# Extinction time of an epidemic with infection age dependent infectivity

Anicet Mougabe–Peurkor 111 Université Félix Houphouet Boigny, Abidjan, Côte
d’Ivoire<EMAIL_ADDRESS>, Ibrahima Dramé 222 Université Cheikh Anta
Diop de Dakar, FST, LMA, 16180 Dakar-Fann, Sénégal<EMAIL_ADDRESS>,
Modeste N’zi 333 Université Félix Houphouet Boigny, Abidjan, Côte d’Ivoire.
<EMAIL_ADDRESS>
and Etienne Pardoux 777 Aix Marseille Univ, CNRS, I2M, Marseille, France .
<EMAIL_ADDRESS>
###### Abstract

This paper studies the distribution function of the time of extinction of a
subcritical epidemic, when a large enough proportion of the population has
been immunized and/or the infectivity of the infectious individuals has been
reduced, so that the effective reproduction number is less than one. We do
that for a SIR/SEIR model, where infectious individuals have an infection age
dependent infectivity, as in the model introduced in the 1927 seminal paper of
Kermack and McKendrick [9]. Our main conclusion is that simplifying the model
as an ODE SIR model, as it is largely done in the epidemics literature,
introduces a biais toward shorter extinction time.

Keywords: Epidemic model; Branching process; Extinction time; Infection age
dependent infectivity; ODE SIR model; Effective reproduction number.

##  1 Introduction

Consider an epidemic which is declining: the number $M$ of infected
individuals is moderate, and decreases, while the total size $N$ of the
population is much larger. In such a phase, the approximation by the
deterministic model is no longer valid. Rather, as the initial phase of an
epidemic, the final phase can be well approximated by a branching process, in
this case a subcritical branching process. The extinction time is thus random.
It is of interest to have some information on the distribution function of
this extinction time. Indeed, if the subcriticality is due in part to some
rules imposed to the population, like mask wearing in public transport,
classrooms, workplace, theaters etc., it is important to evaluate how long
such rules must be maintained.

Our epidemic model is a SIR/SEIR model, i.e. we assume that after having been
infected and having recovered, an individual remains immune to the disease for
ever. This is not quite realistic. However, if the duration of the studied
period is not too long, then the number of individuals who loose their
immunity during that period can be neglected. On the other hand, the
stochastic SIR/SEIR model upon which we base our analysis is non Markov.
Following the ideas of Kermack–McKendrick [9] and Forien, Pang, Pardoux [5],
we consider a model where the infectivity of each infectious individual is
infection age dependent (and random, the realizations corresponding to various
individuals being i.i.d.). We characterize the distribution function of the
extinction time of the approximating non Markov branching process with a
single ancestor as the unique solution of a Volterra–type integral equation,
for which we give a converging numerical approximation. The derivation of the
equation is based upon a methodology introduced by Crump and Mode [3]. From
this result, we deduce in Theorem 3.5 a formula for the time we have to wait
after $t_{0}$ for the epidemic to go extinct, if at time $t_{0}$ we have $M$
infected individuals in a population of size $N$, with $M<<N$.

With the help of a numerical scheme, we compute an approximation of the
distribution function of the time of extinction, and compare the result with
the distribution function of the extinction time of a Markov branching process
which approximates the classical Markov SIR model (whose law of large numbers
limit is the most standard SIR ODE model), which is known explicitly. This
comparison is done between two models which have both the same effective
reproduction number $R_{eff}$ (the mean number of “descendants” one infectious
individual has at this stage of the epidemic), and the same rate $\rho$ of
continuous time exponential decrease. Our conclusion is that the usual ODE SIR
model leads to an underestimation of the extinction time.

Our work was inspired by the recent work of Griette et al. [7], where the
authors neglect the new infections during the final phase. Note that this
approximation is justified by the data, in the case of the end of the Covid
epidemic in Wuhan in 2020. Our work does not make such a simplifying
assumptions, and allows a very general law for the varying infectivity, and a
completely arbitrary law for the duration of the infectious period.

The paper is organized as follows. We present our varying infectivity SIR
model in Section 2, together with its branching process approximation. In
Section 3, we study the distribution function of the extinction time of the
branching process. In section 4, we present several examples of SIR/SEIR
models, including the classical ODE SIR model, ODE SEIR model, and we specify
the type of varying infectivity which we have in mind. In section 5, we
compare the time of extinction of the branching approximations to our varying
infectivity model, and to the ODE SIR model. In section 6, we discuss the
results obtained in that comparison. Finally in section 7 (the Appendix), we
establish the convergence of a numerical approximation scheme of the equation
established in section 3.

Notations In what follows we shall use the following notations:
$\mathbb{Z}=\\{...,-2,-1,0,1,2,...\\}$, ${\mathbb{R}}=(-\infty,\infty)$,
${\mathbb{R}}_{+}=[0,\infty)$ and ${\mathbb{R}}_{-}=[0,\infty)$. For
$x\in{\mathbb{R}}_{+}$, $[x]$ denotes the integer part of $x$ and $\lceil
x\rceil$ ( resp. $\lfloor x\rfloor$) denotes the ceiling function (resp. the
floor function). For $x\in{\mathbb{R}}$, $x^{+}$ (resp. $x^{-}$) denotes the
positive part of $x$ (resp. the negative part of $x$). For
$(a,b)\in{\mathbb{R}}^{2}$, $a<b$, $\mathcal{U}([a,b])$ denotes the uniform
distribution on $[a,b]$. $D([0,\infty))$ denotes the space of functions from
$[0,\infty)$ into $\mathbb{R}$ which are right continuous and have left limits
at any $t>0$. We shall always equip the space $D([0,\infty))$ with the
Skorohod topology, for the definition of which we refer the reader to
Billingsley [2].

##  2 The SIR model with Varying Infectivity

###  2.1 The epidemic model

Let $\\{\lambda_{j}(t),\ t\geq 0\\}$, $j\in{\mathbb{Z}}\backslash\\{0\\}$ be a
collection of mutually independent non negative functions, which are such that
the $\\{\lambda_{j}\\}_{j\geq 1}$ are identically distributed, as well as the
$\\{\lambda_{j}\\}_{j\leq-1}$. We assume that these function belongs $a.s.$ to
$D([0,\infty))$. We consider a SIR model which is such that the $j$–th
initially infected individual has the infectivity $\lambda_{-j}(t)$ at time
$t$, while the $j$–th individual infected after time $0$ has at time $t$ the
infectivity $\lambda_{j}(t-\tau_{j})$, if
$0<\tau_{1}<\cdots<\tau_{\ell}<\cdots$ denote the successive times of
infection in the population. The quantity $t-\tau_{j}$ is the age of infection
of individual $j$ at time $t$. Note that we assume that $\lambda_{j}$ vanishes
on ${\mathbb{R}}_{-}$. The newly infected individual is chosen uniformly at
random in the population, and if that individual is susceptible, then it jumps
from the S to the I compartment at its time of infection while nothing happens
if the individual is not susceptibe. Examples of function $\lambda_{j}(t)$
will be given below. That function can be first zero during the exposed
period, then the individual becomes infectious, and at age of infection
$\eta_{j}=sup\\{t,\ \lambda_{j}(t)>0\\}$, the individual recovers (i.e. jumps
into the R compartment) and is immune for ever. Clearly an important quantity
is the total force of infection in the population at time $t$:
$\mathfrak{F}^{N}(t)$, which is the sum of all the infectivities of the
infected individuals at that time. Here $N$ is the total number of individuals
in the population. The sum of the numbers of individuals in the three
compartments is constant in time : $S^{N}(t)+I^{N}(t)+R^{N}(t)=N$ for all
$t\geq 0$. For $X=S,\mathfrak{F},I,\text{ or }R$, we define the renormalized
quantity $\bar{X}^{N}(t)=X^{N}(t)/N$. The main result of [5] is that as
$N\to\infty$,
$(\bar{S}^{N}(t),\bar{\mathfrak{F}}^{N}(t),\bar{I}^{N}(t),\bar{R}^{N}(t))\to(\bar{S}(t),\bar{\mathfrak{F}}(t),\bar{I}(t),\bar{R}(t))$,
where the limit is the unique solution of the following system of integral
equations, which already appears in the seminal paper of Kermack and
McKendrick [9]:

$\displaystyle\begin{cases}\bar{S}(t)&=\bar{S}(0)-\int_{0}^{t}\bar{S}(s)\bar{\mathfrak{F}}(s)ds,\\\
\bar{\mathfrak{F}}(t)&=\bar{I}(0)\bar{\lambda}^{0}(t)+\int_{0}^{t}\bar{\lambda}(t-s)\bar{S}(s)\bar{\mathfrak{F}}(s)ds,\\\
\bar{I}(t)&=\bar{I}(0)F_{0}^{c}(t)+\int_{0}^{t}F^{c}(t-s)\bar{S}(s)\bar{\mathfrak{F}}(s)ds,\\\
\bar{R}(t)&=\bar{R}(0)+\bar{I}(0)F_{0}(t)+\int_{0}^{t}F(t-s)\bar{S}(s)\bar{\mathfrak{F}}(s)ds\,,\end{cases}$
(2.1)

where $\bar{\lambda}^{0}(t)={\mathbb{E}}[\lambda_{-1}(t)]$ and
$\bar{\lambda}(t)={\mathbb{E}}[\lambda_{1}(t)]$, $F_{0}$ (resp. $F$) is the
distribution function of $\eta_{-1}$ (resp. of $\eta_{1}$) and
$F_{0}^{c}(t)=1-F_{0}(t)$, $F^{c}(t)=1-F(t)$. This convergence holds true
provided that $\lambda\in D$ a.s. and for some $\lambda^{\ast}>0$,
$0\leq\lambda_{j}(t)\leq\lambda^{\ast}$ a.s. for all $j\in\mathbb{Z}$ and
$t\geq 0$, see [6]. The original proof in [5] puts more restrictions on
$\lambda$.

###  2.2 The branching process approximation

Suppose that at time $t_{0}$ only a moderate number $M<<N$ of individuals in
the population is infected, and the mean number
$R_{eff}=\bar{S}(t_{0})\int_{0}^{\infty}\bar{\lambda}(t)dt$ of individuals
which an infected individual infects satisfies $R_{eff}<1$. Then the epidemic
is declining. It can be well approximated by the following non Markovian
continuous time branching process. We will study its extinction time in te
next section, and deduce a good approximation of the time we have to wait
after $t_{0}$ for the epidemic to go extinct. Note that we approximate the
proportion $\bar{S}(t)$ by $\bar{S}(t_{0})$, for any $t\geq t_{0}$.

We consider the branching process $Z(t)$ with (to start with) a single
ancestor at time $0$. The $j$–th individual in the population, independently
from all other individuals, lives for a duration $\eta_{j}$, and during its
life time gives birth to children (one at a time) at rate
$\bar{S}(t_{0})\lambda_{j}(t)$, where $(\eta_{j},\lambda_{j})$ are as above.
This is clearly a non Markov continuous time branching process, which belongs
to the class of Crump–Mode–Jagers branching processes.

##  3 The extinction time of the branching process associated to the varying
infectivity model

For now on, $t\geq 0$ stands for $t-t_{0}$ ($t\geq t_{0}$). We define
$\hat{\lambda}(t):=\bar{S}(t_{0})\lambda(t)$. Let $Z(t)$ denote the number of
descendants at time $t$ of an individual born (i.e. infected) at time $0$, in
the continuous time branching process which describes the number of infected
individuals at time $t$. This ancestor infects susceptible individuals during
the time interval $[0,\eta]$, at the random and varying rate
$\hat{\lambda}(t)$. His descendants have the same behaviour, each one
independently from all the others.

In this paper, we make the following assumption on the infectivity function.

Assumption (H) We shall assume that there exists a constant $\lambda^{*}>0$
such that

$\lambda(t)\leq\lambda^{*}\ \mbox{almost surely, for all }t\geq 0.$

Let $T_{ext}=\inf\\{t>0:Z(t)=0\\}$ denote the extinction time of the epidemic,
$G(s,t)=\mathbb{E}\left(s^{Z_{t}}\right)$, $|s|\leq 1$, denote the probability
generating function of $Z(t)$ and $F(t)=G(0,t)$ the distribution function of
the extinction time.

###  3.1 Distribution function of the extinction time

In this subsection, we will characterize the distribution function of the
extinction time of $Z$ as the unique solution of an integral equation. To this
end, we imitate the computations done in the proof of Theorem 4.1 in [4]. We
first start by determining the generating function $G(s,t)$ of $Z$ in order
next to deduce the distribution function of the extinction time.

Denote by $Z_{0}(t)$ the descendance of the ancestor at time $t$, and for
$j\geq 1$, $Z_{j}(t)$ the descendance of the $j$–th direct descendant of the
ancestor at time $t$ after its birth. Then $\\{Z_{j}(.),j\geq 0\\}$ is a
sequence of independent and identically distributed (i.i.d) random processes
which have the law of $Z$. In order to simplify our notations, we will write
$\hat{\lambda}_{0}$ (resp. $\eta_{0}$) for the value of $\hat{\lambda}$ (resp.
$\eta$) associated with $Z_{0}$ . Formula (3.1) from [3] reads

$Z_{0}(t)=\mathds{1}_{\eta_{0}>t}+\sum_{j=1}^{Q_{0}(t)}Z_{j}(t-t^{j}),$ (3.1)

where $Q_{0}(t)$ is the number of direct descendants of the ancestor born on
the time interval $(0,t]$. Moreover, $Q_{0}(t)$ is a counting process, which
conditionnally upon $\hat{\lambda}_{0}(\cdot)$, is a nonhomogeneous Poisson
process with varying intensity $\hat{\lambda}_{0}(t)$, and
$0<t^{1}<t^{2}<\cdots$ are the successive jump times of the process
$Q_{0}(t)$.

We have

###### Proposition 3.1

The probability generating function $G$ satisfies the following integral
equation

$\displaystyle
G(s,t)={\mathbb{E}}\left[s^{\mathds{1}_{\eta>t}}\exp\Bigg{\\{}\int_{0}^{t}\Big{(}G(s,t-u)-1\Big{)}\hat{\lambda}(u)du\Bigg{\\}}\right].$

Proof. Since $Z$ has the same law as $Z_{0}$, we first to compute
$\mathbb{E}\left[s^{Z_{0}(t)}|\hat{\lambda}_{0}\right]$ in order to deduce the
value of $G$. From (3.1), we deduce that

$\displaystyle\mathbb{E}\left[s^{Z_{0}(t)}|\hat{\lambda}_{0}\right]=$
$\displaystyle\sum_{k=0}^{\infty}s^{\mathds{1}_{\eta_{0}>t}}\mathbb{P}(Q_{0}(t)=k|\hat{\lambda}_{0})\mathbb{E}\left\\{\prod_{j=1}^{k}s^{Z_{j}(t-t^{j})}\Big{|}Q_{0}(t)=k,\hat{\lambda}_{0}\right\\}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}s^{\mathds{1}_{\eta_{0}>t}}\mathbb{P}(Q_{0}(t)=k|\hat{\lambda}_{0})\mathbb{E}\left\\{\prod_{j=1}^{k}G(s,t-t^{j})\Big{|}Q_{0}(t)=k,\hat{\lambda}_{0}\right\\}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}s^{\mathds{1}_{\eta_{0}>t}}\mathbb{P}(Q_{0}(t)=k|\hat{\lambda}_{0})\frac{k!}{\Big{(}\int_{0}^{t}\hat{\lambda}_{0}(v)dv\Big{)}^{k}}\times$
$\displaystyle\int_{0}^{t}\int_{0}^{u_{k}}...\int_{0}^{u_{2}}\prod_{j=1}^{k}G(s,t-u_{j})\hat{\lambda}_{0}(u_{1})...\hat{\lambda}_{0}(u_{k})du_{1}...du_{k}$
$\displaystyle=$ $\displaystyle
s^{\mathds{1}_{\eta_{0}>t}}\exp\Bigg{(}-\int_{0}^{t}\hat{\lambda}_{0}(v)dv\Bigg{)}\times$
$\displaystyle\sum_{k=0}^{\infty}\int_{0}^{t}\int_{0}^{u_{k}}...\int_{0}^{u_{2}}\prod_{j=1}^{k}G(s,t-u_{j})\hat{\lambda}_{0}(u_{1})...\hat{\lambda}_{0}(u_{k})du_{1}...du_{k}$
$\displaystyle=$ $\displaystyle
s^{\mathds{1}_{\eta_{0}>t}}\exp\Bigg{(}-\int_{0}^{t}\hat{\lambda}_{0}(v)dv\Bigg{)}\sum_{k=0}^{\infty}\frac{1}{k!}\Bigg{(}\int_{0}^{t}G(s,t-u)\hat{\lambda}_{0}(u)du\Bigg{)}^{k}$
$\displaystyle=$ $\displaystyle
s^{\mathds{1}_{\eta_{0}>t}}\exp\Bigg{\\{}\int_{0}^{t}\Big{(}G(s,t-u)-1\Big{)}\hat{\lambda}_{0}(u)du\Bigg{\\}}.$

The third equality exploits the well known result on the law of the times of
the jumps of a Poisson process on a given interval, given the number of those
jumps (see Exercise 6.5.4 in [10], which treats the case of a constant rate,
the general case follows via an obvious time change), and the fourth equality
the conditional law of $Q_{0}(t)$, given $\hat{\lambda}_{0}$. We thus obtain
that

$\displaystyle
G(s,t)={\mathbb{E}}\left[s^{\mathds{1}_{\eta_{0}>t}}\exp\Bigg{\\{}\int_{0}^{t}\Big{(}G(s,t-u)-1\Big{)}\hat{\lambda}_{0}(u)du\Bigg{\\}}\right].$

Since $(\hat{\lambda}_{0},\eta_{0})$ has the same law as
$(\hat{\lambda},\eta)$, we can drop the subindices 0 in the last formula,
yielding the formula of the statement. $\blacksquare$

The term $s^{\mathds{1}_{\eta>t}}$ can be written as follows:
$s^{\mathds{1}_{\eta>t}}=\mathds{1}_{\eta\leq t}+s\mathds{1}_{\eta>t}$. From
this, we deduce readily the following Corollary for $F(t)=G(0,t)$.

###### Corollary 3.2

The distribution function $F$ of the extinction time of the branching process
with one unique ancestor born at time $0$ satisfies the following integral
equation:

$\displaystyle F(t)={\mathbb{E}}\left[\mathds{1}_{\eta\leq
t}\exp\Bigg{\\{}\int_{0}^{t}\Big{(}F(t-u)-1\Big{)}\hat{\lambda}(u)du\Bigg{\\}}\right].$
(3.2)

The fact that (3.2) characterizes $F$ follows from the following crucial
result.

###### Proposition 3.3

Equation (3.2) has a unique $[0,1]$-valued solution.

Proof. The distribution function of the extinction time solves this equation.
Let us show that this equation has at most one $[0,1]$-valued solution. To
this end, suppose that the equation has two solutions $F^{1}$ and $F^{2}$
which are upper bounded by $1$. We have

$F^{1}(t)-F^{2}(t)={\mathbb{E}}\Bigg{[}\mathds{1}_{\eta\leq
t}\Bigg{(}\exp\Bigg{\\{}\int_{0}^{t}\Big{(}F^{1}(t-u)-1\Big{)}\hat{\lambda}(u)du\Bigg{\\}}-\exp\Bigg{\\{}\int_{0}^{t}\Big{(}F^{2}(t-u)-1\Big{)}\hat{\lambda}(u)du\Bigg{\\}}\Bigg{)}\Bigg{]}.$

From the fact that $|e^{-x}-e^{-y}|\leq|x-y|$, $\forall x,y>0$, we deduce that

$\displaystyle\Big{|}F^{1}(t)-F^{2}(t)\Big{|}$
$\displaystyle\leq{\mathbb{E}}\left[\int_{0}^{t}\hat{\lambda}(u)\Big{|}F^{1}(t-u)-F^{2}(t-u)\Big{|}du\right]$
$\displaystyle\leq\hat{\lambda}^{*}\int_{0}^{t}\Big{|}F^{1}(u)-F^{2}(u)\Big{|}du,$

where we have used assumption (H) and the notation
$\hat{\lambda}^{\ast}=\bar{S}(t_{0})\lambda^{\ast}$. The desired result
follows by combining this with Gronwall’s lemma. $\blacksquare$

###  3.2 Epidemic starting at time $\chi<0$.

Now let us consider the case where the ancestor has been infected at a random
time $\chi<0$. Then the total progeny at time $t$ of this ancestor can be
written as follows:

$Z_{0}(t)=\mathds{1}_{\eta_{0}>t+\chi}+\sum_{j=1}^{Q_{0}(t)}Z_{j}(t-t^{j}).$

From an easy adaptation of the argument used in the proof of Proposition 3.1,
we deduce the

###### Proposition 3.4

The distribution function $F_{\chi}$ of the extinction time of the epidemic
starting with a unique ancestor at time $0$, who was born at time $\chi(<0)$
satisfies :

$\displaystyle F_{\chi}(t)={\mathbb{E}}\left[\mathds{1}_{\eta\leq
t+\chi}\exp\left\\{\int_{0}^{t}\Big{(}F_{\chi}(t-u)-1\Big{)}\hat{\lambda}(u-\chi)du\right\\}\right].$

###  3.3 Epidemic with multiple infected at the initial time.

In the first two subsections, we have considered an epidemic that starts with
a single infected individual. In this subsection, we consider an epidemic that
starts with $M\in{\mathbb{N}}$ infected individuals at the initial time. The
goal is to determine the distribution of the extinction time. To this end, let
$(\hat{\lambda}_{i},\eta_{i})_{1\leq i\leq M}$ be a sequence of pairs of
random variables where $\hat{\lambda}_{i}$ (resp. $\eta_{i}$) denote the
infectivity (resp. the lifetime) of the ancestor $i$. Let $(u_{i})_{1\leq
i\leq M}$ be a sequence of independent and identically distributed (i.i.d)
random variables with law $\mathcal{U}([0,1])$. Note that the sequence
$\left(\hat{\lambda}_{i},\eta_{i},u_{i}\right)_{1\leq i\leq M}$ is i.i.d and
for each $i$, we assume that $(\hat{\lambda}_{i},\eta_{i})$ and $u_{i}$ are
independent. We assume that the individual $i$ was infected at time
$\chi_{i}=-u_{i}\eta_{i}$, which we believe is the most natural model.

From Proposition 3.4, we know that the distribution of the extinction time of
the epidemic starting with the ancestor $i$, is given by

$\displaystyle\tilde{F}(t)={\mathbb{E}}\left[\mathds{1}_{\tilde{\eta}\leq
t}\exp\left\\{\int_{0}^{t}\Big{(}\tilde{F}(t-r)-1\Big{)}\tilde{\lambda}(r)dr\right\\}\right],$
(3.3)

with $\tilde{\eta}_{i}=\eta_{i}(1-u_{i})$ and
$\tilde{\lambda}_{i}(t)=\hat{\lambda}_{i}(t-\chi_{i})$. Since, as the dynamics
of reproduction remains the same for all infected individuals resulting from
each ancestor, hence from the branching property, we deduce the main result of
this section.

###### Theorem 3.5

The distribution function of the time we have to wait in order to see the
extinction of the epidemic, if at time $t_{0}$ we have $M$ infected
individuals, is well approximated by the following:

$\displaystyle H(t)=\left(\tilde{F}(t)\right)^{M}.$

##  4 Several examples of random function $\lambda(t)$

Our varying infectivity model is in fact a SIR/SEIR, in the sense that it
allows an exposed period just after infection, during which $\lambda(t)=0$.
However, we do not introduce the $E$ compartment ($E$ for Exposed, the status
of an infected individual who is, just after being infected, in a latent
period, not yet infectious), the $I$ compartment including all infected
individuals, whether latent or infectious. In all most used models
$\lambda(t)$ is piecewise constant, the jump times being random, following
most classically an exponential distribution so that the stochastic model is
Markovian and its law of large numbers limit is a system of ordinary
differential equations (in contrast with the integral equation (2.1)).

We now review two classical examples of piecewise constant $\lambda(t)$, which
correspond respectively to the SIR and the SEIR model and finally present the
example of varying infectivity $\lambda(t)$ which we shall use in the next
section for our comparison with the more classical SIR ODE model.

###  4.1 The classical SIR model

The simplest commonly used example of the infectivity $\lambda(t)$ is
$\lambda(t)=\lambda\mathds{1}_{t\leq\eta}$, where $\lambda$ is a positive
constant and $\eta$ is the random duration of the infectious period. In that
case equation (3.2) take the form

$F(t)=\int_{0}^{t}\exp\Bigg{\\{}\lambda\int_{0}^{r}\Big{(}F(t-u)-1\Big{)}du\Bigg{\\}}\mathbb{P}_{\eta}(dr).$

In the particular case of a deterministic $\eta$ ($i.e.$
${\mathbb{P}}_{\eta}=\delta_{a}$, with $a\in{\mathbb{R}}_{+}$), we have

$\displaystyle F(t)$ $\displaystyle=\mathds{1}_{t\geq
a}\exp\left\\{\lambda\int_{0}^{a}\Big{(}F(t-u)-1\Big{)}du\right\\}\quad\mbox{with}\quad
F(0)=0\quad\mbox{and}\quad F(a)=\exp(-\lambda a).$

The most commonly used model corresponds to $\eta$ following an exponential
distribution with parameter $\mu$. In this case, the system of integral
equations (2.1) simplifies as follows :

$\left\\{\begin{array}[]{ll}\frac{dS(t)}{dt}=-\lambda\overline{S}(t)I(t),&\\\
\\\ \frac{dI(t)}{dt}=\left(\lambda\overline{S}(t)-\mu\right)I(t),&\\\ \\\
\frac{dR(t)}{dt}=\mu I(t).&\end{array}\right.$

If we linearize the second equation for $t\geq t_{0}$ by replacing
$\overline{S}(t)$ by $\overline{S}(t_{0})$, we obtain

$I(t)=I(t_{0})\exp\left[\left(\lambda\overline{S}(t_{0})-\mu\right)(t-t_{0})\right].$

From this, it is easy see that

$\rho=\lambda\overline{S}(t_{0})-\mu.$ (4.1)

The fact that the above derivation is correct, although the deterministic
model is not valid for $t\geq t_{0}$, is explained in [5]. Note also that
solving equation (4.6) below gives the same result, as the reader can easily
verify.

Let us now compute $R_{eff}$. An infected individual has infectious contacts
at rate $\lambda\overline{S}(t_{0})$. This means that the expected number of
infectious contacts equals

$R_{eff}=\lambda\overline{S}(t_{0})\times\mathbb{E}[\eta]=\frac{\lambda\overline{S}(t_{0})}{\mu}.$
(4.2)

The approximating branching process is the continuous time Markov branching
process $(X(t))_{t\geq 0}$ which describes the number of descendants alive at
time $t$ of a unique ancestor born at time zero. Every individual in this
population, independently of the others, lives for an exponential time with
parameter $\mu$, and during its lifetime it gives births at rate
$\lambda\overline{S}(t_{0})$. His descendants reproduce according to the same
procedure. We consider the subcritical case $\mu>\lambda\overline{S}(t_{0})$.
Let $G(s,t)=\mathbb{E}\left(s^{X(t)}\right)$, $|s|\leq 1$, be the probability
generating function of $X(t)$. On page 109 of Athreya and Ney [1], or in
formula (5) of Iwasa, Nowak, and Michor [8], we find the explicit form :

$G(s,t)=\frac{\mu(s-1)-e^{-\rho
t}(\lambda\overline{S}(t_{0})s-\mu)}{\lambda\overline{S}(t_{0})(s-1)-e^{-\rho
t}(\lambda\overline{S}(t_{0})s-\mu)}.$

where $\rho$ was defined in (4.1). Let us define
$T_{ext}=\inf\\{t>0:X(t)=0\\}$. We notice that
$F(t)=G(0,t)=\mathbb{P}(X_{t}=0)=\mathbb{P}(T_{ext}\leq t)$ is the
distribution function of the extinction time. From the expression for
$G(s,t)$, we deduce the value of $F(t)$.

###### Proposition 4.1

When starting with a single ancestor at time $0$, the distribution function of
the extinction time is given as :

$\displaystyle F(t)=\frac{1-e^{\rho t}}{1-R_{eff}\times e^{\rho t}},$

where $R_{eff}$ was defined in (4.2).

###  4.2 The classical SEIR model

In this model, upon infection an individual is first exposed (compartment
$E$), during a period $\xi$, during which the individual is not infectious,
then he becomes infectious and stays so for a duration $\eta$, during which he
infects susceptibles at rate $\lambda$, and then finally recovers. In that
case $\lambda(t)$ is $\lambda(t)=\lambda{\bf 1}_{\xi\leq t<\xi+\eta}$ and
equation (3.2) takes the form

$\displaystyle F(t)$
$\displaystyle=\int_{0}^{t}\int_{0}^{t-r}\exp\Bigg{\\{}\lambda\int_{s}^{s+r}\Big{(}F(t-u)-1\Big{)}du\Bigg{\\}}{\mathbb{P}}_{(\xi,\eta)}(ds,dr).$

When $\xi$ and $\eta$ are deterministic, that is to say
${\mathbb{P}}_{(\xi,\eta)}(ds,dr)=\delta_{a}(ds)\delta_{b}(dr)$, with
$(a,b)\in{\mathbb{R}}_{+}^{2}$, we have

$\displaystyle F(t)=\mathds{1}_{t\geq
a+b}\exp\Bigg{\\{}\lambda\int_{a}^{a+b}\Big{(}F(t-u)-1\Big{)}du\Bigg{\\}},\quad\mbox{with}\quad
F(u)=0,\quad\mbox{for all}\ u\in[0,a].$

In case $\xi$ and $\eta$ are independent and follow exponential distributions
with parameters resp. $\gamma$ and $\mu$, the deterministic model obeys the
ODE

$\left\\{\begin{array}[]{ll}\frac{d\overline{S}(t)}{dt}=-\lambda\overline{S}(t)\overline{I}(t),&\\\
\\\
\frac{d\overline{E}(t)}{dt}=\lambda\overline{S}(t)\overline{I}(t)-\gamma\overline{E}(t),&\\\
\\\ \frac{d\overline{I}(t)}{dt}=\gamma\overline{E}(t)-\mu\overline{I}(t),&\\\
\\\ \frac{d\overline{R}(t)}{dt}=\mu\overline{I}(t).&\end{array}\right.$

In this model, again $R_{eff}=\frac{\lambda\overline{S}(t_{0})}{\mu}$. Solving
the equation (4.6) below for $\rho$, we find

$\rho=\frac{1}{2}\left[\sqrt{(\gamma-\mu)^{2}+4\gamma\overline{S}(t_{0})\lambda}-(\mu+\gamma)\right].$
(4.3)

###  4.3 Our varying infectivity model

We again define $\hat{\lambda}(t)=\overline{S}(t_{0})\lambda(t)$. The
infectivity $\hat{\lambda}(t)$ is first zero (corresponding to the latency
period) followed by a gradual increase for some days, and then
$\hat{\lambda}(t)$ starts decreasing down towards $0$ which it hits when the
individual has recovered (see Figure 1).

Figure 1: Example of trajectory of $\hat{\lambda}(t)$.

In the computations of section 5 below, we use a piecewise linear
$\hat{\lambda}(t)$, which allows the function to depend upon a small number of
parameters, see Figure 2.

Figure 2: trajectory of $\hat{\lambda}(t)$ used for the comparisons below.

Here $\tau$ is the duration of the exposed period, $\eta$ that of the
infectious period. We have arbitrarily fixed the length of the period of
increase to $1.5$ days, and taken the maximum value $a$ to be a deterministic
quantity at our disposal. In other word, in this case, we have

$\hat{\lambda}(t)=\begin{cases}0&\text{if}~{}t<\tau\\\
\frac{a}{1.5}\left(t-\tau\right)&\text{if}~{}\tau\leq t<\tau+1.5\\\
a\frac{\tau+\eta-t}{\eta-1.5}&\text{if}~{}\tau+1.5\leq t<\tau+\eta\\\
0&\text{if}~{}\tau+\eta<t\end{cases}$ (4.4)

Let $\mathcal{J}$ be the joint law of $\tau$ and $\eta$. From Corollary 3.2,
we deduce that

$\displaystyle F(t)$ $\displaystyle=\mathbb{E}\Bigg{[}\mathds{1}_{\zeta\leq
t}\exp\Bigg{\\{}\frac{a}{1.5}\int_{\tau}^{\tau+1.5}(F(t-u)-1)(u-\tau)du+\frac{a}{\eta-1.5}\int_{\tau+1.5}^{\tau+\eta}(F(t-u)-1)(\tau+\eta-u)du\Bigg{\\}}\Bigg{]},$

with $\zeta=\tau+\eta$. Thus, we obtain

$\displaystyle F(t)=$
$\displaystyle\int_{0}^{t}\int_{0}^{t}\mathds{1}_{s+r\leq
t}\exp\Bigg{\\{}\frac{a}{1.5}\int_{s}^{s+1.5}(F(t-u)-1)(u-s)du$
$\displaystyle+\frac{a}{r-1.5}\int_{s+1.5}^{s+r}(F(t-u)-1)(s+r-u)du\Bigg{\\}}\mathcal{J}(ds,dr).$

The effective reproduction number is defined by

$R_{eff}=\mathbb{E}\left[\int_{0}^{\infty}\hat{\lambda}(t)dt\right]$ (4.5)

and the rate of decrease $\rho$ of the number of infected individuals is the
unique solution of

$\mathbb{E}\left[\displaystyle\int_{0}^{\infty}e^{-\rho
t}\hat{\lambda}(t)dt\right]=1,$ (4.6)

see Theorem 2.3 in [5].

##  5 Comparison between our Varying infectivity model and an ODE SIR model

In this section, we compare the distribution function of the extinction time
in our varying infectivity model, with that of an ODE SIR model with the same
$R_{eff}$, which is the effective reproduction number at time $t_{0}$, and the
same rate of decrease $\rho$ of the number of infected individuals.

In the following, we assume that the random variables $\tau$ and $\eta$
defined in (4.4) are independent, $\tau\sim\mathcal{U}\left(1.5,2.5\right)$
and $\eta\sim\mathcal{U}\left(7,13\right)$.

###  5.1 Approximation of the distribution function of the extinction time in
the varying infectivity model

Since it is not possible to obtain an explicit solution of (3.2), then we will
use the approximation made in section 7. In other words, we will consider the
following approximate function (whose convergence is established in section 7
below):

$\displaystyle F_{n}\left(\frac{k}{n}\right)$
$\displaystyle=\mathbb{E}\left[\mathds{1}_{\tau+\eta\leq\frac{k}{n}}\exp\left\\{\sum_{\ell=1}^{k}\left(F_{n}\left(\frac{k-\ell}{n}\right)-1\right)\int_{\frac{\ell-1}{n}}^{\frac{\ell}{n}}\hat{\lambda}(u)du\right\\}\right].$

Let us define
$\xi_{n,\ell}=\displaystyle\int_{\frac{\ell-1}{n}}^{\frac{\ell}{n}}\hat{\lambda}(u)du$.
It is easy to see that
$\xi_{n,\ell}\approx\frac{\hat{\lambda}\left(\frac{\ell}{n}\right)}{n}$.
Combining this with (4.4), we deduce that

$\displaystyle\xi_{n,\ell}\approx\frac{a}{1.5}\left(\frac{\ell}{n}-\tau\right)\mathds{1}_{\tau\leq\frac{\ell}{n}<\tau+1.5}+a\left(\frac{\tau+\eta-\frac{\ell}{n}}{\eta-1.5}\right)\mathds{1}_{\tau+1.5\leq\frac{\ell}{n}<\tau+\eta}.$

Now, using the fact that the random variables $\tau$ and $\eta$ are
independent, $\tau\sim\mathcal{U}\left(1.5;2.5\right)$ and
$\eta\sim\mathcal{U}\left(7;13\right)$, we deduce that

$\displaystyle F_{n}\left(\frac{k}{n}\right)$
$\displaystyle\approx\frac{1}{6}\int_{1.5}^{2.5}\int_{7}^{13}\mathds{1}_{x+y\leq\frac{k}{n}}\exp\left\\{\sum_{\ell=1}^{k}\left(F_{n}\left(\frac{k-\ell}{n}\right)-1\right)\frac{a}{1.5}\left(\frac{\ell}{n}-x\right)\mathds{1}_{x\leq\frac{\ell}{n}<x+1.5}\right\\}$
$\displaystyle\quad\times\exp\left\\{\sum_{\ell=1}^{k}\left(F_{n}\left(\frac{k-\ell}{n}\right)-1\right)a\frac{x+y-\frac{\ell}{n}}{y-1.5}\mathds{1}_{x+1.5\leq\frac{\ell}{n}<x+y}\right\\}dxdy$
$\displaystyle\approx\frac{1}{6}\frac{1}{n^{2}}\sum_{j=7n}^{13n}\sum_{i=1.5n}^{2.5n}\mathds{1}_{i+j\leq
k}\exp\left\\{\sum_{\ell=i}^{i+1.5n}\left(F_{n}\left(\frac{k-\ell}{n}\right)-1\right)\left(\ell-i\right)\frac{a}{1.5n^{2}}\right\\}$
$\displaystyle\quad\times\exp\left\\{\sum_{\ell=i+1.5n}^{i+j}\left(F_{n}\left(\frac{k-\ell}{n}\right)-1\right)\frac{i+j-\ell}{j-1.5n}\frac{a}{n}\right\\}.$
(5.1)

###  5.2 Computation of $R_{eff}$

Recall (4.5). We first compute the random quantity
$\displaystyle\int_{0}^{\infty}\hat{\lambda}(t)dt$. This is the surface below
the curve $\hat{\lambda}(t)$, $i.e.$ the surface of the union of two
triangles, and
$\displaystyle\int_{0}^{\infty}\hat{\lambda}(t)dt=\frac{a\eta}{2}$.

Therefore, we have

$\displaystyle R_{eff}=\frac{a}{2}\mathbb{E}[\eta]=\frac{a}{2}\times 10=5a.$

###  5.3 Resolution of equation (4.6)

From (4.4), we have

$\mathbb{E}\left[\int_{0}^{\infty}e^{-\rho
t}\hat{\lambda}(t)dt\right]=a\left(A_{\rho}+B_{\rho}\right),\quad\mbox{with}$
$A_{\rho}={\mathbb{E}}\left(\int_{\tau}^{\tau+1.5}e^{-\rho
t}\frac{t-\tau}{1.5}dt\right)\quad\mbox{and}\quad
B_{\rho}={\mathbb{E}}\left(\int_{\tau+1.5}^{\tau+\eta}e^{-\rho
t}\frac{\tau+\eta-t}{\eta-1.5}dt\right).$

Using the fact that $\tau$ and $\eta$ are independent,
$\tau\sim\mathcal{U}\left(1.5;2.5\right)$,
$\eta\sim\mathcal{U}\left(7;13\right)$, it is easy to check that

$A_{\rho}=\frac{1}{\rho}\left(e^{-1.5\rho}-e^{-2.5\rho}\right)\left[\frac{1}{1.5\rho^{2}}-e^{-1.5\rho}\left(\frac{1}{\rho}+\frac{1}{1.5\rho^{2}}\right)\right],\quad\mbox{and}$
$B_{\rho}=\frac{1}{\rho}\left(e^{-1.5\rho}-e^{-2.5\rho}\right)\left\\{e^{-1.5\rho}\left(\frac{1}{\rho}-\frac{1}{6\rho^{2}}\log\left(\frac{11.5}{5.5}\right)\right)+\frac{1}{\rho^{2}}\mathbb{E}\left[\frac{e^{-\rho\eta}}{(\eta-1.5)}\right]\right\\}.$

Note that the mapping $\rho\mapsto\mathbb{E}\int_{0}^{\infty}e^{-\rho
t}\hat{\lambda}(t)dt$ is decreasing. Consequently, it is easy to compute an
approximate solution of equation (4.6).

###  5.4 Comparison of the distributions and the expectations of the
extinction time between our Varying infectivity model and a ODE SIR model

In what follows, we compare the extinction time in our Varying infectivity
model and in the ODE SIR model with the same $R_{eff}$ and $\rho$. Note that
we compare $F$’s and not $H$’s (see the notations in section 3). We compare
the distribution of the extinction time of our Varying infectivity model given
in (5.1) and of the extinction time of the ODE SIR model given in Proposition
4.1.

Figure 3: Comparison of models with the same $R_{eff}=0.66$ and
$\rho=-0.0683$. Figure 4: Comparison of models with the same $R_{eff}=0.8$ and
$\rho=-0.03816$.

We also compare the expectations of the extinction times of our varying
infectivity model and of an ODE SIR model. To this end, recall that, the
extinction time can be rewrite in the form $T_{ext}=\inf\\{t-t_{0}\ :\
I(t-t_{0})=0\\}$. Thus, for the ODE SIR model, we obtain

$\displaystyle\mathbb{E}[T_{ext}]=\int_{0}^{\infty}\mathbb{P}(T_{ext}>t)dt=\int_{0}^{\infty}\left(1-F(t)\right)dt=\frac{(1-R_{eff})}{\rho
R_{eff}}\ln(1-R_{eff}),$

where we have used the formula of Proposition 4.1 for $F(t)$.

For the varying infectivity model, we obtain

$\displaystyle\mathbb{E}[T_{ext}]=\int_{0}^{\infty}\mathbb{P}(T_{ext}>t)dt=\int_{0}^{\infty}\left(1-F_{n}(t)\right)dt\approx\frac{1}{n}\sum_{k=1}^{n\Lambda}\left(1-F_{n}\left(\frac{k}{n}\right)\right).$

where $\Lambda$ is the point where we stop the calculation of the integral of
$1-F_{n}(t)$.

| $R_{eff}=0.66$ | $R_{eff}=0.8$
---|---|---
| $\rho=-0.0683$ | $\rho=-0.03816$
Varying infectivity model | $\mathbb{E}[T_{ext}]\approx 18.7854$ | $\mathbb{E}[T_{ext}]\approx 22.6568$
ODE SIR model | $\mathbb{E}[T_{ext}]=8.1369$ | $\mathbb{E}[T_{ext}]=10.544$

##  6 Conclusion :

Our comparison shows that, in the final phase of the epidemic, the varying
infectivity SIR model (in fact its branching process approximation) tends to
take more time to extinct then the branching process approximation of the ODE
SIR model. This is not too surprising, since the varying infectivity model has
a memory, contrary to the ODE SIR model. This fact is easily seen when there
is a sudden change of the propagation of the epidemic like the lockdown that
several countries have established during recent Covid epidemic. The authors
who use an ODE model change gradually the infection rate, starting with the
lockdown, while in reality the change of the infection rate was very sudden.
This is a way to compensate the lack of memory of ODE models. We believe that
the fact the varying infectivity SIR model takes more time than the ODE SIR
model to forget its past explains that it takes more time to go extinct. The
varying infectivity SIR model is more complex that the more classical ODE SIR
model, and this probably explains why most authors who quote the seminal
$1927$ paper of Kermack and McKendrick [9] refer only to the very particular
case of constant coefficients, studied in section 3.2 of that paper. Of
course, it is very tempting and sometime preferable to use simple models,
which allow to draw more conclusions. However, it is crucial to understand
which biais the simple model introduce, compared to more realistic models. In
this paper, we have identified one of those biaises, namely the shortening of
the final phase of the epidemic. In future work, we intend to do similar
computations with various classes of varying infectivity models, in order to
confirm these first conclusions.

##  7 Appendix : Approximation of the distribution function of the extinction
time

In this subsection, we define a sequence of functions $\\{F_{n},n\geq 1\\}$
which will allow us to approach the solution of equation (3.2). To this end,
for each $k\in\mathbb{Z}_{+}$, we set

$F_{n}\left(\frac{k}{n}\right)=\mathbb{E}\Bigg{[}\mathds{1}_{\eta\leq\frac{k}{n}}\exp\Bigg{\\{}\sum_{\ell=1}^{k}\left(F_{n}\left(\frac{k-\ell}{n}\right)-1\right)\int_{\frac{\ell-1}{n}}^{\frac{\ell}{n}}\lambda(u)du\Bigg{\\}}\Bigg{]}$
(7.1)

and for each $t\in[\frac{k}{n},\frac{k+1}{n})$,

$F_{n}(t)=\mathbb{E}\Bigg{[}\mathds{1}_{\eta\leq
t}\exp\Bigg{\\{}\sum_{\ell=1}^{k-1}\left(F_{n}\left(\frac{k-\ell}{n}\right)-1\right)\int_{\frac{\ell-1}{n}}^{\frac{\ell}{n}}\lambda(u)du-\int_{\frac{k-1}{n}}^{t}\lambda(u)du\Bigg{\\}}\Bigg{]}.$
(7.2)

The goal of this section is to prove that as $n\longrightarrow+\infty$,
$\\{F_{n}(t),t>0\\}\longrightarrow\\{F(t),t>0\\}$ in $D([0,+\infty))$, where
$F$ is the unique solution of (3.2).

We first check that

###### Lemma 7.1

For any $k\in\mathbb{Z}_{+}$, we have

$F_{n}\left(\frac{k}{n}\right)\leq F\left(\frac{k}{n}\right)\leq 1.$

Proof. Let $k\in\mathbb{Z}_{+}$. We first note that $F(t)\leq 1$ (since $F$ is
a distribution function). To prove the next assertion, we will proceed by
recurrence on $k$. It is clear that $F_{n}(0)=0$. Let us now suppose that
$F_{n}\left(\frac{\ell}{n}\right)\leq F\left(\frac{\ell}{n}\right)$, $\forall
1\leq\ell\leq k-1$. Now let us show that $F_{n}\left(\frac{k}{n}\right)\leq
F\left(\frac{k}{n}\right)$. We have

$\displaystyle F\left(\frac{k}{n}\right)$
$\displaystyle=\mathbb{E}\left[\mathds{1}_{\eta\leq\frac{k}{n}}\exp\left\\{\int_{0}^{\frac{k}{n}}\left(F\left(\frac{k}{n}-u\right)-1\right)\lambda(u)du\right\\}\right]$
$\displaystyle\geq\mathbb{E}\left[\mathds{1}_{\eta\leq\frac{k}{n}}\exp\left\\{\sum_{\ell=1}^{k}\int_{\frac{\ell-1}{n}}^{\frac{\ell}{n}}\left(F\left(\frac{k-\ell}{n}\right)-1\right)\lambda(u)du\right\\}\right]$
$\displaystyle\geq\mathbb{E}\left[\mathds{1}_{\eta\leq\frac{k}{n}}\exp\left\\{\sum_{\ell=1}^{k}\left(F_{n}\left(\frac{k-\ell}{n}\right)-1\right)\int_{\frac{\ell-1}{n}}^{\frac{\ell}{n}}\lambda(u)du\right\\}\right]$
$\displaystyle=F_{n}\left(\frac{k}{n}\right),$

where we have used the fact that $F$ is non-decreasing and the recurrence
assumption. $\blacksquare$

The previous extends to all $t$.

###### Lemma 7.2

For any $t\geq 0$, we have

$F_{n}\left(t\right)\leq F\left(t\right)\leq 1.$

Proof. We first note that

$\int_{0}^{t}\left(1-F(t-u)\right)\lambda(u)du\leq\sum_{\ell=1}^{[nt]}\int_{\frac{\ell-1}{n}}^{\frac{\ell}{n}}\left(1-F(t-u)\right)\lambda(u)du+\int_{\frac{[nt]}{n}}^{t}\lambda(u)du.$

From the fact that $F$ is non-decreasing and $\frac{[nt]}{n}\leq t$, we deduce
that

$\displaystyle\int_{0}^{t}\left(1-F(t-u)\right)du$
$\displaystyle\leq\sum_{\ell=1}^{\lceil
nt\rceil}\int_{\frac{\ell-1}{n}}^{\frac{\ell}{n}}\left(1-F\left(\frac{[nt]-\ell}{n}\right)\right)\lambda(u)du+\int_{\frac{[nt]}{n}}^{t}\lambda(u)du$
$\displaystyle\int_{0}^{t}\left(F(t-u)-1\right)du$
$\displaystyle\geq\sum_{\ell=1}^{[nt]}\left(F_{n}\left(\frac{\lceil
nt\rceil-\ell}{n}\right)-1\right)\int_{\frac{\ell-1}{n}}^{\frac{\ell}{n}}\lambda(u)du-\int_{\frac{[nt]}{n}}^{t}\lambda(u)du,$

where we have used Lemma 7.1 for the last inequality. The desired result
follows by combining the last inequality with (7.2). $\blacksquare$

We have

###### Proposition 7.3

Let $T>0$. Then there exists a constant $C$ such that for all $n\geq 1$ and
$0<s<t<T$,

$-\frac{C}{n}-C(t-s)\leq F_{n}(t)-F_{n}(s)\leq
C(t-s)+\phi(t)-\phi(s)+\frac{C}{n},$

where $\phi(t)={\mathbb{P}}(\eta\leq t)$ the distribution function of $\eta$.

For the proof of this proposition, we will need some several technical lemmas.
In order to simplify the notations below we let

$a_{n}(k)=\left[F_{n}\left(\frac{k+1}{n}\right)-F_{n}\left(\frac{k}{n}\right)\right]^{-}\quad\mbox{and}\quad
b_{n}(k)=\left[F_{n}\left(\frac{k+1}{n}\right)-F_{n}\left(\frac{k}{n}\right)\right]^{+}.$
(7.3)

Let us define, $\forall n\geq 1$, $k\in\mathbb{Z}_{+}$,

$\Lambda_{n}(k)=\sum_{\ell=1}^{k}\bigg{(}F_{n}\Big{(}\frac{k-\ell}{n}\Big{)}-1\bigg{)}\int_{\frac{\ell-1}{n}}^{\frac{\ell}{n}}\lambda(u)du\leq
0,$ (7.4)

(see Lemma 7.1) and let us rewrite (7.1) in the form

$F_{n}\Big{(}\frac{k}{n}\Big{)}=\mathbb{E}\left[\mathds{1}_{\eta\leq\frac{k}{n}}\exp(\Lambda_{n}(k))\right].$
(7.5)

We will need the following lemmas.

###### Lemma 7.4

For any $n\geq 1$, $k\in\mathbb{Z}_{+}$, we have

$A_{1}(n,k)\leq
F_{n}\Big{(}\frac{k+1}{n}\Big{)}-F_{n}\Big{(}\frac{k}{n}\Big{)}\leq
A_{2}(n,k),$

with

$A_{1}(n,k)=\exp\Bigg{\\{}-\frac{\lambda^{*}}{n}\left[\sum_{\ell=0}^{k-1}a_{n}(\ell)+1\right]\Bigg{\\}}-1$
(7.6)

and

$A_{2}(n,k)=\exp\Bigg{\\{}\frac{\lambda^{*}}{n}\sum_{\ell=0}^{k-1}b_{n}(\ell)\Bigg{\\}}-1+\mathbb{P}\Big{(}\frac{k}{n}<\eta\leq\frac{k+1}{n}\Big{)}.$
(7.7)

Proof. Recalling (7.4) and (7.5), we first note that

$\displaystyle\Lambda_{n}(k+1)-\Lambda_{n}(k)=\sum_{\ell=1}^{k}\Big{(}F_{n}\Big{(}\frac{k+1-\ell}{n}\Big{)}-F_{n}\Big{(}\frac{k-\ell}{n}\Big{)}\Big{)}\int_{\frac{\ell-1}{n}}^{\frac{\ell}{n}}\lambda(u)du-\int_{\frac{k}{n}}^{\frac{k+1}{n}}\lambda(u)du.$

It follows that

$\displaystyle-\Big{(}\Lambda_{n}(k+1)-\Lambda_{n}(k)\Big{)}^{-}\geq-\sum_{\ell=1}^{k}\Big{(}F_{n}\Big{(}\frac{k+1-\ell}{n}\Big{)}-F_{n}\Big{(}\frac{k-\ell}{n}\Big{)}\Big{)}^{-}\int_{\frac{\ell-1}{n}}^{\frac{\ell}{n}}\lambda(u)du-\int_{\frac{k}{n}}^{\frac{k+1}{n}}\lambda(u)du$

and

$\displaystyle\Big{(}\Lambda_{n}(k+1)-\Lambda_{n}(k)\Big{)}^{+}\leq\sum_{\ell=1}^{k}\Big{(}F_{n}\Big{(}\frac{k+1-\ell}{n}\Big{)}-F_{n}\Big{(}\frac{k-\ell}{n}\Big{)}\Big{)}^{+}\int_{\frac{\ell-1}{n}}^{\frac{\ell}{n}}\lambda(u)du.$

Thus, we have

$\displaystyle
F_{n}\Big{(}\frac{k+1}{n}\Big{)}-F_{n}\Big{(}\frac{k}{n}\Big{)}$
$\displaystyle=\mathbb{E}\left[\mathds{1}_{\eta\leq\frac{k+1}{n}}\exp(\Lambda_{n}(k+1))-\mathds{1}_{\eta\leq\frac{k}{n}}\exp(\Lambda_{n}(k))\right]$
$\displaystyle=\mathbb{E}\left[\left(\mathds{1}_{\eta\leq\frac{k+1}{n}}-\mathds{1}_{\eta\leq\frac{k}{n}}\right)\exp(\Lambda_{n}(k+1))+\mathds{1}_{\eta\leq\frac{k}{n}}\left(\exp(\Lambda_{n}(k+1))-\exp(\Lambda_{n}(k))\right)\right]$
$\displaystyle\leq\mathbb{E}\left[\left(\mathds{1}_{\eta\leq\frac{k+1}{n}}-\mathds{1}_{\eta\leq\frac{k}{n}}\right)+\mathds{1}_{\eta\leq\frac{k}{n}}\exp(\Lambda_{n}(k))\left(\exp(\Lambda_{n}(k+1)-\Lambda_{n}(k))-1\right)\right]$
$\displaystyle\leq{\mathbb{P}}\left(\frac{k}{n}<\eta\leq\frac{k+1}{n}\right)+{\mathbb{E}}\left(\exp\left[{\left(\Lambda_{n}(k+1)-\Lambda_{n}(k)\right)^{+}}\right]-1\right)$
$\displaystyle\leq\mathbb{E}\left[\exp\Bigg{\\{}\sum_{\ell=1}^{k}\Big{(}F_{n}\Big{(}\frac{k+1-\ell}{n}\Big{)}-F_{n}\Big{(}\frac{k-\ell}{n}\Big{)}\Big{)}^{+}\int_{\frac{\ell-1}{n}}^{\frac{\ell}{n}}\lambda(u)du\Bigg{\\}}\right]-1$
$\displaystyle\quad+\mathbb{P}\Big{(}\frac{k}{n}<\eta\leq\frac{k+1}{n}\Big{)}$
$\displaystyle\leq A_{2}(n,k),$

where we have used (7.3) and (7.7) in the last inequality. We also have

$\displaystyle
F_{n}\Big{(}\frac{k+1}{n}\Big{)}-F_{n}\Big{(}\frac{k}{n}\Big{)}$
$\displaystyle=\mathbb{E}\Big{[}\mathds{1}_{\eta\leq\frac{k+1}{n}}\exp(\Lambda_{n}(k+1))-\mathds{1}_{\eta\leq\frac{k}{n}}\exp(\Lambda_{n}(k))\Big{]}$
$\displaystyle\geq\mathbb{E}\Big{[}\mathds{1}_{\eta\leq\frac{k}{n}}\exp(\Lambda_{n}(k))\Big{(}\exp(\Lambda_{n}(k+1)-\Lambda_{n}(k))-1\Big{)}\Big{]}$
$\displaystyle\geq\mathbb{E}\Big{[}\mathds{1}_{\eta\leq\frac{k}{n}}\exp(\Lambda_{n}(k))\Big{(}\exp(-(\Lambda_{n}(k+1)-\Lambda_{n}(k))^{-})-1\Big{)}\Big{]}$
$\displaystyle\geq\mathbb{E}\Big{[}\exp(-(\Lambda_{n}(k+1)-\Lambda_{n}(k))^{-})-1\Big{]}.$

Combining the above arguments with (7.3) and (7.6), we deduce that

$\displaystyle
F_{n}\Big{(}\frac{k+1}{n}\Big{)}-F_{n}\Big{(}\frac{k}{n}\Big{)}\geq
A_{1}(n,k).$

$\blacksquare$

Recall (7.3). We have

###### Lemma 7.5

Let $T>0$. Then there exists a constant $C$ such that for all $n\geq 1$ and
$0\leq\frac{k}{n}<T$,

$\sum_{\ell=0}^{k}a_{n}(\ell)\leq
C\quad\mbox{and}\quad\sum_{\ell=0}^{k}b_{n}(\ell)\leq C.$

Proof. Let us show the first assertion. For this, we first prove that

$a_{n}(k)\leq r\left(1+r\right)^{k-1},\quad\text{with}\quad
r=\frac{\lambda^{*}}{n}.$

According to Lemma 7.4, we have

$\displaystyle a_{n}(k)\leq-A_{1}(n,k)$
$\displaystyle=1-\exp\Bigg{\\{}-r\left[\sum_{\ell=0}^{k-1}a_{n}(\ell)+1\right]\Bigg{\\}}$
$\displaystyle\leq r\left(\sum_{\ell=0}^{k-1}a_{n}(\ell)+1\right).$

However, it easy to see that $a_{n}(0)=0$ and $a_{n}(1)\leq r$. Let us suppose
$a_{n}(\ell)\leq r\left(1+r\right)^{\ell-1}$, $\forall 1\leq\ell\leq k-1$.
Thus, it is easy to see that,

$\displaystyle a_{n}(k)$ $\displaystyle\leq
r\left(1+r+r(1+r)+...+r(r+1)^{k-2}\right)$
$\displaystyle=r\left(1+r\sum_{i=1}^{k-1}(1+r)^{i-1}\right)=r(1+r)^{k-1}.$

Consequently, since $\frac{k}{n}\leq T$,

$\sum_{\ell=0}^{k}a_{n}(\ell)=\sum_{\ell=1}^{k}a_{n}(\ell)\leq\sum_{\ell=1}^{k}r(1+r)^{\ell-1}=(1+r)^{k}-1\leq
e^{rk}\leq e^{\lambda^{*}T}\leq C_{T}.$

We now show the seond assertion. We first have that
$b_{n}(0)=F_{n}\left(\frac{1}{n}\right)$. Then we have

$\displaystyle\sum_{\ell=0}^{k}b_{n}(\ell)$
$\displaystyle=F_{n}\left(\frac{1}{n}\right)+\sum_{\ell=1}^{k}\left(F_{n}\Big{(}\frac{\ell+1}{n}\Big{)}-F_{n}\Big{(}\frac{\ell}{n}\Big{)}\right)+\sum_{\ell=1}^{k}a_{n}(\ell)$
$\displaystyle=F_{n}\Big{(}\frac{k+1}{n}\Big{)}+\sum_{\ell=1}^{k}a_{n}(\ell).$
$\displaystyle\leq 1+\sum_{\ell=1}^{k}a_{n}(\ell),$

where we have used Lemma 7.1 in the last inequality. The desired result
follows by combining this with the first assertion. $\blacksquare$

We shall need the following

###### Lemma 7.6

Let $T>0$. Then there exists a constant $C$ such that for all $n\geq 1$ and
$0\leq\frac{\ell}{n}<\frac{k}{n}<T$,

$-C\left(\frac{k-\ell}{n}\right)\leq
F_{n}\left(\frac{k}{n}\right)-F_{n}\left(\frac{\ell}{n}\right)\leq
C\left(\frac{k-\ell}{n}\right)+\phi\left(\frac{k}{n}\right)-\phi\left(\frac{\ell}{n}\right),$

where $\phi(t)={\mathbb{P}}(\eta\leq t)$ the distribution function of the
random variable $\eta$.

Proof. Recall (7.6) and (7.7). We have

$\displaystyle-A_{1}(n,k)$
$\displaystyle=1-\exp\Bigg{\\{}-\frac{\lambda^{*}}{n}\left[\sum_{\ell=0}^{k-1}a_{n}(\ell)+1\right]\Bigg{\\}}$
$\displaystyle\leq\frac{\lambda^{*}}{n}\left[\sum_{\ell=0}^{k-1}a_{n}(\ell)+1\right]$
$\displaystyle\leq\frac{C}{n},$

where we have used Lemma 7.5. However, we have

$\displaystyle A_{2}(n,k)$
$\displaystyle=\exp\Bigg{\\{}\frac{\lambda^{*}}{n}\sum_{\ell=0}^{k-1}b_{n}(\ell)\Bigg{\\}}-1+\mathbb{P}\Big{(}\frac{k}{n}<\eta\leq\frac{k+1}{n}\Big{)}$
$\displaystyle\leq
C\frac{\lambda^{*}}{n}\exp\left\\{C\frac{\lambda^{*}}{n}\right\\}+\mathbb{P}\Big{(}\frac{k}{n}<\eta\leq\frac{k+1}{n}\Big{)}$
$\displaystyle\leq\frac{C}{n}+\mathbb{P}\Big{(}\frac{k}{n}<\eta\leq\frac{k+1}{n}\Big{)},$

where we have used the fact that $e^{x}-1\leq xe^{x}$, $\forall x\geq 0$ and
Lemma 7.5. Now combining the above arguments with Lemma 7.4, we deduce that

$-\frac{C}{n}\leq
F_{n}\Big{(}\frac{k+1}{n}\Big{)}-F_{n}\Big{(}\frac{k}{n}\Big{)}\leq\frac{C}{n}+\mathbb{P}\Big{(}\frac{k}{n}<\eta\leq\frac{k+1}{n}\Big{)}.$

However, we note that

$F_{n}\left(\frac{k}{n}\right)-F_{n}\left(\frac{\ell}{n}\right)=\sum_{j=\ell}^{k-1}\left(F_{n}\Big{(}\frac{j+1}{n}\Big{)}-F_{n}\Big{(}\frac{j}{n}\Big{)}\right).$

The desired result follows by combining this with the previous inequalities.
$\blacksquare$

Let us define, $\forall n\geq 1,t>0$, with $k=\lceil nt\rceil$,

$\Lambda_{n}(t)=\sum_{\ell=1}^{k-1}\left(F_{n}\left(\frac{k-\ell}{n}\right)-1\right)\int_{\frac{\ell-1}{n}}^{\frac{\ell}{n}}\lambda(u)du-\int_{\frac{k-1}{n}}^{t}\lambda(u)du,$
(7.8)

(see Lemma 7.1) and let us rewrite (7.2) in the form

$F_{n}(t)=\mathbb{E}\Big{[}\mathds{1}_{\eta\leq
t}\exp\left(\Lambda_{n}(t)\right)\Big{]}.$ (7.9)

We shall need the following

###### Lemma 7.7

Let $T>0$. Then there exists a constant $C$ such that for all $n\geq 1$ and
$0<\frac{\ell-1}{n}<s<\frac{\ell}{n}<\frac{k}{n}<t<\frac{k+1}{n}<T$,

$\left(\Lambda_{n}(t)-\Lambda_{n}(k)\right)^{+}=0,\quad\left(\Lambda_{n}(t)-\Lambda_{n}(k)\right)^{-}\leq
C\left(t-\frac{k}{n}\right),$
$\left(\Lambda_{n}(\ell)-\Lambda_{n}(s)\right)^{+}\leq\frac{C}{n}\quad\mbox{and}\quad\left(\Lambda_{n}(\ell)-\Lambda_{n}(s)\right)^{-}\leq\frac{C}{n}+C\left(\frac{\ell}{n}-s\right),$

where $\Lambda_{n}(.)$ was defined in (7.4).

Proof. From (7.4) and (7.8), we have

$-\lambda^{*}\left(t-\frac{k}{n}\right)\leq\Lambda_{n}(t)-\Lambda_{n}(k)=-\int_{\frac{k}{n}}^{t}\lambda(u)du\leq
0.$

Thus, we obtain the first two assertions. In the same way, from (7.4) and
(7.8) we have

$\displaystyle\Lambda_{n}(\ell)-\Lambda_{n}(s)=$
$\displaystyle\sum_{j=1}^{\ell-2}\left(F_{n}\left(\frac{\ell-j}{n}\right)-F_{n}\left(\frac{\ell-1-j}{n}\right)\right)\int_{\frac{j-1}{n}}^{\frac{j}{n}}\lambda(u)du-\int_{\frac{\ell-1}{n}}^{\frac{\ell}{n}}\lambda(u)du$
$\displaystyle+\left(F_{n}\left(\frac{1}{n}\right)-1\right)\int_{\frac{\ell-2}{n}}^{\frac{\ell-1}{n}}\lambda(u)du+\int_{\frac{\ell-2}{n}}^{s}\lambda(u)du$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{\ell-2}\left(F_{n}\left(\frac{\ell-j}{n}\right)-F_{n}\left(\frac{\ell-1-j}{n}\right)\right)\int_{\frac{j-1}{n}}^{\frac{j}{n}}\lambda(u)du+F_{n}\left(\frac{1}{n}\right)\int_{\frac{\ell-2}{n}}^{\frac{\ell-1}{n}}\lambda(u)du$
$\displaystyle-\int_{s}^{\frac{\ell}{n}}\lambda(u)du.$

Combining this with Lemmas 7.1, 7.5 and (7.3), we deduce that

$\left(\Lambda_{n}(\ell)-\Lambda_{n}(\ell-1,s)\right)^{+}\leq\frac{C}{n}\quad\mbox{and}\quad\left(\Lambda_{n}(\ell)-\Lambda_{n}(\ell-1,s)\right)^{-}\leq\frac{C}{n}+C\left(\frac{\ell}{n}-s\right),$

$\blacksquare$

###### Lemma 7.8

Let $T>0$. Then there exists a constant $C$ such that for all $n\geq 1$ and
$0<\frac{\ell-1}{n}<s<\frac{\ell}{n}<\frac{k}{n}<t<\frac{k+1}{n}<T$,

$-\frac{C}{n}-C\left(\frac{\ell}{n}-s\right)\leq
F_{n}\left(\frac{\ell}{n}\right)-F_{n}(s)\leq\frac{C}{n}+\phi\left(\frac{\ell}{n}\right)-\phi\left(s\right)$

and

$-C\left(t-\frac{k}{n}\right)\leq
F_{n}\left(t\right)-F_{n}\left(\frac{k}{n}\right)\leq
C\left(t-\frac{k}{n}\right)+\phi\left(t\right)-\phi\left(\frac{k}{n}\right),$

where $\phi(t)={\mathbb{P}}(\eta\leq t)$ the distribution function of $\eta$.

Proof. Recall (7.5) and (7.9). From an easy adaptation of the argument of the
proof of Lemma 7.4 and from Lemma 7.7, we have that

$\displaystyle{\mathbb{E}}\left(e^{[\Lambda_{n}(\ell)-\Lambda_{n}(\ell-1,s)]^{-}}-1\right)$
$\displaystyle\leq
F_{n}\left(\frac{\ell}{n}\right)-F_{n}(s)\leq{\mathbb{P}}\left(s<\eta\leq\frac{\ell}{n}\right)+{\mathbb{E}}\left(e^{[\Lambda_{n}(\ell)-\Lambda_{n}(\ell-1,s)]^{+}}-1\right)$
$\displaystyle-{\mathbb{E}}\left([\Lambda_{n}(\ell)-\Lambda_{n}(\ell-1,s)]^{-}\right)$
$\displaystyle\leq
F_{n}\left(\frac{\ell}{n}\right)-F_{n}(s)\leq\phi\left(\frac{\ell}{n}\right)-\phi\left(s\right)+C{\mathbb{E}}\left([\Lambda_{n}(\ell)-\Lambda_{n}(\ell-1,s)]^{+}\right)$
$\displaystyle-\frac{C}{n}-C\left(\frac{\ell}{n}-s\right)$ $\displaystyle\leq
F_{n}\left(\frac{\ell}{n}\right)-F_{n}(s)\leq\frac{C}{n}+\phi\left(\frac{\ell}{n}\right)-\phi\left(s\right).$

In the same way, we get the other assertion. $\blacksquare$

We can now turn to the

$\bf{Proof\ of\ Proposition}$ 7.3 : By combining Lemmas 7.6, 7.8 and the fact
that

$F_{n}(t)-F_{n}(s)=F_{n}(t)-F_{n}\left(\frac{k}{n}\right)+F_{n}\left(\frac{k}{n}\right)-F_{n}\Big{(}\frac{\ell}{n}\Big{)}+F_{n}\Big{(}\frac{\ell}{n}\Big{)}-F_{n}(s),$

we deduce that

$\displaystyle-\frac{C}{n}-C(t-s)$ $\displaystyle\leq F_{n}(t)-F_{n}(s)\leq
C\left(t-\frac{\ell}{n}\right)+\phi(t)-\phi(s)+\frac{C}{n}.$

It follows that

$\displaystyle-\frac{C}{n}-C(t-s)$ $\displaystyle\leq F_{n}(t)-F_{n}(s)\leq
C(t-s)+\phi(t)-\phi(s)+\frac{C}{n}.$

The desired result follows $\hfill\blacksquare$

Recall that the goal of this subsection is to prove the convergence of the
sequence $(F_{n})_{n\geq 1}$ towards $F$, the unique solution of equation
(3.2). For $T>0$, we define $w_{T}^{\prime}(x,.)$ the modulus of continuity of
$x$ $\in$ $D([0,+\infty))$ on the interval $[0,T]$ by

${w}_{T}^{\prime}(x,\delta)=\inf\ \max_{0\leq i<m}\sup_{{t_{i}\leq s<t\leq
t_{i+1}}}|x(t)-x(s)|,$

where the infimum is taken over the set of all increasing sequences
$0=t_{0}<t_{1}<\cdot\cdot\cdot<t_{m}=T$ with the property that $\inf_{0\leq
i<m}|t_{i+1}-t_{i}|\geq\delta$. Let $\\{x_{n},n\geq 1\\}$ be a sequence
function in $D([0,+\infty))$. The following result is a version of Theorem
12.3 from [2] :

###### Proposition 7.9

Let $T>0$. A necessary and sufficient condition for the sequence
$\\{x_{n},n\geq 1\\}$ to be relatively compact in $D([0,+\infty))$ is that

$\quad\quad(i)\ \sup_{n\geq 1}\sup_{0\leq t\leq T}|x_{n}(t)|<+\infty$

and

$\quad\quad(ii)\ \lim_{\delta\rightarrow
0}\limsup_{n\rightarrow+\infty}{w}_{T}^{\prime}(x_{n},\delta)=0.$

We now show that the sequence $(F_{n})_{n\geq 1}$ satisfies the assertions of
the above Proposition.

###### Proposition 7.10

The sequence $(F_{n})_{n\geq 1}$ is relatively compact in $D([0,+\infty))$.

Proof. Condition $(i)$ follows from Lemma 7.2. Hence, it suffices to verify
$(ii)$. To this end, let us define $\psi(t)=\phi(t)+Ct$, $\forall t>0$, where
again $\phi$ is the distribution function on $\eta$. It follows from
Proposition 7.3 that

$|F_{n}(t)-F_{n}(s)|\leq\psi(t)-\psi(s)+\frac{C}{n},\quad\forall t>s>0.$
(7.10)

It is easy to deduce from the definition of $w_{T}^{\prime}(.,.)$ and (7.10)
that

${w}_{T}^{\prime}(F_{n},\delta)\leq{w}_{T}^{\prime}(\psi,\delta)+\frac{C}{n}.$

Note that, since $\psi\in D([0,+\infty))$,
${w}_{T}^{\prime}(\psi,\delta)\rightarrow 0$ as $\delta\rightarrow 0$ (see
Sect. 12, p. 123 in [2]). Thus, the desired result follows. $\blacksquare$

We are now ready to state the main result of this section.

###### Proposition 7.11

As $n\longrightarrow+\infty$,
$\\{F_{n}(t),t>0\\}\longrightarrow\\{F(t),t>0\\}$ in $D([0,+\infty))$, where
$F$ is the unique solution of (3.2).

Proof. From Proposition 7.10, we deduce that at least along a subsequence (but
we use the same notation for the subsequence as for the sequence), $F_{n}$
converges towards a limit denoted by $J$ where $J$ is continuous on the right
and admits a limit on the left. In order to show that $F=J$, it suffices to
prove that $J$ is a solution of equation (3.2) and then use Proposition 3.3.
Indeed, let us rewrite (7.2) in the form

$F_{n}(t)=\mathbb{E}\Bigg{[}\mathds{1}_{\eta\leq
t}\exp\Bigg{\\{}\int_{0}^{\frac{\lfloor
nt\rfloor-1}{n}}\left(F_{n}\left(\frac{\lfloor nt\rfloor-\lceil
nu\rceil}{n}\right)-1\right)\lambda(u)du\Bigg{\\}}-\int_{\frac{\lfloor
nt\rfloor-1}{n}}^{t}\lambda(u)du\Bigg{]}.$

Thus, it only remains to show that

$F_{n}(t)\longrightarrow J(t)={\mathbb{E}}\left[\mathds{1}_{\eta\leq
t}\exp\Bigg{\\{}\int_{0}^{t}\Big{(}J(t-u)-1\Big{)}\lambda(u)du\Bigg{\\}}\right],\
\mbox{as}\ n\rightarrow+\infty$

to obtain the desired result. To this end, we note that

$\displaystyle\int_{0}^{t}\left|F_{n}\left(\frac{\lfloor nt\rfloor-\lceil
nu\rceil}{n}\right)-J(t-u)\right|\lambda(u)du\leq$
$\displaystyle\int_{0}^{t}\left|F_{n}\left(\frac{\lfloor nt\rfloor-\lceil
nu\rceil}{n}\right)-F_{n}(t-u)\right|\lambda(u)du+\int_{0}^{t}|F_{n}(t-u)-J(t-u)|\lambda(u)du$
$\displaystyle\leq\lambda^{*}\int_{0}^{t}\psi(t-u)-\psi\left(t-u-\frac{2}{n}\right)du+\frac{C}{n}\lambda^{*}t+\lambda^{*}\int_{0}^{t}|F_{n}(t-u)-J(t-u)|du,$

where we have used (7.10) in the last inequality. Since $\psi$ is left
continuous and locally bounded, the first term tends to $0$ as $n\to\infty$,
thanks to Lebesgue’s dominated convergence theorem. The second term tends
clearly to $0$. From the convergence in $D$ for the Skorohod topology,
$F_{n}(t-u)\to J(t-u)$ $du$ a.e.. Moreover Lemma 7.2 allows us to use
Lebesgue’s dominated convergence theorem again, and the result follows.
$\blacksquare$

#### Acknowledgement

The authors thank Aurélien Velleret, whose computations confirmed the results
shown in section 5.4, , and the Centre National de la Recherche Scientifique,
which has supported the visits of Anicet Mougabe and Ibrahima Dramé in
Marseille within the program DSCA, jointly with the Institut de Mathématiques
de Marseille.

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