitalised/dead individuals. This ought to apply both to the beginningand the end of policy measures.
Overall, the spreading of the pandemic is exacerbating a series of old inequalitiesand vulnerabilities. If the common perception is that “everybody is equal” infront of the pandemic at closer inspection this is not true. What people do atwork, their contractual framework, and their position in the internal organizationalhierarchy strongly affect the possibility to remote working. Gender and geograph-ical imbalances matter. The digital divide is exacerbating: access to high-speedinternet connection and ICT-devices are the necessary condition to learn at the51-schools. However learning dramatically depends on the education level of theparents themselves. Schools are never been as unequal as nowadays.The coupling of the pandemic and social distancing are making diverse risksconflating: health risk (exposition to social contacts are higher for low incomeoccupations), income risk (probability of job losses is higher for temporary low-income occupations), employment risk (feasibility to remotely work is lower forlow-income occupations).In the following, we shall list a series of policy actions to be undertaken, beyondthe lock-down, in a medium term perspective, to cope with the increasing risk ofwidespread, collective diseases. Overall, the health-management system deeplyaffects the contagion dynamics and therefore needs to be completely reorganised.Let us highlight some crucial reforms to be undertaken in our view:• Increasing the overall public expenditure for the health system by strength-ening local hospitals and laboratories: a capillary hospital system is able tocope with widespread diseases.• Reducing the public subsidies to private clinics, being the latter more inter-ested in profit-seeking activities rather than general medical care assistanceand provision of ICU beds for the general public.• Strengthening the role of GPs, implementing forms of communication andmonitoring activities, fostering at home visits.• Increasing the public financing research.• Compelling the pharmaceutical sector to perform genuinely innovative R & D activities.• Revitalizing national-based laboratories to discover drugs away from themarket system starting with vaccines.• Maintaining inventories of safety devices and instruments necessary to equiphospitals, protect workers and perform testing A systems approach to modelling the contagion and spread over a given territoryhas been developed in our paper, where a multiscale vision has been a key feature52f the approach which includes hospitalization dynamics, and recovery or/deathof patients.An updated version of Fig. 2 is shown in Fig. 14 to account for all specificblocks and related flow chart of the systems approach. Specifically, for spatialdynamics and the role of hospitalization which, in agreement with the reasoningspresented in Sections 4–6, can be viewed as an important feature of the systemsapproach. The figure shows the following blocks:
Block 1:
The dynamics of contagion is at the level of individuals depending onthe level of confinement only in the case of spatial homogeneity, while a deeperunderstanding of these dynamics might account for crowd movement in complexvenues.
Block 2:
The dynamics are generated by the contagion and, subsequently, developinside each individual depending on the interaction at the small scale between virusinfection and immune particles, in host dynamics . The modelling takes account theheterogeneous behaviour of individuals, as well as the heterogeneity, progressionand competition inside each individual entity.
Blocks 3,4:
Show the output of the interactions consisting in recovery or death ofpatients, where this final exit can go through the passage across the hospitalizationwhich is related to the level of the pathology.
Block 5:
Refers to the passage from Block 2 to an organized hospitalizationdynamics. If the dynamics within Block 5 are properly modelled accounting formedical care, the number patients which are recovered should increase, while thatof dead persons should decrease.
Block 6:
Refers to the dynamics by which the contagion spreads over a terri-tory made of a sequence of interconnected areas. The dynamics might includeaggregation through endogenous networks.
Block 7:
Studies the dynamics by which the contagion spreads over a territorythrough long range exogenous networks, where connections between nodes dependon the transportation system.The systems approach has been somehow revisited in Sections 4,5 and 6 by thecontribution from the sciences of virology, immunology, and socio-economics.Sections 4 and 5 mainly, but not exclusively, refer to Block 2, while Section 6mainly refers to the whole systems approach.We do not naively claim that our paper has given an exhaustive description bya differential system of the highly complex dynamics under consideration. Indeed,53n host B B B B B B B κ = α · βγ which refers the intensity of the infection α · β to the immune defence γ . Namely,increasing values of κ denote an increasing level of the infection attack.The following new simulations study the sensitivity to the parameter κ depend-ing on different values of α , β and γ however with the constraint of keeping κ fixed, see Fig. 15. In more detail:• κ = .
2: (a) If β = . α are followed by the same decay of the immune defence due to theconstraint κ = .
2; simulations show that, even if the immune defencedecays, the peak decays and the infection spread is delayed.• κ = .
2: (b) If α = . β are followed by the same decay of the immune defencedue to the constraint κ = .
2; simulations show that, even if the immunedefence decays, the peak decays and the infection spread is delayed, by afaster decay-delay, but the initial infection is much higher than that of case(c).• κ = .
02: (c) The difference with respect to case study (a) is that the decayof the infection rate α only influences the delay.• κ = .
02: (d) The dynamics is analogous to that of the case study (b) butthe height of the peak is lower.A very first, and rapid, biological interpretation is as follows:
The defence of the immune system applies an effective contrast to thevirus progression. However the efficacy of the action is more relevantif the defence keeps a fixed value independently on the level of infectionor progression. If, in addition, the defence increases with increasingvalues of α and β , the efficacy is even higher. Time I n f e c t ed popu l a t i on Time I n f e c t ed popu l a t i on (a) κ = .
2, fixed β = . κ = .
2, fixed α = . Time I n f e c t ed popu l a t i on Time I n f e c t ed popu l a t i on (c) κ = .
02, fixed β = .
01 (d) κ = .
02, fixed α = . κ .(a) Blue: α = . γ = .
2, Red: α = . γ = .
1, Yellow: α = . γ = . β = . γ = .
3, Red: β = . γ = .
2, Yellow: β = . γ = . α = . γ = .
2, Red: α = . γ = .
1, Yellow: α = . γ = . β = . γ = .
3, Red: β = . γ = .
2, Yellow: β = . γ = . T ℓ and T d , respectively, as well as therelated intensity of the actions represented by the contagion coefficients α ℓ and α d ,respectively. The following specific case studies can be simulated by the model:• T ℓ is the lapse of time, after the discovery of the infection, at which lockingbehavioural rules are imposed, corresponding to α ℓ with the aim to keep thenumber of infected people under a threshold level.• T d is the lapse of time from T ℓ to impose less restrictive locking rules(locking-down) corresponding to α d > α ℓ , also in this case with the aim ofkeeping the number of infected people below a threshold level.Figure 16 presents a simulation to investigate the influence of the up-lockingtime, namely the delay to apply the up-locking, on the subsequent dynamics. T T T T d Time I n f e c t ed popu l a t i on T =100T =200T =300 Figure 16: Varying locking times. We take T ℓ = , , T d = α = . t ∈ [ , T ℓ ) ∪ [ T d , T max ] while α = .
25 during thelocking interval. 57igure 17 studies the influence of the lock-down time. Simulations show howdelaying T d reduces the peak, but increases the time interval of the persistence ofthe infection. T l T T T Time I n f e c t ed popu l a t i on T =900T =1200T =1500 Figure 17: Varying de-locking times. We take a fixed locking time T ℓ = T d = , , α = . t ∈ [ , T l ) ∪[ T d , T max ] while α = .
25 during the locking interval. Notice that the three curvescoincide until the first lock-open time, and are consequently represented in blackin that interval.Figure 18 studies the influence of the lock-down level at fixed values of T d and T ℓ . As expected, simulations show how a high level of lock-down can generatehigh level peaks. persistence of the infection.The task of crisis managers consists firstly in providing the strategic plansto reduce and achieve effective values of α ℓ and T ℓ . Subsequently, the problemconsists in referring α d to T d . The strategy depends on a variety of social andpolitical variables which go beyond the aims of our paper. Simulations deliveredin Figures 16–18 do not cover the whole variety of possible case studies. Anexhaustive study can be, however, developed under the direct requirements ofcrisis managers. Therefore, we simply claim that modelling and simulationsprovide support to decision making, while the overall systems approach shown in58 l T d Time I n f e c t ed popu l a t i on α d = 0.3 α d = 0.4 α d = 0.5 Figure 18: Varying the de-locking value α d . We take fixed locking and lock-opentimes T l =
300 and T d = α = . t ∈ [ , T l ) thenreduced to α = .
25 during the locking interval and finally we consider threedifferent lock-open values α d = . , . , .
5. Notice that the three curves coincideuntil the lock-open time, and are consequently represented in black in that interval.Figure 14 leads to study of all different interactive dynamics which appear in thesystem under consideration.Bearing all of the above in mind let us now propose a forward look to researchperspectives by selecting, according to the authors’ bias, five key problems whichare presented in the following, mainly focusing on mathematical topics with theaim of building a bridge between mathematics and biology. However accountingfor the hints, warnings and forward looks on the perspectives proposed in Section 6focused also on the interaction between economic strategies and social dynamics.Indeed, it is not surprising that economists have been induced to develop acritical analysis on the strategies that governments might develop whenever theyaccount for future scenarios and not simply for present situations. As mentionedin Section 1, the event studied in our paper cannot be classified straightforwardlyas a “black swan”, since it could have been predicted, and indeed was predicted bysome, at least up to certain levels. Mathematics offers general explorative models59hat can depict, via computational simulations, the overall collective dynamicsaccounting for the possible actions that the science of medicine can offer. Thiscontribution does not solve the problems of biology and cannot straightforwardlylead to devices of artificial intelligence, but it can help clinicians to develop a carestrategy by offering them additional information to be used alongside their ownclinical experience.Reasonings on the key problems take advantage of the fact that our approach,which does not claim to be exhaustive, appears to be sufficiently flexible to includenew specific features motivated not only by biology, but also by economics.The presentation of our critical analysis refers to the blocks of Figure 14and looks ahead to perspectives to be developed within the multidisciplinary andmultiscale framework proposed in our paper (KP – Key Problem).
KP 1 - Virus progression and immune competition:
This key problem refersto the dynamics inside each individual (Block 2). The modelling approach ofour paper has already introduced some important features of the system underconsideration. For instance, progression of the virus by levels which correspondto hospitalization - followed by medical care - up to the extreme level correspondingto death. The competition between the virus and immune system is also treatedaccounting for the heterogeneous reaction ability of patients.The output of simulations is definitely consistent with the systems vision, butfurther refinements of the model of the dynamics in Block 2 should be developed inorder to contribute to account for the actions of medical care which refer to Block5. Focusing on the objective of designing models with enriched descriptive ability,we remark that the main source towards this development is the literature cited inSections 4 and 5, by which we can learn about firstly about the virus dynamics,subsequently the mechanics of contagion, and finally the different sub-populationswhich play the game and the actions exerted by them. This literature also indicatesthat the biological phenomena related to this specific competition are not fullyunderstood, see [39].Despite this correct statement, we observe that the model of the in-host dy-namics, corresponding to Eqs. (3.3)–(3.5) by the dynamics depicted in Figure 3,is sufficiently flexible to include a more advanced interpretation of biological re-ality. In addition, further dynamics might be taken into account, for instance thatby which the virus chases for foraging and attaches to cells of the lung tissue toexpand by proliferation which ends with further destructive actions over the lungtissue. Analogous care should be focused to model the foraging cells offered by thehost lung which might damage the overall mechanics of the lung in the breathing60ynamics [36].An additional, important refinement consists in subdividing the overall pop-ulations according to age, social and physical state. This subdivision should beuseful not only to refine the modelling of interactions, but also to contribute tothe decision strategy of crisis managers concerning locking, hospitalization and,social support in general.As mentioned, the development of the in-host modelling can take advantage ofSections 4 and 5, starting from their contribution to understanding virus mutationsby which
SARS–CoV–2 is responsible for COVID-19. A pioneer paper devoted tothe modelling of virus mutations followed by a learning dynamics can be foundin [51]. This paper provides some ideas to be developed towards a modellingapproach to depict the complex mutation-learning dynamics specifically referredto COVID-19.
KP 2 - Medical care:
The dynamics of progression of the pathology for hospi-talized patients has not been explicitly treated in our paper as the model simplydescribes how hospitalization can be related to a level of the pathology whichcannot be cared by home confinement. The model computes the flow of infectedpeople towards hospitals and compares it with the effective capacity of them in agiven territory. The model also depicts different scenarios related to the differentlevels by which hospitalization is demanded.This topic deserves further attention towards a detailed study of the role ofmedical cares localized in Block 5 of Figure 14. The modelling approach is not astraightforward generalization of that developed in Block 2 as it requires a detailedmodelling of therapeutical actions. Hence it requires additional interdisciplinarywork from the medical sciences. The representation of Figure 3 is still valid, butprogression of the immune system and regression of the virus should be relatedto well-defined medical actions. The contents of Section 4 and 5 contribute, asmentioned, to model some of the possible therapeutic actions.
KP 3 - Clustered contagion, heterogeneity and monitoring:
As argued in Sec-tion 6 above, the contagion (or not) develops within and across network structureswhich affect who is in contact with whom, for how long, etc. A straightforwarddevelopment ought to address the clustering of interactions in structured spaces -e.g. factories, schools, hospitals, elderly residences, families in primis . It couldwell be for example that above some levels of diffusion, confinement does increasethe spread of the disease in that it confines the interactions but increases their in-teraction rates. Additionally, a necessary extension entails the account for intrinsic61eterogeneity among agents, e.g. young versus old, healthy versus affected byprevious pathology, based in areas with clean versus polluted atmosphere. Finally,in terms of the medical treatment and crisis management, early detection, as homediagnostics by GPs, and continuous monitoring might allow to overturn congestionin hospitals.The model already accounts for heterogeneous behaviours of individuals, but itdoes not yet refer to the aforementioned features. Accounting for them is possiblewithin the framework of the model technically firstly by specializing the overallpopulation into a variety of subsystems and subsequently by referring them to anetwork of aggregation sites. The simulation by agents proposed in [124] movesprecisely towards this target with the aim of providing a broad variety of scenariosto be used by crisis managers. However, it cannot be naively hidden that the priceto pay is an higher level of complexity somehow related to the calibration of themodel treated in the next key problem.Recent empirical investigations have shown that a key role in the in hostdynamics is exerted by the initial virus load in each individual which might beinduced by repeated contacts, e.g. affecting medical staff in hospitals, or simplyby lack of awareness to the risk of contagion. Therefore, further studies shouldbe addressed to derive models where the modelling of the contagion rate includesan additional information on the aforementioned virus load. The main variable incharge of the modelling appears to be the awareness to contagion risks which cancontribute to deeper vision of a lock down strategy. This strategy is not simplyrelated to opening and closing of areas of possible aggregation, but also to the careput on physical distancing and protection.
KP 4 - Calibration of models:
This key problem refers to the identification ofthe parameters, since only if the said parameters can be properly assessed canthe model be used for predictive investigations. This specific aspect of modellinginduces a search for a balance between enriching the complexity of the modeland the difficulty of tuning models with an excess of parameters. A mathematicalmodel has been proposed in Subsection 3.1 corresponding to the flow chart inFig. 2, block i =
2, and the visualization of the progression dynamics and immunereaction shown in Fig. 2, block i =
2, as well as the visualization of the progressiondynamics and immune reaction shown in Fig. 3. The model uses the followingparameters: α modelling the risk of infection related to the level of confinement; β referred to the ability of the virus to progress; and γ modelling the the abilityof the immune system to induce regression of the virus. These parameters havebeen compacted in the key number κ . In addition, the model accounts for the62eterogeneous distribution in the population of the aforementioned ability whichcan be roughly related to the age of the population.Some recent research achievements can contribute to the tuning of specificparameters. A valuable example is the probabilistic study of contagion risk devel-oped in [41] corresponding to different environmental conditions and typology ofinteractions. Simulations have been developed with explorative aims, namely by asensitivity analysis of the dynamical response depending on the parameters of themodel. This output has shown to be useful to crisis managers towards the decisionprocess about the possible strategies. An example has been the study of the up-lockand lock-down strategy by simulations which have shown how far the system issensitive to closing-opening plans and have shown how a wrong decision mightlead to disasters. An exhaustive sensitivity analysis might definitely contribute tothe design of an artificial intelligence system. KP 5 - Dynamics over a globally connected world:
This topic may be the mostchallenging key problem and research perspective. It refers to Blocks 6 and 7,where the dynamics occur involving both healthy and infected people. Someindications have been given towards modelling and simulations accounting bothfor the dynamics in complex venues and for that involving transportation system.The full development of this hint leads to the completion of all the details of theoverall systems approach, where the dynamics in each block contributes to definethe global dynamics in the world.We are aware that the interdisciplinary horizons should be enriched involvingadditional experts in various research fields, e. g. medicine, informatics, physicsand various others. Indeed, we simply wish that our paper is a very first step towardsthis forward-looking scientific dialogue. Therefore all reasonings proposed in thissection allow us to go back to the key target posed at the end of Section 1 byrevisiting the quotation, given in the last part of the section, by the followingconclusive remarks.
The key target of this paper has been the design of a multiscale mod-elling approach suitable to produce simulations mainly with explo-rative ability. This objective has been pursued all along the paperby mathematical tools of the kinetic theory of active particles along asystems approach suitable to account of the dynamics and interactionsin blocks sketched in Figure 14. he systems approach and the multiscale vision appear firstly in thespecific models in each block, where the modelling needs the use morethan one scale, and subsequently in the modelling of interactionsbetween blocks.The critical analysis focused on the key problems has shown howthe various ingredients of the approach can be further developed andimproved. This remark motivates further research activity to updatethe models of each block shown in Fig. 14. This means that weconsider our paper a receptor of future scientific contributions fromthe scientific community somehow motivated also by individual andsocial care. Finally, going back again to mathematical topics, we recall that, concerningthe modelling of smallpox, Daniel Bernoulli wrote (to Euler) [55]:
Dolus an virtus quis in hoste requirat! [135] (‘What matters whetherby valour or by stratagem we overcome the enemy?’)Concerning COVID-19 the correct stratagem, we believe, is to develop a novelmultiscale mathematical framework informed by multidisciplinary data as outlinedin the current paper. The current “crisis” is presenting us, and all mathematicians,with an opportunity to develop new mathematical theories, ideas and techniques,not only to shed light on the spread of pandemics, but also to further develop themathematics itself.It is perhaps fitting to close this paper with the final words of the visionarytalk by David Hilbert in 1900 before the opening of the Mathematical Congress ofMathematicians in Paris [72, 73, 74]:
We also notice that, the farther a mathematical theory is developed,the more harmoniously and uniformly does its construction proceed,and unsuspected relations are disclosed between hitherto separatebranches of the science. So it happens that, with the extension ofmathematics, its organic character is not lost but only manifests itselfthe more clearly. . . .
The organic unity of mathematics is inherentin the nature of this science, for mathematics is the foundation of allexact knowledge of natural phenomena. That it may completely fulfilthis high mission, may the new century bring it gifted masters andmany zealous and enthusiastic disciples. cknowledgements Nicola Bellomo: Support of Granada University, modelling Nature Group, Spain,and Hosting support of the Italian Research Council, IMATI, CNR, Pavia, Italy.Reidun Twarock: Financial support via an EPSRC Established Career Fellowship(EP/R023204/1), a Royal Society Wolfson Fellowship (RSWF\R1\180009), and aJoint Investigator Award to RT and Prof. Peter Stockley (110145 & 110146) isgratefully acknowledged.Mark Chaplain acknowledges the assistance of the Rapid Assistance in Modellingthe Pandemic project coordinated by the Royal Society.Giovanni Dosi and Maria Enrica Virgillito acknowledge support from EuropeanUnion’s Horizon 2020 research and innovation programme under grant agreementNo. 822781 GROWINPRO – Growth Welfare Innovation Productivity.Damian Knopoff: Support of CONICET (Grant Number PIP 11220150100500CO)and Secyt UNC (Grant Number 33620180100326CB).John Lowengrub acknowledges partial support from US National Science Foun-dation grants DMS-1763272 and the Simons Foundation (594598QN) for a NSF-Simons Center for Multiscale Cell Fate Research, as well as grants DMS-1714973and DMS-1936833. JL also thanks the US National Institutes of Health for partialsupport through grants 1U54CA217378-01A1 for a National Center in Cancer Sys-tems Biology at UC Irvine and P30CA062203 for the Chao Family ComprehensiveCancer Center at UC Irvine.
References [1] P. Adhikari, Intra- and intermolecular atomic-scale interaction of the receptorbinding domain in SARS-CoV-2 spike protein: implication for ACE2 receptorbinding, Preprint,(2020).[2] G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. Poyato,and J. Soler, Traffic, crowds, and swarms. From kinetic theory and multiscalemethods to applications and research perspectives,
Mathematical Models andMethods in Applied Sciences , , 1901–2005, (2019).[3] K. G. Andersen, A. Rambaut, W. Ian Lipkin, E. C. Holmes, and R. F. Garry,The proximal origin of SARS-CoV-2, Nature Medicine , , no. 4, 450–452,(2020). 654] R. A. Anderson and R. M. May, Population biology of infectious diseases:Part I, Nature , , 361–367, (1979).[5] R. A. Anderson and R. M. May, Population biology of infectious diseases:Part II, Nature , , 455–461, (1979).[6] R. M. Anderson, The Population Dynamics of Infectious Diseases: Theoryand Application , London: Chapman and Hall, (1982).[7] V. V. Aristov, Biological systems as nonequilibrium structures described bykinetic methods,
Results in Physics , , paper n.102232, (2019).[8] B. Avishai, The pandemic isn’t a black swan but a portent of a more fragileglobal system, The New Yorker
Mathematical Models and Methods in Applied Sciences , , 1–22, (2020).[10] N. T. J. Bailey, Mathematical Tools for Understanding Infectious DiseaseDynamics , London: Griffin, (1975).[11] P. Baldwin and B. W. Di Mauro,
Economics in the Time of COVID-19 ,VoxEU.org Book, (2020).[12] P. Ball,
Why Society is a Complex Matter , Springer-Verlag, Heidelberg,(2012).[13] Y. M. Bar-On, A. Flamholz, R. Phillips, and R. Milo, SARS-CoV-2 (COVID-19) by the numbers, eLife , (2020), 1–15.[14] R. J. Barro, J. F. Ursua, and J. Weng, The Coronavirus and the Great In-fluenza Pandemic: Lessons from the “Spanish Flu” for the Coronavirus’sPotential Effects on Mortality and Economic Activity , NBER WP , 26866,(2020).[15] N. Bellomo and A. Bellouquid, On multiscale models of pedestrian crowdsfrom mesoscopic to macroscopic,
Communications in Mathematical Sciences , Mathematical Models and Methods in Applied Sciences , , 2041–2069,(2016).[17] N. Bellomo, A. Bellouquid, and D. Knopoff, From the micro-scale tocollective crowd dynamics, Multiscale Modelling Simulation , , 943–963,(2013).[18] N. Bellomo, A. Bellouquid, L. Gibelli, and N. Outada, A Quest Towards aMathematical Theory of Living Systems , Birkhäuser, New York, (2017).[19] N. Bellomo and L. Gibelli, Toward a mathematical theory of behavioral-social dynamics for pedestrian crowds,
Mathematical Models and Methods inApplied Sciences , , 2417–2437, (2015).[20] N. Bellomo, L. Gibelli, and N. Outada, On the interplay between behavioraldynamics and social interactions in human crowds, Kinetic Related Models , , 397–409, (2019).[21] N. Bellomo, K. J. Painter, Y. Tao, and M. Winkler, Occurrence vs. absenceof txis-driven instabilities in a May–Nowak Model for virus infection, SIAMJournal Applied Mathematics , , 1990–2010, (2019).[22] A. Bellouquid and M. Delitala, Modelling Complex Biological Systems -A Kinetic Theory Approach , Series: Modeling and Simulation in Science,Engineering and Technology, Birkhäuser, Boston, (2006).[23] D. Bernoulli, Essai d’une nouvelle analyse de la mortalité causée par la petitevérole et des avantages de l’inoculation pour la prévenir,
Mém. MathemmatiqePhysique Academy Sciences, Paris , , (1760/1766)[24] A. L. Bertozzi, J. Rosado, M. B. Short, and L. Wang, Contagion shocks inone dimension, Journal Statistical Physics , , 647–664, (2015).[25] R. Bingham, E. Dykeman, and R. Twarock, RNA virus evolution via aquasispecies-based model reveals a drug target with a high barrier to resistance, Viruses , , no. 11, 347, (2017).[26] G. M. Birchenough, M. E. Johansson, et al., New developments in gobletcell mucus secretion and function, Mucosal Immunology , , n. 4, 712, (2015).6727] B. M. Bolker and B. T. Grenfell, Space, persistence and dynamics ofmeasles epidemics, Philosophycal Transactions Royal Society B , , 309–320, (1995).[28] B. M. Bolker and B. T. Grenfell, Impact of vaccination on the spatial correla-tion and persistence of measles dynamics, Proceedings National AcadademySciences USA , , 12648–12653, (1996).[29] D. Burini and N. Chouhad, A Multiscale view of nonlinear diffusion inbiology: from, cells to tissues, Mathematical Models and Methods in AppliedSciences , , 791–823, (2019).[30] D. Burini and S. De Lillo, On the complex interaction betweencollective learning and social dynamics. Symmetry , , 967, (2019),doi:10.3390/sym11080967.[31] D. Burini, S. De Lillo, and L. Gibelli, Collective learning dynamics modelingbased on the kinetic theory of active particles, Physics of Life Review , ,123–139, (2016).[32] E. Callaway, Coronavirus vaccines: five key questions as trials begin, Nature , no. 7800, 481–481, (2020).[33] V. Capasso, E. Grosso and G. Serio, Mathematical models in epidemiologicalanalysis. 1. Application to cholera pandemic in Bari in 1973,
Annali Sclavo , , 193–208, (1977).[34] V. Capasso and S. L. Paveri-Fontana, A mathematical model for the 1973cholera epidemic in the European Mediterranean region, Review Epidemiqueet Santé Publique , , 121–132, (1979). Errata, Ibidem , 390 (1980).[35] V. Capasso and L. Maddalena, Convergence to equilibrium states for areaction-diffusion system modelling the spatial spread of a class of bacterialand viral diseases, Journal of Mathematical Biology , , 173–184, (1981).[36] A. Carloni, V. Poletti, L. Fermo, N. Bellomo, and M. Chilosi, Heterogeneousdistribution of mechanical stress in human lung: A mathematical approach toevaluate abnormal remodeling, Journal of Theoretical Biology , , 136–140,(2013). 6837] M. Cecconi, G. Forni, and A. Mantovani, COVID-19: An executive reportApril 2020 update, Accademia Nazionale dei Lincei, Commissione Salute
Intereconomics , forthcoming,(2020).[39] Z. J. Cheng and J .Shan, 2019 Novel Coronavirus: Where we are and whatwe know,
Posted: 31 January 2020.doi:10.20944/preprints202001.0381.v1[40] Chun-Yuan Chen, Chung-ke Chang, et. al., Structure of the SARS coro-navirus nucleocapsid protein RNA-binding dimerization domain suggests amechanism for helical packaging of viral RNA,
Journal of Molecular Biol-ogy , , no. 4, 1075–1086, (2007).[41] P. Cirillo and N. N. Taleb, Tail risk of contagious diseases, Nature Physics ,PERSPECTIVE, https://doi.org/10.1038/s41567-020-0921-x, (2020).[42] S. M. Ciupe and J. M. Heffernan, In-host modeling,
Infectious DiseaseModelling , , no. 2, 188–202, (2017).[43] J. M. Conway and A. S. Perelson, A Hepatitis C Virus Infection Model withTime-Varying Drug Effectiveness: Solution and Analysis, PLoS Computa-tional Biology , New York Times 2020
Nature , ,no. 4506, 473–475, (1956). 6947] D. Cyranoski, Profile of a killer: the complex biology powering the coron-avirus pandemic, Nature , , no.7806, 22, (2020).[48] J. le R. d’Alembert, Onzième mémoire, Sur l’application du calcul desprobabilités à l’inoculation de la petite vérole, in: Opuscules Mathématiques,Tome second , David, Paris, pp. 26–95, (1761).[49] M. Day, Covid-19: four fifths of cases are asymptomatic, China figuresindicate,
British Medicine Journal , , m1375, (2020).[50] P. P. Dechant, J. Wardman, T. Keef, and R. Twarock, Viruses and fullerenes- Symmetry as a common thread?, Acta Crystallographica Section A: Foun-dations and Advances , , no. 2, 162–167, (2014).[51] S. De Lillo, M. Delitala and M. Salvatori, Modelling epidemics and virusmutations by methods of the mathematical kinetic theory for active parti-cles, Mathematical Models and Methods in Applied Sciences , , 1405–1425,(2009).[52] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology ofInfectious Diseases : Model Building, Analysis and Interpretation
WileySeries in Mathematical and Computational Biology, (2000).[53] O. Diekmann, J. A. P. Heesterbeek and T. Britton,
Mathematical Tools forUnderstanding Infectious Disease Dynamics
Princeton University Press,(2012).[54] O. Diekmann, Limiting behaviour in an epidemic model,
Nonlinear Analysis,Theory, Methods and Applications , , 459–470, (1977).[55] K. Dietz and J. A. P. Heesterbeek, Daniel Bernoulli’s epidemiological modelrevisited, Mathematical Biosciences , , 1–21, (2002).[56] I. Dubanevics and T. C. B. Mcleish, Computational Analysis of DynamicAllostery and Control in the SARS-CoV-2 Main Protease, Preprint, (2020).[57] E.C. Dykeman, P.G. Stockley, and R. Twarock, Solving a Levinthal’s paradoxfor virus assembly identifies a unique antiviral strategy, Proceedings of theNational Academy of Sciences , , no. 14, 5361–5366, (2014).7058] A. D. Elmezayen, A. Al-Obaidi, U. Alp Tegin, and K. Yelekçi, Drugrepurposing for coronavirus (COVID-19): in silico screening of knowndrugs against coronavirus 3CL hydrolase and protease enzymes, Journalof Biomolecular Structure and Dynamics , (2020), no. 0, 1–13.[59] F. Fenner, D. A. Henderson, et al., Smallpox and its eradication, vol. 6, World Health Organization Geneva , (1988).[60] M. J. Ferrari, B. T. Grenfell and P. M. Strebel, Think globally, act locally: therole of local demographics and vaccination coverage in the dynamic responseof measles infection to control,
Philosophical Transactions Royal Socety B , : 20120141, (2013).[61] R. A. Fridell, C. Wang, et al., Genotypic and phenotypic analysis of vari-ants resistant to hepatitis C virus nonstructural protein 5A replication complexinhibitor BMS-790052 in Humans: In vitro and in vivo correlations, Hepa-tology , Nature , ,no. 7062, 1162–1166, (2005).[65] J. Hadfield, C. Megill, et al., NextStrain: Real-time tracking of pathogenevolution, Bioinformatics , , no. 23, 4121–4123, (2018).[66] B. T. Grenfell, O. N. Bjornstad and J. Kappey, Travelling waves and spatialhierarchies in measles epidemics, Nature , , 716–723, (2001).[67] J. Guedj, H. Dahari, L. Rong, N. D. Sansone, R. E. Nettles, S. J. Cotler,T. J.Layden, S L. Uprichard, and A. S. Perelson, Modeling shows that the NS5Ainhibitor daclatasvir has two modes of action and yields a shorter estimateof the hepatitis C virus half-life, Proceedings of the National Academy ofSciences , , no. 10, 3991–3996, (2013).7168] J. Guedj, L. Rong, H. Dahari, and A. S. Perelson, A perspective on modellinghepatitis C virus infection, Journal of Viral Hepatitis , , no. 12, 825–833,(2010).[69] Groupe d’études géopolitiques de la ENS, La Observatoire du Covid-19, LeGrand Continent , https://legrandcontinent.eu/fr/observatoire-coronavirus/[70] H.L. Hartwell, J.J. Hopfield, S. Leibler, and A.W. Murray, From molecularto modular cell biology,
Nature , , c47–c52, (1999).[71] H. Heesterbeek, R. M. Anderson, V. Andreasen, S. Bansal, D. De An-gelis, C. Dye, K. T. D. Eames, W. J Edmunds, S. D. W. Frost, S. Funk,T. D. Hollingsworth, T. House, V. Isham, P. Klepac, J. Lessler, J. O. Lloyd-Smith, C. J. E. Metcalf, D. Mollison, L. Pellis, J. R. C. Pulliam, M. G. Roberts,C. Viboud, Isaac Newton Institute IDD Collaboration, Modeling infectiousdisease dynamics in the complex landscape of global health, Science , ,aaa4339, (2015).[72] D. Hilbert, Mathematische Probleme, Göttinger Nachrichten : 253–297(1900).[73] D. Hilbert, Mathematische Probleme,
Archiv der Mathematik und Physik :3rd ser., , 44–63, 213–237 (1901).[74] D. Hilbert, Mathematical problems, Bulletin of the American MathematicalSociety , Journal of the Royal Society Interface , , no. 157, (2019).[77] International Monetary Fund, World Economic Outlook, April 2020: TheGreat Lockdown , (2020).[78] M. Hoffmann, H. Kleine-Weber, et al., SARS-CoV-2 cell entry depends onACE2 and TMPRSS2 and is blocked by a clinically proven protease inhibitor,
Cell , , n.2, 271, (2020). 7279] Johns Hopkins University, Mortality Analyses ,https://coronavirus.jhu.edu/data/mortality (2020).[80] R. Lorenzo-Redondo, H.R. Fryer, et al., Persistent HIV-1 replication main-tains the tissue reservoir during therapy,
Nature , , no. 7588, 51–56, (2016).[81] A. Kallen, P. Arcuri and J. D. Murray, A simple model for the spatial spreadand control of rabies, Journal of Theoretical Biology , , 377–393, (1985).[82] T. Keef, R. Twarock, and K. M. Elsawy, Blueprints for viral capsids inthe family of Polyomaviridae, Journal of Theoretical Biology , , no. 4,808–816, (2008).[83] T. Keef, J.P. Wardman, N.A.Ranson, P.G.Stockley, and R. Twarock, Struc-tural constraints on the three-dimensional geometry of simple viruses: Casestudies of a new predictive tool, Acta Crystallographica Section A: Founda-tions of Crystallography , , no. 2, 140–150, (2013).[84] W. O. Kermack and A. G. McKendrick, A contribution to the mathematicaltheory of epidemics, Proceedings of the Royal Society of London. Series A , , 700–721, (1927).[85] Dongwan Kim, Joo-Yeon Lee, Jeong-Sun Yang, Jun Won Kim, V NarryKim, and Hyeshik Chang, The Architecture of SARS-CoV-2 Transcriptome, Cell , , no. 4, 914–921.e10 (2020).[86] D. Kim and A. Quaini, A kinetic theory approach to model pedestriandynamics in bounded domains with obstacles, Kinetic Related Models , ,1273–1296, (2019).[87] D. Kim and A. Quaini, Coupling kinetic theory approaches for pedes-trian dynamics and disease contagion in a confined environment, Mathe-matical Models and Methods in Applied Sciences , , (2020), to appear, arXiv:2003.08357v1 [physics.soc-ph] , 16 Mar 2020, (2020).[88] S.M. Kissler, C. Tedijanto, E. Goldstein, Y.H. Grad, M. Lipsitch, Projectingthe transmission dynamics of SARS-CoV-2 through the postpandemic period, Science , 14 April, eabb5793 DOI: 10.1126/science.abb5793, (2020).[89] D. Knopoff, On the modeling of migration phenomena on small networks,
Mathematical Models and Methods Applied Sciences , , 541–563, (2013).7390] D. Knopoff, On a mathematical theory of complex systems on networks withapplication to opinion formation, Mathematical Models and Methods AppliedSciences , , 405–426, (2014).[91] D. Knopoff and F. Trucco, A compartmental model for antibiotic resistantbacterial infections over networks, International Journal of Biomathematics , (1), 2050001 (16 pages), (2020).[92] K. Kupferschmid, Why do some COVID-19 patients infect manyothers, whereas most don’t spread the virus at all? Science ,doi:10.1126/science.abc8931. Online ahead of print, (2020).[93] K. Kupferschmid, Can plasma from COVID-19 survivors help save others?
Science , doi:10.1126/science.abd0355. Online ahead of print, (2020).[94] H.R. Kwon and E.A. Silva, Mapping the Landscape of Behavioral Theories:Systematic Literature Review.
Journal Plannning Literature , Article Number:UNSP 0885412219881135, (2019).[95] J. Laoukili, E. Perret, et al., IL-13 alters mucociliary differentiation andciliary beating of human respiratory epithelial cells,
Journal Clinical Investi-gation , , n.12, 18174, (2001).[96] Liang Li, Hong Liu, and Yanbin Han, An approach to congestion analy-sis in crowd dynamics models, Mathematical Models and Methods AppliedSciences , , 867–890, (2020).[97] Z. Liu, P. Magal, O. Seydi, and G. Webb, Understanding unreported casesin the 2019-nCov epidemic outbreak in Wuhan, China, and the importance ofmajor public health interventions, 2020 by the author(s). Distributed under aCreative Commons CC BY license.[98] P.M. Matricardi, R.W. Dal Negro, and R. Nisini, The First, Holistic Immuno-logical Model of COVID-19: Implications for Prevention, Diagnosis, andPublic Health Measures, Pediatric Allergy Imunology , doi:10.1111/pai.13271,(2020),[99] O.A. MacLean, R.J. Orton, J.B. Singer, and D.L. Robertson, No evidencefor distinct types in the evolution of SARS-CoV-2,
Virus Evolution , Nature , , 137–142, (1987).[102] J.B. Moore and C.H. June, Cytokine release syndrome in severe COVID-19, Science , , no.6490, 473, (2020).[103] S. Moreno-Gamez, A.L. Hill, D.I.S. Rosenbloom, D.A.Petrov,M.A. Nowak, and P.S. Pennings, Imperfect drug penetration leads to spa-tial monotherapy and rapid evolution of multidrug resistance, Proceedings ofthe National Academy of Sciences , , no. 22, E2874–E2883, (2015).[104] J. D. Murray, Mathematical Biology II: Spatial Models and BiomedicalApplications (3rd Ed.), pp. 661–721, Springer-Verlag, New York, (2003).[105] M. A. Nowak, R. A. Anderson, A. R. McLean, T. F. W. Wolfs, J. Goudsmitand R. M. May, Antigenic diversity thresholds and the development of AIDS,
Science , , 963–969, (1991).[106] M. A. Nowak and R. M. May, Mathematical biology of HIV infections:antigenic variation and diversity threshold, Mathematical Biosciences , ,1–21, (1991).[107] S. Ojosnegros, C. Perales, A. Mas, and E. Domingo, Quasispecies as amatter of fact: Viruses and beyond, Virus Research , , no. 1-2, 203–215,(2011).[108] N.M.A. Okba and M.A. Müller, et al., Antibody Responses in Coron-avirus Disease 2019 Patients, Emerging Infectious Diseases , , no.7, doi:10.3201/eid2607.200841. Online ahead of print (2020).[109] G. Pawelec, Age and Immunity: What Is “Immunosenescence”? Experi-mental Gerontology , , 4, (2018).[110] L. Pareschi and G. Toscani, Interacting Multiagent Systems: KineticEquations and Monte Carlo Methods
Oxford University Press, Oxford,(2013). 75111] A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard and D. D. Ho,HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viralgeneration time,
Science , , 1582–1586, (1996).[112] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamicsin vivo, SIAM Review , , 3–44, (1999).[113] A. S. Perelson Modelling viral and immune system dynamics, NatureReviews Immunology , , 28–36, (2002).[114] H. Rahmandad and J. Sterman, Heterogeneity and Network Structure inthe Dynamics of Diffusion, Management Science , , 998-1014, , 249–278,(2008).[115] I. M. Rouzine, A. D. Weinberger, and L. S. Weinberger, An evolutionary rolefor HIV latency in enhancing viral transmission, Cell , , no. 5, 1002–1012,(2015).[116] Susmita Roy, Dynamical asymmetry exposes 2019-nCoV prefusion spike,bioRxiv, Preprint, 2020.04.20.052290, (2020).[117] R. Sanjuán and P. Domingo-Calap, Mechanisms of viral mutation, Cellularand Molecular Life Sciences , , no. 23, 4433–4448, (2016).[118] E. Schrödinger, What is Life? The Physical Aspect of the Living Cell ,Cambridge University Press, Cambridge, (1944).[119] Service RF, NAS Letter Suggests “Normal Breathing” Can Expel Coron-avirus,
Science , , no.6487, 119, (2020).[120] N. Sfakianakis, A. Madzvamuse, and M.A.J. Chaplain, A hybrid multiscalemodel for cancer invasion of the extracellular matrix, Multiscale ModelingSimulations , (2020).[121] Scienza in Rete,
On line journal of Gruppo2003 per la ricerca scientifica
Viruses , no. 3, 254,(2020).76123] N. N. Taleb, The Black Swan: The Impact of the Highly Improbable ,Random House, New York City, (2007).[124] P. Terna, G. Pescarmona, A. Acquadro, P. Pescarmona, G. Russo, andS. Terna, (2020), An agent-based model of the diffusion of covid-19 usingNetLogo,URL https://terna.to.it/simul/SIsaR.html, (2020).[125] S. Toki, K. Goleniewska, et al., TSLP and IL-33 reciprocally promoteeach other’s lung protein expression and ILC2 receptor expression to enhanceinnate type-2 airway inflammation,
Allergy , doi: 10.1111/all.14196. Onlineahead of print, (2020).[126] R. Twarock, A tiling approach to virus capsid assembly explaining astructural puzzle in virology,
Journal of Theoretical Biology , , no. 4,477–482, (2004).[127] R. Twarock, Mathematical models for tubular structures in the family ofPapovaviridae, Bulletin of Mathematical Biology , , no. 5, 973–987, (2005).[128] R. Twarock, A mathematical physicist’s approach to the structure and as-sembly of viruses, Philosophical Transactions of the Royal Society of London.Series A , , 3357–3374, (2006).[129] R. Twarock, Geometry as a weapon in the fight against viruses, MathematicsToday , 184–187, (2019).[130] R. Twarock and A. Luque, Structural puzzles in virology solved with anoverarching icosahedral design principle,
Nature Communications , , no. 1,4414, (2019).[131] R. Twarock, R. J. Bingham, E. C. Dykeman, and P. G. Stockley, A modellingparadigm for RNA virus assembly, Current Opinion in Virology , , no. Figure1, 74–81, (2018).[132] R. Twarock, German Leonov, and Peter G. Stockley, Hamiltonian pathanalysis of viral genomes, Nature Communications (2018), no. 1, (2021).[133] N. Vabret, G.J. Britton, et al., Immunology of COVID-19: Current state of the science, Immunity 2020 , DOI:https://doi.org/10.1016/j.immuni.2020.05.002 Online ahead of print, (2020).77134] N. van Doremalen, T. Bushmaker, et al., Aerosol and surface stability ofSARS-CoV-2 as compared with SARS-CoV,
New England Journal Medicine , , no. 16, 1564, (2020).[135] P. Vergilius Maro, Aeneid II , line 390.[136] M. Wadman, J. Couzin-Frankel, J. Kaiser, and C. Matacic, A rampagethrough the body,
Science , , no.6489, 356, (20(20).[137] L. Wang, M.B. Short, and A.L. Bertozzi, Efficient numerical methods formultiscale crowd dynamics with emotional contagion, Mathematical Modelsand Methods in Applied Sciences , , 205–230, (2017).[138] W. Wang, Y. Xu Y, et al.,. JAMA 323 (2020) no. 18, 1843. Detection ofSARS-CoV-2 in Different Types of Clinical Specimens, JAMA, Journal of theAmerican Medical Association , , n. 18, 1843, (2020).[139] D. Wrapp, Nianshuang Wang, K. S. Corbett, J. A. Goldsmith, Ching LinHsieh, O. Abiona, B. S. Graham, and J. S.McLellan, Cryo-EM structure ofthe 2019-nCoV spike in the prefusion conformation, Science , , no. 6483,1260–1263, (2020).[140] Zhuo Zhou, Lili Ren, et. al., Heightened innate immune responses in therespiratory tract of COVID-19 patients, Cell Host and Microbes ,DOI: https://doi.org/10.1016/j.chom.2020.04.017 Online ahead of print,(2020).[141] Peng Zhou, Xing Lou Yang, et al. A pneumonia outbreak associated witha new coronavirus of probable bat origin,
Nature , , no. 7798, 270–273,(2020).[142] Yadi Zhou, Yuan Hou, Jiayu Shen, Yin Huang, William Martin, andFeixiong Cheng, Network-based drug repurposing for novel coronavirus 2019-nCoV/SARS-CoV-2, Cell Discovery , , no. 1, 14, (2020).[143] C.G.K. Ziegler, S.J. Allon, et al. SARS-CoV-2 receptor ACE2 is aninterferon-stimulated gene in human airway epithelial cells and is detected inspecific cell subsets across tissues, Cell ,181