Inhomogeneities in Boltzmann-SIR models
IInhomogeneities in Boltzmann–SIR models
A. Ciallella , M. Pulvirenti and S. Simonella . Dipartimento di Ingegneria Civile, Edile – Architettura e Ambientale, andInternational Research Center M&MOCS, Universit´a dell’Aquila,via Giovanni Gronchi 18, 67100, L’Aquila, Italy. . Dipartimento di Matematica, Universit`a di Roma La SapienzaPiazzale Aldo Moro 5, 00185 Rome – Italy, andInternational Research Center M&MOCS, Universit`a dell’Aquila,Piazzale Ernesto Pontieri 1, Monteluco di Roio, 67100 L’Aquila – Italy. . UMPA UMR 5669 CNRS, ENS de Lyon46 all´ee d’Italie, 69364 Lyon Cedex 07 – France
Abstract.
We investigate, by means of numerical simulations, the qualitative properties ofa Boltzmann equation for three species of particles introduced in previous work, capturingsome features of epidemic spread. a r X i v : . [ m a t h - ph ] F e b Introduction
In a recent contribution [5], we presented a kinetic model for mixtures of three species ofparticles, or “agents”, labelled S , I , R . This stays for susceptible, infected and recovered, asinspired from the basic SIR system in epidemiology. In this model, together with collisionsand transport, a reaction S + I → I + I takes place with constant rate β ∈ (0 , S and aparticle of type I . No other reactions occur, but particles of type I decay as I → R witha constant rate γ >
0. The one-particle distribution functions depend on time t , position x and velocity v : f S = f S ( t, x, v ) f I = f I ( t, x, v ) f R = f R ( t, x, v ) f := f S + f I + f R where f is the total density. The microscopic model is further based on a few elementaryfeatures: ∗ the interactions are binary, and localized; ∗ the number of interactions per unit time is expected to be finite; ∗ the qualitative behaviour is independent of the number of particles N , provided that thisis large in a suitable scaling limit; ∗ a statistical description is appropriate.This leads to a Boltzmann equation which reads (in two dimensions): ( ∂ t + v · ∇ x ) f S = Q ( f S , f S ) + Q ( f S , f R ) + (1 − β ) Q ( f S , f I ) − βQ − ( f S , f I )( ∂ t + v · ∇ x ) f I = Q ( f I , f I ) + Q ( f I , f R ) + Q ( f I , f S ) + βQ + ( f S , f I ) − γf I ( ∂ t + v · ∇ x ) f R = Q ( f R , f ) + γf I (1.1)where Q = Q + − Q − ,Q + ( f, g )( v ) := (cid:90) R (cid:90) S B ( ω ; v − v ∗ ) f ( v (cid:48) ) g ( v (cid:48)∗ ) dω dv ∗ ,Q − ( f, g )( v ) := f ( v ) (cid:90) R (cid:90) S B ( ω ; v − v ∗ ) g ( v ∗ ) dω dv ∗ , is a given interaction kernel (see the next section for specific choices) and typically v (cid:48) = v − ω ( v − v ∗ ) · ωv (cid:48)∗ = v ∗ + ω ( v − v ∗ ) · ω (1.2)are the outgoing velocities for a collision, preserving momentum and energy. In particular, f is governed by the standard Boltzmann equation:( ∂ t + v · ∇ x ) f = Q ( f, f ) . (1.3)The formal link with the classical theory of epidemics is obtained for spatially ho-mogeneous distributions (no dependence on x ), looking at averaged fractions of agents A ( t ) := (cid:82) f A ( t, v ) dv in the species A ∈ { S, I, R } . Performing the integral with respect to v of Eq.s (1.1) and using that (cid:82) Q + = (cid:82) Q − , we find ˙ S = − β Q − ( f S , f I )˙ I = β Q − ( f S , f I ) − γI ˙ R = γI . Such a set of equations is not closed but, when dealing with “Maxwellian molecules” definedby the requirement that (cid:82)
B dω = 1, one gets Q − ( f S , f I ) = S ( t ) I ( t ) and therefore ˙ S = − βIS ˙ I = βIS − γI ˙ R = γI ; (1.4)namely the simplest SIR model equations [4]. The latter have been considerably usedand extended, for almost a century; see e.g. [2] for a recent overview of mathematicalepidemiology, or [6] for a case in the huge amount of studies on the current COVID-19pandemic.A stochastic N -particle system can be constructed, with distribution functions converg-ing to the solution of (1.1) as N → ∞ ([5]) and corresponding, numerically, to the DSMC(direct Simulation Monte Carlo) method. In the present paper, we adopt the Boltzmann-SIR equations (1.1) as toy model, and the underlying particle system as a tool to studythe qualitative behaviour. This allows to reinterpret some features of SIR type models, interms of spatial inhomogeneities.As in [5] we stress that we do not pretend the kinetic model to provide any realisticprediction in epidemiology, as of course real agents do not interact as elastically colliding2articles in a rarefied gas. Realistic interactions are obviously difficult to be describedin mathematical terms. Individual strategies might play a critical role and, in essence,the interactions might be not even necessarily binary (e.g. a single agent infecting manysusceptible agents almost simultaneously). Motivated by the simple connection with (1.4),we are rather interested in capturing behaviour which has only little dependence on thedetails of the microscopic interaction.More precisely, we perform numerical simulations of system (1.1), with the followingplan. In Section 2, we consider several kernels B and verify that: (i) the macroscopicevolution for ( S ( t ) , I ( t ) , R ( t )) is rather insensitive to the choice of the cross-section; (ii)the evolution can be significantly sensitive to spatial non-uniformity of labels, even when f has reached global equilibrium. In Section 3, we perturb the model by external actionsmimicking, roughly, meeting points with (airport, travel stations) or without (supermarket)injection of agents. We observe how the local concentration of densities enhances thetransient of I ( t ), and identify regimes for the external flows producing nontrivial asymptoticvalues, and possibly recurrent waves. This section is devoted to the basic properties of Eq. (1.1), referred to as “free model”(model without external actions), which we recall (in more compact form): ( ∂ t + v · ∇ x ) f S = Q ( f S , f ) − βQ + ( f S , f I )( ∂ t + v · ∇ x ) f I = Q ( f I , f ) + βQ + ( f S , f I ) − γf I ( ∂ t + v · ∇ x ) f R = Q ( f R , f ) + γf I . . We shall consider three different cross–sections, namely:
1) Hard spheres , as for the mechanical system of N billiard balls, from which the Boltzmannequation is obtained in the Boltzmann–Grad limit (see e.g. [3]). The collision law is givenby (1.2) and the interaction kernel is B ( ω ; v − v ∗ ) = ω · ( v − v ∗ ) χ ( ω · ( v − v ∗ ) ≥ , (2.1)where χ ( E ) is the characteristic function of the event E .
2) Semidiscrete model , which is again a hard–sphere type system, but with particle veloc-3ties of modulus 1, i.e. v ∈ S . The collision law is v (cid:48) = v − ω ( v · ω ) v (cid:48)∗ = v ∗ − ω ( v ∗ · ω ) . That is, each particle is reflected against the line orthogonal to the versor ω joining thetwo centers. Energy is conserved but not momenta. B is still given by (2.1).
3) Maxwellian molecules . A popular simple model for the Boltzmann equation [1], forwhich the collision law is given by (1.2) while B satisfies (cid:90) dω B ( ω ; v − v ∗ ) = const. . The right hand side is remarkably independent of the relative velocity v − v ∗ .Since the full probability density satisfies (1.3), in cases and if the mean–free pathis small f ( x, v ) ≈ | Λ | M ( v )after a brief transient, where | Λ | is the measure of the domain Λ and M is a Maxwellianvelocity distribution M ( v ) = e − ( v − u )22 σ (2 πσ ) , v ∈ R (2.2)with u ∈ R and σ > S , I , R is also independent of x , then f A ( t, v ) ≈ A ( t ) | Λ | M ( v ) , A = S, I, R .
Even if the full system is at equilibrium, the dynamics of particle labels (state of the agents)may well be active and we find ˙ S = − βmIS ˙ I = βmIS − γI ˙ R = γI (2.3)where m = c (cid:90) dv (cid:90) dv ∗ | v − v ∗ | b M ( v ) M ( v ∗ ) , c > b = 1 in case , b = 0 in case . In case one has similar behaviour, but M is replacedby the uniform distribution f ( x, v ) ≈ | Λ | | S | . Therefore the kinetic picture plays a role for a short transient only and, for a largerscale of time, it does not say more than the standard SIR model, if the distributions oflabels are spatially homogeneous.We recall that (1.4) is almost explicitly solvable. The asymptotic distribution is foundby setting R ( t ) = R + γ (cid:90) t I ( τ ) dτ (showing I ( t ) → t → ∞ ) and dSdR = − βγ S . Setting ¯ A = lim t →∞ A ( t ), using ¯ R + ¯ S = 1 and assuming R = R (0) = 0 (no recoveredagents at time zero), one gets ¯ S = S e − βγ ¯ R = S e − βγ e βγ ¯ S , hence e − βγ ¯ S βγ ¯ S = (1 − I ) βγ e − βγ . Since max ye − y = e , given β and γ one can find non vanishing solutions for ¯ S .In the numerical simulations of the Boltzmann-SIR model, the above asymptotic isdetermined, roughly, by a “herd immunity” situation which is reached when susceptibleagents are surrounded by a sufficiently large fraction of recovered agents, shielding themfrom the infected population.Inspired by the fact that problems of interest are frequently non–homogeneous, we willfocus now on profiles where particle labels are not uniformly distributed in space, so thatthe kinetic model is indeed more detailed than (2.3). Consider, for instance, the case ofan initial distribution of infected agents concentrated in a small region. Even when f hasreached global equilibrium, the system as a whole can still be far from uniform (in space)for quite a long time f A ( t, x, v ) ≈ A ( t, x ) | Λ | M ( v ) , A = S, I, R and ( S ( t ) , I ( t ) , R ( t )) can be notably different from the solution of (2.3). Such a behaviourwill be discussed in the next subsection. 5 .1 Description of the simulations The numerical simulations are based on the DSMC method. The details of this methodcan be found for instance in Chapter 10 of [3] or in [7]. Here we just describe the setting.The system consists of N point particles moving in a square Λ with side length L = 10 and periodic boundary conditions. An equally spaced grid partitions the domain intoidentical square cells of size δ × δ (a total of (cid:0) Lδ (cid:1) cells). N is constant in time, no agent isintroduced or removed from the system. Each particle moves with constant velocity up tothe next collision instant. The mean free path, i.e. the average distance travelled by eachagent between two consecutive collisions, will be denoted by λ , the mean free time by τ .Time is discretized and the evolution of the system is divided into a free evolution stepwhere all particles move freely for a discrete time unit ¯ t , and a collision simulation step,where pairs of particles lying in the same cell are randomly chosen to perform a binarycollision.For each simulation presented, we report the relevant parameters in the captions of thefigures. We list now the choices that are common to all simulation runs.The particles are initially distributed uniformly in space, while velocities are distributeduniformly on S . We assign to each particle a label, S , I , or R , that can be distributed inboth uniform or non-uniform way, as specified in each case. As the energy of the systemis fixed, in the case of hard spheres and Maxwellian molecules the velocity distributionquickly converges to a Maxwellian as (2.2) with u = 0 and σ = 1 /
2. Hence τ (cid:39) λ (cid:104) v (cid:105) wherethe mean scalar velocity (cid:104) v (cid:105) can be explicitly calculated. The cells side is δ (cid:39) λ and thediscrete time step is ¯ t (cid:39) τ . The number of agents N is always such that, on average, atleast 20 particles lie in each cell. The prescribed rule for the dynamics, following (1.1), isthat a collision of an I (infected) and an S (susceptible) particle produces two I particleswith probability β , and that an I particle becomes an R (recovered) particle after anexponential time with rate γ . The fractions at time 0 are assumed to be I (0) = 0 . S (0) = 1 − I (0) = 0 . R (0) = 0.In the following, a few examples of numerical experiments of the system with differentcross–sections are presented. The evolution of the fractions of the three populations S , I ,and R is plotted for the particle system, and compared with the solution of the SIR modelEq. (2.3).We consider two different situations. In the first one, the initial distributions of agentsare all uniform. In this setting, we want to test the consistency of the kinetic model withthe SIR model, and check that results turn out to be independent of the choice of thecross–section. In the second case, the initial datum is such that all the infected agents6re contained in a small disk of area 0 . · L . This is used to show that, even in asimple setting, the average description produced by the SIR model can lose quantitativeand qualitative information related to spatial patterns.The case of Maxwellian molecules is reported in Fig. 1. For uniform initial distributions,the correspondence between particle system (left panel, solid lines) and SIR (right panel) isclear. In the left panel, we show also the case of concentrated initial distribution of infectedagents (dashed lines). In this run, the mean free path is sufficiently small ( λ (cid:39) L ) toproduce an apparent difference. time f r a c t i on Evolution of S,I,R fractions S h I h R h S c I c R c f r a c t i on Figure 1: Left panel: evolution of S , I , R fractions for a particle system with Maxwellianmolecule cross–section simulated via DSMC. The solid and dashed lines represent, respec-tively, the case of homogeneous and concentrated initial population of I agents. Parame-ters: N = 1800000, β = 1, γ = , λ (cid:39) . τ (cid:39) .
05. Right panel: numerical solution of(2.3) with m = τ , and the same β and γ as in the left panel case.The case of hard sphere cross–section is reported in Fig. 2. For uniform distributions(left panel, solid lines), we observe a small quantitative difference in the asymptotic fractionof the susceptible population (therefore of the recovered one) with respect to the solutionof (2.3) (right panel). Indeed, the DSMC tends to select colliding particles with largevelocities: infected agents travelling with high speed are likely to transmit the infection.This leads to a slightly wider diffusion of the I population with respect to the system ofODEs (2.3) (in the experiment presented in Fig. 2, ¯ S is estimated to be 0 .
318 by the DSMCmethod while its actual value is 0 . λ (cid:39) L ), starting from a concentrated initialdatum does not change considerably the quantitative behaviour.Finally, the case of semidiscrete model cross-section is reported in Fig. 3. We find7xcellent agreement between homogeneous particles system and SIR. time f r a c t i on Evolution of S,I,R fractions S h I h R h S c I c R c f r a c t i on Figure 2: Left panel: evolution of S , I , R fractions for a particle system with hard spherecross–section simulated via DSMC. The solid and dashed lines represent, respectively,the case of homogeneous and concentrated initial population of I agents. Parameters: N = 180000, β = 0 . γ = , λ (cid:39) . τ (cid:39) .
85. Right panel: numerical solution of(2.3) with m = τ , and the same β and γ as in the left panel case. time f r a c t i on Evolution of S,I,R fractions S h I h R h S c I c R c f r a c t i on Figure 3: Left panel: evolution of S , I , R fractions for a particle system with semidiscretecross–section simulated via DSMC. The solid and dashed lines represent, respectively,the case of homogeneous and concentrated initial population of I agents. Parameters: N = 300000, β = 0 . γ = , λ (cid:39) . τ (cid:39) .
0. Right panel: numerical solution of (2.3)with m = τ , and the same β and γ as in the left panel case.8s expected, the larger asymptotic value of S particles in the non-uniform cases isdue to the time needed for the system to mix the populations, the difference being moreimportant for λ small. In this section we study three types of perturbation of the Boltzmann-SIR model, favour-ing non-equilibrium regimes. Symbolically, we call them “supermarket”, “airport”, and“diffuse jet”.
Let D ⊂ Λ be a box (the supermarket). In addition to the dynamics described by the freemodel, we assume that each particle jumps instantaneously in D at an exponential timeof rate γ . After the jump, the particle is uniformly distributed in D . Then it moves withunchanged velocity. The kinetic equations are: ( ∂ t + v · ∇ x ) f S = Q ( f S , f ) − βQ + ( f S , f I ) − γ ( f S − χ D | D | g S )( ∂ t + v · ∇ x ) f I = Q ( f I , f ) + βQ + ( f S , f I ) − γf I − γ ( f I − χ D | D | g I )( ∂ t + v · ∇ x ) f R = Q ( f R , f ) + γf I − γ ( f R − χ D | D | g R ) . (3.1)Here χ D is the characteristic function of D and | D | its area; g A with A = S, I, R is thevelocity distribution of A i.e. g A ( v ) = (cid:82) dx f A ( x, v ). The equation for f = (cid:80) A f A is( ∂ t + v · ∇ x ) f = Q ( f, f ) − γ ( f − χ D | D | g )where g is total velocity distribution (cid:82) dx f . The density ρ = (cid:82) dv f satisfies( ∂ t ρ + div x ( uρ )) = − γ ( ρ − χ D | D | )where ρu = (cid:82) dv v f . As we are not able to characterize explicitly the stationary solutionsto Eq. (3.1), we turn to numerical investigation.The local higher density (in D , and in a neighbourhood of it) makes more likely theincrease of I particles. Ultimately, this leads to an asymptotic behaviour with lower numberof susceptible agents (with respect to the free model). Two parameters contribute tomagnify the effect: the intensity of jumps γ and the smallness of the box | D | , as reportedin Fig 4. Note that here a smaller D produces a more significant effect on the difference in9he asymptotics with respect to the different initial configurations. This is at variance withthe free model, where uniform initial data provide a wider diffusion than a concentratedinitial population I . time f r a c t i on Evolution of S,I,R fractions S h I h R h S I R S I R S I R time f r a c t i on Evolution of S,I,R fractions S c I c R c S I R S I R S I R Figure 4: Evolution of S , I , R fractions for a particle system with hard sphere cross–section simulated via DSMC, Eq. (3.1). Parameters: N = 180000, β = 0 . γ = , λ (cid:39) . τ (cid:39) .
8. Left panel: | D | = | Λ | , homogeneous initial datum. Subscript h refers to the free model (no jumps), subscript i refers to assumed value γ = ( i · ) − , i = 1 , ,
3. Right panel: | D | = | Λ | , concentrated initial datum. In addition to a density concentration, we consider now the action of an external flow. Asbefore, agents S and R jump in D with a rate γ (infects are not allowed to fly). Thenthey disappear and are simultaneously replaced by an equal number of agents. The injectedagents are either S or R , with equal probability (1 − α ) /
2, or I with probability α . Theextreme case is α = 1 (maximal flux of infects). The equations are : ( ∂ t + v · ∇ x ) f S = Q ( f S , f ) − βQ + ( f S , f I ) − γ ( f S − (1 − α ) χ D | D | ( g S + g R ))( ∂ t + v · ∇ x ) f I = Q ( f I , f ) + βQ + ( f S , f I ) − γf I + γ α χ D | D | ( g S + g R )( ∂ t + v · ∇ x ) f R = Q ( f R , f ) + γf I − γ ( f R − (1 − α ) χ D | D | ( g S + g R )) . (3.2)In this case, f does not solve a closed equation. An immediate generalization of Eq. (3.2) is obtained by considering different fractions α , α and α (with α + α + α = 1) in place of α , − α and − α . This does not change the qualitative behaviour. I particles in D crucially determines the long time behaviour. In the extreme case α = 1, nosusceptible agent survives. In the opposite case α = 0 (no infected are ever introduced), ifthe jump rate is sufficiently intense, the infection may be never extinguished: the source ofsusceptible agents leads to a stationary configuration where all three populations are non–zero; see Fig. 5, γ = ( i · ) − , i = 2 ,
4. If instead the jump rate is low, the asymptoticvalues are ¯ S = ¯ R = 0 . I (Fig. 5, γ = (6 · ) − ). If, additionally, a small fraction of agents I is injected,recurrent small waves arise; see Figures 6 and 7, corresponding respectively to cases withextinction and without extinction of I (depending on α, γ ). time f r a c t i on Evolution of S,I,R fractions S I R S I R S I R time f r a c t i on Evolution of S,I,R fractions S c I c R c Figure 5: Evolution of S , I , R fractions for a particle system with hard sphere cross-sectionsimulated via DSMC, Eq. (3.2). Parameters: N = 180000, β = 0 . γ = , λ (cid:39) . τ (cid:39) . | D | = | Λ | , α = 0, concentrated initial datum. Left panel: subscript i refersto assumed value γ = ( i · ) − , i = 2 , ,
6. Right panel: free model ( γ = 0), sameparameters. 11 time f r a c t i on Evolution of S,I,R fractions
SIR time f r a c t i on Evolution of I fractions I Figure 6: Particle system with hard sphere cross-section simulated via DSMC, Eq. (3.2).Parameters: N = 180000, β = 0 . γ = , λ (cid:39) . τ (cid:39) . | D | = | Λ | , α = 2 · − , γ = 10 − , concentrated initial datum. Left panel: Evolution of S , I , R fractions. Rightpanel: detail of the I fraction. time f r a c t i on Evolution of S,I,R fractions S c I c R c S I R S I R S I R S I R time f r a c t i on Evolution of I fractions I c I I I I Figure 7: Particle system with hard sphere cross–section simulated via DSMC, Eq. (3.2).Parameters: N = 180000, β = 0 . γ = , λ (cid:39) . τ (cid:39) . | D | = | Λ | , α = 10 − ,concentrated initial datum. Subscript c refers to the free model ( γ = 0), subscript i refersto assumed value γ = ( i · ) − , i = 1 , , ,
4. Left panel: Evolution of S , I , R fractions.Right panel: detail of the I fraction. Finally, we consider the effect of an external flow, as in the previous section, but withoutdensity localization. This corresponds to simple random diffuse replacement of non infected12gents by infected agents with probability α , and by susceptible and recovered agents withequal probability − α : ( ∂ t + v · ∇ x ) f S = Q ( f S , f ) − βQ + ( f S , f I ) − γ ( f S − (1 − α )2 ( f S + f R ))( ∂ t + v · ∇ x ) f I = Q ( f I , f ) + βQ + ( f S , f I ) − γf I + γ α ( f S + f R )( ∂ t + v · ∇ x ) f R = Q ( f R , f ) + γf I − γ ( f R − (1 − α )2 ( f S + f R )) . (3.3)In this case, one has a close system of equations for averaged fractions (holding in case ofhomogeneous solutions for Maxwellian molecules, or after thermalization), reminiscent ofSIR-like models with more possible reactions: ˙ S = − βmIS + γ (cid:0) − α S + − α R (cid:1) ˙ I = βmIS − γI + γ α ( S + R )˙ R = γI + γ (cid:0) − α R + − α S (cid:1) . (3.4)For m = γ = 1, it reduces to ˙ S = − βIS + R − S − α ( S + R )˙ I = βIS − γI + α ( S + R )˙ R = γI + S − R − α ( S + R ) , where it is easier to recognize the competing terms effect.For α = 1 we have, again, extinction of the susceptible agents; for α (cid:54) = 1 non–trivialstationary solutions exist for the three populations. In the case α = 0, for γ sufficientlysmall, the simulation of (3.3) displays an instant of total vanishing of the infection: I ( t ) isequal to 0 from this time on, and the asymptotic values are ¯ S = ¯ R = . The ODE system(3.4), instead, has solution I ( t ) (cid:54) = 0 for every time t ≥
0. The asymptotic values are hereclose to the case α (cid:54) = 0 very small and the values ¯ S and ¯ R are not 1 / time f r a c t i on Evolution of S,I,R fractions S h I h R h S c I c R c f r a c t i on time f r a c t i on Evolution of S,I,R fractions S h I h R h S c I c R c f r a c t i on Figure 8: Evolution of S , I , R fractions. Top panels: γ = , bottom panels γ = .Left panel: particle system with hard spheres cross–section simulated via DSMC. The solidand dashed lines represent, respectively, the case of homogeneous and concentrated initialpopulation of I agents. Parameters: N = 200000, β = 0 . γ = , λ (cid:39) . τ (cid:39) . α = 2 · − . Right panel: numerical solution of (3.4) for m = τ , and the same β , γ , γ , α as the particle system. 14e conclude with a remark on the solution to (3.4). The asymptotic fraction ¯ S is veryweakly dependent on γ , provided that γ has the same order of magnitude of β · m and γ ,or smaller. Indeed, changing γ influences how long the solution stays close to the solutionto (2.3), and it changes the asymptotic fractions ¯ I and ¯ R that sum 1 − ¯ S , but it does notperturb significantly ¯ S . We report on this in Table 1 and Fig. 9.( γ ) − ¯ S ¯ I ¯ R
10 0.373059 0.157384 0.46955620 0.385659 0.101855 0.51248450 0.394638 0.0508194 0.554544100 0.397964 0.0278859 0.574149300 0.400297 0.00996697 0.5897361000 0.401138 0.00306936 0.5957935000 0.401429 0.000619549 0.59795110000 0.401466 0.000310134 0.598224Table 1: Asymptotic values of S , I , R as a function of γ , Eq. (3.4) with β = , γ = , α = 0 .
01. 15
100 200 300 400 5000.00.20.40.60.81.0 time f r a c t i on f r a c t i on f r a c t i on f r a c t i on f r a c t i on f r a c t i on f r a c t i on f r a c t i on f r a c t i on f r a c t i on f r a c t i on f r a c t i on f r a c t i on f r a c t i on f r a c t i on e.d.c.b.a.Figure 9: Evolution of S , I , R for β = , γ = , α = 0 .
01. Rows: a. γ = ; b. γ = ;c. γ = ; d. γ = ; e. γ = .Left column: solutions to free SIR (2.3) for t ∈ [0 , t ∈ [0 , t ∈ [0 , eferences [1] A.V. Bobylev. The theory of the nonlinear spatially uniform Boltzmann equationfor Maxwell molecules. In: Math. Phys. Rev. , Soviet Sci. Rev. Sec. C Math. Phys.Rev. (1988).[2] F. Brauer and C. Castillo-Ch´avez. Mathematical Models in Population Biologyand Epidemiology. Springer (2001).[3] C. Cercignani, R. Illner and M. Pulvirenti. The Mathematical Theory of DiluteGases. Applied Mathematical Sciences , Springer–Verlag, New York (1994).[4] W. O. Kermack and A. G. McKendrick. Contribution to the mathematical theoryof epidemics.
Proc. Roy. Soc. Lond A , 700-721 (1927).[5] M. Pulvirenti and S. Simonella. A kinetic model for epidemic spread. M&MOCS :3, 249-260 (2020).[6] N. Parolini, L. Ded`e, P.F. Antonietti, G. Ardenghi, A. Manzoni, E. Miglio, A.Pugliese, M. Verani, A. Quarteroni. SUIHTER: A new mathematical model forCOVID-19. Application to the analysis of the second epidemic outbreak in Italy. arXiv:2101.03369arXiv:2101.03369