The field theoretical ABC of epidemic dynamics
Giacomo Cacciapaglia, Corentin Cot, Michele Della Morte, Stefan Hohenegger, Francesco Sannino, Shahram Vatani
TThe field theoretical ABC of epidemic dynamics
Giacomo Cacciapaglia , , Corentin Cot , , Michele Della Morte , Stefan Hohenegger , ,Francesco Sannino , , Shahram Vatani , Institut de Physique des 2 Infinis (IP2I), CNRS/IN2P3, UMR5822, 69622 Villeurbanne, France Universit´e de Lyon, Universit´e Claude Bernard Lyon 1, 69001 Lyon, France IMADA & CP -Origins. Univ. of Southern Denmark, Campusvej 55, DK-5230 Odense, Denmark CP -Origins and D-IAS, Univ. of Southern Denmark, Campusvej 55, DK-5230 Odense, Denmark Dipartimento di Fisica, E. Pancini, Univ. di Napoli, Federico II and INFN sezione di NapoliComplesso Universitario di Monte S. Angelo Edificio 6, via Cintia, 80126 Napoli, Italy
Abstract:
We go beyond a systematic review of several main mathematical models em-ployed to describe the diffusion of infectious diseases and demonstrate how the differentapproaches are related. It is shown that the frameworks exhibit common features such ascriticality and self-similarity under time rescaling. These features are naturally encodedwithin the unifying field theoretical approach. The latter leads to an efficient description ofthe time evolution of the disease via a framework in which (near) time-dilation invarianceis explicitly realised. When needed, the models are extended to account for observed phe-nomena such as multi-wave dynamics. Although we consider the COVID-19 pandemic as anexplicit phenomenological application, the models presented here are of immediate relevancefor different realms of scientific enquiry from medical applications to the understanding ofhuman behaviour. a r X i v : . [ q - b i o . P E ] J a n ontents ζ and Constant (cid:15) A.1 2-dimensional Lattice 41A.2 Numerical Simulation 42– i –
The SIR Model as an RG Equation: COVID-19 and Constant RecoveryRate 43C Check of Solution (5.13) 45
Several mathematical models have been designed to describe epidemic dynamics, such asthat of COVID-19. These stem from processes involving individuals to statistically basedapproaches aimed at capturing the global properties of the system. Examples include thetime-honoured compartmental models, such as SIR and its variations (see e.g. [1–3]), theepidemiological Renormalisation Group (eRG) framework [4, 5], and first principles fieldtheoretical [6] and percolation lattice methods (see e.g. [7]).The aim of our work is to summarise, review and connect the various approaches inorder to provide a useful dictionary for understanding the current and future pandemics. Infact, the study via various mathematical approaches of the spread of communicable diseaseshas a long history, dating back almost a century (see the pioneering 1927 paper [8]). Thegoal of all approaches is to describe (and predict) the spread of a disease as a function oftime: very generally speaking, new infected individuals can appear when an uninfected one(usually called a susceptible individual) gets in contact with an infectious individual such thatthe disease is passed on. After some time, infected individuals may turn non-infectious (atleast temporarily) via recovering or dying from the disease or by some other means of removal from the actively involved population. Very broadly speaking, there are two different ways ofmathematically modelling these processes (see Fig. 1 for a schematic overview): • Stochastic approach: all (microscopic) processes between individuals are of a prob-abilistic nature, i.e. the contact between a susceptible and an infectious individual hasa certain probability to lead to an infection of the former; infected individuals have acertain probability of removal after a certain time, etc . In these approaches, time isunderstood as a quantised variable and time-evolution is typically described in the formof differential-difference equations (called master equations ). The solutions depend ona set of probabilities ( e.g. the probability of a contact among individuals leading to aninfection), geometric parameters (such as the number of ’neighbouring’ individuals thata single infected can potentially infect) as well as the initial conditions. Furthermore,in order to make predictions or to compare with deterministic approaches, some sort ofaveraging process is required. • Deterministic approach: the time evolution of the number of susceptible, infectedand removed individuals is understood as a fully predictable process and is typicallydescribed through systems of coupled, ordinary differential equations in time (the latter– 1 – eterministic models (compartmental models) e.g.: • SIR • SIS stochastic models e.g.: • percolation models • random walks • diffusion models field theoretical models e.g. • epidemic FT models • eRG mean field appr., averaging b e t a - f un c t i o n D o i - P e li t i a pp r . Figure 1 : Schematic overview of different approaches to describe the time evolution ofpandemics and their relation to field theoretical methods.is understood as a continuous variable). Solutions of these systems are therefore de-termined by certain parameters (such as infection and recovery rates) as well as initialconditions ( e.g. the number of infectious individuals at the outbreak of the disease).While particular approaches following either of these two strategies can be very different,their solutions typically exhibit several common features: (i) Criticality: depending on the parameters of the model and of the initial conditions,the solutions feature either a quick eradication of the disease where the total numberof infected ( i.e. the cumulative number of individuals that get infected over time)remains relatively low, or a fast and wide spread of the disease, leading to a muchlarger total number of infected. Which of these two classes of solutions is realised isusually governed by a single ordering parameter ( e.g. the average number of susceptibleindividuals infected by a single infectious), and the transition from one type to the othercan be very sharp. (ii) Self-similarity and waves: depending on the disease in question, solutions may exhibitdistinct phases in their time evolution in the form of a wave pattern, where phases ofexponential growth of the number of infected individuals are followed by intermediateperiods of near-linear growth. Each wave typically looks similar to the previous ones.Furthermore, certain classes of solutions may also exhibit spatial self-similarities, i.e. the solutions describing the temporal spread of the disease among individuals followsimilar patterns as the spread among larger clusters ( e.g. cities, countries etc. ). (iii) Time-scale invariance: several solutions exhibit a (nearly) time-scale invariant be-haviour, which is a symmetry under rescaling of the time variable and of the rates (in-fection, removal, etc. ). If the solution exhibits a wave-structure, these near-symmetric– 2 –egions can appear in specific regimes, e.g. in between two periods of exponentialgrowth.These properties are familiar in field theoretical models in physics, e.g. in solid state andhigh energy physics, which deal with phase transitions. Indeed, over the years, it has beendemonstrated that the various approaches mentioned above can be reformulated (or at leastrelated to) field theoretical descriptions. The latter are typically no longer sensitive to mi-croscopic details of the spread of the disease at the level of individuals, but instead capture universal properties of their solutions. They are therefore an ideal arena to study propertiesof the dynamics of diseases and the mechanisms to counter their spread.In the following we shall start by presenting examples of deterministic and stochastic ap-proaches and show how they can be related to field theoretical models. We start in Section 2with analysing the direct percolation approach, which is based on a microscopic stochasticdescription of the diffusion processes. We shall see that the approach, in the mean field ap-proximation, naturally leads to compartmental models. The latter (as well as generalisationsthereof) are reviewed in Section 3: we commence this investigation with a basic review ofthe SIR model and then investigate how to incorporate multi-wave epidemic dynamics payingparticular attention to the inter-wave period. In the context of the COVID-19, the latter hasrecently been shown to be crucial to tame the next wave of pandemic, as was first discoveredwithin the complex eRG (CeRG) framework [9, 10] .As natural next step we summarise the most recent approach to epidemic dynamics, i.e. the eRG [4, 11] in Section 4. The latter is inspired by the Wilsonian renormalisation groupapproach [12, 13] and uses the approximate short and long time dilation invariance of thesystem to organise its description. for the COVID-19, the eRG has been shown to be veryefficient when describing the epidemic and pandemic time evolution across the world [14] andin particular when predicting the emergence of new waves and the interplay across differentregions of the world [15, 16]. We finally provide a map between the traditional compartmentalmodels and the eRG.The discussion of sections 2, 3 and 4 is general in the sense that the methods apply togeneral infectious diseases and populations. In Section 5 we consider particular features ofthe current ongoing COVID-19 epidemic, and discuss how the different approaches can beadapted to it.Several excellent reviews already exist in the literature [1–3, 17]. Our work complementsand integrates them, adds to the literature on the field theoretical side and further incorpo-rates more recent approaches. Arguably the most direct way to (theoretically) study the spread of a communicable diseaseis via systems that simulate the process of infection at a microscopic level, i.e. at the level of– 3 –ndividuals in a (finite) population. The most immediate such models are lattice simulations,in which the individuals are represented by the lattice sites, some of which may be infectedby the disease. These lattice sites can spread the disease with a certain probability to neigh-bouring sites, following an established set of rules. Lattice models, therefore, allow to trackthe spread of the disease in discretised time steps and, after taking the average of severalsimulations, allow to make statements about the time evolution (and asymptotic values) ofthe number of infected individuals. As we shall see in the following, even simple models ofthis type show particular time-scaling symmetries, as well as criticality ( i.e. the fact that theasymptotic number of infected individuals changes rapidly, when a certain parameter of themodel approaches a specific critical value).A larger class of models that work with a discrete number of individuals (as well asdiscretised time) consists of percolation models , which broadly speaking consist of points(sites) scattered in space that can be connected by links. Depending on the specific details,one distinguishes: • Bond percolation models : in this case the points are fixed and the links between themare created randomly. Examples of this type are (regular) lattices in various spatialdimensions with nearest neighbour sites being linked. • Site percolation models : in this case the position of the points is random, while the linksbetween different points are created based on rules that depend on the positions of thepoints.More complex models can also incorporate both aspects. An important quantity to compute inany percolation model is the so-called pair connectedness , i.e. the probability that two pointsare connected to each other (through a chain of links with other points). Assuming the systemto extend infinitely ( i.e. there are infinitely many sites), we can importantly distinguishwhether it is made of only local clusters (in which finitely many sites are connected) orwhether it is in a percolating state (where infinitely many sites are connected). The probabilityof which of these two possibilities is realised usually depends on the value of a single parameter(typically related to the probability p that a link exists between two ‘neighbouring’ sites),in such a way that the transition from local connectedness to percolation can be describedas a phase transition (see e.g. [18]). The system close to this critical value p c lies in thesame universality class of several other models in molecular physics, solid state physics andepidemiology: this implies that the behaviour of certain quantities follows a characteristicpower law behaviour that is the same for all the theories in the same universality class. Forexample, the probability P ( p ) for a system of be in the percolating state (as a function of p )takes the form lim p → p c P ( p ) ∼ ( p − p c ) ν , (2.1)where ν is called a critical exponents . Models within the same universality class share thesame critical exponents despite the fact that the concrete details of the theory (in particular– 4 –he concrete meaning of the quantity P in Eq. (2.1)) may be very different. This connectionmakes percolation models very versatile and many of them have been studied extensively (see[7] and references therein).In the following, we shall first present a simple lattice simulation model, which allows us toreveal important properties of the time evolution of the infection (notably criticality and time-rescaling symmetry). Furthermore, we shall discuss a percolation model that, near criticality,is in the same universality class as time-honoured epidemiological models, along with some ofits extensions and generalisations. Furthermore, we shall consider numerical simulations on asimple lattice model to illustrate its critical behaviour (and also to illuminate several aspectsthat play an important role in the epidemiological process). The simplest (and most direct) way to study percolation models is to simulate the timeevolution of the spread of a disease via stochastic processes on a finite dimensional lattice.The detailed rules for a simple model in 2 dimensions (on a cubic lattice Γ ( N )2 of (2 N + 1) × (2 N + 1) lattice sites, generated by an orthonormal set of basis vectors ( e , e )) are describedin appendix A. These rules put the following two basic mechanisms into an algorithm thatsimulates the spread of the disease throughout the finite and isolated population in discretisedtime steps: the infection of other individuals in the vicinity of an infectious one and theremoval (recovery) of an infectious individual (so that it can no longer infect other individuals).In the following we shall highlight some of the key-features of this model as functions of threeparameters: • The infection probability g ∈ [0 ,
1] for an infectious individual to infect a neighboursite. In practice, the probability of a single individual in the neighbourhood (defined interms of the coordination radius, see below) to be infected is equal to g divided by thenumber of sites within a radius r from the infectious one. This choice, as we shall wee,allows us to draw a more direct relation between g and the infection rate parameterdefined in Compartmental Models. • The removal probability e ∈ [0 ,
1] for an infectious individual to be removed from theactive population. • The coordination radius r ∈ R + , which is a measure for the distance (on the lattice)over which direct infections between individuals can take place, i.e. only sites within adistance r from the infectious one can be infected.A plot of the evolution of the number of infected as a function of the discretised time-stepsis shown in Fig. 2 for a sample choice of the parameters. At large t , the number of infectedapproaches an asymptotic value, which is a function of ( g , e ) as well as of the coordinationradius r . Varying these parameters leads to substantially different asymptotic values, as isshown in Fig. 3: in the four panels, we plot the asymptotic values as a function of the infectionprobability g . We used a lattice with N = 100 and fixed e = 0 .
1. For each point, we repeated– 5 – igure 2 : Number of infected individuals as a functionof the discretised time for a lattice with N = 100, g =0 . e = 0 . r = 1.the process 50 times to compute theshown mean and standard devia-tion. As expected, the larger g thehigher the fraction of infected casesat the end of the process. The plotsalso show the critical behaviour ofthe system, as the asymptotic valuejumps from a very small value atsmall g to a value of the same or-der of the total population ( i.e. thenumber of sites in the lattice). Foreach value of r , one can define acritical value g c ( r ): increasing r re-duces the value of g c .In the simulations in Fig. 3 weuse the same initial condition, where all the sites within a radius 5 (in lattice units) from thecentre of the lattice are set to the infectious state, thus having initially 81 cases. Due to thestochastic nature of the process, the final number of infected cases does depend non-triviallyon the initial state, especially for small coordination radius r . For r = 1 and e = 0 .
1, thisdependence on the initial infected N I is shown in the left panel of Fig. 4, where we plot theasymptotic value of infected as a function of N I , randomly distributed on the lattice. Weplotted the results for three different values of g = 0 . , . .
7, where the middle valueis close to the critical g c . The critical behaviour described above seems also sensitive in N I .This could be due to finite volume effects, as the evolution of the infections is expected todepend crucially on the density of initial infected cases in the lattice and on their spatialdistribution. This effects should disappear at infinite volume. Especially near the criticalvalue, we observe a large spread of the values of the asymptotic values. This is particularlyevident for small densities of initial infections, where stochastic effects become relevant. Asan example, we show a bundle of 50 solutions near the critical value in the right panel ofFig. 4. Here we briefly summarise the percolation approach and the derivation via field theory ofthe reaction diffusion processes. We follow Pruessner’s lectures [19] and borrow part of hisnotation. The overarching goal is to reproduce and extend the action given in the seminalwork of Cardy and Grassberger [6].We, therefore, consider a model of random walkers described by a field W diffusingthrough a lattice, reproducing themselves and dropping some poison P as they stroll around.The poison field P does not diffuse but kills walkers if they hit a poisoned location. Interpret-ing the positions of the walkers as infected sites and those of the poison as simultaneouslyrepresenting either the immune or removed individuals, the model effectively describes a– 6 – a) r = 1 (b) r = 2(c) r = 5 (d) r = 50 Figure 3 : Evolution of the final number of infected as a function of the infection probability g for different coordination radii r . The removal probability is fixed to e = 0 . W → W + W , with rate σ ,W → W + P , with rate α ,W + P → P , with rate β . (2.2)The first branching process corresponds to infection, while the last two processes describe im-munisation. In addition we will consider a process of spontaneous creation, by which infectedcan appear at one site independently from the presence of other infected at neighbouringsites, with a rate ξ . – 7 – igure 4 : Left panel: Evolution of the final number of infected as a function of the initialinfected. The mean and the standard deviation are computed over 50 simulations for eachpoint with e = 0 . g = 0 .
5. Right panel: Time evolution of the infected cases for 50simulations with e = 0 . g = 0 . N I = 2 and r = 1. n W x , n P x e e Figure 5 : Schematic presenta-tion of the state { n W x , n P x } with e i the basis vectors of Γ.The field theory is derived from a discretised versionof the model, eventually taking the continuum limit. Thestarting point is a Master Equation which will directly leadto the action through a process of second-quantisation. LetΓ ⊂ Z d be a d -dimensional hypercubic lattice with coordi-nation number q , which is generated by a set of vectors e .We denote by { n W x , n P x } a state with site x occupied by n W x and n P x particles of type W and P ∀ x ∈ Γ (for a schematicrepresentation see Fig. 5). The probability that such stateis realised at time t is denoted by P ( { n W x , n P x } ; t ). Configu-rations can change via the different mechanisms describedabove. The probability thus satisfies the first order differ-ential equation (Master Equation): dP ( { n W x , n P x } ; t ) dt = Hq (cid:88) y ∈ Γ (cid:88) e ∈ e (cid:2) ( n Wy + e + 1) P ( { n Wy − , n Wy − e + 1 , n P x } ; t ) − n Wy P ( { n W x , n P x } ; t ) (cid:3) + σ (cid:88) y ∈ Γ (cid:2) ( n Wy − P ( { n Wy − , n P x } ; t ) − n Wy P ( { n W x , n P x } ; t ) (cid:3) + α (cid:88) y ∈ Γ (cid:2) n Wy P ( { n W x , n Py − } ; t ) − n Wy P ( { n W x , n P x } ; t ) (cid:3) + β (cid:88) y ∈ Γ (cid:2) ( n Wy + 1) n Py P ( { n Wy + 1 , n P x } ; t ) − n Wy n Py P ( { n W x , n P x } ; t ) (cid:3) + ξ (cid:88) y ∈ Γ (cid:2) P ( { n Wy − , n P x } ; t ) − P ( { n W x , n P x } ; t ) (cid:3) . (2.3)– 8 – W y − , n P y n W y + e + 1 , n P y + e e H/q n W y , n P y n W y + e , n P y + e e Figure 6 : Schematic representation of the process leading to the first line of Eq.(2.3): asingle walker moving to a neighbouring lattice site (with n W y ≥ n P y , n W y + e , n P y + e ≥ n W y − , n P y σ n W y , n P y Figure 7 : Schematic representation of the branching process leading to the second line of(2.3): a single walker creating a copy of itself at the the site y (with n W y ≥ n P y ≥ n W y , n P y − α n W y , n P y Figure 8 : Schematic representation of the branching process leading to the third line of(2.3): a walker ’drops’ poison at the lattice site y (with n P y ≥ n W y ≥ n W y + 1 , n P y β n W y , n P y Figure 9 : Schematic representation of the branching process leading to the fourth line of(2.3): a single walker ’dying’ from poison at the lattice site y (with n P y , n W y ≥ q nearestneighbours with frequency H/q . This process is schematically shown in Fig. 6. There { n Wy − , n Wy − e + 1 , n P x } denotes the state differing from { n W x , n P x } by having a walker lessat y and a walker more at y − e . The second and third lines produce the first two branchingprocesses in Eq. (2.2) respectively and are schematically shown in Figs 7 and 8. The fourthline accounts for the third process there and is graphically represented in Fig. 9. Finallythe last line gives spontaneous creation of one walker at site y and is schematically shown inFig. 10.In view of a second quantisation, following the Doi-Peliti approach [22–24] it is naturalto interpret the state { n W x , n P x } as obtained by the action of creation operators a † ( x ) (for W ) and b † ( x ) (for P ) on a vacuum state. One introduces also the corresponding annihilation– 9 – W y − , n P y ξ n W y , n P y Figure 10 : Schematic representation of the branching process leading to the fifth line of (2.3):a single walker is spontaneously created at the lattice site y (with n W y ≥ n P y ≥ a † ( x ) |{ n W x , n P x }(cid:105) = |{ n W x + 1 , n P x }(cid:105) , (2.4) a ( x ) |{ n W x , n P x }(cid:105) = n W x |{ n W x − , n P x }(cid:105) , (2.5) (cid:104) a ( x ) , a † ( y ) (cid:105) = δ x , y , (2.6)and similarly for the b -operators, which commute with the a -operators. The field theory isrealised by considering the time-evolution of the state | Ψ( t ) (cid:105) = (cid:88) { n W x ,n P x } P ( { n W x , n P x } ; t ) |{ n W x , n P x }(cid:105) , (2.7)which can be derived from the Master Equation (2.3). Upon mapping each operator toconjugate fields a → W , ˜ a = a † − → W + ,b → P , ˜ b = b † − → P + , (2.8)where the tilded operators are known as Doi-shifted operators, one obtains that the evolutionis controlled by exp {− (cid:82) d d xdt S ( W + , W, P + , P ) } , with the action density S given by S = W + ∂ t W + P + ∂ t P + D ∇ W + ∇ W − σ (1 + W + ) W + W − α (1 + W + ) P + W + β (1 + P + ) W + W P − ξW + , (2.9)where D = lim a → H a /q is the hopping rate in the continuum ( a is the lattice spacing). Theaction in Eq. (2.9) corresponds to the result in [6] augmented here by the last source termdue to spontaneous generation. This produces a background of infected and it is responsiblein this approach for the strolling dynamics, as we motivate in the next section and illustrateby numerical studies in the last section of the paper.The renormalisation group equations stemming from the action (2.9), which follow closelyto that of other theories (such as directed percolation models or reggeon field theory [25, 26]),have been analysed in [6]. In particular, the Fourier transform of the correlation function ofa field W and a field W + was computed and shown to satisfy the following scaling law nearcriticality F (cid:0) (cid:104) W ( (cid:126)x, t ) W + (0 , (cid:105) (cid:1) ( ω, (cid:126)k ) = | (cid:126)k | η − Φ( ω ∆ ν t , (cid:126)k ∆ ν ) , (2.10)– 10 –or some function Φ. Here ∆ is a measure for the proximity to criticality ( i.e. it is proportionalto p − p c of Eq. (2.1) in the context of the percolation model) and ( η, ν t , ν ) are critical exponentsdetermining the universality class of the model. The quantity above is a measure for theprobability of finding a walker at some generic time and position ( (cid:126)x, t ) ∈ R if there was oneat the origin, where d = 6 corresponds to the critical dimension of the system [6]. As mentioned before, the model described by the action in Eq.(2.9) is in the same universalityclass as numerous other models that are directly relevant for the study of epidemic processes.As shown in [6] the particular choice ξ = 0, in fact, includes the SIR model, which is the mostprominent representative of compartmental models. To make the connection more concrete,we return to studying the time evolution of a disease on a lattice Γ and divide the individualsthat are present at a given lattice site x ∈ Γ into three classes (compartments): • Susceptible : these are individuals that are currently not infectious, but can contractthe disease. We do not distinguish between individuals who have never been infectedand those who have recovered from a previous infection, but are no longer immune. Weshall denote n S x the number of susceptible individuals at x . • Infectious : these are individuals who are currently infected by the disease and canactively transmit it to a susceptible individual. We shall denote n I x the number ofinfectious individuals at x . • Removed (recovered) : these are individuals who currently can neither be infected them-selves, nor can infect susceptible individuals. This comprises individuals who have(temporary) immunity (either natural, or because they have recovered from a recent in-fection), but also all deceased individuals. We shall denote n R x the number of removedindividuals at x .Concretely, for ξ = 0, the model in [27] is very suitable for numerical Markovian simula-tions and can be connected to the SIR model. The processes of the model in [27] are n S x + n I x (cid:48) → n I x + n I x (cid:48) , infection with rate ˆ γ ,n I x → n R x , recovery with rate ˆ (cid:15) , (2.12)where x and x (cid:48) are nearest neighbour sites on Γ ( i.e. x (cid:48) = x + e for some basis vector e ∈ e ). As discussed in [27], treating the process as deterministic (in particular, interpreting In a dimensional regularisation scheme, they were found to be η = − (cid:15) , ν t = 1 + (cid:15) , ν = 12 − (cid:15) . (2.11)in [6], where (cid:15) = 6 − d . The occupation numbers ( n S x , n I x , n R x ) are denoted ( X ( x ) , Y ( x ) , Z ( x )) respectively in [27]. – 11 – n S x , n I x , n R x ) as continuous functions of time) one obtains the following equations of motion dn S x dt ( t ) = − ˆ γ n S x ( t ) (cid:88) e ∈ e n I x + e ( t ) ,dn I x dt ( t ) = ˆ γ n S x ( t ) (cid:88) e ∈ e n I x + e ( t ) − ˆ (cid:15) n I x ( t ) ,dn R x dt ( t ) = ˆ (cid:15) n I x ( t ) , (2.13)where the sums on the right hand side extend over the nearest neighbours of x . Since thesum of all three equations in (2.13) implies ddt ( n S x + n I x + n R x )( t ) = 0, the total number ofindividuals is conserved and we denote its value by N = (cid:88) x ∈ Γ ( n S x ( t ) + n I x ( t ) + n R x ( t )) . (2.14)Furthermore, we introduce the relative number of susceptible, infectious and removed indi-viduals respectively S ( t ) = 1 N (cid:88) x ∈ Γ n S x ( t ) , I ( t ) = 1 N (cid:88) x ∈ Γ n I x ( t ) , R ( t ) = 1 N (cid:88) x ∈ Γ n R x ( t ) , (2.15)which satisfy S ( t ) + I ( t ) + R ( t ) = 1 . (2.16)Finally, by taking a mean-field approximation for the infected field in (2.13) ( i.e. replacing n I x by I ( t ) ∀ x ∈ Γ, such that the sums (cid:80) e ∈ e n I x + e in (2.13) are replaced by qN (cid:80) x ∈ Γ n I x = qI ( t )) and summing over all x ∈ Γ, one obtains the following coupled first order differentialequations: dSdt ( t ) = − q ˆ γ S ( t ) I ( t ) , dIdt ( t ) = q ˆ γ S ( t ) I ( t ) − ˆ (cid:15) I ( t ) , dRdt ( t ) = ˆ (cid:15) I ( t ) , (2.17)where q is the coordination number, i.e. , the number of nearest neighbours for each site (4 ina two-dimensional rectangular lattice). As we shall discuss in the next section, this system ofdifferential equations, which has to be solved under (2.16) and with suitable initial conditions,is structurally of the same form as the SIR model [8], one of the oldest deterministic modelsto describe the spread of a communicable disease.Spontaneous generation can be included in (2.17) as an additional process n S x → n I x , with rate ˆ ξ . (2.18)In the deterministic and mean-field equations, this amounts to a term − ˆ ξS ( t ) in the firstequation in (2.17), and the corresponding, opposite in sign, one in the second equation, as weshall discuss in the context of the SIR model in the following section.– 12 – Compartmental Models
Independently of percolation models and epidemic field theory descriptions, the differentialequations (2.17) have been proposed as early as 1927 to describe the dynamic spread ofinfectious diseases in an isolated population of total size N (cid:29)
1. It is the first (and relativelysimple) example of a class of deterministic approaches that are called compartmental models ,whose hallmark is to divide the population into several distinct classes. As the name indicates,in the context of the SIR model these have already been described in detail in subsection 2.4: • S usceptible: the total number of susceptible individuals at time t shall be denoted N S ( t ). • I nfectious: the total number of infectious individuals at time t shall be denoted N I ( t ). • R emoved (recovered): the total number of removed individuals at time t shall be denoted N R ( t ).We assume that the total size of the population remains constant, i.e. we impose the algebraicrelation 1 = S ( t ) + I ( t ) + R ( t ) , ∀ t ∈ R + , (3.1)where (without restriction of generality), we assume that the outbreak of the epidemics startsat t = 0. We shall also refer to S , I and R as the relative number of susceptible, infectiousand removed individuals respectively. Furthermore, we assume that N is sufficiently largesuch that we can treat S , I and R as continuous functions of time: S , I , R : R + −→ [0 , . (3.2)While in section 2.4 the differential equations (2.17) are a consequence of the basic microscopicprocesses in Eq.(2.12) on the lattice Γ, within the SIR model they are independently arguedon the basis of dynamical mechanisms that change ( S, I, R ) as functions of time: • Infectious individuals can infect susceptible individuals, turning the latter into infectiousindividuals themselves. We call an ‘infectious contact’ any type of contact that resultsin the transmission of the disease between an infectious and a susceptible and we denotethe average number of such contacts per infectious individual per unit of time by γ . Inthe classical SIR model [8], γ is considered to be constant ( i.e. it does not change overtime), however, in the following sections we shall not limit ourselves to this restriction.The total number of susceptible individuals that are infected per unit of time (and thusbecome infectious themselves) is thus γ N S I .– 13 – Infectious individuals can be removed by recovering (and thus gaining temporary immu-nity) or by being given immunity ( e.g via vaccinations), by death or via any other formof removal. We shall denote (cid:15) the rate at which infected individuals become removed.As before, we consider (cid:15) as a function that may change with time. • Removed individuals may become susceptible again after some time or, conversely,susceptible individuals may become directly removed. In both cases we shall denote therespective rate by ζ , which, however, may be positive or negative. If removed individualsare only temporarily immune against the disease, they can become susceptible again.In this case ζ >
0, which corresponds to the rate at which removed individuals becomesusceptible again. Susceptible individuals may become immunised against the disease( e.g. through vaccinations). In this case ζ <
0. We remark that this is not the onlyway to implement vaccinations to compartmental models, as the most direct way is toadd a specific compartment.The flow among susceptible, infectious and removed is schematically shown in Fig. 11. Fur-thermore, we denote the number of relative susceptible, infectious and removed individualsat time t = 0 as S ( t = 0) = S , I ( t = 0) = I , R ( t = 0) = 0 , (3.3)where S , I ∈ [0 ,
1] are constants that satisfy S + I = 1. Without loss of generality westart with zero removed at the initial time. With this notation, the time dependence of S , I and R as functions of time is described by the following set of coupled first order differentialequations γ N I SN S N I (cid:15)N I N Rζ N R Figure 11 : Flow between susceptible, in-fectious and removed individuals. dSdt = − γ I S + ζ R ,dIdt = γ I S − (cid:15) I ,dRdt = (cid:15) I − ζ R , (3.4)together with the algebraic constraint (3.1) andthe initial conditions (3.3). For ζ = 0, this isindeed the same model as described in Section 2.4. The Eqs (3.4) can be solved analytically for ζ = 0 as we will discuss in the next subsection.First, we shall present some qualitative remarks that can be obtained by considering numericalsolutions, which we obtained by using a simple forward Euler method. We first consider ζ = 0, These equations coincide to Eq.(2.17) upon identifying q ˆ γ ≡ γ , ˆ (cid:15) ≡ (cid:15) , and for ζ = 0. Spontaneousgeneration of infectious individuals can be added straightforwardly. – 14 – R e ,0 = S [ t ] I [ t ] R [ t ]
50 100 150 200 250 300 350 t0.20.40.60.8 R e ,0 = S [ t ] I [ t ] R [ t ] Figure 12 : Numerical solution of the differential equations (3.4) for S = 0 . γ = 0 . ζ = 0 for two different choices of (cid:15) : (cid:15) = 0 . R e, = 0 .
919 (left) and (cid:15) = 0 .
05 suchthat R e, = 1 .
84 (right).for which the temporal evolution of (
S, I, R ) is illustrated in Fig. 12 in two qualitativelydifferent scenarios, depending on the value of the initial effective reproduction number R e, ,that we define as [28] R e, = S σ , σ = γ(cid:15) . (3.5)The quantity σ , often called basic reproduction number ( R ), can be interpreted as the averagenumber of infectious contacts of a single infectious individual during the entire period theyremain infectious, in other words, the average number of susceptible individuals infected bya single infectious one. In the left panel of Fig. 12, ( γ, (cid:15), S ) have been chosen such that R e, <
1: in this case, even though at initial time a significant fraction of the population (8%)is infectious, the function I ( t ) decreases continuously, leading to a relatively quick eradicationof the disease. In the right panel of Fig. 12, we chose R e, >
1: the number of infectiouscases grows to a maximum and starts decreasing once only a small number of susceptibleindividuals remain available.This behaviour is more clearly visible in the asymptotic number of susceptible ( i.e. S ( ∞ ) = lim t →∞ S ( t )) or (equivalently) the cumulative number of individuals that havebecome infected throughout the entire epidemic. Both quantities are a measure of howfar the disease has spread among the population. For later use, we define the function I c ( t ) : [0 , ∞ ) (cid:55)→ [0 , N ] as I c ( t ) = N I + (cid:90) t dt (cid:48) γ N I ( t (cid:48) ) S ( t (cid:48) ) . (3.6)It quantifies the cumulative total number of individuals who have been infected by the diseaseup to time t . The definition (3.6) can be used for generic ζ as a function of time. For ζ = 0,– 15 –sing Eqs (3.4), we obtain the identity γ I S = ddt ( I + R ) that allows to simplify Eq.(3.6) to: I c ( t ) = N ( I ( t ) + R ( t )) = N (1 − S ( t )) , for ζ = 0 . (3.7)For ζ = 0, we also have that lim t →∞ I ( t ) →
0, thus we find the following relations at infinitetime: I c ( ∞ ) = lim t →∞ I c ( t ) = 1 − S ( ∞ ) = R ( ∞ ) = lim t →∞ R ( t ) . (3.8)The limit S ( ∞ ) can be computed analytically, by realising that G ( t ) = S ( t ) e σ R ( t ) , (3.9)is conserved, i.e. dGdt ( t ) = 0 ∀ t ∈ R . This implies that S can be written as S ( t ) = S e − σ (1 − I ( t ) − S ( t )) . (3.10) R e ,(cid:0) S [ ] I C [ ∞ ] N Figure 13 : Asymptotic number of sus-ceptible and cumulative number of infec-tious as a function of R e, for S = 1 − − .With lim t →∞ I ( t ) = 0, this equation can besolved for the asymptotic number of susceptiblein the limit t → ∞ S ( ∞ ) = − S R e, W ( − R e, e − Re, S ) , (3.11)where W is the Lambert function. The limitingvalues S ( ∞ ) and I c ( ∞ ) /N are shown in Fig. 13 asfunctions of R e, for the initial conditions of S =1 − − , i.e. a starting configuration with oneinfectious individual per million. A kink seems toappear for R e, = 1, however both functions aresmooth (continuous and differentiable) for S <
1, as highlighted in the subplots. In the limit S →
1, the solutions discontinuously jump toconstant, as the absence of initial infectious individuals prevents the spread of the disease.Qualitatively, this plot shows that for R e, <
1, the disease becomes eradicated before asignificant fraction of the population can be infected. However for R e, > ζ (cid:54) = 0, we can distinguish two different cases, depending on the sign: • Re-infection ζ >
0: a positive ζ implies that removed individuals become suscepti-ble again after some time. This can be interpreted to mean that recovery from thedisease only grants temporary immunity, such that a re-infection at some later time ispossible. At large times t → ∞ , the system enters into an equilibrium state, such that– 16 – S ( t ) , I ( t ) , R ( t )) approach constant values ( S ( ∞ ) , I ( ∞ ) , R ( ∞ )). To find the latter, weimpose the equilibrium conditionslim t →∞ d n Sdt n ( t ) = lim t →∞ d n Idt n ( t ) = lim t →∞ d n Rdt n ( t ) = 0 , ∀ n ∈ N , (3.12)which have as solution( S ( ∞ ) , I ( ∞ ) , R ( ∞ )) = (1 , ,
0) if σ ≤ S = 1 , (cid:16) (cid:15)γ , ( γ − (cid:15) ) ζγ ( (cid:15) + ζ ) , ( γ − (cid:15) ) (cid:15)γ ( (cid:15) + ζ ) (cid:17) if σ > , for ζ > . (3.13)Here we have used that 0 ≤ ( S ( t ) , I ( t ) , R ( t )) ≤ S ( t ) , I ( t ) , R ( t ))cannot become negative) as well as the fact that the equilibrium point (1 , ,
0) cannotbe reached for S < γ > (cid:15) : indeed, this would require S ( t ) > (cid:15)γ , and dIdt ( t ) < , (3.14)which are not compatible with (3.4). The two qualitatively different solutions of (3.4)that lead to the asymptotic equilibria (3.13) are plotted in Fig. 14: for σ < σ > S (except for the trivial initial condition S = 1) but only on thebasic reproduction number σ . This fact can be intuitively understood as the rate ζ dynamically increases the number of susceptible individuals, thus the regime becomesindependent on the initial condition. • Direct immunisation ζ <
0: a negative ζ implies the possibility that over time suscep-tible individuals can become removed and thus immune to the disease, proportionallyto the number of removed individuals. The immunisation mechanism could be due to, e.g. , vaccinations. Schematically, different solutions are shown in Fig. 15. For ζ < S ( ∞ ) , I ( ∞ ) , R ( ∞ )) = (0 , ,
1) atlarge t → ∞ . The relation between Compartmental Models and Percolation Field Theory has already beenestablished. However it is also possible to link the numerical simulations to the SIR model, as Furthermore, the only solutions of the conditions d Sdt ( t ) = dIdt ( t ) = d Rdt ( t ) = 0 are in fact the twoequilibrium points (3.13) (where in fact all derivatives of ( S , I , R ) vanish). This therefore suggests that thereare no solutions that are continuous oscillations with non-decreasing amplitudes and the system indeed reachesan equilibrium at t → ∞ . This is indeed what is found by the numerical solutions in Fig. 14. – 17 –
00 200 300 400 500 t0.20.40.60.81.0 σ = < S [ t ] I [ t ] R [ t ]
100 200 300 400 500 600 700 t0.20.40.60.8 σ = > S [ t ] I [ t ] R [ t ] Figure 14 : Numerical solution of the differential equations (3.4) for S = 0 . γ = 0 . ζ = 0 .
01 for two different choices of (cid:15) : (cid:15) = 0 . σ = 0 . (cid:15) = 0 .
05 implying σ = 2 (right).
50 100 150 200 t0.20.40.60.81.0 σ = < S [ t ] I [ t ] R [ t ]
50 100 150 t0.20.40.60.81.0 σ = > S [ t ] I [ t ] R [ t ] Figure 15 : Numerical solution of the differential equations (3.4) for S = 0 . γ = 0 . ζ = − .
01 for two different choices of (cid:15) : (cid:15) = 0 . σ = 0 . (cid:15) = 0 .
05 implying σ = 2 (right).the rules they follow are very similar to the microscopic process of the field theory approach.To visualise this we used the results in Fig. 3, where the lattice is of size 201 ×
201 ( i.e. apopulation of 40401) and the recovery probability is fixed to 0 .
1. Once the recovery rateand the initial number of susceptible individuals S is fixed, in the SIR model the value ofthe infection rate completely determines the asymptotic number of total infected (3.11). Foreach coordination radius, we look for the best rescaling of the infection probability that couldreproduce the behaviour in Fig. 3, i.e. we compute the optimal ρ such that changing g −→ ρ g gives the best fit of the numerical results. We show the solution in Fig. 16.– 18 – a) r = 1 (b) r = 2(c) r = 5 (d) r = 50 Figure 16 : Evolution of the final number of infected cases as a function of the infectionprobability for different coordination radii r , compared to the asymptotic solution of theSIR model. The optimal factor found for the cases (a),(b),(c) and (d) are respectively: ρ = 0 . , . , . , . Apart from the numerical solutions, we can also gain insight into analytical aspects by dis-cussing a parametric solution of the classical SIR model [29]. For simplicity, we assume ζ = 0,– 19 –uch that the system (3.4), (3.1) and (3.3) reduces to dSdt ( t ) = − γ I ( t ) S ( t ) , dIdt ( t ) = γ I ( t ) S ( t ) − (cid:15) I ( t ) , dRdt ( t ) = (cid:15) I ( t ) , with ( S + I + R )( t ) = 1 and S ( t = 0) = S > ,I ( t = 0) = I > ,R ( t = 0) = 0 . (3.15)Since (3.1) allows to solve, e.g. , for R ( t ) = 1 − S ( t ) − I ( t ), it is sufficient to consider thedifferential equations for S and I . Dividing the latter by the former, we obtain a differentialequation for I as a function of S dIdS = − σ S , (3.16)which can be integrated to I ( S ) = − S + 1 σ ln S + c , for c ∈ R . (3.17)The parameter σ is defined in (3.5) and the constant c in (3.17) can be fixed by the initialconditions at t = 0 and gives c = I + S − σ ln S , such that I ( S ) = 1 − S + 1 σ ln SS . (3.18)A plot of this function in the allowed region P = { ( S, I ) ∈ [0 , × [0 , | S + I ≤ } , (3.19)for different initial conditions and σ = 0 . σ = 3 is shown in Fig. 17. An interestingfeature of the SIR model, which is visible from these graphs, is the fact that the solution I ( S ) in (3.18) has a maximum at I max = 1 − σ (1 + ln( σS )), which lies inside of P only ifthe initial effective reproduction number defined in Eq. (3.5) is R e, ≡ σS ≤
1. Since S ( t ) isa monotonically decreasing function of time, as demonstrated in [29], this implies: • If R e, ≤
1, then I ( t ) tends to 0 monotonically for t → ∞ . • If R e, > I ( t ) first increases to a maximum equal to 1 − σ (1 + ln( σS )) and thendecreases to zero for t → ∞ . The limit S ( ∞ ) = lim t →∞ S ( t ) is the unique root of1 − S ( ∞ ) + 1 σ ln (cid:18) S ( ∞ ) S (cid:19) = 0 , (3.20)in the interval [0 , σ ], which is explicitly given in terms of the Lambert function in (3.11).Furthermore, inserting the solution (3.18) into (3.15) we obtain the following non-linear, firstorder differential equation for S (as a function of time) dSdt = γ S ( S − − γ Sσ ln SS . (3.21)The latter can be solved numerically using various methods.– 20 – .0 0.2 0.4 0.6 0.8 1.0 S0.20.40.60.81.0I 0.0 0.2 0.4 0.6 0.8 1.0 S0.20.40.60.81.0I Figure 17 : Relative number of infectious I as a function of the relative number of susceptible S for S ∈ { . , . , . , . , . , . , . , . , . } and σ = 0 . σ = 3 (right).Curves with a local maximum are drawn in blue while curves which are monotonically growingwithin P are drawn in red. The SIR model, with 3 compartments (
S, I, R ) and constant rates γ , (cid:15) and ζ furnishes asimple, but rather crude description of the time evolution of an epidemic in an isolated pop-ulation. This description can be refined and extended in various fashions. The most commonway consists in adding more compartments, with more refined properties, like SIRD (includ-ing Deceased separately), SEIR (including Exposed individuals, in presence of a substantialincubation period), SIRV [30] (including vaccinated individuals), an so on [17]. Here, we aremainly interested in modifications of the rates, or couplings, between the three SIR compart-ments. In the following we indicate some of these and along with certain aspects of theirsolutions. In the classical SIR model (3.4), the rates ( γ, (cid:15), ζ ) are considered to be constant in time.This assumption is difficult to justify, in particular for epidemics that last over a longerperiod of time: for example, even in the absence of an effective vaccine, populations maytake measures to prevent the spread of the disease by imposing social distancing rules orquarantine procedures, thus changing the (effective) infection rate γ . Pathogen mutationsand various forms of immunisations (including vaccines) can also increase of reduce the valueof γ over time. With a prolonged duration of an epidemic, more data about the disease can– 21 –e collected, leading to better ways to fight it on a biological and medical level, thus changingthe recovery rate (cid:15) . Similarly, the disease may mutate and bypass previous immunisationstrategies, thus changing the rate ζ at which removed individuals may become susceptibleagain. Modelling such effects and gauging their impact on the time evolution of an epidemicsrequires ( γ, (cid:15), ζ ) to change over the duration of the pandemic. This can either be achieved byinterpreting them as (explicit) functions of t ∈ R ( i.e. ( γ ( t ) , (cid:15) ( t ) , ζ ( t ))), or (as a particularcase) to consider them to be functions of the relative number of susceptible and/or infectiousindividuals ( i.e. ( γ ( S, I ) , (cid:15) ( S, I ) , ζ ( S, I ))). Since (
S, I ) themselves are functions of time, thelatter possibility induces an implicit dependence on t . The functional dependence can beused, for example, to model population-wide lockdowns, i.e. quarantine measures that areimposed if the relative number of infectious individuals exceeds a certain value.In the following we shall provide a simple (numerical) example of how the time dependence
20 40 60 80 100 120 t0.20.40.60.81.0 S [ t ] I [ t ] I no - q [ t ] R [ t ] Figure 18 : Numerical solution of the SIRequations (3.4) for the time-dependent in-fection rate (3.22) with S = 0 . (cid:15) =0 . ζ = 0, γ = 0 . w = 0 . I = 0 . γ =const., but assume thatthe population takes measures (social distancing,lockdowns, etc. ) to ensure that the actual infec-tion rate γ ( t ) is reduced by a percentage w if thenumber of (active) infectious individuals exceedsa certain value ∆ I . To model such social dis-tancing measures in a very simplistic fashion, wecan for example introduce the following implicittime-dependence: γ ( I ) = γ [1 − w θ ( I ( t ) − ∆ I )] , (3.22)where θ is the Heaviside theta-function. We has-ten to add that (3.22) is evidently only a very crude depiction of lockdown and quarantinemeasures taken by societies in the real-world: indeed, decisions on whether or not to imposea lockdown (or other social distancing measures) are usually based on numerous indicatorswhich would (at least) require a more complicated dependence of γ on I ( e.g. its derivativesor averages of I over a certain period of time prior to t ). Furthermore, the conditions whena lockdown is lifted are typically independent of those when it is imposed.An exemplary numerical solution of (3.4) for the particular γ in (3.22) is shown in Fig. 18.For better comparison we have also plotted I no-q ( t ), which is the solution for I ( t ) in the caseof constant γ = γ = const. ( i.e. with no reduction of the infection rate) and all remaining To be mathematically rigorous, since θ is not a continuous function, using such an infection rate in (3.4),would require to interpret ( S ( t ) , I ( t ) , R ( t )) as distributions. This can be circumvented by replacing θ ( I ( t ) − ∆ I )by 1+tanh( κ ( I ( t ) − ∆ I )) with κ a parameter that ’smoothens’ the step function. For the following discussion,however, this point shall not be relevant. – 22 –arameters chosen the same. Despite its simplicity and shortcomings, the model allows tomake a few basic observations: the plot shows that the time-dependent infection rate leads toa reduction of the maximum of infectious individuals (’flattening of the curve’). Moreover, thissimple model allows to compare the effectiveness of the quarantine measures as a functionof w and ∆ I . To gauge this effectiveness, we consider the cumulative number of infectedindividuals, which is plotted for different values of w and ∆ I in Fig. 19. These plots, confirmthe intuitive expectation that lockdown measures are the more effective the stronger thereduction of the infection rate is and the earlier they are introduced. However, due to itssimplicity, the model also misses certain aspects compared to the time evolution of real-worldcommunicable diseases in the presence of measures to prevent its spread: for example, possiblydue to non-zero incubation time of most infectious diseases, the effect of quarantine measureson the number of infectious individuals can be detected only a certain time after the measureshave been imposed (see [31–34] where this has been established for the COVID-19 pandemic).To include the latter would require a refinement of the model. I C / N w = = = = I C / N Δ I = Δ I = Δ I = Δ I = Figure 19 : Numerical solution of the SIR equations (3.4) for the time-dependent infectionrate (3.22), with S = 0 . (cid:15) = 0 . ζ = 0, γ = 0 . w, ∆ I ): w ∈ { . , . , . } and ∆ I = 0 .
05 (left) and w = 0 .
25 and ∆ I ∈ { . , . , . , } (right). In Section 2.3, in the context of percolation models, we have discussed (microscopic) processesthat correspond to the ’spontaneous’ creation of infected individuals. Such processes can,for example, simulate the infection of individuals through external sources ( e.g. pathogensources, contaminated food sources, wildlife, etc .), but may also be used to model the infectionof susceptible individuals through asymptomatic infectious individuals or the appearance ofinfectious individuals from outside of the population through travel. How to introduce thisprocess in SIR-type models has been discussed at the end of section 2.4. Mathematically, the– 23 –IR equations (3.4) can be extended to (where the rate ξ = ˆ ξ of section 2.4): dSdt = − γ I S + ζ R − ξ S , dIdt = γ I S − (cid:15) I + ξ S , dRdt = (cid:15) I − ζ R , (3.23)which still needs to be solved with the initial conditions (3.3). Here ξ ∈ R + is a constantthat governs the rate at which new infectious individuals appear in the population. The lat-ter corresponds to a qualitative change in the basic infection mechanisms: since susceptibleindividuals can contract the disease even if there are no infectious individuals present in thepopulation, the epidemic can not be stopped before the entire population becomes infected.As a consequence, the cumulative number of infected tends to N for t → ∞ . This is schemat-
50 100 150 200 t0.20.40.60.81.0 S [ t ] I [ t ] R [ t ] I C [ t ]/ N
50 100 150 200 t0.20.40.60.81.0 S [ t ] I [ t ] R [ t ] I C [ t ]/ N Figure 20 : Numerical solution of the differential equations (3.23) for S = 0 . γ = 0 . ζ = 0 .
045 for two different choices of ξ : ξ = 0 (left) and ξ = 0 .
002 (right).ically shown in Fig. 20, where we show the solutions for ξ = 0 (left panel) compared to thesolution for ξ (cid:54) = 0 (right panel). In the former case, the number of cumulative infected tendsto a finite value, while in the latter case, lim t →∞ S ( t ) →
50 100 150 200 t0.20.40.60.81.0 S [ t ] I [ t ] R [ t ] I C [ t ]/ N Figure 21 : Numerical solution of thedifferential equations (3.23) for S =0 . γ = 0 . ζ = 0 .
045 and ξ =0 . (cid:12)(cid:12) sin (cid:0) πt (cid:1)(cid:12)(cid:12) .dependent rate ξ ( t ). This can be used to model atime-dependent rate of the spontaneous creationof new infectious individuals, e.g. induced byquarantine measures or geographical restrictionsof the population. As a simple example, we haveplotted the numerical solution for a periodic func-tion ξ in Fig. 21. Since ξ does not remain zeroafter finite time, the relative number of suscep-tible tends to 0 (indicating that the entire pop-ulation is infected for t → ∞ ). Moreover, thesolution features oscillations in time, which couldbe interpreted as different waves of the epidemicspreading in the population. – 24 – .5.3 Superspreaders Another generalisation of the SIR models consists in adding multiple compartments of in-fectious individuals, i.e. new subgroups that allow to refine the study of the disease spreadin a not-so-uniform population. These additional compartments can, therefore, distinguishindividuals based on biological/medical indicators ( e.g. gender, age, preexistent medical con-ditions, etc.), geographic distribution, social behaviour and/or may be used to introduceadditional stages in the progression of the disease, such as latency periods or different stagesof symptoms. Inclusion of more compartments naturally renders the relevant set of differen-tial equations more complicated and is more demanding in terms of computational costs (see[35] as an example).In the following we shall present one simple example of including an additional class ofindividuals which is useful when modelling different (social) behaviour among individuals.Indeed, in general, the infection rate γ is not homogeneous throughout the entire population, N S N I N I N R β ( γ I + γ I ) N S (1 − β )( γ I + γ I ) N S (cid:15) N I (cid:15) N I ζ N R Figure 22 : Flow between susceptible, 2 compart-ments of infectious and removed individuals.since it depends on various factorssuch as geographical mobility, socialbehaviour etc. , which may vary con-siderably. A particular effect in thisregard is the existence of so-called su-perspreaders . These are individualswho are capable of transmitting thedisease to susceptible individuals at arate that significantly exceeds the av-erage. This effect could also be dueto a mutation in the pathogen caus-ing the disease. The presence of su-perspreaders can be described by in-troducing two groups of infectious individuals I , , with different infection rates γ , andappearing with a relative ratio β ∈ [0 , ζ = 0), and can be describedby the following differential equations [36]: dSdt = − ( γ I + γ I ) S , dI dt = β ( γ I + γ I ) S − (cid:15) I ,dI dt = (1 − β )( γ I + γ I ) S − (cid:15) I , dRdt = (cid:15) ( I + I ) , (3.24)together with the algebraic relation 1 = S + I + I + R , (3.25)as well as the initial conditions S ( t = 0) = S , I ( t = 0) = I , , I ( t = 0) = I , , R ( t = 0) = 0 , (3.26)– 25 –ith 0 ≤ S , I , , I , ≤ , S + I , + I , . (3.27)In [36] the parameters β , γ , , and (cid:15) were assumed to be constant in time. By defining aneffective infectious population J = ( γ I + γ I ) /λ , we can extract the following differentialequations for ( S, J ) dSdt = − λ J S , dJdt = λ J S − (cid:15) J , with λ = γ β + (1 − β ) γ . (3.28)Thus, for S and J we obtain the same equations as in the classical SIR model, which can besolved along the lines of section 3.4: we extract the following non-linear first-order equationfor S : dSdt = λ S − (cid:15) S ln S + c S , with c = (cid:15) ln S − λ S − ( γ I , + γ I , ) . (3.29)which leads to the asymptotic number of susceptible S ( ∞ ) implicitly given by0 = λ S ( ∞ ) − (cid:15) ln S ( ∞ ) + c . (3.30)As was pointed out in [36], the SIR model with superspreaders leads to the same dynamicsas the classical SIR models, albeit with a larger-than-average infection rate λ , due to thecontribution of superspreaders. With constant infection and recovery rates and monoton-ically diminishing number of susceptible ( i.e. for ζ = 0), the impact of superspreaders isconceptually not detectable. Nevertheless, from the perspective of the total number of in-fected, superspreaders may have a significant impact in driving the epidemics. In Fig. 23(left) we have plotted the time evolution of a typical solution, which indeed follows the samepattern as the usual SIR model. However, as visible from Fig. 23 (right), even the presenceof a relatively small number of superspreaders can have a strong impact on the cumulativenumber of infected.Finally, it was argued in [36] that in situations in which the number of susceptible in-dividuals is no longer a monotonic function (which can for example be achieved by allowingfor a non-trivial ζ ), the time evolution of the SIR model looks qualitatively different in thepresence of superspreaders. As we have seen from simple numerical studies in section 3.2, solutions ( S ( t ) , I ( t ) , R ( t ))of the classical SIR equations (3.4) exhibit interesting properties as functions of time, whichstructurally remain valid for many of the generalisations discussed in section 3.5. In particular, Note that our definition of J differs from the definition of the infective potential J = γ I + γ I in [36]by a constant normalisation. – 26 – S [ t ] I [ t ] I [ t ] R [ t ]
50 100 150 200 250 t0.10.20.30.40.50.60.7 I C / N I C [ t ]/ N I C ( (cid:1) = ) [ t ]/ N Figure 23 : Numerical solution of the SIR equations in the presence of superspreaders (3.24):time evolution for S = 0 . I , = 0 . I , = 0, γ = 0 . γ = 1, (cid:15) = 0 .
05 and β = 0 . i.e. β = 1) (right).the solutions show a qualitatively different behaviour when a key parameter (in the classicalSIR model, the initial effective reproduction number R e, = S σ ) exceeds a critical value.This seems to play a similar role to an ordering parameter in physical systems undergoing aphase transition. A further related observation is the fact that Eqs (3.4) are invariant undera re-scaling of the time-variable, if simultaneously all the rates are also re-scaled: t → µ t , γ → µ γ , (cid:15) → µ (cid:15) , ζ → µ ζ , ∀ µ ∈ R \ { } . (3.31)This rescaling of the time-variable is structurally not unlike the change of the energy scalein (quantum) field theories that is used to describe the Wilsonian renormalisation of thecouplings among elementary particles [12, 13]. The renormalisation flow can also featuresimilar symmetries to the ones of the solutions of the SIR equations. Compartmental modelscan be formulated in a way that is structurally similar to Renormalisation Group Equations(RGEs) [4, 5], and this analogy lead to the formulation of an effective description called epidemiological Renormalisation Group [4, 11], which we will introduce in the next section.To understand the analogy, we recall that most (perturbative) quantum field theories areeffective models: they are typically based on an action that encodes fundamental interactionsof certain ’bare’ fundamental fields, whose strength is described by a set of coupling constants { λ i } (where i takes values in a suitable set {S} ). Each effective description, however, is gen-erally well adapted only at a certain energy scale, beyond which new degrees of freedom aremore appropriate and new interactions may become important. In practice, one introduces acut-off parameter (or some other regularisation form), beyond which the effective descriptionis no longer valid. The Lagrangian can thus be interpreted as encoding all effective interac-tions, after having integrated out all interactions at an energy scales higher than the cut-off.– 27 –rom this perspective it is clear that changing the energy scale (and thus the cut-off) will leadto different interactions being integrated out and thus has a strong impact on the Lagrangian(along with the fundamental degrees of freedom used to describe it). The process of arrivingat the new effective theory is called renormalisation . To describe this process, we study uni-versal quantities that are invariant under the renormalisation, first and foremost the partitionfunction Z ( { λ i } ), which depends on the set of coupling constants mentioned earlier. If { λ (cid:48) a } (with a taking values in a new set {S (cid:48) } ) is the new set of renormalised couplings and Z (cid:48) thepartition function of the renormalised theory, invariance of the partition function implies Z ( { λ i } ) = Z (cid:48) ( { λ (cid:48) a } ) . (3.32)Thus continuously changing the energy scale, the theory will sweep out a trajectory in thespace of all possible effective theories, called the renormalisation group flow , which is governedby the invariance (3.32). From the perspective of the Lagrangian, the theory sweeps out atrajectory in the space of all couplings λ i . This is governed by the beta-functions β i ( λ k ),defined as the derivatives of the couplings λ i with respect to the logarithm of the cut-offparameter, and are functions of the couplings λ i . The flow is thus described in terms of asystem of differential equations, like the SIR model does, whose fixed points ( i.e. zeros of thebeta functions) denote critical ( i.e. scale invariant) points of the theory.Before making the connection to epidemiology, we remark that physical theories in generalallow for field redefinitions, which means that they can equivalently be formulated usingdifferent bare fields. This implies that the coupling set { λ i } is not unique, but should ratherbe thought of as a (local) choice of basis in the space of couplings. A specific choice of aset of { λ i } is called a (renormalisation) scheme . While a priori the specific form of the beta-functions depend on the scheme (in particular their perturbative expansions as functions ofthe { λ i } ), a change of scheme can be understood as an analytic transformation in the spaceof couplings.In [4], and subsequent works [9, 11, 14], it was suggested to interpret the time evolutionof the spread of a disease (specifically COVID-19) within the framework of the Wilsonianrenormalisation group equation. We shall explain this description in more detail in section 4.In the following, however, we shall show how such a description can at least qualitatively beobtained from the SIR equations by allowing time-dependent infection and removal rates, asfirst pointed out in [11]. In preparation of section 4, we notice that the SIR model (with ζ = 0, but time-dependentinfection and recovery rates γ ( t ) and (cid:15) ( t )) can be written in a form which is strongly remi-niscent of a renormalisation group equation. To this end, we return to (3.15) and repeat thesame steps as in section 3.4 except for allowing σ : [0 , → R + to be a priori a function of S .Thus, we can integrate equation (3.16) in the following form I ( S ) = 1 − S + (cid:90) SS duu σ ( u ) , (3.33)– 28 –hich is compatible with the initial conditions in Eq.(3.15) at t = 0. Inserting this relationinto the first equation of (3.4) (for ζ = 0) yields dSdt = − γ ( t ) S ( t ) (cid:20) − S + (cid:90) SS duu σ ( u ) (cid:21) . (3.34)Instead of the relative number of susceptible, this equation can be re-written in terms of thecumulative number of infected individuals I c , as defined in Eq. (3.6). Thus Eq.(3.34) can berewritten as dI c dt = N γ (cid:18) − I c N (cid:19) (cid:34) I c N + (cid:90) − I c N S duu σ ( u ) (cid:35) . (3.35)Next, generalising what was proposed in [4, 9], we define α ( t ) = φ ( I c ( t )) , (3.36)where φ : [0 , N ] → R is a strictly monotonically growing, continuously differentiable function(with non-vanishing first derivative). In [4] (in the context of the COVID-19 pandemic) φ waschosen to be the natural logarithm, while in [9, 10] φ ( x ) = x was chosen. For the moment, weshall leave φ arbitrary, which mimics the liberty to choose different renormalisation schemesin the framework of the Wilsonian approach. Upon defining formally the β -function as β ( I c ( t )) = − dαdt , (3.37)Eq. (3.35) can be re-formulated as − β = (cid:18) dφdI c (cid:19) dI c dt = (cid:18) dφdI c (cid:19) N γ (cid:18) − I c N (cid:19) (cid:34) I c N + (cid:90) − I c N S duu σ ( u ) (cid:35) . (3.38)An explicit example that is designed to make contact with the work in [9] is discussed inAppendix B. Eq.(3.38), at least structurally, resembles a RGE and has several intriguingproperties to support this interpretation. Note that with Eq.(3.6), we can also write β ( t ) = − (cid:18) dφdI c (cid:19) dI c dt = − (cid:18) dφdI c (cid:19) N γ ( t ) I ( t ) S ( t ) , (3.39)which vanishes when: • the infection rate vanishes γ ( t ) = 0, • or there are no susceptible individuals left S ( t ) = 0, • or the number of active infected vanishes I ( t ) = 0 and the disease is eradicated. A priori, φ could also explicitly depend on t (not only through I c ( t )). In the following we shall not explorethis possibility. – 29 –urther (structural) evidence can be given by considering concrete solutions. A concreteexample for the interplay between the beta-function and σ is provided in appendix B. Fur-thermore, independently of its connection to compartmental models, a renormalisation groupapproach can be used to model and describe the dynamics of an epidemic, as we discuss inthe following Section. As already mentioned in the previous section, it has been proposed [4, 14] to study thespread of a communicable disease within the framework of the Wilsonian renormalisationgroup. We have already pointed out that the SIR model (3.4) can be formulated in a fashionthat is structurally similar to a RGE. In this section we will, therefore, review the newframework proposed in [4, 14], dubbed epidemic Renormalisation Group (eRG) . The latteris an effective description in which microscopic interactions have been “integrated out”, anddetails of the other approaches are taken into account in an effective way. This leads to amuch more economical description in terms of calculation complexity. This has been usedto to characterise a single epidemic wave, to study inter-region propagation of the disease[14–16] or the effect of non-pharmaceutical interventions [34].The main hint for this new framework came from data of the Hong Kong (HK) Sars-2003 outbreak, as well as the COVID-19 pandemic during the spring of 2020: as pointed outfirstly in [4], the time dependence of the cumulative total number of infected cases in variousregions of the world shows the same characteristic behaviour. In this framework, therefore,the relevant quantity is the number of all individuals that have been infected with the diseaseuntil the given time. However, the same framework can be also applied to the number of
10 20 30 40 50 60 70 t5000100001500020000 I C Figure 24 : The logistic functionschematically representing the cumulativenumber of infected as a function of time.With regards to (4.1) we have A = 20 . B = 1 . .
000 and κ = 0 .
2. hospitalisations or the number of deceased in-dividuals. In [4], it was shown that the time-dependent cumulative number of infected individ-uals, I c ( t ) ∼ f ( t ), can be described in terms of a logistic function f : R −→ [0 , A ] t (cid:55)−→ f ( t ) = A B e − κt , (4.1)where A, B, κ ∈ R + \ { } .This function shows a characteristic ’S’-shape(see Figure 24 for a schematic representation) andis a solution of the non-linear first order differen-tial equation dfdt ( t ) = κA f ( t ) ( A − f ( t )) . (4.2)– 30 –he solution has the following asymptotic valueslim t →−∞ f ( t ) = 0 , lim t →∞ f ( t ) = A , (4.3)corresponding to the zeros of the derivative dfdt = 0. The parameter A corresponds, therefore,to (a function of) the asymptotic number of infected cases during the epidemic wave. On theother hand, κ , which has dimension of a rate, measures how fast the number of infectionsincrease, while B corresponds to a shift of the entire curve in time and determines the be-ginning of the infection increase. More details about the properties of this function and itsepidemiological interpretation can be found in [4] and will not be repeated here. It is, how-ever, important to notice that the parameters κ and A can be removed from the differentialequation by a simple rescaling of the function and of the time variable: d ˜ fdτ = ˜ f ( τ ) (1 − ˜ f ( τ )) , τ = κt , ˜ f ( τ ) = f ( τ /κ ) A . (4.4)While A is a mere normalisation, κ can be thought of as a ’time dilation’ parameter. Oncethe normalised solutions are shown in the ’local time’ τ , therefore, all epidemic waves revealthe same temporal shape.Qualitatively, the solution f ( t ) interpolates between a fixed point at t → −∞ (whichcorresponds to the absence of any infected) to a fixed point at t → ∞ (which corresponds tothe asymptotic value of all infections after the eradication of the disease). Furthermore, thefixed point at t → −∞ is repulsive, i.e. a single infectious individual will trigger the spread ofthe entire disease, while the fixed point at t → ∞ is attractive and represents the end of theepidemic. In [4, 9, 14] therefore the following dictionary between the spread of an epidemicand the Wilsonian renormalisation group was proposed: • The time variable is identified with the (negative) logarithm of the energy scale µtt = − ln (cid:18) µµ (cid:19) , (4.5)where t / µ set the scale for the time and energy (for simplicity, and without loss ofgenerality, we will fix t = 1). With this identification, Eq. (4.2) is similar to the RGEfor the gauge coupling in a theory that features a Banks-Saks type fixed point [37], i.e. an interactive fixed point at low energies (Infra-Red). • The solution can be associated to a coupling constant in the high energy physics RGEs, f ≡ α . The epidemic coupling strength is defined as a monotonic, derivable and bijec-tive, function φ of the cumulative number of infected cases α ( t ) = φ ( I c ( t )) . (4.6)In [4, 9] φ was chosen as the natural logarithm φ ( x ) = ln( x ), while in [9, 10] it was chosen φ ( x ) = x . The choice was justified by a better fit to the actual data of the COVID-19 pandemic, while from the perspective of the Wilsonian renormalisation group, thedifference corresponds to a different choice of scheme.– 31 – The beta function is defined as the time-derivative of the epidemic coupling strength β ≡ dαd ln (cid:16) µµ (cid:17) = − dαdt = − dφdI c dI c dt ( t ) . (4.7)Since φ is a monotonic function, fixed points of the β -function correspond to zeroes of thederivative of I c , which we denote I ∗ c . They can be characterised through the so-called scalingexponents : ϑ = ∂β∂α (cid:12)(cid:12)(cid:12)(cid:12) α ∗ = (cid:40) − κ for α ∗ = 0 ,κ for α ∗ = A , (4.8)where α ∗ = φ ( I ∗ c ) is the epidemic coupling constant at the fixed point. Negative (positive)scaling exponents correspond to a repulsive (attractive) fixed point.In order to better model the respective data of various countries during the COVID-19pandemic, it was furthermore proposed in [9, 10] to consider the more general beta-function − β ( α ) = dαdt ( t ) = λ α (cid:16) − αA (cid:17) p , (4.9)for p ∈ [1 / , ∞ ] and λ, A ∈ R + . The role of the exponent p is to smoothen the ’S’-shape ofthe solution when it approaches the attractive fixed point at α ∗ = A . The approach discussed so far assumes an isolated population of sufficient size. However, thesimplicity of the eRG approach allows for a simple generalisation to study the interactionbetween various regions of the world [14] via the travel of individuals. For M separated pop-ulations (labelled by i = 1 , . . . , M ) of size N i whose cumulative number of infected is denoted I c ,i , it was shown in [14] that infections can be transmitted between these populations bytravellers. Thus, the epidemic diffusion can be described by M coupled differential equations,in the form of Eq.(4.9) for each population, with the addition of the following term: − β ( α i ) = λ α i (cid:16) − α i A (cid:17) p + dφdI c ,i M (cid:88) j =1 k ij N i ( I c ,j ( t ) − I c ,i ( t )) , (4.10)where k ij ∈ R is a measure for the number of travellers between populations i and j . Thecontribution to the beta function can be obtained by replacing I c ,i → φ − ( α i ), where α i isthe epidemic coupling in each population. For more details, see Ref. [14].This new term couples the epidemic couplings in each region, and can explain the diffusionof the virus among regions. It has been validated by predicting the second wave of COVID-19that has hit Europe in the fall of 2020 [15] and by explaining the wave pattern observed inthe United States [16]. In order to avoid confusion with the parameters appearing in the SIR model (notably Eq. (3.4)), we haverenamed the overall coefficient in (4.9) λ instead of γ . – 32 – = δ =- δ max λ t0.00.51.01.5 α ( t ) p = / = - - - δ λ Δ t endemic Figure 25 : Right: solutions of the CeRG equation, normalised to A = 1 and with time inunits of λ , for − δ = 0 , − , − , − and δ max , for p = 0 .
55. Left: Estimated duration ofthe linear growth phase, in units of λ , as a function of − δ for p = 0 .
5, 0 .
6, 0 .
7, 0 .
8, 0 . δ = − δ max . Although the beta-function in Eq. (4.9) is relatively simple and contains only two parameters,it describes the time evolution of short-time epidemics (such as HK SARS-2003 and eachwave of COVID-19) quite efficiently, as the flow from a repulsive to an attractive fixed point(or from an ultraviolet to an infrared fixed point in the language of high-energy physics).However, this beta-function is too simple to describe correctly longer lasting pandemics witha more intricate time-evolution (such as subsequent waves of COVID-19): the attractive fixedpoint at t → ∞ corresponds to a complete eradication of the disease and (4.9) thus describesoutbreaks that follow a single wave.In particular, COVID-19 epidemiological data has clearly shown that most waves, definedas periods of exponential growth in the number of new infected cases, are followed by periodswhere the number of new cases remains constant. This leads to a linear growth in the cumu-lative number of infections, I c . In [9] it was proposed that this linear phase is evidence fora near time-scale invariance symmetry in the dynamics governing the diffusion of the virus.The time-evolution of pandemics can still be described within the framework of a renormali-sation group equation, however with a more complicated beta-function that features a richerstructure of (complex) fixed points. The new framework was called the Complex epidemicRenormalisation Group (CeRG) . In the CeRG approach, the beta function of Eq. (4.9) ismodified as follows: − β ( I c ) = dI c dt = λ I c (cid:34)(cid:18) − I c A (cid:19) − δ (cid:35) p = λ I c (cid:18) I c A − √ δ (cid:19) p (cid:18) I c A − − √ δ (cid:19) p , (4.11)where the additional parameter δ ∈ R − , i.e. δ = −| δ | . While this equation can be written forany epidemic coupling α , here we commit to the case α ( t ) = I c ( t ) for reasons that will be clear– 33 –n the next subsection. The eRG equation can be recovered for δ →
0. For non-vanishing δ ,instead of only two asymptotic fixed points, this functions has three fixed points I c , = 0 , I c , ± = A (cid:16) ± i (cid:112) | δ | (cid:17) . (4.12)with I c , ± ∈ C complex. Besides the (repulsive) fixed point at I ∗ c = 0 which remains, theattractive fixed point splits into two complex fixed points. Since the (cumulative) numberof infected individuals is a strictly real number, the system cannot actually reach the latterfixed points and thus cannot exactly enter into a time-scale invariant regime at infinite time.Instead, for small | δ | , when the solution approaches the would-be fixed point at I c ≈ A , thetime evolution will be strongly slowed down due to the effect of the nearby complex fixedpoint. This results in a near-linear behaviour of the solution, as shown in the left panel ofFig. 25. Thus, the new beta function (4.11) realises an approximate time-scale symmetry inthe solution. Concretely, the precise form of the flow in the vicinity of these (complex) fixedpoints depends on | δ | : • For | δ | < δ max = p p , the beta-function has a local maximum and I c enters into aregime of near linear growth characterised by dI c dt ( t ) ∼ const. (4.13)In the context of epidemics, the linear growth phase can be associated to an endemicphase of the disease, when the virus keeps diffusing within the population without anexponential growth in the number of new infected (this corresponds to a situation withreproduction number R = 1, which keeps the number of infectious cases constant). • In the CeRG, the linear growth is only an intermediate phase, which preludes to a newexponential increase in the number of infections. The duration depends on | δ | , and cancan be estimated as [9] ∆ t endemic = − (cid:90) ∞ A dI c β ( I c ) . (4.14)This time is plotted for different values of p as a function of δ in the right panel ofFig. 25. • For | δ | ≥ δ max the beta-function no longer has a local maximum and I c keeps growingexponentially, without a linear growing phase.The endemic linear-growing phase, therefore, is the prelude of a new wave of the epidemicdiffusion. The CeRG approach can describe this endemic phase and the beginning of the nextwave, however the number of infections would continue to grow indefinitely. In the followingsection we will further extend the approach to take into account the multi-wave pattern.– 34 – .3 Multi-wave pattern explained Pandemics like the 1918 Spanish flu [38] and COVID-19 have shown the appearance of multipleconsecutive waves of exponential increase in the number of infections. In the case of COVID-19, the data support the fact that an endemic linearly-growing phase is always present inbetween two consecutive waves [10]. The CeRG model can be extended to take into accountthis structure, in a way that reproduces nicely the current data [16].The multi-wave beta function, for an epidemic with w consecutive waves, can be writtenas: − β multi − waves ( I c ) = λI c w (cid:89) ρ =1 (cid:34)(cid:18) − ζ ρ I c A (cid:19) − δ ρ (cid:35) p ρ , (4.15)with ζ ρ ≤ | δ ρ | (cid:28) p ρ > ρ ∈ { , . . . , w } . The normalisation A can be fixed tomatch the first wave, so that 0 < ζ w < · · · < ζ < ζ = 1 . (4.16)Besides the repulsive fixed point at I ∗ c = 0, the equation has a series of complex fixed pointsruled by the parameters δ ρ . Without loss of generality, we can fix δ w = 0 so that the disease isextinguished after the last wave, and the total number of infection during the whole epidemicis given by lim t →∞ I c ( t ) = A/ζ w . This description, however, only works for α ( t ) ∝ I c ( t ), forwhich the value of the various fixed points are well separated [10], but not for α ( t ) ∝ ln I c ( t ). The methods that we have discussed so far are in principle applicable to a large numberof different diseases. The main differences are in key parameters (method of transmission,incubation time, mortality rate etc. ) and have an impact on the resulting time evolution ofthe epidemic (such as total duration of the epidemic, total number of infected and fatalities, etc .). The ongoing COVID-19 pandemic has shown interesting features concerning its timeevolution, namely a distinct multi-wave structure of repeated phases of exponential growth inthe number of infected individuals interspersed with phases of (quasi-)linear growth. Thesephases can be modelled (and studied) within some of the models discussed above such as theeRG framework that makes use of near fixed point dynamics. After recalling the data forCOVID-19 we will further discuss how SIR-like models can take into account the inter-wavedynamics.
As large scale testing is at the heart of many countries’ strategies to combat the COVID-19pandemic, there are large amounts of data available documenting the spread of the corona-virus across the globe. These data reveal universal trends, which we shall discuss briefly inthe following. We shall limit ourselves to the time-evolution before the advent of vaccinations.– 35 –he outbreak of the epidemic is usually characterised by an exponential growth of thenumber of infections, followed by an endemic phase of (quasi-)linear growth. The latter in turnis not a stable phase, but typically leads to another exponential growth phase, resulting in awave-like time evolution of the spread of COVID-19. As a prototypical example, the epidemicdata for South Africa are shown in Figure 26: the exponential growth of infections leads toa local maximum at around calendar week 31 (left panel), followed by a quasi-linear rise ofthe cumulative number of infected (right plot), which in turn anticipates a new exponentialgrowth phase.
Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan
South Africa
10 20 30 40 50020 00040 00060 00080 000100 000120 000 Calendar week N e w I n f e c t ed Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan
South Africa
10 20 30 40 500200 000400 000600 000800 0001 × Calendar week C u m u l a t i v e I n f e c t ed Figure 26 : Two-wave structure of the spread of COVID-19 in South Africa: number of newinfections per day (left) and cumulative number of infected (right).The onset and duration of each phase, as well as their precise shape and even the numberof waves, differ from country to country: for comparison, in Fig. 27 the corresponding datafor the number of new infections and the cumulative number of infected for Japan are shownas functions of time.
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan
Japan
10 20 30 40 50010 00020 00030 00040 000 Calendar week N e w I n f e c t ed Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan
Japan
10 20 30 40 50050 000100 000150 000200 000250 000 Calendar week C u m u l a t i v e I n f e c t ed Figure 27 : Three-wave structure of the spread of COVID-19 in Japan: number of newinfections per day (left) and cumulative number of infected (right).– 36 – .2 Analytic Solution during Linear Growth Phase
In [9, 10] it has been pointed out (and reviewed in the previous subsection) that, in mostcountries, the data reflecting the evolution of the COVID-19 pandemic show a particularphase during which the cumulative number of infected grows linearly as a function of time.This phenomenon has been discussed already within the effective description of the eRG inSection 4.2. Here we try to understand it from the perspective of the compartmental models:indeed, we have seen from the explicit solutions in section 2 and 3 that such a behaviour isnot found in simple percolation and compartmental models in which, notably, the probabilityor rate of infection remains constant throughout the entire pandemic. However, more generalapproaches and extensions of these simple models can have solutions that exhibit such lineargrowth phases. Since the phenomenon is seen in the cumulative number of infected (which isa ’global’ key figure pertaining to the entire population), we shall in the following analyse itfrom the perspective of a SIR model, with time-dependent infection and recovery rate.
We consider a SIR model described by the equations (3.4) and the constraint (3.1) as wellas the initial conditions (3.3) with time-dependent γ , (cid:15) and ζ (see Section 3.5.1). We defineas a linear growth regime a period [ t , t ] for which the cumulative number of infections I c ,defined in Eq.(3.6) as: I c ( t ) = N I + (cid:90) t dt (cid:48) γ ( t (cid:48) ) N I ( t (cid:48) ) S ( t (cid:48) ) , (5.1)is a linear function of time, i.e. ddt I c ( t ) = N f = const. ∀ t ∈ [ t , t ] , (5.2)while 0 ≤ S ( t ) , I ( t ) , R ( t ) ≤
1, with f ∈ R + . This implies γ ( t ) I ( t ) S ( t ) = f ∀ t ∈ [ t , t ] . (5.3)The condition above allows to analytically solve the SIR equations (3.4) ∀ t ∈ [ t , t ] with theinitial conditions at the beginning of the linear growth S ( t = t ) = S s , I ( t = t ) = I s , R ( t = t ) = R s , with 0 ≤ S s , I s , R s ≤ ,S s + I s + R s = 1 . (5.4)To see this, we define D (cid:15) ( t ) = e (cid:82) tt (cid:15) ( t (cid:48) ) dt (cid:48) , and D ζ ( t ) = e (cid:82) tt ζ ( t (cid:48) ) dt (cid:48) , (5.5)which have the properties dD (cid:15) dt ( t ) = (cid:15) ( t ) D (cid:15) ( t ) , dD ζ dt ( t ) = ζ ( t ) D ζ ( t ) , D (cid:15) ( t = t ) = 1 = D ζ ( t = t ) . (5.6)– 37 –ext, we insert (5.3) into (3.4) to obtain the equation dIdt = − (cid:15) I + f , ∀ t ∈ [ t , t ] . (5.7)which is only an equation for I (and is decoupled from S and R ). Multiplying by D (cid:15) ( t ), wefind (cid:20) dIdt + (cid:15) I (cid:21) D (cid:15) ( t ) = f D (cid:15) ( t ) ddt [ I ( t ) D (cid:15) ( t )] = f D (cid:15) ( t ) (5.8)which can be directly integrated, with the initial conditions (5.4), as: I ( t ) = 1 D (cid:15) ( t ) (cid:20) f (cid:90) tt D (cid:15) ( t (cid:48) ) dt (cid:48) + I s (cid:21) , ∀ t ∈ [ t , t ] . (5.9)For the relative number of recovered, R , we can integrate the last equation of (3.4) dRdt ( t ) + ζ ( t ) R = (cid:15) ( t ) I ( t ) , (5.10)where, inserting the solution for I ( t ) in (5.9), the right hand side can be understood as aninhomogeneity. Multiplying by D ζ we obtain, as before, ddt [ R ( t ) D ζ ( t )] = (cid:15) ( t ) I ( t ) D ζ ( t ) , (5.11)which can be directly integrated, with the initial conditions (5.4), to give R ( t ) = R s D ζ ( t ) + I s (cid:90) tt dt (cid:48) (cid:15) ( t (cid:48) ) D (cid:15) ( t (cid:48) ) D ζ ( t (cid:48) ) D ζ ( t ) + f (cid:90) tt dt (cid:48) (cid:90) t (cid:48) t dt (cid:48)(cid:48) (cid:15) ( t (cid:48) ) D (cid:15) ( t (cid:48)(cid:48) ) D (cid:15) ( t (cid:48) ) D ζ ( t (cid:48) ) D ζ ( t ) , ∀ t ∈ [ t , t ] . (5.12)Finally, S ( t ) is obtained through the constraint (3.1): S ( t ) = 1 − I ( t ) − R ( t ). Notice that thesolutions (5.9) and (5.12) remain valid as long as 0 ≤ S ( t ) , I ( t ) , R ( t ) ≤ ζ and Constant (cid:15) To simplify the solutions found above, we can adapt the functions ζ and (cid:15) to reflect moreclosely the COVID-19 pandemic: since currently only very few cases of patients contractingCOVID-19 twice are known in the medical literature [39], we can set ζ ( t ) = 0 to simplify thesolutions (5.9), (5.12). Since ζ = 0 also implies D ζ ( t ) = 1, we find for these solutions S ( t ) = S s − f ( t − t ) ,I ( t ) = I s D (cid:15) ( t ) + f (cid:90) tt D (cid:15) ( t (cid:48) ) D (cid:15) ( t ) dt (cid:48) ,R ( t ) = R s + I s (cid:90) tt dt (cid:48) (cid:15) ( t (cid:48) ) D (cid:15) ( t (cid:48) ) + f (cid:90) tt dt (cid:48) (cid:90) t (cid:48) t dt (cid:48)(cid:48) (cid:15) ( t (cid:48) ) D (cid:15) ( t (cid:48)(cid:48) ) D (cid:15) ( t (cid:48) ) , ∀ t ∈ [ t , t ] . (5.13)– 38 –e have verified in appendix C that this is indeed a solution of (3.4) that satisfies the correctinitial conditions.Furthermore, since the recovery rate in most cases depends on the disease in questionand/or the state of medical advancement to cure it, (cid:15) is difficult to change throughout apandemic without significant effort. For simplicity, we therefore consider it in the followingto be constant, i.e. (cid:15) = const. (in addition to ζ = 0), such that D (cid:15) ( t ) = e (cid:15) ( t − t ) . In this case,we can perform the integrations in (5.13) I ( t ) = e − (cid:15) ( t − t ) (cid:20) f (cid:90) tt dt (cid:48) e (cid:15) ( t (cid:48) − t ) + I s (cid:21) = e − (cid:15) ( t − t ) I s + f(cid:15) (cid:16) − e − (cid:15) ( t − t ) (cid:17) , ∀ t ∈ [ t , t ] , (5.14)as well as the relative number of removed R ( t ) = R s + I s (cid:15) (cid:90) tt dt (cid:48) e − (cid:15) ( t (cid:48) − t ) + (cid:15)f (cid:90) tt dt (cid:48) e − (cid:15)t (cid:48) (cid:90) t (cid:48) t dt (cid:48)(cid:48) e (cid:15)t (cid:48)(cid:48) = R s + f ( t − t ) + (cid:18) I s − f(cid:15) (cid:19) (cid:16) − e − (cid:15) ( t − t ) (cid:17) , ∀ t ∈ [ t , t ] . (5.15)One can directly verify that these expressions satisfy (3.4) along with S ( t ) + I ( t ) + R ( t ) = S s + I s + R s , ∀ t ∈ [ t , t ] . (5.16)For some (random) values of (cid:15) , f , S s , I s and R s , the functions S ( t ), I ( t ) and R ( t ) (for theregion where 0 ≤ S ( t ) , I ( t ) , R ( t ) ≤
1) are plotted in the left panel of Fig. 28, while theassociated γ ( t ) = fS ( t ) I ( t ) is plotted in the right panel.
100 200 300 400 t0.20.40.60.81.0 S [ t ] I [ t ] R [ t ]
100 200 300 400 t0.10.20.30.40.50.60.7 γ [ t ] Figure 28 : Solutions (5.13) and γ ( t ) for (cid:15) = 0 . f = 0 . S s = 0 . I s = 0 . R s = 0 and t = 0 as a function of time t . During the linear growth periods, the COVID-19 data also shows that the number of activeinfectious individuals remains constant. Intriguingly, this feature is also observed in the– 39 –olutions in the left panel of Fig. 28. In this section, we will seek a solution of the SIR modelwith this property, i.e. I ( t ) = f = const. ∀ t ∈ [ t , t ] , (5.17)for some f ∈ [0 , ddt I ( t ) = 0 , ∀ t ∈ [ t , t ] . (5.18)Injecting this into (3.4) we obtain (assuming that I ( t ) (cid:54) = 0 ∀ t ∈ [ t , t ]) S = (cid:15)γ , ∀ t ∈ [ t , t ] , (5.19)and thus for ζ (cid:54) = 0 ddt (cid:18) (cid:15)γ (cid:19) = − (cid:15) f + ζ R , = ⇒ R = 1 ζ (cid:20) ddt (cid:18) (cid:15)γ (cid:19) + (cid:15) f (cid:21) , ∀ t ∈ [ t , t ] . (5.20)For ζ = 0 we obtain the following constraint for the infection and recovery rate ddt (cid:18) (cid:15)γ (cid:19) = − (cid:15) f , ∀ t ∈ [ t , t ] . (5.21)For the classical SIR model (for which (cid:15) and γ are time-independent ∀ t and ζ = 0), assumingthat γ (cid:54) = 0, the constraint (5.21) implies that either • f = 0, which however is excluded since I (cid:54) = 0; • or (cid:15) = 0, in which case dRdt = 0 ∀ t ( i.e. not just t ∈ [ t , t ]). However, with the initialconditions (3.3) this implies R ( t ) = 0 and thus ddt S ( t ) = − γ f S = ⇒ S = c e − γ f t , ∀ t ∈ [ t , t ] , (5.22)for c ∈ [0 , dSdt = 0 and thus (with γ (cid:54) = 0 and f (cid:54) = 0) S = 0 (consistent with (5.19)), in which case I = f = 1 and the entirepopulation is infected (and stays infected for all times).Thus, within the classical SIR model, the only solution with I ( t ) = f (cid:54) = 0 constant is (cid:15) = 0( i.e. instead of the SIR model we only consider the SI model) and I = 1. This corresponds tothe late phase of the SI model, where the entire population is infected. We demonstrated thatthe traditional SIR model cannot account for the linear growth of the cumulative number ofinfected related to (5.18) and observed in the COVID-19 data.– 40 – Outlook and Conclusions
In this work we go beyond a systematic review of the main mathematical models used todescribe the diffusion of infectious diseases by showing how the different approaches arerelated. We also show how to extend the models to account for observed phenomena, likemulti-wave dynamics and the emergence of time-dependent symmetries such as approximatetime-dilation invariance. The models are, at a more fundamental level, either of stochastic ordeterministic nature and we observe that field theory emerges as a unifying framework.We start with percolation models and, via numerical analyses, we show that near criti-cality they merge into a field theoretical description as envisioned by Cardy and Grassberger.The results seed the link to traditional compartmental models that are ubiquitously found inepidemiology. We provide an in-depth review of SIR-like compartmental models that, from atheoretical vantage point, elucidates their mechanics and dynamics. We analyse, review andextend the models to take into account single-wave dynamics, multi-wave patterns and evensuperspreaders. Last but not least, in percolation and compartmental models we identify theemergence of approximate time-scale invariance of the diffusion solutions. This fact allowsus to naturally introduce and review the most recent entry in mathematical modelling of in-fectious diseases, i.e. the epidemic Renormalisation Group framework. The latter efficientlyorganises the diffusion of diseases around symmetry principles and it yields a novel mathe-matical understanding of multi-wave dynamics that stems from concepts, such as complexfixed points, introduced to describe (quantum) phase transitions.There are a large number of potential spinoffs that one can imagine branching out intoother realms of science: from medical applications that take into account, for example, theimpact of mutations and vaccination campaigns [40, 41] to quantitative studies of the impacton human behaviour [42, 43].Although the models are universally applicable to any diffusion mechanism, from in-fectious diseases to chemical reactions and other realms of social dynamics, we mentionedCOVID-19 to calibrate and exemplify their power and applicability.
A Basic Percolation Model and Numerical Simulations
A.1 2-dimensional Lattice
Let Γ ⊂ Z be a 2-dimensional hypercubic lattice, which is generated by a set of orthonormalvectors e = { e , e } (with e i · e j = δ ij for i, j ∈ { , } ), such that any lattice site can bewritten in the form x = x e + x e ∈ Γ with x , ∈ Z . At any given time t , each lattice siterepresents an individual, which can be in one of three different states that is characterised bythe discrete function f : Γ × R → { , , } (see Figure 29) • f ( x , t ) = 0: susceptible individuals (drawn as blue sites in Figure 29), representingindividuals that are not infected with the disease, but can contract it– 41 – f ( x , t ) = 1: infectious individuals (drawn as red sites in Figure 29), representing individ-uals that are infected with the disease and are capable of infecting nearby susceptiblesites • f ( x , t ) = 2: removed individuals (drawn as green sites in Figure 29), representing in-dividuals that have been infected with the disease (at some prior stage), but are notlonger capable of infecting nearby susceptible sitesThese definitions are the same that are used to describe the various compartments of the SIR...... · · · · · ·• • • • •• • • • •• • • • •• • • • •• • • • • e e − e − e Figure 29 : 2-dimensional lat-tice generated by the basis vectors( e , e ). Red lattice sites representinfectious individuals (value 1), bluesusceptible (value 0) and green recov-ered ones (value 2).model in section 3. An important question is which sites of suscep-tible individuals can become infected by a ’nearby’infectious. Most percolation models, allow infectionof nearest neighbour sites, i.e. if f ( x , t ) = 1 at sometime t , then susceptible individuals at x ± e i ∀ i = 1 , r ∈ R + (which we call the co-ordination radius) of a single infectious to potentiallybecome infected. Specifically, for an infectious at alattice site x = x e + x e (with f ( x , t ) = 1, repre-sented by the red dot in Figure 30) all susceptible atthe lattice sites N x = { y e + y e ∈ Γ | ( x − y ) + ( x − y ) ≤ q } (represented by the solid blue dots in Figure 30) may become directly infected. All other sites(represented by blue circles in Figure 30) cannot be infected by x (but may become infectedthrough other sites). A.2 Numerical Simulation
In order to perform numerical simulations, we shall restrict the lattice to be of finite size withperiodic boundary conditions:Γ ( N )2 = { x e + x e ∈ Γ | x , ∈ {− N − , − N, . . . , N, N + 1 }} . (A.1) We shall discuss the relation between the current model and the SIR model in Section 2.4. – 42 –n this lattice we prepare a starting configuration at t = t = 0 of susceptible, infectious andremoved individuals, i.e. we define f ( x , ∀ x ∈ Γ ( N )2 . We then consider a discretised timeevolution, i.e. , we define a discrete ∆ t and (given the configuration f ( x , t ) ∀ x ∈ Γ ( N )2 ), wecompute the configuration f ( x , t + ∆ t ) according to a number of stochastic rules shown in thefollowing: ...... · · · · · · e e − e − e Figure 30 : Schematic representa-tion of susceptibles that may becomeinfected by a single infectious. 1. ∀ x ∈ S ( t ) : let n x = |{ y ∈ {N x | f ( y , t ) = 1 }}| , let k ( x , t ) ∈ [0 , k ( x , t ) ≤ − (1 − g / A ) n x set f ( x , t + ∆ t ) = 1, else f ( x , t + ∆ t ) = 02. ∀ x / ∈ S ( t ) : if f ( x , t ) = 2 set f ( x , t + ∆ t ) = 23. ∀ x / ∈ S ( t ) if f ( x , t ) = 1 let (cid:96) ( x , t ) ∈ [0 , (cid:96) ( x , t ) ≥ e set f ( x , t + ∆ t ) = 1, if (cid:96) ( x , t ) < e set f ( x , t + ∆ t ) = 2where S ( t ) = { x ∈ Γ ( N )2 | f ( x , t ) = 0 } is the ensembleof all lattice suceptible sites at time t , while g ∈ [0 , e ∈ [0 ,
1] are fixed real numbers that represent theprobabilities of infection (if a susceptible site is in prox-imity to an infectious site) and removal respectively. A is the number of sites inside the circle of radius r calledcoordination radius. It is important to realise that theabove rules are stochastic in nature in the sense that k ( y , t ) and (cid:96) ( x , t ) are randomly generated real num-bers, that generate the time evolution of the configuration. This in particular means that byapplying these rules twice to the same configuration at time t will (in general) lead to twodifferent configurations at times t + ∆ t . Thus, in order to obtain meaningful results of e.g. how many lattice sites are infected at time t + n ∆ t in the limit of n very large, requires torun the simulation based on the above rules many times with equivalent initial conditionsand compute an average value at the very end. In this way, we can study the impact of theparameters g and e , as well as the coordination number q on the spread of the disease on thelattice Γ ( N )2 . B The SIR Model as an RG Equation: COVID-19 and Constant RecoveryRate
In this appendix we provide a concrete example of how to formulate a SIR model (withtime-dependent σ ) in a way that highlights similarities to a renormalisation group equation,following the logic outlined in Section 3.6.2. In particular, we show how a particular beta-function (which is discussed in detail in Section 4 can be obtained from a time-dependent σ ,– 43 –sing eq. (3.38). Concretely, we make contact with the β -function in (4.11) − β ( I c ) = λ I c (cid:34)(cid:18) − I c A (cid:19) − δ (cid:35) p , (B.1) φ ( I c ) = I c , and p, δ, A constant. Furthermore, for simplicity, we shall assume that (cid:15) isconstant, i.e. the rate of recovery remains constant throughout the pandemic , while γ and σ = γ(cid:15) are continuous functions of S . Finally (to make contact with (4.11)) we shall considerthe asymptotic limit S → β in (3.39) with β , leads to an integral equation. For (cid:15) = const. we can turnthe latter into a differential equation for σ (recall S = 1 − I c N ) ddI c (cid:34) β ( I c ) (cid:15) σ (cid:0) − I c N (cid:1) (cid:35) = 1 − (cid:0) − I c N (cid:1) σ (cid:0) − I c N (cid:1) , (B.2)which can be brought into the form0 = σ (cid:48) ( S ) + g ( S ) σ ( S ) + g ( S ) σ ( S ) , with g ( S ) = S − Nβ ( N (1 − S )) ( (cid:15) − β (cid:48) ( N (1 − S ))) ,g ( S ) = N(cid:15)Sβ ( N (1 − S )) . (B.3)In the above and following equations, the prime indicated a derivative with respect to theargument of the function. The general solution of this first order, non-linear differentialequation can be written as σ ( S ) = D ( S ) σ + (cid:82) SS dx D ( x ) g ( x ) , with D ( S ) = exp (cid:20) − (cid:90) SS g ( x ) dx (cid:21) . (B.4)Here σ is an integration constant, which can be determined by comparing the first derivativeof β and β at S = S → i.e. at I c = N (1 − S ) = 0). Indeed, β (cid:48) (0) = β (cid:48) (0) implies σ (1) = σ = 1 − (cid:15) β (cid:48) (0) = 1 + λ(cid:15) (1 − δ ) p . (B.5)With β given in (B.1), the integral over g can be performed analytically (involving an Appellhypergeometric function). However, using this result to insert D ( S ) into the first expressionin (B.4), the integral in the denominator is more involved and we could only find analytic (cid:15) depends on biological properties of the virus as well medical and pharmaceutical means of the populationto cure it. Since these are difficult to change without significant effort, the value of (cid:15) is difficult to change. – 44 –olutions for generic λ, (cid:15) for ( p = , δ = 0) and ( p = , δ = 0), whose limit S → S → σ (1 − I c N ) (cid:12)(cid:12)(cid:12)(cid:12) p = 14 δ =0 = λN(cid:15) ( N − I c ) (cid:113) − I c A − (cid:15)λ A(cid:15)I c ( λ + (cid:15) ) (cid:18)(cid:113) − I c A − (cid:19) (cid:18)(cid:113) − I c A + 1 (cid:19) (cid:15)λ F (cid:32) (cid:15)λ , λ + (cid:15)λ ; (cid:15)λ + 2; − (cid:113) − I c A (cid:33) , lim S → σ (1 − I c N ) (cid:12)(cid:12)(cid:12)(cid:12) p = 12 δ =0 = N ( A − I c )( λ + (cid:15) ) (cid:0) − I c A (cid:1) − (cid:15)λ A(cid:15) ( N − I c ) F (cid:0) (cid:15)λ , λ + (cid:15)λ ; 2 + (cid:15)λ ; I c A (cid:1) . (B.6)However, the integration can be performed numerically, and for different values of ( p, δ ), σ as a function of I c is shown in Figure 31. We note that for p ≤ /
2, Im( σ ) (cid:54) = 0 for I c > A ,thus indicating that the solution does not extend beyond the maximal number of cumulativeinfected I c = A (see Figure 32). Similar plots for δ (cid:54) = 0 are shown inFinally, we also remark that the numerical integration allows us to include δ < β -function proposed in [9] − β ( I c ) = λ I c (cid:34)(cid:18) − I c A (cid:19) − δ (cid:35) p (1 − ζI c ) , (B.7)as shown in Figure 33. In the case ζ > σ ) (cid:54) = 0 for I c > ζ − , indicatingas above the breakdown of the assumptions. C Check of Solution (5.13)
We check explicitly that (5.13) is a solution of (3.4) for ζ = 0. We start with dS ( t ) dt = − f , (C.1)which is indeed the first equation of (3.4) (taking into account (5.3)). We next compute dI ( t ) dt = − I s (cid:15) ( t ) D (cid:15) ( t ) + f − f (cid:15) ( t ) (cid:90) tt D (cid:15) ( t (cid:48) ) D (cid:15) ( t ) dt (cid:48) = f − (cid:15) ( t ) (cid:20) I s D (cid:15) ( t ) + f (cid:90) tt D (cid:15) ( t (cid:48) ) D (cid:15) ( t ) dt (cid:48) (cid:21) = f − (cid:15) ( t ) I ( t ) , (C.2)which is indeed the second equation of (3.4) (taking into account (5.3)). where we used (5.6).Finally, we consider dR ( t ) dt = I s (cid:15) ( t ) D (cid:15) ( t ) + f (cid:90) tt dt (cid:48) (cid:15) ( t ) D (cid:15) ( t (cid:48) ) D (cid:15) ( t ) = (cid:15) ( t ) (cid:20) I s D (cid:15) ( t ) + f (cid:90) tt D (cid:15) ( t (cid:48) ) D (cid:15) ( t ) dt (cid:48) (cid:21) = (cid:15) ( t ) I ( t ) , (C.3)which is indeed the third equation of (3.4). Furthermore, we can directly check that (5.13)satisfies the initial conditions S ( t = t ) = S s , I ( t = t ) = I s , R ( t = t ) = R s . (C.4)Thus, (5.13) is indeed the unique solution of (3.4) for ζ = 0 that satisfies (5.4). We remark in passing that we were able compute analytic solutions for other combinations of ( p, δ ) forspecific combinations of ( λ, (cid:15) ), i.e. for certain fixed ratios λ(cid:15) . – 45 – σ [ - Ic N ] ( p = ,(cid:1) = ) ϵ = ϵ = ϵ = ϵ = ϵ = ϵ = ϵ = σ [ - Ic Nc ] ( p = , (cid:1) = ) ϵ = ϵ = ϵ = ϵ = ϵ = ϵ = ϵ = σ [ - Ic N ] ( p = , (cid:1) = ) ϵ = ϵ = ϵ = ϵ = ϵ = ϵ = ϵ = σ [ - Ic N ] ( p = (cid:1) = ) ϵ = ϵ = ϵ = ϵ = ϵ = ϵ = ϵ = σ [ - Ic N ] ( p = , (cid:1) = ) ϵ = ϵ = ϵ = ϵ = ϵ = ϵ = ϵ = σ [ - Ic N ] ( p = , (cid:1) = ) ϵ = ϵ = ϵ = ϵ = ϵ = ϵ = ϵ = Figure 31 : σ as a function of I c for different values of p and δ = 0 in the limit S → N = 1 . . A = 50 .
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