key: cord-0594832-6e1l5nx4
authors: Pang, Guodong; Pardoux, Etienne
title: Functional law of large numbers and PDEs for epidemic models with infection-age dependent infectivity
date: 2021-06-07
journal: nan
DOI: nan
sha: 40d07ea02af5b5d02ec0d8f581599e331ae7f979
doc_id: 594832
cord_uid: 6e1l5nx4

We study epidemic models where the infectivity of each individual is a random function of the infection age (the elapsed time of infection). To describe the epidemic evolution dynamics, we use a stochastic process that tracks the number of individuals at each time that have been infected for less than or equal to a certain amount of time, together with the aggregate infectivity process. We establish the functional law of large numbers (FLLN) for the stochastic processes that describe the epidemic dynamics. The limits are described by a set of deterministic integral equations, which has a further characterization using PDEs under some regularity conditions. The solutions are characterized with boundary conditions that are given by a system of Volterra equations. We also characterize the equilibrium points for the PDEs in the SIS model with infection-age dependent infectivity. To establish the FLLNs, we employ a useful criterion for weak convergence for the two-parameter processes together with useful representations for the relevant processes via Poisson random measures.

Kermack and McKendrick pioneered the introduction of PDE models to describe the epidemic dynamics for models with infection-age dependent (variable) infectivity in 1932 [14] . The underlying assumption of their model is that the infectious periods have a general distribution with density which is modeled through an infection-age dependent recovery rate, the infectious individuals having an infection-age dependent infectivity, and the recovered ones a recovery-age susceptibility. In the present paper, we do not consider possible loss of immunity. We defer to a work in preparation the study of variable susceptibility. In the present paper, we mainly consider the SIR model (although we can allow for an exposed period, as will be explained below) and the SIS model. This work is a continuation of our first work on non-Markov epidemic models [20] , and our work on varying infectivity models [6] , see also [19] . In those papers, we show that certain deterministic Volterra type integral equations are Functional Law of Large Numbers (FLLN) limits of adequate individual based stochastic models. An important feature of our stochastic models is that they are non-Markov (since the infectious duration need not have an exponential distribution), and as a result the limiting deterministic models are equations with memory. Note that as early as in 1927, Kermack and McKendrick introduced in their seminal paper [13] a SIR model with both infection-age dependent infectivity and infection-age dependent recovery rate, the latter allowing the infectious period to have an arbitrary absolutely continuous distribution (the infection-age dependent recovery rate is the hazard rate function of the infectious period). One part of that paper is devoted to the simpler case of constant rates, and apparently most of the later literature on epidemic models has concentrated on this special case, which leads to simpler ODE models, the corresponding stochastic models being Markov models, at the price of the models being far from the reality of epidemics.

In this paper, we go back to the original model of Kermack and McKendrick [13] , with two new aspects. First, as in our previous publications, we want to obtain the deterministic model as Key words and phrases. Functional law of large numbers, deterministic Volterra integral equations, PDEs, non-Markovian epidemic models, infection-age dependent (varying) infectivity, Poisson random measure, SIR, SIS, equilibrium in the SIS model. a law of large numbers limit of stochastic models, and second, we distribute the various infected individuals at time t according to their infection-age, and establish a PDE for the "density of individuals" being infected at time t, with infection-age x.

In our stochastic epidemic model, each individual is associated with a random infectivity, which varies as a function of the age of infection (elapsed time since infection). The random infectivity functions, effective during the infected period, are assumed to be i.i.d. for the various individuals, and will also generate the infectious period. The infectivity function is assumed to be càlàg with a given number of discontinuities, and upper bounded by a deterministic constant. In particular, the law of the infectious period can be completely arbitrary. Our modeling approach allows the random infectivity functions to have an initial period of time during which they take zero values, corresponding to the exposed period. Thus our model generalizes both the classical SIR and SEIR models. To describe the epidemic dynamics of the model, we use a (two-parameter or measurevalued) stochastic process that tracks at each time t the number of individuals that have been infected for a duration less than or equal to a certain amount of time x, and an associated aggregate infectivity process which at each time t sums up the infectivities of all individuals who are infected. From these processes, we can describe the cumulative infection process, the total number of infected individuals as well as the number of recovered ones at each time. We use similar processes to describe the epidemic dynamics for the SIS model with infection-age dependent infectivity.

In the asymptotic regime of a large population (i.e., as the total population size N tends to infinity), we establish the FLLN for the epidemic dynamics. The limits are characterized by a set of deterministic Volterra-type integral equations (Theorem 2.1). Under certain regularity conditions, the density function of the two-parameter (calendar time and infection age) limit process can be described by a one-dimensional PDE (Proposition 2.1). Its solution is characterized with a boundary condition satisfying a one-dimensional Volterra-type integral equation. The aggregate infectivity limit process can be described by an integral of the average infectivity function with respect to the limiting two-parameter infectious process (Corollary 2.1, see also Remark 2.6) . For the classical SIR model, we recover the well-known linear PDE first proposed by Kermack and McKendrick [14] . We also characterize the PDE solution when the infectious periods are deterministic (Proposition 2.2). For the SIS model, we also describe the limiting epidemic dynamics and the PDE representations, and derive the equilibrium quantities associated with the PDE and total count limit.

1.1. Literature review. Non-Markov stochastic epidemic models lead (via the FLLN) to deterministic models, which either low dimensional evolution equation with memory (i.e., Volterra type integral equations), or to coupled ODE/PDE models, where the two variables are the time and the age of infection (time since infection). The first paper of Kermack and McKendrick [13] adopts the first point of view, and the two next [14, 15] the second one. In our recent previous work on this topic [20, 6] , we have adopted the first description. The goal of the present paper is to show that in the limit of a large population, our stochastic individual based model with age of infection dependent infectivity and recovery rate converges as well to a limiting system of PDE/ODEs. While the general model from [13] was largely neglected until rather recently, most of the literature concentrating on the particular case of constant rates, there has been since the 1970s some papers considering age dependent epidemic models, see in particular [9] . More recently, several papers have introduced coupled PDE/ODE models for studying age of infection dependent both infectivity and recovery rate, see in particular [23, 11, 24, 17, 5] and chapter 13 in [18] . Since the beginning of the Covid-19 pandemic, a huge number of papers have been produced, with various models of the propagation of this disease. Most of them use ODE models, but a few, notably [12, 8, 7] consider age of infection dependent infectivity, and possibly recovery rate. The last one is to our knowledge the only one which derives the ODE/PDE model as a law of large numbers limit of stochastic individual based models. However, they do not really consider an epidemic model but rather a branching process approximation of the early phase of an epidemic, and the way they model the dependence of the rate of infection w.r.t. the age of infection is less general than in our model.

We also like to mention the relevant work in queueing systems where the elapsed service times are tracked using two-parameter or measure-valued processes. The most relevant to us are the infinite-server (IS) queueing models studied in [21, 22, 1] , where FLLN and FCLT are established for two-parameter processes to tracking elapsed and residual service times. However, the proof techniques we employ in this paper are very different from those papers. Here we exploit the representations with Poisson random measures and use a new weak convergence criterion (Theorem 4.1). In addition, despite similarities with the IS queueing models, the stochastic epidemic models have an arrival (infection) process that depend on the state of the system. As a consequence, the limits in the FLLNs result in PDEs while the IS queueing models do not.

Organization of the paper. The paper is organized as follows. In Section 2, we describe the stochastic epidemic model with infection-age dependent infectivity, and state the FLLN and the PDE, and we also characterize the solution properties of the PDE. The limits and PDE for the SIS model are presented in Section 3, which also includes the equilibrium behavior. In Sections 4, we prove the FLLN. The Appendix gives the proof of the convergence criterion in Theorem 4.1.

1.3. Notation. All random variables and processes are defined on a common complete probability space (Ω, F, P). The notation ⇒ means convergence in distribution. We use 1 {·} for the indicator function. Throughout the paper, N denotes the set of natural numbers, and R k (R k + ) denotes the space of k-dimensional vectors with real (nonnegative) coordinates, with R(R + ) for k = 1. For x, y ∈ R, we denote x ∧ y = min{x, y} and x ∨ y = max{x, y}. Let D = D(R + ; R) denote the space of R-valued càdlàg functions defined on R + . Throughout the paper, convergence in D means convergence in the Skorohod J 1 topology, see chapter 3 of [4] . Also, D k stands for the k-fold product equipped with the product topology. Let C be the subset of D consisting of continuous functions. Let C 1 consist of all differentiable functions whose derivative is continuous. Let D ↑ denote the set of increasing functions in D. Let D D = D(R + ; D(R + ; R)) be the D-valued D space, and the convergence in the space D D means that both D spaces are endowed with the Skorohod J 1 topology. The space C C is equivalent to C(R 2 + ; R + ). Let C ↑ (R 2 + ; R + ) denote the space of continuous functions from R 2 + into R + , which are increasing as a function of their second variable. For any R-valued càdlàg function φ(·) on R + , the integral b a φ(x)dx represents (a,b] φ(x)dx for a < b. For any increasing càdlàg function F (·) : R + → R + , abusing notation, we write F (dx) by treating F (·) as the positive (finite) measure on R + whose distribution function is F .

We consider an epidemic model in which the infectivity rate depends on the age of infection (that is, how long the individuals have been infected). Specifically, each individual i is associated with an infectivity process λ i (·), and we assume that these random functions are i.i.d.. Let η i = inf{t > 0 : λ i (r) = 0, ∀r ≥ t} be the infected period corresponding to the individual that gets infected at time τ N i . The η i 's are i.i.d., with a cumulative distribution function

Individuals are grouped into susceptible, infected and recovered ones. Let the population size be N and S N (t), I N (t) and R N (t) denote the numbers of the susceptible, infected and recovered individuals at time t. We have the balance equation: N = S N (t) + I N (t) + R N (t), t ≥ 0. Assume that S N (0) > 0, I N (0) > 0 and R N (0) = 0. Let I N (t, x) be the number of infected individuals at time t that have been infected for a duration less than or equal to x. Note that for each t, I N (t, x) is nondecreasing in x, which is the distribution of I N (t) over the infection-ages. Let A N (t) be the cumulative number of newly infected individuals in (0, t], with the infection times {τ N i : i ∈ N}.

Let {τ N j,0 , j = 1, . . . , I N (0)} be the times at which the initially infected individuals at time 0 became infected. Thenτ N j,0 = −τ N j,0 , j = 1, . . . , I N (0), represent the amount of time that an initially infected individual has been infected by time 0, that is, the age of infection at time 0. WLOG, assume that 0 > τ N 1,0 > τ N 2,0 > · · · > τ N I N (0),0 (or equivalently 0 <τ N 1,0 <τ N 2,0 < · · · <τ N I N (0),0 ). Set τ N 0,0 = 0. We define I N (0, x) = max{j ≥ 0 :τ N j,0 ≤ x}, the number of initially infected individuals that have been infected for a duration less than or equal to x at time 0. Assume that there exists 0 ≤x < ∞ such that I N (0) = I N (0,x) a.s.

Each initially infected individual j = 1, . . . , I N (0), is associated with an infectivity process λ 0 j (·), and we assume that they are also i.i.d., with the same law as λ i (·). This is reasonable since it is for the same disease, and the infectivity for the initially and newly infected individuals with the same infection age should have the same law. The infectivity processes take effect at the epochs of infection. For each j, let η 0 j = inf{t > 0 : λ 0 j (τ N j,0 + r) = 0, ∀r ≥ t} be the remaining infectious period, which depends on the elapsed infection timeτ N j,0 , but is independent of the elapsed infection times of other initially infected individuals. In particular, the conditional distribution of η 0 j given thatτ N j,0 = s > 0 is given by

Note that the η 0 j 's are independent but not identically distributed. For an initially infected individual j = 1, . . . , I N (0), the infection age is given byτ N j,0 + t for 0 ≤ t ≤ η 0 j , during the remaining infectious period. For a newly infected individual i, the infection age is given by t − τ N i , for τ N i ≤ t ≤ τ N i + η i during the infectious period. Note that λ i (·) and λ 0 j (·) are equal to zero on R − . The aggregate infectivity process at time t is given by

2) (Note that the notation I N was used for the infectivity process in [6, 19] .) The instantaneous infection rate at time t can be written as

The infection process A N (t) can be written as

where Q is a standard Poisson random measure on R 2 + . Among the initially infected individuals, the number of individuals who have been infected for a duration less than or equal to x at time t is equal to

Recall the age limit of the initially infected individualsx at time zero. Thus, the number of the initially infected individuals that remain infected at time t can be written as

Among the newly infected individuals, the number of individuals who have been infected for a duration less than or equal to x at time t is equal to

Thus, the number of newly infected individuals that remain infected at time t can be written as

We also have the total number of individuals infected at time t that have been infected for a duration which is less than or equal to x:

Note that for each t, I N 0 (t, ·) has support over [0, t +x] and I N 1 (t, ·) has support over [0, t]. Thus

Here we occasionally use ∞ in the second component for convenience with the understanding that

We remark that the sample paths of I N (t, x) belong to the space D D , denoting D(R + ; D(R + ; R)), the D-valued D space, but not in the space D(R 2 + ; R). We prove the weak convergence in the space D D where both D spaces are endowed with the Skorohod J 1 topology. Note that the space D(R 2 + ; R) is a strict subspace of D D , although they are equivalent in the continuous cases, that is, C(R 2 + ; R) = C C . See more discussions on these spaces in [21, 22, 2, 3] . Remark 2.1. The SEIR model. Suppose that λ i (t) = 0 for t ∈ [0, ξ i ), where ξ i < η i , and denote I as the compartment of infected (not necessarily infectious) individuals. An individual who gets infected at time τ N i is first exposed during the time interval [τ N i , τ N i + ζ i ), and then infectious during the time interval (τ N i + ζ i , τ N i + η i ). One may state that the individual is infected during the time

All what follows covers perfectly this situation. In other words, our model accomodates perfectly an exposed period before the infectious period, which is important for many infectious diseases, including the Covid-19. However, we distinguish only three compartments, S for susceptible, I for infected (either exposed or infectious), R for recovered.

In the sequel, the time interval [τ N i , τ N i + η i ) will be called the infectious period, although it might rather be the period during which the individual is infected (either exposed or infectious).

Define the fluid-scaled processesX N = N −1X N for any processes X N . We make the following assumptions on the initial quantities.

Assumption 2.1. There exists a deterministic continuous nondecreasing functionĪ(0, x) for x ≥ 0 withĪ(0, 0) = 0 such thatĪ N (0, ·) →Ī(0, ·) in D in probability as N → ∞. LetĪ(0) =Ī(0,x). Then (Ī N (0),S N (0)) → (Ī(0),S(0)) ∈ (0, 1) 2 in probability as N → ∞ whereS(0) = 1 −Ī(0) ∈ (0, 1).

Suppose now that the r.v.'s {τ N j,0 } 1≤j≤N are not ordered, but rather i.i.d., with a common distribution function G which we assume to be continuous. It then follows from the law of large numbers that Assumption 2.1 holds in this case.

We make the following assumption on the random function λ.

Assumption 2.2. Let λ(·) be a process having the same law of {λ 0 j (·)} j and {λ i (·)} i . Assume that there exists a constant λ * such that for each 0 < T < ∞, sup t∈[0,T ] λ(·) ≤ λ * almost surely. Assume that there exist an integer k, a random sequence 0 = ζ 0 ≤ ζ 1 ≤ · · · ≤ ζ k and associated random functions

(2.11)

In addition, we assume that there exists a deterministic nondecreasing function ϕ ∈ C(R + ;

almost surely for all t, s ≥ 0 and for all ℓ ≥ 1.

Recall that the basic reproduction number R 0 is the mean number of susceptible individuals whom an infectious individual infects in a large population otherwise fully susceptible. In the present model, clearly

We obtain the same formula if the deterministic functionλ(t) is replaced by a process λ i (t) independent of η i , with meanλ(t). More precisely, in that case the sequence (λ i (t), η i ) i≥1 is assumed to be i.i.d., and for each i, λ i and η i are independent.

The proof of the following Theorem, which is the main result of this section, will be given in section 4. where the limits are the unique continuous solution to the following set of integral equations, for t, x ≥ 0,S

14)

17) The functionĪ(t, x) is nondecreasing in x for each t, the integrals w.r.t. d xĪ (0, y) and d xĪ (t, x) are Lebesgue-Stieltjes integrals with respect to the measure which coincides with the distributional derivative ∂ xĪ (0, ·) =Ī x (0, ·) (resp. ∂ xĪ (t, ·) =Ī x (t, ·)). As a consequence,Ī N →Ī in D in probability as N → ∞ wherē

x) the density function ofĪ(t, x) with respect to x. Note thatS(t) = 0 for t < 0 andī(t, x) = 0 both for t < 0 and x < 0. If the density functionī(t, x) exists, we obtain the following PDE representation. We shall denote by µ(x) the hazard function of the r.v. η, i.e.,

Proposition 2.1. Suppose that F is absolutely continuous, with the density f , and thatĪ(0, x) is differentiable with respect to x, with the density functionī(0, x). Then for t > 0, the increasing functionĪ(t, ·) is absolutely continuous, and (t, x) a.e. in (0, +∞) 2 ,

is the unique non-negative solution of the following Volterra equation

Moreover, the PDE (2.28) has a unique solution which is given as follows. For x ≥ t, Remark 2.5. We remark that the PDE given in [14] resembles that given in (2.19) , see equations (28)-(29), see also equation (2.2) in [10] . In particular, the function µ(x) is interpreted as the recovery rate at infection age x. Equivalently, it is the hazard function of the infectious duration.

Proof. By the fact that F has a density, we see that the two partial derivatives ofĪ exist (t, x) a.e. From (2.15), they satisfȳ

Thus, summing up (2.24) and (2.25), we obtain for t > 0 and x > 0,

By taking the derivative on both sides of (2.26) with respect to x (possibly in the distributional sense for each term on the left), we obtain for t > 0 and x > 0,

For the boundary conditionī(t, 0), by (2.14) and (2.17), we havē

Thus we obtain the expression in (2.20) . We next prove that equation (2.20) has a unique nonnegative solution.

Observe that x(t) =ī(t, 0) is also a solution to

and any non-negative solution of (2.20) solves (2.29). First, note that for any t ≥ 0, 0 ≤ x 0λ (y + t)ī(0, y)dy ≤ λ * Ī (0) and 0 ≤λ(t) ≤ λ * , from which we conclude that t 0ī (s, 0)ds ≤S(0). Indeed, if that were not the case, there would exist a time TS (0) < t such that

0ī (s, 0)ds =S(0), hence t 0ī (s, 0)ds ≥S(0) and from (2.29), we would have x(t) = 0 for any t ≥ TS (0) . Under Assumption 2.2, if x 1 (t) and x 2 (t) are two nonnegative integrable solutions, then

which, combined with Gronwall's Lemma, implies that x 1 ≡ x 2 . Now existence is provided by the fact that the functionī(t, 0) is a non-negative solution of (2.29). Note also that clearly, using a combination of an argument similar to that used for uniqueness, and of the classical estimate on Picard iterations for ODEs, one could establish that the sequence defined by x (0) (t) ≡ 0 and for n ≥ 0,

givenī(0, ·), is a Cauchy sequence in C(R + ), hence existence. We next derive the explicit solution expressions in (2.22) and (2.23) . We note that an immediate consequence of (2.28) is that for x ≥ 0,

(2.22) follows from the first identity, and (2.23) from the second. It is then immediate that (2.19) follows from these expressions and (2.28).

Corollary 2.1. The formula (2.14) for I(t) can be rewritten

. Inserting the resulting formula forῩ in the second integral of the right hand side of (2.14), and then exploiting (2.22) in order to modify the first integral, we obtain

from which the result follows.

In that case, (2.30) reduces to the very simple formula

(

A similar formula holds if we replace the deterministic functionλ(t) by a copy λ i (t) of a random function, which is independent of η i , as discussed in Remark 2.3, and whose expectation isλ(t). Then, we haveῩ

Sinceī(t, 0) =Ῡ(t), the results above can be stated using this expression ofῩ. In the literature of PDE epidemic models, the formula for the instantaneous infectivity rateῩ(t) is usually stated in the form of (2.32). This expression has clearly a very intuitive interpretation. In particular, the identityī

is often imposed.ī(t, 0) is the instantaneous rate for an individual to get infected at time t (resulting in a newly infectious individual with a zero age of infection), while the right hand side is the instantaneous infection rate by the existing infectious population at time t, which depends on all the infectious individuals with all ages of infection. This of course includes time t = 0, which formulates a constraint on the initial condition {Ī(0, x)} 0≤x≤x . Our general formulation can be also expressed in an analogous way as shown in Corollary 2.1.

In the special case of exponentially distributed infectious periods, i.e. µ(x) ≡ µ, we obtain the following well known results, see, e.g., [23, 11, 17] .

with the boundary conditionsī(0, x) for x ∈ [0,x] andī(t, 0) as given in (2.20) .

Proof. In this case, the above proof simplifies. Indeed, we have for t ≥ 0 and x ≥ 0,

By taking derivative with respect to x when t > 0 and x > 0, we obtain that equation (2.25) becomesī

By (2.28), we have for t > 0 and x > 0,

The boundary conditions follow in the same way as in the general model.

Suppose that the infectious periods are deterministic and equal to t i , i.e., F (t) = 1 t≥t i . Then we have

with δ t i (x) being the Dirac measure at t i , and with the boundary conditionsī(0,

Note that in this caseī(t, x) = 0 for x ≥ t i , and (2.20) reads

The PDE (2.34) has a unique solutionī, which is given as follows.

The boundary conditionī(t, 0) solves the following Volterra equation. If 0 < t < t i ,

Proof. Herex = t i and F (t) = 1 t≥t i . Moreoverλ(t) = 0 for t > t i . By (2.15), we obtain

By taking partial derivatives with respect to t, using (2.17), we obtain for t ≥ 0 and x ≥ 0,

Note that both sides of (2.36) are equal to zero for any t > 0 and x = 0, and also that when t = 0, for any x ≥ 0, the RHS of equation (2.36) reduces toĪ(0, x ∧ t i ). By taking derivative with respect to x on both sides of (2.36), we obtain for t > 0 and x > 0,

Summing up the equations (2.37) and (2.38), we obtain

By taking derivative with respect to x, usingī(t, x) =Ī x (t, x), we obtain (2.34).

The identity (2.35) is obtained in the same way as (2.20) in Proposition 2.1, noting that in the present case there is no infectious individual with infection age greater than t i , henceī(t, x) = 0 for x ≥ t i . 

(noting that they are not continuous unless F 0 is continuous), and assuming the density functions exist, we obtain the PDE:

with the same boundary conditions as in Proposition 2.1. The proofs for these results follow from a similar but simpler argument and are thus omitted.

Remark 2.8. The total fraction of the population infected during the epidemic is given by 0) is the solution to (2.20) . We also refer the reader to equation (12) in Kaplan [12] , based on his constructed "Scratch" model.

In the SIS model, the infectious individuals become susceptible once they recover. Since S N (t) + I N (t) = N for each t ≥ 0 with a population size N , the epidemic dynamics is determined by the process I N (t) alone, and we have the same representations of the processes I N 0 (t, x) and I N 1 (t, x) in where

2) 

However, the formula forS(t) is different in the case of the SIS model, that is, (2.21) does not hold. Instead, we havē

Thus, the Volterra equation on the boundary reads

whose form is similar to the one for the SIR model.

It is also clear that if the c.d.f.'s F (t) = 1 − e −µt , we have the same PDE forī(t, x) as given in (2.33) with the boundary condition:

If the infectious periods are deterministic and equal to t i , then we have the following PDE for i(t, x):

with the boundary conditions given in Proposition 2.2, that is, for 0 < t < t i ,

Recall that the standard SIS model has a nontrivial equilibrium pointĪ * = 1 − µ/λ if µ < λ, where λ is the infection rate (the bar over λ is dropped for convenience), and 1/µ is the mean of the infectious periods. See Section 4.3 in [20] for the account of the SIS model with general infectious periods. Here we consider the model in the generality of infection-age dependent infectivity. 

The density functionī(t, x) has an equilibriumī * (x) in the age of infection x, given bȳ

where µ −1 = ∞ 0 F c (t)dt is the expectation of the duration of the infectious period. If F has a density f , then the equilibrium densityī * (x) satisfies

Proof. Assume that the equilibriumĪ * (x) :=Ī(∞, x) exists. We deduce from (3.3), combined with (2.30) thatĪ * (x) must satisfȳ

where F e (x) = µ Plugging this formula in the previous identity, we deduce that

Then the formula (3.6) can be directly deduced from this equation. The formula (3.7) follows by taking the derivative with respect to x in (3.9).

Remark 3.1. If the distribution F is exponential, that is, F (x) = 1 − e −µx , then we obtain

whereĪ * is given in (3.6).

is a deterministic function, as in Remark 2.6. Thenλ(t) = λ(t)F c (t). If λ(t) ≡ λ is a constant and F has mean µ −1 , thenĪ * in (3.6)

which reduces to the well known result for the standard SIS model with constant rates, assuming µ < λ. Note that the last expressions on the right in (3.6) and (3.10) coincide.

If the initial conditions are assumed such that the remaining infectious periods of the initially infectious individuals are i.i.d. with F 0 = F e , then we also obtain the same equilibrium point. In this case, we havē

It can be checked that if this equation has an equilibrium point, it is also equal to that given above.

In this section, we prove Theorem 2.1. We will use the following theorem in the proofs below (see Theorem 3.5.1 in Chapter 6 of [16] for the pre-tightness criterion, which extends that in the Corollary on page 83 of [4] to the space C([0, 1] k , R)). The proof can be easily extended to the space D D . For the convenience of the reader, we give a proof of the following result in section 5.1 below. 

then X N (t, s) → 0 in probability, locally uniformly in t and s.

We shall also use repeatedly the following Lemma.

Lemma 4.1. Let f ∈ D(R + ) and {g N } N ≥1 be a sequence of elements of D ↑ (R + ) which is such that g N → g locally uniformly, where g ∈ C ↑ (R + ). For for any T > 0,

Proof. The assumption implies that the sequence of measures g N (dt) converges weakly, as N → ∞, towards the measure g(dt). Since moreover f is bounded, and the set of discontinuities of f is of g(dt) measure 0, this is essentially a minor improvement of the Portmanteau theorem.

Convergence of I N 0 (t, x). We first treat the process I N 0 (t, x) in (2.5). 

in probability, where the limitĪ 0 (t, x) is given bȳ

Note that the pair of variables (τ N j,0 , η 0 j ) satisfies (2.1), and I N (0, (x − t) + ) = max{j ≥ 1 :τ N j,0 ≤ (x − t) + }. Let

SinceĪ N (0, ·) →Ī(0, ·) in D in probability, andĪ(0, ·) is continuous, the convergence holds locally uniformly in t in probability and from Lemma 4.1 and the continuous mapping theorem, we deduce that in probability

. We first check condition (i) from Theorem 4.1. We have

where the second term in the first equality is equal to zero by the independence of η 0 j and η 0 j ′ given the timesτ N j,0 andτ N j ′ ,0 and by using a conditioning argument. This implies that as N → ∞, sup t≥0 sup x≥0 E V N (t, x) 2 → 0, and thus condition (i) in Theorem 4.1 holds.

We next show condition (ii) from Theorem 4.1, that is, for any ǫ > 0, as δ → 0, 

We first prove (4.4). We have

For the first term,

By the conditional independence of the η 0 j 's, the first term on the right of (4.7) is bounded by

which converges to zero as N → ∞. Since by Assumption 2.1Ī(0, ·) is continuous, thanks to Lemma 4.1, lim sup N of the second term is upper bounded by

which is zero for δ > 0 small enough. The second term on the right of (4.6) is treated exactly as the last term we have just analyzed. Finally for the third term, we note that P sup 

for δ > 0 small enough, sinceĪ(0, ·) is continuous. Thus we have shown (4.4).

We next prove (4.5). Let

.

This implies that for ǫ > 0,

Note that for each fixed t and x, Z N 

For v > u, we have

For v ≤ u, we have

At this point, we can choose δ ′ = δ. We have

In view of (4.8) and (4.9), (4.5) follows from the fact that

Hence in view of (4.10), (4.5) will follow from the fact that for i = 1, 2, 3,

Exploiting again the conditional independence of the η 0 j 's, we obtain that

which tends to 0 as N → ∞ uniformly for x ∈ [0, T ′ ], hence (4.11) for i = 1. Thanks to Assumption 2.1 and Lemma 4.1, we deduce that, as N → ∞, uniformly in x ∈ [0, T ′ ],

which is continuous w.r.t. δ and equals 0 at δ = 0. Hence for any ǫ > 0, if δ is small enough, then

which shows (4.11) for i = 2. The same argument clearly gives (4.11) for i = 3. where Q(ds, du) = Q(ds, du) − dsdu is the compensated PRM.

The quadratic variation of M N A (t) is given by

It suffices to verify the martingale property: for t 2 > t 1 ≥ 0, and

Proof. It is clear that under Assumption 2.2, ifΛ N (t) := t 0Ῡ In the following of this section, we consider a convergent subsequence ofĀ N . Recall thatĪ where the limitĪ 1 (t, x) is given bȳ

We can write (from now on, b a stands for (a,b] )

Then from Lemma 4.1, we deduce that for any t, x ≥ 0,

We will next show that for any ǫ > 0, there exists δ > 0 such that the following holds for any (t, x):

It is not hard to deduce from (4.22) and (4.23), by a two-dimensional extension of the argument of the Corollary on page 83 of [4] , that as N → ∞,Ȋ N 1 (t, x) ⇒Ī 1 (t, x) locally uniformly in t and x. Whenever t ≤ t ′ ≤ t + δ and x ≤ x ′ ≤ x + δ, we have

SinceĀ N (t) ⇒ t 0Ῡ (s)ds locally uniformly in t, andῩ(s) ≤ λ * , the limit in law of the right hand side of the last inequality is bounded by

which is less than ǫ for δ > 0 small enough. Hence, (4.23) follows.

Let now

To prove (4.19) , it remains to show that, as N → ∞,

We apply Theorem 4.1. By Markov's inequality and the decomposition of

The result then follows from the next two lemmas. 

Proof. For u < x,

In the case of u > x,

Then we obtain P sup

LetQ(ds, du, dz) denote a PRM on R 3 + with mean measure ν(ds, du, dz) = dsduF (dz) and Q denote the associated compensated PRM. By the Markov inequality, we obtain the first term is bounded by 9ǫ −2 times

where the last inequality follows from (4.17). The first term converges to zero as N → ∞, and we note that

For the second term in (4.26), we have

where the first term converges to zero as N → ∞ by the convergenceM N A (t) → 0 in probability, locally uniformly in t, while the second term is bounded as in (4.27) .

For the last term in (4.26), we use the martingale decomposition ofĀ N and the bound forῩ N in (4.17) , and obtain sup 0≤t≤T

which, sinceM N A (t) → 0 locally uniformly in t, implies that, provided δ < ǫ/λ * ,

Thus we have shown that (4.25) holds. 

Thus, we obtain for ǫ > 0,

(4.31)

The sum of the two expectations is bounded by

Hence if we choose δ ′ = √ δ, we have proved that, as δ → 0,

as requested. For the second term in (4.30), if u < v, we have

provided v ≤ δ and u ≤ δ ′ . For u > v, and again v ≤ δ and u ≤ δ ′ ,we have

Thus we obtain that

It then follows from (4.28) that provided δ > 0 and δ ′ > 0 are small enough,

Thus we have shown that (4.29) holds.

Convergence of the aggregate infectivity process. Recall I N in (2.2), and let I N := 

Proof. We write

We first consider Ξ N 0 (t). For each fixed t, by conditioning on σ{I N (0, y) : 0 ≤ y ≤x} = σ{τ N j,0 , j = 1, . . . , I N (0)}, we obtain

We then have for t, u > 0,

Then by Assumption 2.2, writing λ 0

Both terms on the right hand side are increasing in u, and thus, we have

Here for the second term, we have

The first term on the right of (4.35) tends to 0 as N → ∞, since by conditioning on σ{I N (0, y) : 0 ≤ y ≤x} = σ{τ N j,0 , j = 1, . . . , I N (0)}, and since the ζ ℓ j 's are mutually independent and globally independent of theτ N j,0 's, we obtain

The second term on the right of (4.35) equals

is continuous and equals 0 at δ = 0, for any ǫ > 0, ther exists δ > 0 small enough such that the above quantity vanishes. Thus, we have shown that

Next, consider ∆ N,2 0 (t, u), which is ∆ N,1 0 (t, u), with the j-th term in the absolute value being replaced by its conditional expectation givenτ N j,0 . The computations which led above to (4.34) give

So the same arguments as those used above yield that (4.36) holds with ∆ N,1 0 (t, u) replaced by ∆ N,2 0 (t, u).

Thus we have shown that in probability, Ξ where I(t) is given by

Proof. By the above lemma, it suffices to show that

The expression of I N in (4.33) can be rewritten as

It follows from Lemma 4.1 that for any t > 0, as N → ∞, I N (t) ⇒ I(t). It remains to show that the sequence I N is tight in D. For that purpose, exploiting the Corollary on page 83 of [4] , it suffices to show that for any ǫ > 0, 

NowĪ(0, dy) a.e., G δ (y +t) → 0, and since 0 ≤ G δ (y +t) ≤ λ * , it follows from Lebesgue's dominated convergence that

x 0 G δ (y + t)Ī(0, dy) → 0, as δ → 0,hence for δ > 0 small enough, this quantity is less than ǫ, and the indicator vanishes.

It remains to establish (4.42). We have

The result follows since the sum of the two first terms on the right are less than ǫ/2 for δ > 0 small enough, while the two last terms tend to 0, as N → ∞.

Completing the proof of Theorem 2.1. By Lemmas 4.2 and 4.5, we have that, along a subsequence,

x) ⇒Ī(t, x) =Ī 0 (t, x) +Ī 1 (t, x) ∈ D D as N → ∞, whereĪ 0 (t, x) andĪ 1 (t, x) are given in (4.2) and (4.20), respectively. Also recall thatS N = S N (0) −Ā N by (2.9). We need to show the joint convergence By independence of the variables associated with the initially and newly infected individuals, it suffices to show the joint convergence (Ā N ,Ī N 1 ) ⇒ (Ā,Ī 1 ) in D × D D as N → ∞. In fact by (4.24) , it suffices to show the joint convergence

forȊ N 1 (t, x) given in (4.21) . This follows from the continuity in the Skorohod J 1 topology of the mapping

where D ↑ denotes the set of increasing elements of D, and the continuity is easy to establish by integration by parts. Next, we also have the joint convergence By Lemma 4.8, it suffices to show that the joint convergence of (Ā N ,Ȋ N 1 ) with I N , which follows from applying the continuous mapping theorem. Again by the independence ofĪ 0 (t, x) andĪ 1 (t, x) and usingȊ N 1 (t, x) given in (4.21) as above, we obtain the joint convergence by applying the continuous mapping theorem.

Recall the expression of Υ N (t) =S N (t)I N (t). Applying the continuous mapping theorem again, we obtain that Υ N (t) ⇒Ῡ(t) =S(t) I(t) in D as N → ∞.

Thus by (4.16), we conclude that

(s)ds = · 0S (s) I(s)ds in D as N → ∞.

Therefore, the limit (S,Ī) satisfies the set of integral equations in (2.13), (2.15), and the limit I coincides with I defined by (2.14). The limitsĪ in (2.18) andR in (2.16) then follow immediately. The set of integral equations has a unique deterministic solution. Indeed, it is easy to see that the system of equations (2.13) and (2.14) (together with the first part of (2.17)) has a unique solution (S, I), given the initial valuesĪ(0, ·). The other processesĪ,Ī,R are then uniquely determined. Hence the whole sequence converges in probability. From (2.15), we deduce that for all t > 0,

This prove the second equality in (2.17) . It remains to prove the continuity. The continuity in t ofS(t) is clear. Let us prove that t → I(t) is continuous. Since λ i is càdlàg and bounded, it is easily checked that t →λ(t) = E[λ(t)] is also càdlàg. In fact it is continuous if all the F ℓ 's for 1 ≤ ℓ ≤ k are continuous. The points of discontinuity ofλ(t) are the points where one of the laws of the ζ ℓ has some mass. The set of those points is at most countable. Consequently, if t n → t, the set of y's whereλ(t n + y) may not converge toλ(t + y) is at most countable, and this is a set of zeroĪ(0, dy) measure. Since moreover 0 ≤λ(t n + y) ≤ λ * , t → x 0λ (y + t)Ī(0, dy) is continuous. Let us now consider the second term in (2.14). We first note that sinceλ(t − s) ≤ λ * andS(t) ≤ 1, it follows from (2.14), (2.17) and Gronwall's Lemma that I(t) ≤ λ * e λ * t . Let t n → t. We have t 0λ (t − s)Ῡ(s)ds − tn 0λ (t n − s)Ῡ(s)ds ≤ t 0 |λ(t − s) −λ(t n − s)|Υ(s)ds + (λ * ) 2 e λ * (t∨tn) |t − t n |.

Clearly the above right hand side tends to 0, as n → ∞. A similar argument shows thatR and I are continuous, and that (t, x) →Ī(t, x) is continuous. Finally, since the convergence holds in D × D × D D × D and the limits are continuous, the convergence is locally uniform in t and x. This completes the proof of Theorem 2.1. For any t ∈ [0, T ], we define γ T,δ (t) to be the element of Γ T,δ such that γ T,δ (t) ≤ t < γ T,δ (t) + δ, and for any s ∈ [0, S], we define γ S,δ (s) to be the element of Γ S,δ such that γ S,δ (s) ≤ s < γ S,δ (s) + δ. Let (t, s) and (t ′ , s ′ ) be two points in [0, T ] × [0, S] such that |t − t ′ | ∧ |s − s ′ | ≤ δ. We have X N (t, s) − X N (t ′ , s ′ ) = X N (t, s) − X N (t, γ S,δ (s)) + X N (t, γ S,δ (s)) − X N (γ T,δ (t), γ S,δ (s)) + X N (γ T,δ (t), γ S,δ (s)) − X N (γ T,δ (t ′ ), γ S,δ (s)) + X N (γ T,δ (t ′ ), γ S,δ (s)) − X N (γ T,δ (t ′ ), γ S,δ (s ′ )) + X N (γ T,δ (t ′ ), γ S,δ (s ′ )) − X N (t ′ , γ S,δ (s ′ )) + X N (t ′ , γ S,δ (s ′ )) − X N (t ′ , s ′ ). 

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