Stochastic epidemic models with varying infectivity and waning immunity RAPHA EL FORIEN GUODONG PANG ETIENNE PARDOUX AND ARSENE BRICE ZOTSANGOUFACK

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Stochastic epidemic models with varying infectivity

and waning immunity

RAPHA¨

EL FORIEN, GUODONG PANG, ´

ETIENNE PARDOUX, AND ARSENE BRICE ZOTSA–NGOUFACK

Abstract.

We study an individual-based stochastic epidemic model in which infected individuals

become susceptible again, after each infection. In contrast to classical compartment models, after each

infection, the infectivity is a random function of the time elapsed since infection. Similarly, recovered

individuals become gradually susceptible after some time according to a random susceptibility

function.

We study the large population asymptotic behaviour of the model: we prove a functional law of

large numbers (FLLN) and investigate the endemic equilibria of the limiting deterministic model.

The limit depends on the law of the susceptibility random functions but only on the mean infectivity

functions. The FLLN is proved by constructing a sequence of i.i.d. auxiliary processes and adapting

the approach from the theory of propagation of chaos. The limit is a generalisation of a PDE model

introduced by Kermack and McKendrick, and we show how this PDE model can be obtained as a

special case of our FLLN limit.

is less than (or equal to) some threshold, the epidemic does not last forever and eventually

disappears from the population, while if

is larger than this threshold, the epidemic will not go

extinct and there exists an endemic equilibrium. The value of this threshold turns out to be the

harmonic mean of the susceptibility a long time after an infection.

1. Introduction

Many infectious diseases are such that the immunity acquired after recovery from the illness is

eventually lost, after a period whose length varies both with the individual and with the illness.

The huge majority of the literature on mathematical epidemiology which considers models with loss

of immunity assumes that this loss happens instantaneously (see for example [

]). However, it is

rather clear that, in reality, immunity wanes progressively over some period, which can vary from

one individual to another (see for example [

]). In their pioneering 1927 paper [

], Kermack

and McKendrick introduced the ﬁrst mathematical model of epidemic propagation in which the

infectivity of individuals depends on the time elapsed since their infection (this time is often called

the age of infection). This model was deterministic and assumed that recovered individuals can no

longer be infected. In two subsequent papers, [

], Kermack and McKendrick introduced another

deterministic model in which recovered individuals eventually lose their immunity. They studied the

conditions under which an infectious disease can become endemic, i.e., reach a stable equilibrium

where some macroscopic fraction of the population is infected. This kind of equilibrium for the

deterministic system is called an endemic equilibrium, as opposed to the disease-free equilibrium, in

which there are no infected individuals.

Our aim in the present paper is to revisit these questions with the use of a general stochastic

model, following the recent developments carried out by the ﬁrst three authors. In [

], the 1927

model of Kermack and McKendrick was obtained as the inﬁnite population limit of an individual-

based stochastic model in which the infectivity of individuals is a random function of their age of

infection. More precisely, it was shown that the fraction of susceptible individuals and the average

Date: March 28, 2025.

Key words and phrases. Stochastic epidemic model, varying infectivity, varying immunity/susceptibility, functional

law of large numbers, integral equation, infection-age and recovery-age dependent PDE model.

arXiv:2210.04667v2 [math.PR] 27 Mar 2025

2 RAPHA¨

EL FORIEN, GUODONG PANG, ´

ETIENNE PARDOUX, AND ARSENE BRICE ZOTSA–NGOUFACK

Exposed Infectious Immune

Susceptible

Figure 1. Illustration of a typical realization of the random infectivity and suscep-

tibility functions of an individual from the time of infection to the time of recovery,

and then to the time of losing immunity and becoming fully susceptible (or in general,

partially susceptible).

infectivity (also called the force of infection) converge to a deterministic limit which solves a system

of non-linear integral equations already introduced in [

]. Note that an alternative to the integral

equations model is a PDE model, see [29]. The widespread ordinary diﬀerential equations (ODEs)

compartmental SIR model is only a very special case of these equations, in which neither the

infectivity nor the recovery rate depends upon the age of infection. Such models are less realistic as

they do not reproduce the dependence of the dynamics of the epidemic on its past [

], and

thus neglect the inertia of the evolution of the epidemic. Note that ODE models are law of large

numbers limit of Markov stochastic models, in which individuals move from one compartment to

the next after an exponentially distributed time.

In the present paper, we introduce a very general model in which individuals are characterized by

their infectivity and susceptibility, which are assumed to be given as random functions of their age

of infection. These random functions are assumed to be i.i.d. among the various individuals in the

population, and a new independent pair of random functions is drawn at each new infection. See

Figure 1for a realization of the infectivity and susceptibility of an individual after his/her infection.

Note however that, in contrast with the model in [

], we consider a closed population, in

which there are no birth and no death (we do not exclude deaths due to infections, represented by

individuals whose susceptibility remains equal to zero after the fatal infection).

We also assume (as in [

]) that all pairs of individuals have contacts at the same rate. In

other words, we assume a situation of homogeneous mixing. Of course, this is not quite satisfactory.

One may wish to take into account the spatial distribution of the population, as well as the variety of

social behaviors of the individuals. However, mathematical models involve necessarily a simpliﬁcation

of the complex reality. We believe that the results presented here constitute a signiﬁcant progress

over the classical models where all rates are constant and immunity is lost instantaneously. In future

works, we do intend to combine the complexity of the present work with that of inhomogeneous

models, such as spatial models or models on graphons.

Besides the fact that we prefer to present our model in the form of a system of integral equations

rather than a system of partial diﬀerential equations, our deterministic model is more general than

the model of [

]. The reason is the following. While the law of large numbers limit of the model

with infection age dependent infectivity depends only on the mean of the random infectivity function

[

], the deterministic limit of the model with random susceptibility depends on the distribution of

this random susceptibility function in a much more complicated way. To see this, suppose that we

wish to compute the average susceptibility of the individuals at some time

. This average can be

obtained by summing the contributions of individuals that have not been infected on the interval

, t

), and of individuals that have been infected at some time

s∈

, t

) and have not yet been

reinfected again on the interval [

s, t

). But the probability that such an individual has not been

reinfected by time

depends on the full trajectory of their susceptibility, as well as on the trajectory

of the force of infection, on the interval [

s, t

) (see

(3.5)

below). This explains why the deterministic

limit is more complex than that in [

], and shows that, in their 1932-33 model, Kermack and

McKendrick implicitly assumed that the only random component of the susceptibility function

is the time of recovery (see the discussion in Section 5.2). As a consequence, we obtain a strict

generalization of the model in [24,25] as the deterministic limit of our stochastic model.

Since in our model there is a ﬂux of new susceptibles, due to waning of immunity, one expects

that under certain conditions, there may be a stable endemic equilibrium. We manage to study

the existence, uniqueness and some stability properties of an endemic equilibrium (as well as the

stability or instability of the disease–free equilibrium) in our deterministic limit. We identify a

threshold, which is the harmonic mean of the large time limit of the susceptibility (which is 1 in

Figure 1, but may be less than 1 in our general model). If the basic reproduction number

(deﬁned below by

(4.1)

) is smaller than this threshold, then the process converges to the disease-free

equilibrium. We also show that under appropriate assumptions, if

is larger than the threshold,

and the model converges as

t→ ∞

to some limit, then this limit is the unique endemic equilibrium,

which is fully characterized. We conjecture that under appropriate assumptions, when

is larger

than the threshold, any solution of the deterministic limit starting with a non zero force of infection

at time t= 0 does converge to the endemic equilibrium.

Let us comment on the method used to prove our law of large numbers result. One key argument

is based upon a coupling of the processes which count the numbers of infections of the various

individuals up to time

with i.i.d. counting processes, where the renormalized force of infection

is replaced by its deterministic limit. This approach, for which the inspiration came from [

], is

an application of ideas from the theory of propagation of chaos, see Sznitman [

]. Note both

that we were unable to adapt the methodology of [

] to the setting of the present paper, and

that the assumptions made here on the random infectivity function are weaker than those made in

[

]. Recently, the methodology of the present paper has been applied to the model in [

], thus

resulting in the same result under weaker assumptions, see [

]. We also note that our approach

diﬀers signiﬁcantly from the techniques classically used for age-structured population models, as

in [

]. In these papers, the authors describe the model as a branching process

(sometimes with interaction), which becomes an inﬁnite dimensional Markov process if one adds all

the age structure in the present state of the system, in which the lifespan, birth rate and death rate

depend on the ages of all individuals in the population. The state of such a model is then described

by the empirical measure of the ages of the individuals. These works combine inﬁnite-dimensional

stochastic calculus and measure-valued Markov processes analysis in order to prove the convergence

of the model. In contrast, our stochastic models are non Markovian, and their deterministic limit

does have a memory.

Using random infectivity and susceptibility functions allows us to build a very general model

which is both versatile and tractable. It captures the eﬀect of a progressive loss of immunity, and

this loss is allowed to be diﬀerent from one individual to another. The integral equations that we

obtain to describe the large population limit of our model are both compact and extremely general,

since most epidemic models with homogeneous mixing and a closed population can be written in

this form. The eﬀect of the variability of susceptibility on the endemic threshold has received very

little attention in the literature, despite some profound implications which we outline in the present

4 RAPHA¨

EL FORIEN, GUODONG PANG, ´

ETIENNE PARDOUX, AND ARSENE BRICE ZOTSA–NGOUFACK

work. The fact that the threshold depends on the harmonic mean of the susceptibility reached

after an infection shows that the heterogeneity of immune responses in real populations should

not be neglected in public health decisions. Similarly, the variability of the immune response after

vaccination (both in time and between individuals) should aﬀect the eﬃcacy of vaccination policies

in non-trivial ways, although these questions are outside the scope of the present work.

We want to comment on the terminology which can be used for our model as a compartmental

model. Recall that the compartments most classically used in epidemic models include S for

Susceptible individuals, E for Exposed (those infected individuals who are not yet infectious), I for

infectious and R for Recovered. We claim that our model can be classiﬁed as an SEIRS, SIRS, or

SIS model. Indeed, referring to Figure 1, we can consider that a given individual passes from the

S to the E compartment when he/she becomes infected, then into the I compartment when the

attached infectivity ﬁrst becomes positive, into the R compartment when the infectivity reaches 0

and remains null, and into the S compartment when the attached susceptibility becomes positive.

However, without modifying the dynamics of the epidemic, we can merge the E and I compartments

into a compartment I (for infected), where the infectivity need not be positive all the time. Similarly,

we can merge the R and S compartments into the S compartment (or U, for uninfected, as we suggest

below), whose members may have a susceptibility equal to zero. As a matter of fact, our model is

vey general and includes most of the existing homogeneous epidemic models without demography

as particular examples. It is non Markov. All the rates are not only infection age dependent, but

also random, i.e., diﬀerent from one individual to another, which we believe reﬂects the reality of

epidemics.

Models with gradual waning of immunity have been studied since Kermack and McKendrick

by only a handful of authors, including Inaba, who in a series of works, see [

], has

performed a careful mathematical study of the PDE model from [

], as well as Breda et al. [

]

who have considered an integral equation version of the same model. Other authors have pursued

the study of the system of ODE/PDEs, see in particular Thieme and Yang [

], Barbarossa and R¨ost

[

] and Carlsson et al. [

]. More recently, Khaliﬁ and Britton [

], compare the level of immunity in

a model with gradual vs. sudden loss of immunity. We also mention the recent work [14], where a

similar stochastic model of varying infectivity and waning immunity with vaccination is studied.

We notice the diﬀerence from our modeling approach besides the vaccination aspect: the random

susceptibility function and the random infectivity function are assumed mutually independent in

each infection, contrary to what we assume here. In [

] Zotsa-Ngoufack establishes the central

limit theorem for the model described in the present paper.

Organization of the paper. The rest of the paper is organized as follows. In Section 2, we deﬁne

the model in detail. In Section 3, we state the assumptions and the functional law of large numbers

(FLLN) and discuss how the results reduce to already known results when we restrict ourselves

to the classical SIS and SIRS models. The results on the endemic equilibrium are presented in

Section 4. In Section 5, we focus on the generalized SIRS model with a particular set of infectivity

and susceptibility random functions and initial conditions, and show how the limit relates to the

Kermack and McKendrick PDE model with the corresponding infection-age dependent infectivity

and recovery-age dependent susceptibility. The proofs for the FLLN are given in Section 6and

those for the endemic equilibrium in Section 7.

Notation. Throughout the paper, all the random variables and processes are deﬁned on a common

complete probability space (Ω

,F,P

). We use

−−−−−→

N→+∞

to denote convergence in probability as the

parameter

N→ ∞

. Let

denote the set of natural numbers and

(

) the space of

-dimensional

vectors with real (nonnegative) numbers, with

(

R+

) for

= 1. We use

1{·}

for the indicator

function. Let

(

R+

;

) be the space of

-valued c`adl`ag functions deﬁned on

R+

, with

convergence in

meaning convergence in the Skorohod

topology (see, e.g., [

, Chapter 3]). Also,

we use

to denote the

-fold product with the product

topology. Let

be the subset of

consisting of continuous functions and D+the subset of Dof c`adl`ag functions with values on R+

2. Model description

2.1. Deﬁnition of the model. We consider a population of ﬁxed size

. Let (

λ0, γ0

) and (

λ, γ

)

be two random variables taking values in

(

R+,R+

)

, and let

{

(

λk,0, γk,0

)

≤k≤N}

be a

family of i.i.d. copies of (

λ0, γ0

) and

{

(

λk,i, γk,i

)

, i ≥

≤k≤N}

be a family of i.i.d. copies of

(

λ, γ

), independent of the previous family. The function

λk,0

(resp.

γk,0

) is the infectivity (resp.

susceptibility) of the

-th individual between time 0 and the time of his/her ﬁrst (re)-infection, and

λk,i

(

) denotes the infectivity of the

-th individual, at time

after his/her

-th infection (where

we count here only the infections after time 0), and

γk,i

(

) denotes the susceptibility of the

-th

individual, at time tafter his/her i-th infection, given that this is his/her most recent infection.

We can think of the law of (

λ0, γ0

) as being a mixture of the law for the initially infected

individuals, who have been infected before time 0 and for which

λ0

(0)

≥

0 and

γ0

(0) = 0, and the

law for the initially susceptible individuals, for which

λ0

(0) = 0 and

γ0

(0)

0 (possibly

γ0

(0) = 1).

We assume that (λ0, γ0) and (λ, γ) satisfy the following assumption.

Assumption 2.1. We assume that:

(i)

≤γ0

(

)

≤

1and 0

≤γ

(

)

≤

1almost surely and there exists a deterministic constant

λ∗<∞such that for all t≥0,0≤λ0(t)≤λ∗and 0≤λ(t)≤λ∗almost surely.

(ii) Almost surely,

sup{t≥0, λ0(t)>0} ≤ inf{t≥0, γ0(t)>0}and

sup{t≥0, λ(t)>0} ≤ inf{t≥0, γ(t)>0}.(2.1)

Assumption 2.1-(ii) implies that, as long as an individual has not recovered, he/she cannot be

reinfected. Hence in each infected-immune-susceptible cycle, the infected and susceptible periods do

not overlap. Note that only (i) is necessary for the process to be well deﬁned, and we discuss the

consequences of removing part (ii) of the above assumption in Remark 3.2 below. It would be rather

natural to assume that

γ0

(

) and

(

) are a.s. non decreasing. We do not make this restriction at

this stage, since we do not need it. We shall make this assumption in Section 4below.

For i≥0 and 1 ≤k≤N, we deﬁne

ηk,0:= sup{t≥0 : λk,0(t)>0},and ηk,i := sup{t≥0 : λk,i(t)>0}.

We will also use the notations

η0= sup{t > 0, λ0(t)>0}and η= sup{t > 0, λ(t)>0}.(2.2)

Let

(

) be the number of times that the individual

has been (re)–infected on the time interval

, t

]. The time elapsed since this individual’s last infection (or since time 0 if no such infection has

occurred), is given by

ςN

k(t) := t−sup{s∈(0, t] : AN

k(s) = AN

k(s−)+1} ∨ 0,

where we use the convention

sup ∅

−∞

. With this notation, the current infectivity and suscepti-

bility of the k-th individual are given by

λk,AN

k(t)(ςN

k(t)),and γk,AN

k(t)(ςN

k(t)).

Figure 2shows a realization of these processes for two individuals. Let us now deﬁne

(

) and

SN(t) as the average infectivity and susceptibility in the population, i.e.,

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StochasticepidemicmodelswithvaryinginfectivityandwaningimmunityRAPHA¨ELFORIEN,GUODONGPANG,´ETIENNEPARDOUX,ANDARSENEBRICEZOTSA–NGOUFACKAbstract.Westudyanindividual-basedstochasticepidemicmodelinwhichinfectedindividualsbecomesusceptibleagain,aftereachinfection.Incontrasttoclassicalcompartmentmodels,af...

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Stochastic epidemic models with varying infectivity and waning immunity RAPHA EL FORIEN GUODONG PANG ETIENNE PARDOUX AND ARSENE BRICE ZOTSANGOUFACK

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