Epidemic response to physical distancing policies and their impact on the outbreak risk
EEpidemic response to physical distancing policies and theirimpact on the outbreak risk
Fabio Vanni ∗†‡
David Lambert § ‡
Luigi Palatella ¶ Abstract
We introduce a theoretical framework that highlights the impact of physical distancingvariables such as human mobility and physical proximity on the evolution of epidemics and,crucially, on the reproduction number. In particular, in response to the coronavirus disease(CoViD-19) pandemic, countries have introduced various levels of ’lockdown’ to reduce thenumber of new infections. Specifically we use a collisional approach to an infection-age struc-tured model described by a renewal equation for the time homogeneous evolution of epidemics.As a result, we show how various contributions of the lockdown policies, namely physical prox-imity and human mobility, reduce the impact of SARS-CoV-2 and mitigate the risk of diseaseresurgence. We check our theoretical framework using real-world data on physical distanc-ing with two different data repositories, obtaining consistent results. Finally, we propose anequation for the effective reproduction number which takes into account types of interac-tions among people, which may help policy makers to improve remote-working organizationalstructure.
Keywords:
Renewal equation, Epidemic Risk, Lockdown, Social Distancing, Mobility, SmartWork.
As the coronavirus disease (CoViD-19) epidemic worsens, understanding the effectiveness of publicmessaging and large-scale physical distancing interventions is critical in order to manage the acute ∗ Sciences Po, OFCE , France † Institute of Economics, Sant’Anna School of Advanced Studies, Pisa, Italy ‡ Department of Physics, University of North Texas, USA § Department of Mathematics, University of North Texas, USA ¶ Liceo Scientifico Statale “C. De Giorgi”, Lecce, Italy a r X i v : . [ phy s i c s . s o c - ph ] J u l nd the long-term phases of the spread of the epidemic. The CoViD-19 epidemic has forced manycountries to react by imposing strategies primarily based on mobility and physical lockdowns to-gether with intranational and international border limitations. Data regarding these interventionscan help refine future efforts by providing near real-time information about changes in patternsof human movement. In particular, estimates of aggregate flows of people are incredibly valu-able. Infectiousness depends on the frequency of contacts and on the level of infection within eachindividual. In airborne infections, the former can be decomposed as a product of mobility andphysical proximity, interpreted broadly as an effective distance measure which also includes thedegree of physical protection used by individuals aware of the risk of infection. The latter involvesan internal micro-scale competition between the virus and the immune system which depends onenvironmental factors like pollution levels and repeated viral exposure, which can modify the viralload shed by infectious individuals. Moreover, one should distinguish between random movements(as supposed in our collision approach) and structured contacts (as in crowds or workplaces) sothat interactions can be shaped by packing density of individuals. In the example we analyzebelow, we assume these factors to be homogeneous in space (within each region of interest), whichrepresents only an approximation of the real infection transmission. In reality, these factors arespatially heterogeneous and the interactions involve decision processes which change the directionand speed of the individual so changing the kinematical dynamics of the epidemic.The economic impact of the CoViD-19 pandemic is mainly due to indirect effects related to policiesof physical distancing, see Bellomo et al. [6]. So it would be desirable to show how the risk ofcontagion is related to heterogeneous exposure to infection which is more concentrated in certainworker categories such as those in the health sectors and less concentrated for workers able towork remotely. Indirect effects of the mitigatory response to the CoViD-19 are primarily due tothe mobility side of the lockdown. This is because reducing personal mobility implies closing manybusinesses and services. However, many jobs have been converted into remote-work from home.This conversion is in a proportion which is higher than what is commonly supposed to be affordableby the job system without decreasing the national economic productivity.Mechanistic models of disease transmission are often used to forecast disease trajectories and likelydisease burden but are currently hampered by substantial uncertainty in disease epidemiology.Models of disease transmission dynamics are hindered by uncertainty in the role of asymptomatictransmission, the length of the incubation period, the generation interval, and the contribution ofdifferent modes of transmission. Phenomenological models provide a starting point for estimationof key transmission parameters, such as the reproduction number, and forecasts of epidemic impact.They represent promising tools to generate early forecasts of epidemic impact particularly in thecontext of substantial uncertainty in epidemiological parameters, Breda et al. [7], Chowell et al.[11], Flaxman et al. [28], Metz [47], Nishiura [49], Yan and Chowell [63]. From a practical pointof view, it is fundamental to understand which approach best permits one to forecast epidemicdynamics in the presence of incomplete data that, due to overload in the healthcare system, veryoften are only available during the early phases of disease spread. For CoViD-19 this is especially2rue due to the number of undetected cases, since the total daily number of tests that can beperformed is limited. In our study, we focus our attention on the contribution of asymptomaticor undiagnosed individuals to the propagation of the contagion, as well as the impact of physicaldistancing policies in response to the epidemic outbreak of CoViD-19 in Italy and the US. First, weintroduce the renewal equation approach to the evolution of epidemics to estimate several crucialvariables from data. We use it in the evaluation of the reproduction number of SARS-CoV-2 underan active containment strategy, and investigate implications for future risk. Later, we focus onthe dynamic response of the epidemic to mobility lockdown and make an estimate of the hazardin relaxing this policy alongside other factors. In conclusion, the approach we follow concentrateson the fraction of people which are infectious but have not been detected, i.e., not reported asinfected. We assume these hidden infectious agents have the ability to spread the disease in anenvironment where susceptible agents are present and all the individuals have certain mobilityand physical distance parameters. We interpret this approach in terms of a macroscopic collisiontheory of infected individuals in a region with a given susceptible population, taking into accountthe mobility of individuals as well as their radii of interaction as a proxy of physical distancing.In the next section, we introduce the renewal theory of epidemic dynamics and embed within ita kinetic theory of collisional contacts among individuals with the contribution of two effects ofphysical distancing: mobility and interpersonal proximity. In section 3, we depict and discussthe data we used and the results we obtain using the collisional theory of epidemics so as toexpress the effective reproduction number in terms of physical distancing variables. In Section4, we introduce a reproduction number as combinations of various categories which contributewith different weights to the overall epidemic trend. We show how this definition could be usedfor policy purposes in managing mobility of workers in a smart-working perspective. Finally, wediscuss the possible future lines of research on top of our approach to epidemics. The most important assumptions in our use of phenomenological models are (1) Short time scale ofthe epidemic (much shorter than the characteristic birth and death time scales of the population)(2) Well mixed population (force of infection homogeneously the same for all ages, sexes, etc.) (3)closed population (no immigration or emigration) (4) initial small shock (the initial infected groupextremely small with respect to the size of the susceptible population). The renewal equation wasintroduced in the context of population dynamics studies. Later it was reinterpreted along thelines of stochastic processes, as in Fraser [32], where transmission occurred via a Poisson infectionprocess. This process is such that the probability that, between time t and t + δt , someone infecteda time τ ago successfully infects someone else is A ( t, τ ) δt , where δt is a very small time interval.As a consequence, the predicted mean infectious incidence at time t follows the so-called renewal3 a) CoViD-19 infection incidence by onset of symptoms. Source: ISS [41]. (b)
Mobility trend. Source: Google [35] . (c)
Individual protection, source: DPC [19].
Figure 1:
Social distancing data in Italy: (a) CoViD-19 incidence . (b) percent change in visits to different placeswith respect to baseline. (c) Number of respiratory protectors (face masks) as a proxy measure of increaseddistancing precautions. Vertical lines represent the beginning and the end of the lockdown policy in Italy. j ( t ) = (cid:90) ∞ A ( t, τ ) j ( t − τ ) dτ (1) j ( t ) = − ddt n s ( t ) , (2)where τ is the generation time, which describes the duration from the onset of infectiousness inthe primary case to the onset of infectiousness in a secondary case (infected by the primary case),and j ( t ) is the rate of production of infectious individuals. The kernel A ( t, τ ) is the average rateat which an individual infected τ time units earlier generates secondary cases. In other words A ( t, τ ) is the expected infectivity of an individual with infection-age τ , it can be interpreted asthe reproduction function for new infections at time t . A practical issue concerns the extrinsicdynamics (e.g., public health interventions) of time inhomogeneities highlighting the depletion ofsusceptible individuals when contact tracing, quarantine, and isolation are implemented during thecourse of an epidemic. The kernel A can be factorized as A ( t, τ ) = n s ( t ) β ( t, τ )Γ( t, τ ) , where β ( t, τ ) is the product of the contact rate and the risk of infection (i.e., the effective contact rate), and Γ( t, τ ) is the probability of being infectious at infection age τ . So, reduction in contact frequencywith calendar time t affects β ( t, τ ) while early removal of infectious individuals at calendar time t changes the form of Γ( t, τ ) . An earlier average infection age at first transmission of the diseasewill result from contact tracing and isolation. However, the classic approach to renewal equationsfor epidemics assumes, as in Breda et al. [7], Champredon et al. [10], Nishiura [50], that the non-linearity of an epidemic is characterized by the depletion of susceptible individuals alone, so that A ( t, τ ) = n s ( t ) β ( τ )Γ( τ ) . Finally, the number of infected individuals is called prevalence, whichindicates the proportion of persons who have the ability to infect at a given calendar time. It canbe written as: p ( t ) = (cid:90) ∞ Γ( τ ) j ( t − τ ) dτ. (3)Notice that p ( t ) is not the number of active infected individuals generally reported in epidemicdata published by different national health services. This is because the officially detected casesare actively confined (in hospitals or at home) and so their contribution to epidemic spreading isnot so relevant. On the contrary p ( t ) represents the infected people that are still conducting theirlives as usual, possibly infecting other people.An important variable is the incidence-persistence ratio IPR t at time t :IPR t = j ( t ) p ( t ) . (4)This is important, as it indicates the propensity of currently infected individuals to infect suscep-tibles. Let D := (cid:82) ∞ Γ( τ ) dτ be the average infectious period (mean generation time). Taking β independent of the calendar time t , the actual (or effective) reproduction number can be writtenas the incidence-prevalence ratio: R ( t ) = IPR t · D. (5)5he effective reproduction number represents the average number of secondary infections generatedby each new infectious case (assuming n s and other environmental variables retain their currentvalues forever). It has been employed in interpreting the course of an epidemic .Let us assume that during an outbreak, only a certain fraction of infected persons are observedthrough direct testing, other infectious individuals are not observed, e.g., because of lack of symp-toms or the mildness of their illness. In particular, asymptomatic secondary transmissions, causedby those who have been infected and have not developed symptoms yet, and also by those whohave been infected and will not become symptomatic throughout the course of infection, must beconsidered. At a given calendar time, t , we imagine that the important new cases are not theobserved newly infected (which are put in a position so as not to infect, via safety protocols), butrather the fraction of newly infected that are not observed. These unobserved infected (or at leastsome of them) spread the disease around. Therefore, we split the incidence into two parts: j o ( t ) = λ t j ( t ) j ( t ) = j o ( t ) + j x ( t ) , where λ t is the rate of detection which can change over time depending on the details of and degreeof adherence to testing protocols and medical screenings. Thus, the relation between non-detectedinfected and detected infected individuals is: j x ( t ) = 1 − λ t λ t j o ( t ) . (6)If the population screening procedure is effective, we have λ = 1 . This could happen, for example,if the infected group is made up of only symptomatic persons which are infectious only after theonset of symptoms. This is the perfect situation for stopping the outbreak, as all of the infectedindividuals are detected and (ideally) contained. We relate the variable λ to contact tracingtechnologies which can be used to make λ closer and closer to the ideal value of , as shown byFerretti et al. [26]. Now, let us imagine that some infectious individuals have not been detected andisolated, so that some of them are free to move and have contacts with the susceptible population.We wish to evaluate a measure of risk of exposure for a given susceptible individual. We assume akinetic approach to the evaluation of this risk, where unobserved spreaders are free to infect otherindividuals. We imagine that the contagion acts within a certain radius of an infected individual.In a gas, this radius would correspond to the interaction crossection of a gas particle. We imaginean environment in which two types of individuals are present at a calendar time t : N s is the numberof susceptible individuals in a region along with another J x individuals, which are infectious buthave not been detected and are free to move in that region. Diseases with long generation times usually exhibit strong dependency of infectiousness on infection-age, in-dicating that the effective reproduction number might not be as useful as the instantaneous reproduction number.Although it appears that the effective reproduction number may not be useful for a disease with a long generationtime (e.g. HIV/AIDS), it might be extremely useful for a disease with acute course of illness, especially when wehave both prevalence and incidence in hand, see Yan and Chowell [63]. The subscript o stands for “observed”. a) For a given mobility, red circles are the undetected persons thatcould interact with susceptible individuals (black circles) which areat risk of infection. (b)
Temporal intervals between consecutive generationsof infectious individuals.
Figure 2:
Frequency of contacts of infectious agents with susceptible individuals in a given region during timeintervals of size equal to the infection-generation timescale, τ g We consider the regional mobility, µ , to be the average distance explored by each individual duringthe time interval, t , (usually daily). We define the distance, r , to be the maximum distance thatan infected person can be from a susceptible person (in the model) and still cause them to becomeinfected. This distance depends, for example, on the virus’ infectiousness as a function of distanceand on the viral load. Physical distancing regulations and hygienic norms (such as mask wearing)will result in a decrease in r . Using the collision theory for chemical reactions in solution withtwo types of molecules, we can write down the rate of contacts between the two types in a givenvolume, per unit time: z = n s j x πrµ. (7)Where we have assumed that all agents are ideal point particles that do not interact directly, andtravel through space in straight lines. We further assume that the collisions are instantaneous andelastic. However, not all contacts will result in secondary infectious, rather only those contactsthat have sufficient viral load so as to surmount a certain threshold for triggering the infection.Such transmission efficacy should depend inversely on the physical distance between individuals.Moreover, the collision rate, in reality, depends on time and, in general, on the epidemic’s evolution.This is because the total number of agents changes over time. As an approximation, we embed allof these complexities in the choice of r , so to maintain the simplest form of eq.(7).As discussed in Champredon et al. [10], Gielen [34], a phenomenological approach to epidemicsbased on a renewal equation for the incidence (number of newly infected individuals), is definedusing the rate of secondary transmissions at a time, t , and infection-age, τ , for a populationconcentrated in the unit of area dA : j ( t, x ) = j o ( t, x ) λ ( t ) = (cid:90) t Γ( τ ) (cid:90) A Z ( t, τ, x , x ) j o ( t − τ, x ) dAdτ + i ( t ) . (8)Where Γ( τ ) is the survival function of infectiousness, i.e., the probability of being infectious throughat least an infection-age of τ . The non-linearity of an epidemic is characterized by the secondarytransmission rate Z ( t, τ, x ) , that is the integral kernel informing the rate of secondary transmissionsper single primary case at infection-age τ , at the position x of the region of area A . The quantity7 o ( t − τ ) is the rate of infection of new cases (incidence) at a time τ before the present. Finally, i ( t ) is a function that describes the effects of an external source of infected persons. For the specialcase i ( t ) = Aδ ( t ) , it encodes the number of initially imported infected individuals. The propagator kernel Z can be expressed in terms of the collision theory of non-interacting spheresof radius R in eq.(8) as: j o ( t, x ) = n s ( t, x ) λ ( t ) (cid:90) A Θ [ r ( t, x ) − | x − x | ] µ ( t, x ) (cid:90) t − λ ( t − τ, x ) λ ( t − τ, x ) j o ( t − τ, x ) η ( t, τ )Γ( τ − τ g ) dτ d x . (10)Here, η ( t ) captures the probability that a contact will transmit the infection, and r is the infec-tious radius. The latter is inversely related to the minimum mandated physical distance betweenindividuals, ρ , so that the greater the physical distance between individuals is the smaller theinfectious zone r = (2 πρ ) − is. Note that if it were possible to detect, track, and isolate everynewly infected individual, an epidemic could be stopped within a time τ A . It is useful to note thatthe detection rate, λ , is essentially a scaling factor for the survival probability Γ( τ ) . The value of λ could change with infection age as well as t during the disease outbreak. These changes mightdepend, for example, on the ability to detect and isolate individuals, or the efficiency of contacttracing during the epidemic. The contact tracing efficiency varies with the characteristics of theinfection and the speed and coverage of the tracing process. At the beginning of a large outbreak,testing and manual tracing quickly becomes an unmanageable strategy and a lockdown to reducephysical contact may then become a more efficient and effective means of controlling the epidemic.However, lockdowns aren’t sustainable in the long term because of their social, economic, phys-ical, and mental health effects. Lockdown policies have reduced the spread of CoViD-19, but asrestrictions are relaxed transmission may go up again. Hopefully, with a testing, tracking, andtracing strategy, and additional hygienic precautions in place it will still be possible to keep theepidemic under control. In table 1, we make a summary of the typical factors which contribute tothe transmission of a disease. The biological and environmental properties are accounted for in theTransmissivity variable η . Physical proximity, viral load, and environmental conditions determinethe infectious dose necessary to trigger the infection in a new host. For example, closed environ-ments such as workplaces and schools correspond to higher η values in the model as compared toan outdoor space. Another important component is the temporal duration of a contact that in ourmodel is considered negligible. This approximation is then collapsed into η as an average exposureto the viral particles so as to determine an infection after the existence of a contact has beenestablished. The availability of reliable data over time is key to lifting containment measures. Inparticular, there needs to be sufficient monitoring of the progression of the coronavirus pandemic,including through large-scale testing. There are two main types of CoViD-19 tests. Swab tests, Note that one could completely disregard the external source of infectious individuals, by modelling an infinitelyold epidemic where τ ∈ [0 , ∞ ) in the renewal integral: j ( t, x ) = (cid:90) ∞ Γ( τ ) (cid:90) A Z ( t, τ, x , x ) j o ( t − τ, x ) dAdτ. (9) able 1: Parameters of kinetic approach to infectious contacts. Mobility times proximity gives the rate ofinteractions.
Collision variable Description
Mobility µ Movement trends over time
Social Movements
Average velocity and the path lengthof individual trajectories
Infectious Zone r The area in which contact with an infectious individualcan trigger a secondary infection in airborne diseases
Physical Proximity
Average distance between personsto be considered for airborne diseases(interpersonal distance, protection devices and hygienic procedures affect it)
Transmissibility η The chance that a contact results in an infection
Viral Load
Concentration of viral particlesin the material being shed by an infected patient
Contact duration
Period of time of a collision
Environment
Air flow, UV exposure, climate factors such astemperature and humidity that influence infectiousness which usually take a sample from the throat or nose, to detect viral RNA. These determine ifone currently has CoViD-19. The other type is blood tests, which detect antibodies. This typeof test can provide evidence to determine whether one has had CoViD-19, and is now immune.Although tests can perform well in ideal laboratory conditions, in practice lots of other factorsaffect accuracy including: the timing of the test, how the swab was taken, and the handling of thespecimen. The meaning of a test result for a given person depends not only on the accuracy of thetest, but also on the estimated risk of disease before testing.Making reasonable assumptions, we can write a mean-field approximation of the renewal equationfor incidence evolution over time. This equation is useful for evaluating the growth rate of epidemicsdirectly from the reported data of the disease. Firstly, we assume the spatial homogeneity of everyvariable. In particular, we consider the average distance, ρ ( t ) , between individuals, their averagemobility, µ ( t ) , and the fraction of missed cases, λ , to be spatially homogeneous and constantwith respect to infection age. Moreover, we define a typical time interval necessary to detect anindividual to be infected. We conservatively assume it to be equal to its maximum possible value,the serial interval which (we assume for simplcity) equals the generation time, τ g . The detectiontime can, in general, depend on calendar time (as screening procedures improve over time). We9onsider the detection-age to be our time-scale for the evolution of the observed infected individuals.The detection time is always at least equal to the latent period and can be thought of as equivalentto the incubation time plus the time needed to screen for the infection and isolate the infectedindividual. This is why we take the mean detection time to have the same value as the generationtime and the serial interval of the contagion. Thus, we can assume a window of infectiousnesswhich takes into account the fact that the secondary infected person has to be infected at least τ g days after contact with the primary infected person to be infectious. Consequently, we considerthe survival probability to be windowed between τ g and τ A . Therefore, eq.(10) can be rewritten interms of the expected number of new cases J o ( t ) = j o ( t ) N : J o ( t ) ≈ η ( t ) N s ( t − τ g ) r ( t − τ g ) µ ( t − τ g ) τ A (cid:88) τ = τ g λ ( t ) 1 − λ ( t − τ ) λ ( t − τ ) J o ( t − τ ) . (11)We assume that the mobility (and infectious zone size) of an infector in the past doesn’t affecttheir ability to infect someone, except on the day the person is actually infected. Furthermore, thesum over τ is there to account for the fact that people with a range of infection ages can infectsomeone. The actual reproduction number can be used as a predictive tool to track the epidemic’s evolution.It is also a measure of epidemic risk, in the sense that if it is above one for an extended timeperiod, then an outbreak is possible. Thus, by linking a dynamical model with time-series data,one obtains a measure of epidemic risk. This risk can be derived from eq.(5), leading to the effectivereproduction number R ( t ) ∼ n s ( t − τ g ) n s ( t − τ g ) η ( t − τ g ) η ( t − τ g ) ρ ( t − τ g ) ρ ( t − τ g ) µ ( t − τ g ) µ ( t − τ g ) R ( t ) , (12)where t is an initial (or calibration) time. The above equation represents the change in theaverage number of secondary cases caused by a single primary case throughout the course ofinfection at calendar time t with respect to an initial value (for example, before the lockdown).All variables should be interpreted as average values. Note that the expression for R ( t ) does notdepend explicitly on the distribution of infection survivial times, Γ . It only depends on the typicaltime between infection and detection. So, the Γ distribution can be any distribution as long asthat timescale does not change perceptibly.From eq.(12) it is evident that as the epidemic evolves, the force of infection is reduced for variousreasons, primarily due to physical distancing policies adopted by most countries in the form ofa lockdown of the population’s mobility. Since it is not practical to reduce physical distancingbeyond a certain socially and economically acceptable level, the only possible ultimate reasons forthe end of an epidemic are the depletion of susceptible population (immunization), a change in theintrinsic infectiousness of the virus, a sustained change in public hygiene habits (mask wearing, If we were taking into account variations in the time between infection and detection, we would have to integrateover changes in mobility during that time period. However, that would not be the same sum as the sum over τ . In the present section, we apply our theoretical approach to real-world data. The data repositoriesused to obtain our results are listed in Table 2. This table includes both the epidemiologicaland social distancing data sources. We follow two approachs in evaluating proxies for physicalproximity. The first approach is in regards to the Italian evolution of the CoViD-19 epidemic alongthe lines of social distancing trends. The rest of the studies address the course of the epidemicin some US states and three European countries: the UK, Ireland, and the Czech Republic, forwhich data on physical proximity are available to us.
Table 2:
Repositories used for epidemiological and social distancing data.
Data Repositories
Epidemiological
Johns Hopkins University [59]CoViD Tracking Project [16]Dip. Protezione Civile [20]Epicentro ISS [41]Gov.UK CoViD-19 [4, 38] R t estimations Rt live [56]Epiforecast [23]CoViD-19 Projections [14]
Social Distancing
Google Mobility [35]Voxel51 Physical Distancing Index[61]Analisi Distribuzione Aiuti [19]Unacast Social Distancing Dataset [58]Physical distancing (also known as social distancing) is a practice recommended by public healthofficials to stop or slow down the spread of contagious diseases. It requires the creation of physicalspace between individuals who may spread certain infectious diseases. The use of cloth facecoverings should reduce the transmission of CoViD-19 by individuals who do not have symptoms11nd may reinforce physical distancing. Public health officials also caution that face coveringsmay increase risk if users reduce their use of strong defenses such as physical distancing andfrequent hand washing. We split physical distancing into two components: movement (mobility)and distance (proximity, interpreted as radius of contact).As regards to the former, we essentially have relied on Google [35] mobility open source data.We interpret that data as movement trends, reflected in changes in numbers of visits to variouscategories of places (parks, public transportation hubs, residences, etc.). Google reports thesechanges relative to a average baseline on the same day of the week evaluated before the pandemicoutbreak. We have used the the average mobility across all the places (residential included). Themobility is not intended to be the average speed of a given active particle. Since we are interested inthe rate of collisions, the speed of a particle approaching the particle of interest is just as importantas the speed of the particle of interest. So, we take mobility to be the average relative speed of thegroup. If the velocities of all individuals are uncorrelated, the mobility, i.e., the relative velocityin a certain society is proportionally related to the mobility of each individual .As regards the physical proximity we use two different proxies according to the availability ofdata in the aforementioned repositories. For the case of Italy, we use data on the number of facialcoverings and masks distributed among the population which we interpret as inversely proportionalto physical proximity in the model. This can be considered reasonable at least during the period oflockdown. In the work of Chu et al. [12], MacIntyre and Wang [44], the authors show that the riskof getting infected drops by half for every additional meter of distancing . Essentially, we assumethat the use of face coverings in this population corresponds proportionally to an awareness of theimportance of increasing interpersonal distance. In particular, we assume that the number of peoplewearing a face mask is consistent with an increase of human distance. If this tendency is constantover time, the ratio between the sizes of infectious zones at two different periods of time will cancelout the proportionality factors, thus accounting for the share of face masks used. Regarding theUS and certain other European countries (Ireland and the Czech Republic), we use an inferredmeasure of average proximity among individuals, as explained in the following paragraphs. Deep There are many definitions of mobility and there are multitudinous data repositories. Mobility trends aresubdivided into movements, distances traveled, and shelter-in-place trends. The first accounts for changes inmovement fluxes, as in the data from Google and Apple [3]. The second type is the change in average distancetraveled by individuals. This type of mobility data is available from Safegraph Inc. [40], or other repositories such asCuebiq [17], Unacast [58] for the US. These repositories also include shelter-in-place analysis, and contact/encounterdensity. As for international data, other repositories provide similar data. For example, Facebook Data for GoodMobility Dashboard [25], which gives the change in frequency of travel and the percentage of people staying put.When kilometers traveled per day is used as a mobility proxy, one can use the principle of maximum entropy, underthe assumptions that the individuals are independent particles and that there exists an average daily trip length inthe population, see Bazzani et al. [5]. Thus, the distribution of distances travelled is exponential, p ( L ) = L e − L/L .Here L is the characteristic daily path length reported in mobility data. We checked that distance traveled perday and movement requests show almost exactly the same trend. In particular, by keeping a physical distance from another person of 1 meter, or 3.3 feet, the chance oftransmission falls to 12.8 percent, and a distance over 3.3 feet reduces the chance to 2.6 percent t , is an average measureof peak activity during a window of time (a few days) around that time. So, a large PDI valuemeans that there were a lot of people out and about around that time, at some point. Voxel51’sPDI website presents measures of activity at a single location in the city. While these measuresare likely correlated with overall trends in activity in the city, and are thus an interesting proxy forpublic behavior, correlation is not guaranteed. This project uses Artificial Intelligence (AI) andcamera feeds to compute a Physical Distancing Index to track physical distancing behaviors in real-time in cities around the world. We know from data the concentration of people in a certain area,and so we have an estimation of the density of individuals in a two-dimensional plane. From thatinformation, one can infer the mean distance among individuals, assuming random positions. Themaximum entropy distribution would be a uniform probability density of particle positions. Thiscorresponds to an exponential distribution of inter-particle distances. The characteristic lengthscale (equal to the mean distance between particles) is the inverse of the square root of the particlenumber density (up to a factor of order π ).Consequently the average time between collisions is τ µ = (cid:96)/µ , where (cid:96) is the mean free path, definedto be the average distance traveled by the particle between each collision (cid:96) = 1 / πrd , where r isthe radius of the particle and d is the particle density ( d = N/A ) over a region of area A . Supposethat the probability that a molecule undergoes a collision between a time, t , and a time, t + dt ,is given by γdt ,for some constant γ (the collision rate). If we assume γ is a constant, this impliesthat no memory of previous collisions remains at the time of any later collision. Calling P ( t ) theprobability that the particle has not undergone to a contact from time up until time t , then theprobability that it further makes it to a time t + dt without collision is P ( t + dt ) = P ( t )(1 − γdt ) so that P ( t ) = γe − γt . Yielding the result that the collision rate is the inverse of the collision time γ = 1 /τ µ .In the general analysis of epidemic data we refer to reported infected persons by their datesof diagnosis via laboratory test. However, some countries also report infections by the date offirst symptoms reported by patients. In particular, we have used the latter type of data whenpossible (Italy) and inferred it in the case of the USA and the UK via an analysis of the effectivereproduction number assessed by Epiforecast [23], Covid19Projections [14] and Rtlive [56]. Thisis significant, since up to three weeks can pass between the day of infection and that of diagnosiswith CoViD-19 . We use epidemiological data at the level of states, meanwhile we use mobility Furthermore, reports by first date of symptoms do not suffer from the strong weekly periodicity due to lowerrates of processing of tests (and other data) on weekends (and holidays). The delay between infection and a positivetest result comes from the incubation time of the disease (up to fourteen days) and from the time periods between
Italy
March 09 May 18 June 03New York (US) March 22 May 15 May 28Florida (US) April 03 May 04California (US) March 18 May 24Nevada (US) April 01 May 09London (UK) March 23 May 11 June 23Dublin (IR) March 27 May 18 June 08Prague (CZ) March 16 July 01
Table 3:
Lockdown policy dates in states and regions in our study. Reopening stands for easing shelter policies beforethe very end of a lockdown. We use data from the cities as representative of the physical distancing behavior for theentire regions they are in, for which estimations of reproduction number are available. Since the lockdown policiesare made up of progressive enforcement or easing of restrictions, we consider these dates as reference time-stampsfor government actions to combat the spreading of the disease. show the two derivations of the effective reproduction number R ( t ) . The first is found usingan ensemble estimation with the convolutional renewal approach described in Appendix A) andEpiforecast [23]. The other derivation is assessed using the physical distancing approach viathe above described kinetic theory of collisional infections. We see how the two behaviors arecompatible in describing the behavior of the epidemic. We also call attention to the fact thatmobility alone is not sufficient to explain the dynamics of epidemics, as discussed in Cintia et al.[13]. We see how physical proximity is crucial in determining why, despite an increase in mobilityafter the end of the lockdown period, a relatively stable R t below persists. Recall that, in the caseof Italy, the number of face masks distributed to the populations have been considered as proxyof physical proximity. On the other hand, one should subtract from the susceptible populationthe number of asymptomatic or undetected infected individuals, which are not counted in officialreports. We provide an estimate for this number in the next section. For the other regions understudy, we use physical proximity in terms of population density able to move. In particular, for USstates we use [56] as estimation of the reproduction number as well as the estimation of susceptiblepopulation considered. This is the context for figure Fig.4. When analyzing the UK, Ireland,and the Czech republic, we use an ensemble of R t estimates, averaging Epiforecast [23], Covid19projections [14], and our convolution estimation. the beginning of symptoms, seeking of medical assistance, and the completion of diagnostic laboratory tests. As aconsequence, tracking the evolution of epidemics from data can suffer from a delay of about fifteen days. igure 3: Actual reproduction number for Italy during the lockdown period (March 9th to May 18th). We comparethe traditional derivation with the analytical derivation from the collisional approach of this paper. We have takeninto account data on the depletion of the susceptible population and physical distancing policies made up of socialmobility and physical proximity. In the inset we show a lockdown only on mobility without any physical proximitychange. We see that without maintaining physical distance or any protections such as facial masks, the epidemicgrowth increases with the disinhibition of social movements. In reality this has not happened.
In figure 4, we show the hardest hit states in the US at the date of our work, New York andFlorida. The degree to which this figure show that one can reconstruct the reproduction numberusing the social mobility approach is remarkable. Note that for New York state an important causefor the reduction R t is due to the depletion of the susceptible population, while physical distancinghas a smaller impact. Meanwhile, in Florida, the behavior is mainly due to physical distancingrestrictions taken up at the end of the shelter-at-home policy. We perform the same analysis theUS states of California and Nevada, in Fig.5 obtaining similarly accurate results.If one does not have reliable data to estimate the epidemic risk through the reproduction number,one can look for a proxy of this variable that is less sensitive to a lack of physical distancing data.In fact, in the absence of physical distancing data, the crude reproduction number can be verynoisy. This leads us to consider a proxy of the reproduction number that can predict a generationtime period ahead. So we introduce the growth factor which is far more stable with respect to alack of physical distancing data as long as the epidemiological data is still good, as explained indepth in Appendix D.In the analysis presented in this work, we need to calibrate the measure of the reproductionobtained via physical distancing and the one obtained by the traditional methods (the renewal ap-proach, SIR model, curve fitting, and machine learning) reported in literature. So we evaluate thepre-multiplicative scaling factor in the reproduction number eq.(12), using linear regression withan intercept coefficient fixed to be zero. Moreover, when plotting the reproduction number andgrowth rate, to visualize the trend, we use non-parametric regression analysis with LoWeSS (Lo-cally Weighted Scatterplot Smoothing) surrounded by a confidence interval obtained throughbootstrapping. 15 a)(b) Figure 4:
Reproduction number estimates for two US states. Comparison between the reproduction number cal-culated from symptom onset data as in literature Rtlive [56] (red line) and the reproduction number computedaccording our kinetic approach, using data from Google [35] for mobility, Voxel51 [61] for the social proximity andCovidTracking [16] for epidemic data. Ribbons are the credible interval obtained via bootstrapping. In theinsets, the black solid line is R ( t ) using physical distancing variables only, meanwhile the dashed black line is R ( t ) due to the depletion of susceptibles only. Epidemiological data and R ( t ) estimation are from the date of onset ofsymptoms (indirectly calculated). The changing trends of the reproduction number may be due to several interrelated reasons apartfrom physical distancing policies. These reasons can be collected into two groups. The first has todo with the virus itself and its capacity to spread. An increase in temperature or the development16 a) (b)
Figure 5:
Reproduction number estimates for two US states: California (evaluated in Los Angeles) (a) and Nevada(evaluated in Las Vegas) (b). In the latter case the camera was in a location devoted to leisure activities, this yieldsa biased estimate of the mean interpersonal distance as compared with rest of the state. of less dangerous strains can decrease the effective infectiousness of the contagion. The other groupof reasons is connected to the decrease in the number of susceptible individuals. Supposing thatthe latter is the actual reason, we made a linear fit of R as a function of the fraction of the totalpopulation infected. Shown in Fig. 7 using a very simple ansatz: R ( t ) = ˜ R (cid:16) − cλ (cid:17) . (13)In this way, when the number of susceptibles is zero R ( t ) → . So, the value of the officiallydetected fraction of the population leading to R ( t ) = 0 , c null = λ , gives the ratio, λ , betweenthe number of officially detected and the number of actual cases (supposing that this ratio isapproximately constant in time). We perform the fit only using points 20 days after the lockdown(day 35 in the figures) for Italian regions and US states. We find that the critical official fractionof infected persons is c null = 4 . , both for Italian regions and US states. The data and thebest fit are shown in Fig. 7. This means that the actual infected population fraction should beobtained by multiplying the officially detected cases by a factor of Λ = 1 /c null (cid:39) . These resultsare a-posteriori partially confirmed by some preliminary results from antibody testing performedin Italy. They are also in line with the estimated test fraction, Λ , found through models used byofficials to decide on policies like shelter-in-place orders, such as the CovidActNow [15] initiativeand Flaxman et al. [28].As a further step to check the robustness of our findings, we have analyzed an independent datasetprovided by Unacast [58]. These data offer metrics for physical distancing based on GPS devices.They provide proxies for social mobility (interpreted as average distance traveled) and interper-sonal proximity (human encounters), calculated with respect to the 4 weeks before the CoViD-1917 a) (b)(c) Figure 6:
Reproduction number estimates for London, UK (a), Prague, Czech Republic (b), and Dublin, Ireland(c). Comparison between the reproduction number ensemble average of Epiforecast [23] and Covid19Projections[14] (red line) and the reproduction number computed according our kinetic approach, using data from Google [35]mobility, Voxel51 [61] for social proximity and CovidTracking [16] for epidemic data. Ribbons are the credibleinterval obtained via bootstrap. Epidemiological data and R ( t ) estimation are from data referring to the date oflab diagnosis. outbreak. The former is the percent reduction in the total distance traveled per device, averagedacross all devices located in given US state. The latter is an estimate of close encounters betweentwo devices per square kilometer, expressed as a fraction of the baseline. This estimate counts twousers from the same province as having come in contact if they were within a circle of radius of 50meters of each other within a 1-hour period. This data provides a check on the accuracy of ourprevious results and allows us to extend them to all 50 US states. A selection of these results isshown in Figs.8-11. These figures indicate that the results using Unacast data are consistent withthe other databases for social mobility and human proximity.18 R t c(t) LombardiaPiemonteVenetoEmilia-Romagnac null = 21.1 (a) Reproduction number, R , calculated via convolution as afunction of the fraction of officially infected population. R t c(t) USA - all statesc null = 21.1 (b) R calculated via convolution as a function of the fractionof officially infected population Figure 7:
Estimation of asymptomatic, mildly symptomatic, and undetected individuals in some key Italian regions(a) and the USA (b). The errors in the estimation are about of the fitted value of the slope.
The outbreak of the CoViD-19 pandemic has pushed many countries towards a response that en-visages social distancing policy the implementation of which has an important social and economicimpact on the organization of production and of the work process. Working from home has been anecessary practice for many firms and workers during the lockdown period of the CoViD-19 crisis,so it represents a crucial measure for sustaining production during the crisis, even if its effects onproductivity are unclear, as discussed by OECD [53]. Recent analyses reveal that in the rangeof − of all the job occupations in western countries could be performed from home (onaverage if one accounts for different teleworkability of specific activities) as studied in differentcountries: for Italy in Cetrulo et al. [9], Duranti et al. [22], for UK in for National Statistics [30],for the US by Gould and Shierholz [37], U.S. Bureau Labor Statistics [60]. As a consequence,lockdown measures should take into account such limitations and allow for other activities so as toadopt all the possible social distancing measures that can be effective for each workplace (air ven-tilation, face masks and coverings, etc.), see Chu et al. [12]. On top of that, a more complex taskconsists of also considering the demand side of the lockdown, with many people avoiding spendingmoney in activities that are considered to potentially increase risk of infection. It is beyond thescope of this paper to explore the economic loss and social damage due to pandemics. However,one might account for the heterogeneity of types of contacts by writing the reproduction numberin eq.(12) as a sum of different reproduction numbers across categories of individuals evolving intime according to the state of the infection: R ( t ) = W (cid:88) α =1 R α ( t ) = W (cid:88) α =1 R α ( t ) n α ( t − τ g ) η α ( t − τ g ) µ α ( t − τ g ) ρ α ( t − τ g ) c α ( t ) , (14)where the normalization constant (set at t = t , the initial time) is c α ( t ) = ρ α ( t − τ g ) n α ( t − τ g ) η α ( t − τ g ) µ α ( t − τ g ) .Note that in eq.(14), the subscript α indexes the types of contact, which are assumed to be mutually19 a) (b)(c) (d) Figure 8:
Effective reproduction number data courtesy of Rtlive [56] and social distancing data courtesy of Unacast[58]. exclusive and exhaustive. In general, these assumptions are not necessarily fulfilled , but they areplausible if we consider the two categories to be working activities, for example. Since differentcategories of contacts can have, e.g., different average proximity, mobility and duration of contact(or even environmental conditions), the number of susceptibles corresponds to the number of peopleat risk in that particular category . For example, one could consider the impact of lockdownrestrictions on the occupational structure of a country, and quantify the jobs that can be doneat home, defining the composition of the underlying labour force in terms of occupational, wage One person might go to work as well as to a restaurant, to a bar, to the shopping center, then on vacation, etc. The effective reproduction number may vary as well, because the communities in different locations may differin their level of immunity. Similarly, the basic reproduction number, which is the reproduction number when thereis no immunity from past exposures, may vary across locations because contact rates among people may differ dueto differences in population density and cultural differences. a) (b)(c) (d) Figure 9:
Effective reproduction number data courtesy of Rtlive [56] and social distancing data courtesy of Unacast[58]. and contractual distributions. So, in principle, we could split R ( t ) into two categories accordingthe teleworkability of the activity, and consequently estimate the risk of an epidemic outbreakaccording to lockdown policies. As shown in Fig.12a, during the period before the outbreak ofCoViD-19, the diffusion of teleworking and smart working in Europe was very unequal . Asreported by Eurofound [24], the average is about (sum of regular and occasional teleworkers),but in some countries there are peaks over and in others -including Italy -teleworkers wereless than of the workforce. Turning to the smart working activities during the period oflockdown, European countries have responded with a strong increase in teleworking in differing We use the terms teleworking and smartworking interchangeably. However, teleworking is a way to workindependently of the geographic location of the office or business. On the other hand, smartworking is a newversion of telework, with an innovative workflow based on a strong element of flexibility in terms of hours andlocation of job, which applies to companies with flexible organizational models. a) (b)(c) (d) Figure 10:
Effective reproduction number data courtesy of Rtlive [56] and social distancing data courtesy of Unacast[58]. degrees, ranging from near for Finland to for Spain. This is because each country hasa different share of workers who can telework, depending on the activity and the organization oflabour, as discussed in Fig.12b for the US. For simplicity, we imagine two general categories ofinteractions outside the home. We classify these interpersonal contacts into occasional contactsand structured ones as described in Appendix C. The first represent the erratic movement of anindividual used in our kinetic framework, the latter is the situation of allocation of individualsto bounded areas such as schools, workplaces, or hospitals. Let us define a unique sector of theeconomy where α is represented as teleworking activity which mainly follow a structured contactpattern, while the remaining − α of the productive activity cannot be performed remotely. Thenthe reproductive number can be expressed as: R ( t ) = R α ( t ) + R − α ( t ) ∼ R ( t ) η ( t − τ g ) η ( t − τ g ) (cid:16) αm n α ( t − τ g ) n α ( t − τ g ) + (1 − α ) p n − α ( t − τ g ) n − α ( t − τ g ) (cid:17) . (15)22 a) (b) (c)(d) (e) (f)(g) (h) (i) Figure 11:
Effective reproduction number data courtesy of Rtlive [56] and social distancing data courtesy of Unacast[58].
Here m ≈ m α is the residual mobility for people working remotely, but still moving for ordinaryneeds ( m ≈ . is the residential movement trend from Google’s mobility data, which is multipliedby α , i.e., the share of people working remotely ). The variable p is the average distance betweenindividuals in workplaces, taking into account all types of individual protection measures (masks or Recall that the Google dataset measures visitor numbers to specific categories of locations every day andcompares the change in this number relative to an average baseline day before the pandemic outbreak. igure 12: Remote-working from home in the EU and US. (a) Teleworking with daily frequency in some Europeancountries before (dark gray tick bars) and after (light gray thin bars) the outbreak of CoViD-19. Source: Eurofound[24]. (b) Share of US workers who can telework, by industry, source: U.S. Bureau Labor Statistics [60]. other devices that together reduce the probability to be infected) . For example, in an workplaceor a school, where structured contacts occur, one might approach the problem of reducing therisk of disease transmission using individual position allocation according to physical distancingprescriptions.A crude estimate can be made by considering a constant biological-environmental state, a numberof population at risk equally distributed among the two categories, and at the beginning of anepidemic, so that R t ∼ R (cid:0) m α + p (1 − α ) (cid:1) . The share of teleworking activity α in order tominimized the epidemic risk is given by α = p/ m . This value of α yields a reproduction numberequal to R mint = 2 m α (1 − α/ R . Now consider a realistic estimated teleworking percentageof α = 30% , and a best-case scenario where physical distancing protocols allow one to reducethe contact risk to p = 0 . . In this scenario, the minimum value of the reproduction numberis R t ∼ R / ≈ . , where we consider the basic reproduction number at the beginning of theLockdown to be R = 3 , as also discussed in D’Arienzo and Coniglio [18], Park et al. [54], Zhuanget al. [64]. The value of R t ≈ . we have estimated, incidentally, is the minimal value reachedduring the lockdown in Italy. This was acheived with a percentage of remote working of about , compared to a capacity of allowed by remote systems organization. Moreover, the If one follows the research from Chu et al. [12] and considers all the individual protection procedures, one canobtain a minimal value of p ≈ . , as best-case scenario. of theworkforce. This excessive amount of remote working may have impacted the productivity of theItalian economy. Note that the estimations above are considered in the best-case scenario whereindividual safe position allocation and both the occasional and structured contacts are assessed inthe most precise possible way and under favorable environmental conditions. As a consequence,such reasoning represents only a theoretical attempt pointing towards future directions for policymaking strategies. The renewal equation is a powerful tool for analyzing and modelling epidemic data. We have foundit to be both practically and conceptually useful. In combining the renewal equation with a kineticcollisional model for infection propagation, we have been able to derive a set of predictive equationsfor the short-to-medium-term behavior of an epidemic. These tools allowed us to disentangle theeffects of population mobility, physical proximity, and depletion of susceptibles. Knowing theeffects of each of these components of the response of the government and society to the CoViD-19 epidemic should allow for less costly and more effective strategies for defeating epidemics. Inparticular, the collision model approach to estimation of infection risk should allow local, regional,and national governments to better assess the continuing threat of CoViD-19 to the public welfare.Some future directions for this research are: extension of the model to be more realistic, extensionof the analysis to obtain more useful information about the propagation of the epidemic, andincorporation of the lessons learned into more comprehensive methods for combating CoViD-19.This analysis has focused on the lockdown, but the same theoretical tools along with additionaltechnology and data resources show promise for the analysis of the post-lockdown response andfurther mitigation of this disease.We have mainly focused our study on the spread of a contagion in a homogeneous population, andat this stage, we do not investigate the dynamics of the severity of the disease. This is interrelatedwith the mechanisms of immune response to the SARS-CoV-2 infection. In order to examine thesedynamics, we would need to focus our attention on the microscale corresponding to viral particlesand immune cells. Since these agents induce the dynamics of the varying intensities of the diseaseobserved at the macroscopic scale of the human population. Furthermore, to assess the severity ofan epidemic in a population, one should take into account both the reproduction number R ( t ) andthe absolute number of cases. A high R ( t ) is manageable in the very short run as long as thereare not many people sick to begin with. An important aspect of R ( t ) is that it represents only anaverage across a region. This average can miss regional clusters of infection. Another subtlety notcaptured by R ( t ) is that many people never infect others, but a few ’superspreaders’ pass on thedisease many more times than average, perhaps because they mingle in crowded, indoor events25here the virus spreads more easily. This means that bans on certain crowded indoor activitiescould have more benefit than blanket restrictions introduced whenever the R ( t ) value hits one. Inconclusion, in addition to R ( t ) one should look at trends in numbers of new infections, deaths,hospital admissions, and cohort surveys to see how many people in a population currently have thedisease, or have already had it. Fatality rates and intensive care hospitalization rates are relatedto disease severity. Virulence increases with repeated contacts, since it is related to the numberof exposures to the virus and the infectious dose. Our estimation of the rate of collision is equalto γ = 4 πdµr . In fact, mask wearing, physical distancing and hygiene may also be reduce theinfectious dose that people encounter in the population at large. Fabio Vanni acknowledges support from the European Union’s Horizon 2020 research and innova-tion programme under grant agreement No.822781 GROWINPRO - Growth Welfare InnovationProductivity.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
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Quantitative methods for investigating infectious diseaseoutbreaks , volume 70. Springer, 2019.[64] Zian Zhuang, Shi Zhao, Qianying Lin, Peihua Cao, Yijun Lou, Lin Yang, Shu Yang, DaihaiHe, and Li Xiao. Preliminary estimating the reproduction number of the coronavirus disease(covid-19) outbreak in republic of korea and italy by 5 march 2020.
International Journal ofInfectious Diseases , 2020. 31 ppendixA A semi-analytical estimate of R t and data calibration The classic approach to the renewal equation for epidemics, cf. Breda et al. [7], Champredonet al. [10], Nishiura [50], in its common version assumes that the non-linearity of an epidemic ischaracterized by the depletion of susceptible individuals alone (i.e., contact and recovery rates areindependent of calendar time). Under such assumptions, the kernel A can be described in termsof the so called generation-time distribution g ( τ ) which measures the time between when a persongets infected and when they subsequently infect other people. The generation-time distribution ismade up of two factors: the first is the probability of being infectious τ time units after initiallybecoming infectious. The second contribution is given by the “transmission potential”, that is, theaverage number of secondary infections at “infection age” τ .Then, the instantaneous reproduction number is defined as: R i ( t ) := (cid:90) ∞ A ( t, τ ) dτ, (16)which is understood as the expected number of secondary infections transmitted by a typicalinfectious individual at calendar time t . For t = 0 , it corresponds to R , which is the basicreproduction number. We highlight that the instantaneous reproduction number is different fromthe effective one, at least in principle, but under certain assumptions they are relatively similar,and such conditions are particularly fulfilled if the lockdown policy are in action, so that the twomeasures for the reproduction number coincide, when the number of cases per day is roughlyconstant, both definitions should give R = 1, or rather if β = β .In general, the infectivity kernel A ≥ is an arbitrary function of the infection age τ , but as shownin Fraser [32], Magal et al. [45], Meehan et al. [46] it can be interpreted as the probability to beinfectious (capable of transmitting the disease) with age of infection τ . This probability is variableas the disease progresses within an infected individual. Following the renewal equation approachto estimation of the reproduction number as in Yan and Chowell [63, Ch.8] and used in Aleta et al.[2], Flaxman et al. [28], Nouvellet et al. [52], let us call τ A and τ g the maximum and the minimuminfection age at which an average individual can contribute to the force of infection, respectively.A general approach to estimation of R ( t ) , widely adopted in epidemiological modelling, inverts thegeneral convolution equation eq.(10) supposing the kernel distribution to be known. We, instead,obtain information about the infection distribution, Γ( τ ) , from the response of the epidemic’sevolution to a decrease in β during lockdown. We do this by seeking the form of Γ( t ) that leads toa stable and step-like behavior of β ( t ) resembling the behavior of the population’s mobility (see,e.g., Fig.13). Notice that eq.(10) is linear in j ( t ) and the nonlinearity due to saturation phenomenais hidden inside n s ( t ) . This means that the results we obtain are also valid if we rescale j ( t ) by a32onstant, λ (due, e.g., to the asymptomatic and other non-detected cases). Under the assumptionthat β is independent of the infection-age, we invert eq.(10), the discrete anti-convolution equation,obtaining: β i n s,i = j i (cid:32) i (cid:88) k =1 Γ k j t − k (cid:33) − . (17)Trying various forms of Γ i , we observe that the key ingredient to obtain a flatter behavior of β i n s,i ,is an abrupt transition of Γ i to zero after a time τ A . In particular, after the date correspondingto 15 days after the lockdown in Italian regions, yielding τ A = 15 days. The time evolution of R estimated from our approach, supposing n s equals (total population susceptible) is: R i = β i ∞ (cid:88) k =1 Γ k . (18)The simplest choice of the infection-age probability is: Γ( t ) = (cid:40) τ g ≤ t < τ A otherwise (19)This function produces a regular and intuitive behavior of the reproduction number R of CoViD-19that resembles very nearly the mobility pattern (see Fig.13 and 14) with τ A = 15 days and where τ g = 4 . , see Du et al. [21], Ganyani et al. [33], Lauer et al. [42]. Let us notice that, the R t doesdepend on Γ( τ ) , but only through its integral over all possible values of tau. As a consequence, themost changing Γ can do is change R(t) by a re-scaling. If the mean infectious period is the samefor two generation-time distributions, then R will be the same. However, effectively, infectious agedistribution depends on t . Since contact tracing, testing, and isolation (as well as treatments) willtend to reduce the active infectious period (and their use depends on t). However, the scale of R t is important, since the value of 1 is a fixed point that one would like to be below.In the renewal equation approach, A ( t, τ ) is usually decomposed as A ( t, τ ) = R t g ( τ ) , where g ( τ ) is the generating-time distribution, so that j ( t ) = R t (cid:90) ∞ g ( τ ) j ( t − τ ) dτ. (20)Typically, the generation distribution is unknown, though it can be approximated by assuming itis the same as the serial-interval distribution, which refers to the time between successive cases ina chain of transmission (the time interval between infection and subsequent transmission). .If we look at the data regarding j ( t ) , we see that, after that the effect of the lockdown becomesapparent, j ( t ) shows an exponential decay. This is especially clear in the case of the data referredto the date of symptom onset. We similarly evaluated a gamma distributed Γ k with mean . and scale factor of . , obtaining equivalentresults for the time period of the lockdown. The serial interval may be the same as the generation time if the onset of symptoms is the same as the onsetof infectiousness and the latent period is constant. This is not the case when the incubation period of the primarycase depends on the time from onset to secondary transmission igure 13: R t values obtained from cases reported by date of diagnosis described in the appendix. Source: DPC[20] If we follow this approach, the value of R t can be evaluated analytically. Indeed we have that, if j ( t ) = j l exp( − ηt ) j l exp( − ηt ) = R t (cid:90) ∞ g ( τ ) j l exp( − ηt ) exp( ητ )d τ. (21)Cancelling j l exp( − ηt ) , and multiplying both sides by (cid:82) ∞ Γ( τ )d τ , we have R t = ˆΓ(0)ˆΓ( − η ) . (22)Here ˆΓ( s ) is the Laplace transform of Γ( t ) . In the case of a window-like Γ( t ) between τ g and τ A we have R t = η ( τ A − τ g )exp( τ A η ) − exp( τ g η ) . (23)Setting η = 1 / . (days) − here yields R t (cid:39) . .Now, let us use instead the gamma distribution Γ (cid:48) ( t ) . Recall that its Laplace transform is ˆΓ (cid:48) ( s ) = β α ( s + β ) α . (24)Using the values of the parameters found in the literature [28], namely α = 1 . , β = 0 . ,and η = 1 / . , we obtain R t (cid:39) . . This value is slightly larger than that found with thewindow-like approach we followed in the main part of the paper.34 igure 14: R t values computed for cases reported by date of onset of symptoms. Source: ISS [41]. The time shiftwith respect to Fig.13 is due to the fact that the onset of symptoms is typically one week before the official detectionof the case. The renewal approach can be connected to deterministic, compartmental models such as SIRmodels. For the SIR model, β is considered constant with respect to infection age and is calledthe transmission rate. The infectious survival probability is Γ( t ) = e − γt , with γ = D − , defined tobe the recovery rate, which is the same as the inverse of the mean infectious time D . Consideringthe renewal collision equation eq.(8), we substitute the collisions of individuals moving in a regionwith a rate of contacts β ( t, τ ) (an average value representative of the whole region), so that wecan neglect geographic factors. Thus, we have a correspondence with compartmental deterministicmodels considered in terms of the integral kernel: A ( t, τ ) = βe − γτ SIR (25) A ( t, τ ) = β σγ − σ ( e − στ − e − γτ ) SEIR , (26)both for the SIR and for the corresponding SEIR model, with average duration of latency equal to /σ . Examining the basic reproduction number at the beginning of the epidemic ( t → ) we findthe popular basic reproduction number R , R := (cid:90) A ( t = 0 , τ ) dτ = βγ . (27)This is equivalent to the reproduction number obtained in the SIR model, where R is the ratiobetween effective contact rate and the removal rate (i.e., the inverse of the expected duration ofinfection). 35o, as regards the effective reproduction number we can write for the SIR model: R ( t ) = IPR t · D (28) = βγ n s ( t ) . (29)For the SIR model, the reproduction number is evaluated as R SIR ( t ) = j o ( t ) I ( t ) . (30)In several regions and countries we observe that after the lockdown the behavior of the numberof daily infections has responded to abrupt changes with a delay of approximately 15 days. Afterthat, the time evolution of j ( t ) is very similar to an exponential decay, to a new steady value.See, for example, the growth and decrease exponential decays fitted to the Italian national datasupplied by the ISS [41], referred to the date of the onset of symptoms in Fig.15. The same data,
10 100 1000 1000001/30 02/13 02/27 03/12 03/26 04/09 04/23 05/07 05/21 0 0.5 1 1.5 2 c a s e s w i t h sy m p t o m s / da y R t case by daybest fit η =1/22.14R t (istantaneous)semi-analytical R t Figure 15:
Daily cases reported to the date of the onset of symptoms (source ISS [41]) in lin-log scale. The greenline is the best exponential fit to the data between March 13th and May 5th. The black line is the instantaneous R t while the horizontal line refers to the semi-analytical estimate corresponding to η = 22 . , τ g = 4 , and τ A = 14 . if interpreted as the output of an SIR or SEIR-like model, leads to a value of of the reproductionnumber R SIR = j o ( t ) I ( t ) (where I ( t ) is the number of active infected persons reported at time t )which decays approximately exponentially, beginning just after the abrupt decrease in mobilitydue to the lockdown effect. Note that in our renewal description, the active cases are the totalnumber of infected people which are not isolated, and consequently have the same mobility asthe rest of the population. We call the ratio of the number of non-isolated infected people to thetotal population the prevalence p ( t ) (cid:54) = I ( t ) as used in the SIR description. This behavior is quite36urprising because, as shown in Fig.1b, the mobility remains quite constant after the lockdown, soit is not clear why β SIR decays until the end of May 2020. For this reason we think that, due to theobvious incompleteness of data, it is better to use our approach in evaluating the epidemiologicallysignificant value of R t . B Temporal intervals in epidemics
Understanding the time intervals between successive generations of infected individuals is crucial toappropriately quantifying the transmission dynamics of infectious diseases. As discussed in Brittonand Scalia Tomba [8], Champredon et al. [10], Fine [27], Svensson [57], there are three fundamentaltime periods that determine transmission from one individual to another for directly-transmittedinfectious diseases: the latent ( l ), incubation ( n ), and infectiousness periods, as summarized in4. Calling w the interval of time between the end of the infector’s latent period and the time Table 4: typical time periods in infectious disease evolution
Term Description in clinical analysis
Incubation period time from infection to first clinical symptoms(one can be infectious in this period)
Latent period time from infection to onset of infectiousness(one cannot transmit yet)
Infectious period time during which an infected individualcan transmit a pathogen to others(An infectious individual may not show symptoms) in epidemic modeling
Generation time the time duration from the onset of infectiousnessin a primary case to the onset of infectiousness in a secondary caseinfected by the primary case. (not easily observable)
Serial interval duration of time between the onset of symptomsin a primary case and the onset of symptoms in a secondary caseinfected by the primary case (a readily observable period). of disease transmission to an infectee, the difference between the latent and incubation periodsis noted d = l − n . In particular, as in figure 16, we define the generation time as g = l + w and the serial interval as s = g − n + n = ( l + w ) + ( d − d ) . Assuming l and d to beidentically distributed, the generation-interval distribution and serial-interval distribution have thesame mean. If all the time periods are also uncorrelated in addition to being identically distributed:37ar ( s ) = var ( g ) + 2 var ( d ) . Thus, if the variance of the difference between the latent and incubationperiods is small, the variance of the serial and generation intervals are similar. Recent studies Figure 16:
Clinical and epidemiological periods and parameters. Figure modified from Champredon et al. [10]. on CoViD-19,as in Ferretti et al. [26], Park et al. [55], have highlighted that there are significantcontributions of different transmission routes to the distribution of generation times (time frominfection to onward transmission), and consequently the basic reproduction number R and thedisease dynamics. The different routes hinge mainly on the role of non-symptomatic carriers intransmission. In particular, if the generation-interval distribution of asymptomatic transmissiondiffers from that of symptomatic transmission, then estimates of the basic reproduction numberwhich do not explicitly account for asymptomatic cases may be systematically biased. C Packing occasional and structured contacts
Let us briefly discuss a naive way to minimize the transmission of airborne diseases occurring viahuman contacts, which can be split into two categories: structured and occasional contacts. Inour collisional kinetic framework we have considered contacts among individuals to be random or,in other words, occasional. In addition to these erratic contacts (happening, for example, in thestreets), one can consider structured contacts occurring at home, in hospitals, workplaces, andschools, just to mention a few of the possibilities. For structured contacts, we should consider theuse of a different approach than collision theory.To address the case of structured contacts, we consider a geometrical viewpoint known as circlepacking theory. Circle packing theory studies the arrangement of individual circular zones (infec-tious zones in our case) on a given surface such that no overlap occurs and so that no circle can beenlarged without creating an overlap. In order to minimize exposure to the virus, and so reducethe viral load in closed and clustered spaces like workplaces and hospitals, it should preferable to38rrange individuals inside a given boundary such that no two infectious zones overlap and some(or all) of them are mutually tangent. Using the most efficient hexagonal packing one can obtain apacking density of π/ so that . of the working area will be covered by workers. The actualpacking density will be less than this value because of boundaries which will force one to eitheruse a sub-optimal packing method or leave gaps at the edges. If each person should be at least r meters away from others to be a safe distance of infection,in an environment box with side length L , the number of workers (or students in a class) that could be allocated is: n = 112 (cid:18) Lr (cid:19) . However, using circle packing theory to arrange people in bounded area is only a part of the strategyto reduce the infectious dose. Other effects are also important, for example, the same arrangementof people can have different effects depending on whether it occurs indoors or outdoors. However,a better packing can help to use the same spaces and buildings in a more efficient way, which isespecially relevant for schools and workplaces.Now we turn to the occasional contacts among individuals which makes the path of each individualerratic. Such random movements could also be organized in order to reduce the relative velocityof individuals. For example, when workers have to move it would be beneficial choose repellingpaths so as to avoid contact or collisions. In our kinetic framework, particle interactions, whetherrepulsive or attractive, are so weak that they are also negligible. Nevertheless, such repulsivebehavior can result in very erratic walks. Thus, increasing the relative velocity of active particlesand consequently increasing the unpredictability of the trajectories. Nevertheless, one can imaginepersons to move in a coordinated way so as to minimize their relative speeds. Again, accordingto kinetic theory, the particles of a gas are in state of continuous random motion. The particlesmove in different directions with different speeds and we use the mean relative speed betweenparticles. In any case, we violate the hypothesis that the velocities of particles are uncorrelated.The best way to reduce the relative velocity while allowing the particles to keep on moving is anordered motion. For example, in a box one can move in parallel on overlapping lanes with identicalseeds. Apart from some remaining fluctuations the mobility as relative velocity is minimized withan optimal packing allocation of space. In this way we could reach the maximal reduction ofcollisions without blocking the overall social movements.This sort of coordinated spatial allocation might be inspired by crown shyness in trees, see Franco[31], Goudie et al. [36], Hastings et al. [39]. Networks of treetop chasms have been documented inforests around the world in which canopies maintain gaps, in a phenomenon called crown shyness,that may help trees share resources and stay healthy. This phenomenon consists of limitationson the growth of the canopy the trees, in such a way that the leaves and branches of adjacenttrees do not touch each other. This effect allows a greater penetration of light into the forest andpermits forest plants to perform photosynthesis more efficiently. Moreover, it avoids damaging thebranches and leaves in case of storm or gusts of wind and prevents diseases, larvae, and insects thatfeed on leaves from spreading easily from one tree to another. This example of a natural social39 a) Circle Packing (b)
Workplace office (c)
Crown-Shyness
Figure 17:
Physical distance in structured environments. (a) Packing optimum: side length of square is L = 40 ,infectious zone for individuals has radius r = 2 . Thus we can fit n = 442 persons in the region, using the hexagonalpacking arrangement. This wastes only . of the available space. (b) Example of a office workplace where eachlocation is surrounded by a circle meters in diameter, source: Flerlage [29]. However, in addition to physicaldistance, one should consider other factors such as air movement and ventilation, shared spaces and face coverings,to produce safe working environments. (c) crown delineation, which distinguishes crowns and identifies the speciesof each crown, source: Goudie et al. [36]. distancing strategy makes us suggest the possibility of building models of mobility inspired bythis and similar phenomena with the addition of a transport component to accomodate movementneeds. D Predicting ahead one generation time
We examine the growth rate of epidemics considered as a nonlinear system given by: X ( t + 1) = F ( X ( t ) , p ) , t ∈ Z , (31)where X ( t ) ∈ R is the state vector representing the total number of cases of infected individuals, p ∈ R l is a parameter vector, and F : R n × R l → R n is a continuously differentiable function. Thegrowth factor is defined as: G ( t ) = ∆ X ( t + 1)∆ X ( t ) = | f (cid:48) ( X ( t ) , p ) | . (32)40hich indicates the tendency of the epidemic to increase ( G ( t ) > ) or decrease ( G ( t ) < ).Geometrically, this critical value of the growth factor is an inflection point of the cumulativenumber of cases. The growth factor is also a proxy of the basic reproduction number, because itis approximately monotonically related to the true reproduction number and crosses when thetrue reproduction number crosses , as explained in Wallinga and Lipsitch [62].Since ∆ X ( t ) = J ( t ) , the renewal equation allows us to estimate the generational growth rate tobe: G ( t + τ g ) = J ( t + τ g ) J ( t ) ≈ η ( t ) N s ( t ) η ( t − τ g ) N s ( t − τ g ) r ( t ) r ( t − τ g ) µ ( t ) µ ( t − τ g ) τ A − τ g (cid:80) τ =0 J o ( t − τ ) τ A (cid:80) τ = τ g J o ( t − τ ) . (33)By definition, G ( t ) calculates the multiplicative increase in the number of incident cases over ap-proximately one serial interval, but without requiring one to specify the serial interval distribution.We also notice that for CoViD-19, the detection time (estimated using the serial interval) averagesabout τ g = 5 days, while the maximum infection age is estimated to be τ A = 14 days, cf. Ganyaniet al. [33], Li et al. [43], Nishiura et al. [51].We see that the mobility is an essential piece of information when social and governmental forces,like the onset of lock-downs or other social and economical actions, modify the spread of anepidemic. Of current interest are the effects when lockdown policies are abandoned or weakened,so that mobility begins to increase. We have supposed that the number of susceptible individualshas minimally changed. However, it is possible to estimate from data the individuals which are atrisk of infection by considering the number of undetected individuals, as shown below. We usedmobility data from Google [35] and Apple [3] as well as from forthcoming tracking systems.Finally,in order to evaluate the impact of different components of social distancing in the Growth ratedynamics (33) we use Dynamic Time Warping (DTW). DTW is used to quantify the similarity orcalculate the distance between two time series with different lengths. In time series analysis, DTWis one of the algorithms for measuring similarity between two temporal sequences, which may varyin speed. DTW indicates that even by introducing only the mobility data we could improve ourforecast by about 10 percent. 41 a)(b) (c) Figure 18:
Infected individuals reported by date of laboratory diagnosis: Italy a)(b) (c) Figure 19:
Infected individuals reported by date of onset of symptoms: Italy a)(b) (c) Figure 20:
Infected individuals by date of laboratory diagnosis: USA. Loess regression with confidence intervalhas been used to highlight the growth trend. here are seeds of many things that fly about that areboth sources of life and sources of death, and when, bysome chance , the latter are gathered together and disturbthe sky, the air below becomes diseased. Lucretius , De Rerum Natura, book VI, De Rerum Natura, book VI