# Threshold dynamics of SAIRS epidemic model with Semi-Markov switching

Stefania Ottaviano  Stefania Ottaviano
University of Trento, Dept. of Civil, Environmental and Mechanical
Engineering, Via Mesiano 77, Trento, 38123, Italy<EMAIL_ADDRESS>
###### Abstract.

We study the threshold dynamics of a stochastic SAIRS-type model with
vaccination, where the role of asymptomatic and symptomatic infectious
individuals is explicitly considered in the epidemic dynamics. In the model,
the values of the disease transmission rate may switch between different
levels under the effect of a semi-Markov process. We provide sufficient
conditions ensuring the almost surely epidemic extinction and persistence in
time mean. In the case of disease persistence, we investigate the omega-limit
set of the system and give sufficient conditions for the existence and
uniqueness of an invariant probability measure.

###### Key words and phrases:

Susceptible–Asymptomatic infected–symptomatic Infected–Recovered–Susceptible,
Vaccination, Semi-Markov switching, Stochastic Stability, Invariant
probability measure

##  1\. Introduction

Starting with the research of Kermack and McKendrick [13], in the last century
a huge amount of mathematical epidemic models have been formulated, analysed
and applied to a variety of infectious diseases, specially during the recent
Covid-19 pandemic.

Once an infectious disease developed, the main goal is containing its spread.
Several control strategies may be applied, such as detection and isolation of
infectious individuals, lockdowns or vaccination. However, the detection of
infectious individuals is far from being easy since they may not show
symptoms. The presence of asymptomatic cases allows a wide circulation of a
virus in the population, since they often remain unidentified and presumably
have more contacts that symptomatic cases. The contribution of the so called
“silent spreaders” to the infection transmission dynamics are relevant for
various communicable diseases, such as Covid-19, influenza, cholera and
shigella [12, 27, 19, 23, 1, 22, 20]; hence, asymptomatic cases should be
considered in such mathematical epidemic models.

The containment of the disease with multiple lockdowns or isolation processes
affects the transmission of the disease through the population. Moreover, in
real biological systems, some parameters of the model are usually influenced
by random switching of the external environment regime. For example, disease
transmission rate in some epidemic model is influenced by random
meteorological factors linked to the survival of many bacteria and viruses
[24, 25]. Thus, the transmission rate as the ability of an infectious
individual to transmit infection and also as expression of the contact rate
between individuals can be subject to random fluctuations. Hence, the choice
of fixed deterministic parameters in models is unlikely to be realistic. In
epidemiology, many authors have considered random switching systems (also
called hybrid systems), whose distinctive feature is the coexistence of
continuous dynamics and discrete events (random jumps at points in time). In
particular, many works consider regime switching of external environments
following a homogeneous continuous-time Markov chain [7, 8, 14, 9, 28, 21].
The Markov property facilitates the mathematical analysis, although it can be
a limitation as the sojourn time in each environment is exponentially
distributed, which yields constant transition rates between different regimes.
However, in reality, the transition rates are usually time-varying, hence, in
each environmental state the conditional holding time distribution can be not
exponential. For example, as shown in [24, 25] and reported in [15], the dry
spell (consisting of consecutive days with daily rain amount below some given
threshold) length distribution is better modeled by Pearson type III
distribution, gamma distribution or Weibull distribution.
In this work, in order to include random influences on transmission parameters
and overcome the drawback of the Markov setting, we use a semi-Markov process
for describing environmental random changes. Semi-Markov switching systems are
an emerging topic from both theoretical and practical viewpoints, able of
capturing inherent uncertainty and randomness in the environment in many
applied fields, ranging from epidemiology to DNA analysis, financial
engineering, and wireless communications [30]. Compared to the most common
Markov switching systems, they better characterize a broader range of
phenomena but brings more difficulties to their stability analysis and
control. Recently, a semi-Markov switching model has been used to analyze the
coexistence and competitiveness of species in ecosystems [16]. In
epidemiology, to the best of our knowledge, there are only very few semi-
Markov switching models [15, 29, 4] and no one of these considers the role of
the asymptomatic individuals in the disease dynamics. Thus, in this paper, we
want to fill this gap and improve our understanding of these types of hybrid
systems.

Precisely, we study a SAIRS-type model with vaccination, where the total
population $N$ is partitioned into four compartments, namely $S$, $A$, $I$,
$R$, which represent the fraction of Susceptible, Asymptomatic infected,
symptomatic Infected and Recovered individuals, respectively, such that
$N=S+A+I+R$. The infection can be transmitted to a susceptible through a
contact with either an asymptomatic infectious individual, at rate
$\beta_{A}$, or a symptomatic individual, at rate $\beta_{I}$. Once infected,
all susceptible individuals enter an asymptomatic state, indicating a delay
between infection and symptom onset if they occur. Indeed, we include in the
asymptomatic class both individuals who will never develop the symptoms and
pre-symptomatic who will eventually become symptomatic. From the asymptomatic
compartment, an individual can either progress to the class of symptomatic
infectious $I$, at rate $\alpha$, or recover without ever developing symptoms,
at rate $\delta_{A}$. An infected individuals with symptoms can recover at a
rate $\delta_{I}$. We assume that the recovered individuals do not obtain a
long-life immunity and can return to the susceptible state after an average
time $1/\gamma$. We also assume that a proportion $\nu$ of susceptible
individuals receive a dose of vaccine which grants them a temporary immunity.
We do not add a compartment for the vaccinated individuals, not distinguishing
the vaccine-induced immunity from the natural one acquired after recovery from
the virus. We consider the vital dynamics of the entire population and, for
simplicity, we assume that the rate of births and deaths are the same, equal
to $\mu$; we do not distinguish between natural deaths and disease related
deaths [20].
Moreover, we assume that the environmental regimes (or states) influence the
infectious transmission rates $\beta_{A}$ and $\beta_{I}$, and that may switch
under the action of a semi-Markov process. Accordingly, the values of
$\beta_{A}$ and $\beta_{I}$ switch between different levels depending on the
state in which the process is.
The paper is organized as follows. In Section 2, we provide some basic
concepts of semi-Markov processes and determine the SAIRS model under study.
We show the existence of a unique global positive solution, and find a
positive invariant set for the system. In Section 3, we investigate the
threshold dynamics of the model. Precisely, we first consider the case in
which $\beta_{A}(r)=\beta_{I}(r):=\beta(r)$,
$\delta_{A}(r)=\delta_{I}(r):=\delta(r)$ and find the basic reproduction
number $\mathcal{R}_{0}$ for our stochastic epidemic model driven by the semi-
Markov process. We show that $\mathcal{R}_{0}$ is a threshold value, meaning
that its position with respect to one determines the almost surely disease
extinction or the persistence in time mean. Then, we investigate the case
$\beta_{A}(r)\neq\beta_{I}(r)$ or $\delta_{A}(r)\neq\delta_{I}(r)$. First, we
find two different sufficient conditions for the almost surely extinction,
that are interchangeable, meaning that it is sufficient that one of the two
are verified to ensure the extinction. After, we find a sufficient condition
for the almost surely persistence in time mean of the system. Thus, we have
two not adjacent regions depending on the model parameters, one where the
system goes to extinction almost surely, and the other where it is persistent.
In Section 5, as well as in Section 6, for simplicity, we restrict the
analysis to the case of a semi-Markov process with two states. Under the
disease persistence condition, we investigate the omega-limit set of the
system. The introduction of the backward recurrence time process, that keeps
track of the time elapsed since the latest switch, allows the considered
stochastic system to be a piecewise deterministic Markov process [15]. Thus,
in Section 6, we prove the existence of a unique invariant probability measure
by utilizing an exclusion principle in [26] and the property of positive
Harris recurrence.
Finally, in Section 7, we validate our analytical results via numerical
simulations and show the relevant role of the mean sojourn time in each
environmental regime in the extinction or persistence of the disease.

##  2\. Model description and basic concepts

###  2.1. The Semi-Markov process.

Let $\\{r(t),t\geq 0\\}$ be a semi-Markov process taking values in the state
space $\mathcal{M}=\\{1,\ldots,M\\}$, whose elements denote the states of
external environments influencing the transmission rates value of the model.
Let

$0=\tau_{0}<\tau_{1}<\ldots<\tau_{n}<\ldots$

be the jump times, and

$\sigma_{1}=\tau_{1}-\tau_{0},\;\sigma_{2}=\tau_{2}-\tau_{1},\;\ldots,\;\sigma_{n}=\tau_{n}-\tau_{n-1},\ldots$

be the time intervals between two consecutive jumps. Let $(p_{i,j})_{m\times
m}$ denote the transition probability matrix and $F_{i}(t)$, $t\in[0,\infty)$,
$i=1,\ldots,M$, the conditional holding time distribution of the semi-Markov
process, then

$\mathbb{P}(r(\tau_{n+1})=j,\sigma_{n+1}\leq
t|r(\tau_{n})=i)=p_{i,j}F_{i}(t),$

and the embedded chain $\\{X_{n}:=r(\tau_{n}),n=0,1,\ldots\\}$ of
$\\{r(t),t\geq 0\\}$ is Markov with one-step transition probability
$(p_{i,j})_{m\times m}$. Moreover, let $f_{i}(t)$ represents the density
function of the conditional holding time distribution $F_{i}(t)$, then for any
$t\in[0,\infty)$ and $i,j\in\mathcal{M}$, we define

$\displaystyle q_{i,j}(t)$
$\displaystyle:=p_{i,j}\frac{f_{i}(t)}{1-F_{i}(t)}\geq 0\qquad\forall i\neq
j,$ $\displaystyle q_{i,i}(t)$ $\displaystyle:=-\sum_{j\in\mathcal{M},j\neq
i}q_{i,j}(t)\qquad\forall i\in\mathcal{M}.$

We give the following same assumptions as in [15], which will be valid
throughout the paper.

Assumptions (H1):

  * (i)

The transition matrix $(p_{i,j})_{m\times m}$ is irreducible with $p_{i,i}=0$,
$i\in\mathcal{M}$;

  * (ii)

For each $i\in\mathcal{M}$, $F_{i}(\cdot)$ has a continuous and bounded
density $f_{i}(\cdot)$, and $f_{i}(t)>0$ for all $t\in(0,\infty)$;

  * (iii)

For each $i\in\mathcal{M}$, there exists a constant $\varepsilon_{i}>0$ such
that

$\frac{f_{i}(t)}{1-F_{i}(t)}\geq\varepsilon_{i}$

for all $t\in[0,\infty)$.

In [15], the authors provide a list of some probability distributions
satisfying the assumption (H1), and show that the constraint conditions of
(H1) are very weak. Specifically, they provide the phase-type distribution
(PH-distribution) of a nonnegative random variable, and prove that this PH-
distribution, or at most an approximation of it, satisfies the conditions in
(H1). Thus, they conclude that essentially the conditional holding time
distribution of the semi-Markov process can be any distribution on
$[0,\infty)$.

######  Remark 1.

Let us note that in the case of exponential (memoryless) sojourn time
distribution, the semi-Markov process $\\{r(t),t\geq 0\\}$ degenerates into a
continuous time Markov chain. That is, if $F_{i}(t)=1-e^{-q_{i}t}$ for some
$q_{i}>0$, $i\in\mathcal{M}$, then

$\frac{f_{i}(t)}{1-F_{i}(t)}\geq q_{i}$

for all $t\in[0,\infty)$, from which

$q_{i,j}:=q_{i,j}(t)=\begin{cases}q_{i}p_{i,j}&\text{if }i\neq j\\\
-q_{i}&\text{if }i=j\end{cases}$

where $q_{i,j}$ is the transition rates from state $i$ to state $j$, and
$q_{i,j}\geq 0$ if $i\neq j$, while $-q_{i}=q_{i,i}=-\sum_{i\neq j}q_{i,j}$.
Thus, the matrix $Q=(q_{i,j})_{m\times m}$ generates the Markov chain
$\\{r(t),t\geq 0\\}$, i.e.,

$\mathbb{P}\\{r(t+\Delta t)=j|r(t)=i\\}=\begin{cases}q_{i,j}\Delta t+o(\Delta
t),\qquad\text{if}\quad i\neq j,\\\ 1+q_{i,j}\Delta t+o(\Delta
t),\qquad\text{if}\quad i=j,\end{cases}$

where $\Delta t>0$ represents a small time increment.

By the assumptions (H1) follows that the matrix $Q$ is irreducible. Under this
condition, the Markov chain has a unique stationary positive probability
distribution $\pi=(\pi_{1},\ldots,\pi_{m})^{T}$ which can be determined by
solving the following linear equation $\pi^{T}Q=0$, subject to
$\sum_{r=1}^{M}\pi_{r}=1$, and $\pi_{r}>0$, $\forall r\in\mathcal{M}$.

Let us introduce the process

$\eta(t)=t-\sup\\{u<t:r(u)\neq r(t)\\},$

which represents the amount of time the process $\\{r(t),t\geq 0\\}$ is at the
current state after the last jump. It is also denoted as the backward
recurrence time process. The pair $\\{(\eta(t),r(t)),t\geq 0\\}$ satisfies the
Markov property [17], moreover it is strong Markov [10, Chapter 6].

###  2.2. Model description.

Let us consider a SAIRS model with vaccination, as in [20].

$\begin{split}\frac{dS(t)}{dt}&=\mu-\bigg{(}\beta_{A}A(t)+\beta_{I}I(t)\bigg{)}S(t)-(\mu+\nu)S(t)+\gamma
R(t),\\\
\frac{dA(t)}{dt}&=\bigg{(}\beta_{A}A(t)+\beta_{I}I(t)\bigg{)}S(t)-(\alpha+\delta_{A}+\mu)A(t),\\\
\frac{dI(t)}{dt}&=\alpha A(t)-(\delta_{I}+\mu)I(t),\\\
\frac{dR(t)}{dt}&=\delta_{A}A(t)+\delta_{I}I(t)+\nu
S(t)-(\gamma+\mu)R(t).\end{split}$ (1)

Let us now incorporate the impact of the external random environments into
system (1). We assume that the external random environment is described by a
semi-Markov process. We only consider the environmental influence on the
disease transmission rate since it may be more sensitive to environmental
fluctuations than other parameters of model (1). Thus, the average value of
the transmission rate may switch between different levels with the switching
of the environmental regimes. As a result, the SAIRS deterministic model (1)
evolves in a random dynamical system with semi-Markov switching of the form

$\begin{split}\frac{dS(t)}{dt}&=\mu-\bigg{(}\beta_{A}(r(t))A(t)+\beta_{I}(r(t))I(t)\bigg{)}S(t)-(\mu+\nu)S(t)+\gamma
R(t),\\\
\frac{dA(t)}{dt}&=\bigg{(}\beta_{A}(r(t))A(t)+\beta_{I}(r(t))I(t)\bigg{)}S(t)-(\alpha+\delta_{A}+\mu)A(t),\\\
\frac{dI(t)}{dt}&=\alpha A(t)-(\delta_{I}+\mu)I(t),\\\
\frac{dR(t)}{dt}&=\delta_{A}A(t)+\delta_{I}I(t)+\nu
S(t)-(\gamma+\mu)R(t),\end{split}$ (2)

Let us introduce $\bar{\beta}:=(\beta_{A},\beta_{I})$. If the initial
conditions of the driving process $\\{(\eta(t),r(t)),t\geq 0\\}$ are
$\eta(0)=0$ and $r(0)=r_{0}$, then system (2) starts from the initial
condition $(S(0),A(0),I(0),R(0))$ and follows (1) with
$\bar{\beta}=\bar{\beta}(r_{0})$ until the first jump time $\tau_{1}$, with
conditional holding distribution $F_{r_{0}}(\cdot)$. Then, the environmental
regime switches instantaneously from state $r_{0}$ to state $r_{1}$; thus, the
process restarts from the state $r_{1}$ and the system evolves accordingly to
(1) with $\bar{\beta}=\bar{\beta}(r_{1})$ and distribution $F_{r_{1}}(\cdot)$
until the next jump time $\tau_{2}$. The system will evolve in the similar way
as long as the semi-Markov process jumps. This yields a continuous and
piecewise smooth trajectory in $\mathbb{R}^{4}$. Let us note that the solution
process $\\{x(t)=(S(t),A(t),I(t),R(t)),t\geq 0\\}$ that records the position
of the switching trajectory of (2) is not Markov. However, by means of
additional components, $\\{(x(t),\eta(t),r(t)),t\geq 0\\}$ is a homogeneous
Markov process.

In this paper, unless otherwise specified, let
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq 0},\mathbb{P})$ be a
complete probability space with a filtration $\\{\mathcal{F}_{t}\\}_{t\geq 0}$
satisfying the usual conditions (i.e. it is right continuous and
$\mathcal{F}_{0}$ contains all $\mathbb{P}$-null sets).

Since $S+A+I+R=1,$ system (2) is equivalent to the following three-dimensional
dynamical system:

$\begin{split}\frac{dS(t)}{dt}&=\mu-\bigg{(}\beta_{A}(r(t))A(t)+\beta_{I}(r(t))I(t)\bigg{)}S(t)-(\mu+\nu+\gamma)S(t)+\gamma(1-A(t)-I(t)),\\\
\frac{dA(t)}{dt}&=\bigg{(}\beta_{A}(r(t))A(t)+\beta_{I}(r(t))I(t)\bigg{)}S(t)-(\alpha+\delta_{A}+\mu)A(t),\\\
\frac{dI(t)}{dt}&=\alpha A(t)-(\delta_{I}+\mu)I(t),\end{split}$ (3)

with initial condition $(S(0),A(0),I(0))$ belonging to the set

$\Gamma=\\{(S,A,I)\in\mathbb{R}_{+}^{3}|S+A+I\leq 1\\},$

where $\mathbb{R}_{+}^{3}$ is the non-negative orthant of $\mathbb{R}^{3}$,
and initial state $r(0)\in\mathcal{M}$.

System (3) can be written in matrix notation as

$\frac{dx(t)}{dt}=g(x(t),r(t)),$ (4)

where $x(t)=(S(t),A(t),I(t))$ and
$g(x(t))=(g_{1}(x(t)),g_{2}(x(t)),g_{3}(x(t)))$ is defined according to (3).
In the following, for any initial value $x(0)$, we denote by
$x(t,\omega,x(0))=(S(t,\omega,x(0)),A(t,\omega,x(0)),I(t,\omega,x(0)))$, the
solution of (4) at time $t$ starting in $x(0)$, or by $x(t)$ if there is no
ambiguity, for the sake of simplicity, and by $x_{r}(t)$ the solution of the
subsystem $r$.

In the following theorem we ensure the no explosion of the solution in any
finite time, by proving a somehow stronger property, that is $\Gamma$ is a
positive invariant domain for (3).

######  Theorem 1.

For any initial value
$(x(0),\eta(0),r(0))\in\Gamma\times\mathbb{R}^{+}\times\mathcal{M}$, and for
any choice of system parameters $\beta_{A}(\cdot)$, $\beta_{I}(\cdot)$, there
exists a unique solution $x(t,\omega,x(0))$ to system (3) on $t\geq 0$.
Moreover, for every $\omega\in\Omega$ the solution remains in $\Gamma$ for all
$t>0$.

###### Proof.

Let $0=\tau_{0}<\tau_{1}<\tau_{2}<\ldots,<\tau_{n}<\ldots$ be the jump times
of the semi-Markov chain $r(t)$, and let $r(0)=r_{0}\in\mathcal{M}$ be the
starting state. Thus, $r(t)=r_{0}$ on $[\tau_{0},\tau_{1})$. The subsystem for
$t\in[\tau_{0},\tau_{1})$ has the following form:

$\frac{dx(t)}{dt}=g(x(t),r_{0}),$

and, for [20, Thm 1], its solution $x(t)\in\Gamma$, for
$t\in[\tau_{0},\tau_{1})$ and, by continuity for $t=\tau_{1}$, as well. Thus,
$x(\tau_{1})\in\Gamma$ and by considering $r(\tau_{1})=r_{1}$, the subsystem
for $t\in[\tau_{1},\tau_{2})$ becomes

$\frac{dx(t)}{dt}=g(x(t),r_{1}).$

Again, $x(t)\in\Gamma$, on $t\in[\tau_{1},\tau_{2})$ and, by continuity for
$t=\tau_{2}$, as well. Repeating this process continuously, we obtain the
claim.

As the switching concerns only the infection rates $\beta_{A}$ and
$\beta_{I}$, all the subsystems of (3) share the same disease-free equilibrium
(DFE)

$x_{0}=\left(S_{0},A_{0},I_{0}\right)=\left(\frac{\mu+\gamma}{\mu+\nu+\gamma},0,0\right).$

Now, we report results related to the stability analysis of each deterministic
subsystems of (3) corresponding to the state $r$, $r=1,\ldots,M$. The proof of
the following results can be found in [20], where the global stability of the
deterministic model (1) is investigated.

######  Lemma 2.

The basic reproduction number $\mathcal{R}_{0}$ of the subsystem of (3)
corresponding to the state $r$ is given by

$\mathcal{R}_{0}(r)=\left(\beta_{A}(r)+\dfrac{\alpha\beta_{I}(r)}{\delta_{I}+\mu}\right)\dfrac{\gamma+\mu}{(\alpha+\delta_{A}+\mu)(\nu+\gamma+\mu)},\qquad
r=1,\ldots,M.$ (5)

Now, let us define

$(F-V)(r)=\left(\begin{matrix}\beta_{A}(r)S_{0}-(\alpha+\delta_{A}+\mu)&\beta_{I}(r)S_{0}\\\
\alpha&-(\delta_{I}+\mu)\end{matrix}\right),$ (6)

where $F$ and $V$ are the matrices in equations $(6)$ and $(7)$ defined in
[20].

######  Lemma 3.

Let us fix $r\in\mathcal{M}$. The matrix $(F-V)(r)$ related to the subsystem
$r$ of (3) has a real spectrum. Moreover, if $\rho(FV^{-1}(r))<1$, all the
eigenvalues of $(F-V)(r)$ are negative.

######  Theorem 4.

Let us fix $r\in\mathcal{M}$. The disease-free equilibrium $x_{0}$ is globally
asymptotically stable for the subsystem $r$ of (3) if $\mathcal{R}_{0}(r)<1$.

######  Lemma 5.

Let us fix $r\in\mathcal{M}$. The endemic equilibrium
$x^{*}_{r}=(S^{*}(r),A^{*}(r),I^{*}(r))$ exists and it is unique in
$\mathring{\Gamma}$ for the subsystem $r$ of (3) if $\mathcal{R}_{0}(r)>1$.
Moreover, $x^{*}_{r}$ is locally asymptotically stable in $\mathring{\Gamma}$.

######  Theorem 6.

Let us fix $r\in\mathcal{M}$ and assume that
$\beta_{A}(r)=\beta_{I}(r)=:\beta(r)$ and $\delta_{A}=\delta_{I}=:\delta$. The
endemic equilibrium $x^{*}_{r}=(S^{*}(r),A^{*}(r),I^{*}(r))$ is globally
asymptotically stable in $\mathring{\Gamma}$ for the subsystem $r$ of (3) if
$\mathcal{R}_{0}(r)>1$.

######  Theorem 7.

Let us fix $r\in\mathcal{M}$, and consider $\beta_{A}(r)\neq\beta_{I}(r)$ or
$\delta_{A}\neq\delta_{I}$. Assume that $\mathcal{R}(r)_{0}>1$ and
$\beta_{A}(r)<\delta_{I}$. Then, the endemic equilibrium $x^{*}$ is globally
asymptotically stable in $\mathring{\Gamma}$ for the subsystem $r$ of(2).

##  3\. Threshold dynamics of the model

By the assumptions (H1) the embedded Markov chain
$\\{X_{n},n\in\mathbb{N}\\}$, associated to the semi-Markov process
$\\{r(t),t\geq 0\\}$ has a unique stationary positive probability distribution
$\pi=(\pi_{1},\ldots,\pi_{M})$. Let

$m_{i}=\int_{0}^{\infty}[1-F_{i}(u)]du$

be the mean sojourn time of $\\{r(t),t\geq 0\\}$ in state $i$. Then, by the
Ergodic theorem [6, Thm 2, p. 244], we have that for any bounded measurable
function $f:(E,\mathcal{E})\to(\mathbb{R}_{+},\mathcal{B}(\mathbb{R}^{+})),$

$\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}f(r(s))ds=\frac{\sum_{r\in\mathcal{M}}f(r)\pi_{r}m_{r}}{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}}\qquad\text{a.s.}$
(7)

Hereafter, we denote

$\widecheck{\beta}:=\max_{r\in M}\\{\beta_{A}(r),\beta_{I}(r)\\}.$

###  3.1. $\beta_{A}(r)=\beta_{I}(r):=\beta(r)$, $r=1,\ldots,M$,
$\delta_{A}=\delta_{I}:=\delta$.

######  Theorem 8.

Let us assume $\beta_{A}(r)=\beta_{I}(r):=\beta(r)$,
$\delta_{A}=\delta_{I}:=\delta$ in each subsystem $r=1,\ldots,M$. If

$\sum_{r\in\mathcal{M}}\pi_{r}m_{r}\left(\beta(r)\frac{\gamma+\mu}{\nu+\gamma+\mu}-(\delta+\mu)\right)<0,$

then the solution of system (3) with any initial value
$(x(0),\eta(0),r(0))\in\Gamma\times\mathbb{R}^{+}\times\mathcal{M}$ satisfies

$\displaystyle\lim_{t\to+\infty}S(t)$
$\displaystyle=\frac{\gamma+\mu}{\nu+\gamma+\mu}=:S_{0}\qquad\text{a.s.},$ (8)
$\displaystyle\lim_{t\to+\infty}A(t)$ $\displaystyle=0\qquad\text{a.s.},$ (9)
$\displaystyle\lim_{t\to+\infty}I(t)$ $\displaystyle=0\qquad\text{a.s}.$ (10)

###### Proof.

We know that for all $\omega\in\Omega$, it holds

$\frac{dS(\omega,t)}{dt}\leq\mu+\gamma-(\mu+\nu+\gamma)S(\omega,t).$

For different selections of sample point $\omega\in\Omega$, the sample path
$S(\omega,t)$ may have different convergence speeds with respect to time $t$.
Thus, for any $\omega\in\Omega$ and any constant $\varepsilon>0$, by the
comparison theorem, there exists $T(\omega,\varepsilon)>0$, such that for all
$t>T$

$S(\omega,t)\leq S_{0}+\varepsilon,$

hence

$\limsup_{t\to\infty}S(t)\leq S_{0},\qquad\text{a.s.}$ (11)

Based on this consideration, we shall prove assertions (9) and (10). We have
that for all $\omega\in\Omega,$ and $t>T$

$\displaystyle\frac{d\ln(I(t)+A(t))}{dt}$
$\displaystyle=\beta(r(t))S(t)-(\delta+\mu)$
$\displaystyle\leq\beta(r(t))(S_{0}+\varepsilon)-(\delta+\mu).$

This implies that

$\ln(I(t)+A(t))\leq\ln(I(T)+A(T))+\int_{T}^{t}(\beta(r(u))(S_{0}+\varepsilon)-(\delta+\mu))\;du$

from which, by the ergodic result for semi-Markov process (7), we get

$\displaystyle\limsup_{t\to\infty}\frac{\ln(I(t)+A(t))}{t}$
$\displaystyle\leq\limsup_{t\to\infty}\frac{1}{t}\int_{T}^{t}(\beta(r(u))(S_{0}+\varepsilon)-(\delta+\mu))\;du$
$\displaystyle=\frac{1}{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}}\bigg{[}\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\beta(r)(S_{0}+\varepsilon)-(\delta+\mu))\bigg{]}\qquad\text{a.s.}$

If
$\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\beta(r)(S_{0}+\varepsilon)-(\delta+\mu))<0$,
then for sufficiently small $\varepsilon>0$, we have
$\sum_{r\in\mathcal{M}}\pi_{r}(\beta(r)(S_{0}+\varepsilon)-(\delta+\mu))<0$,
and consequently

$\lim_{t\to+\infty}A(t)=0,\qquad\text{and}\qquad\lim_{t\to+\infty}I(t)=0\qquad\text{a.s}.$
(12)

Now, we shall prove assertion (8). Let
$\bar{\Omega}=\\{\omega\in\Omega:\lim_{t\to+\infty}A(t)=0\\}\cap\\{\omega\in\Omega:\lim_{t\to+\infty}I(t)=0\\}$.
Then, from (12), $\mathbb{P}(\bar{\Omega})=1$. Then, for any
$\omega\in\bar{\Omega}$ and any constant $\varepsilon>0$, there exists
$T_{1}(\omega,\varepsilon)>0$, such that for all $t>T_{1}$

$A(\omega,t)<\varepsilon,\qquad I(\omega,t)<\varepsilon.$

Thus, we have for all $\omega\in\bar{\Omega}$, and $t>T_{1}$

$\displaystyle\frac{dS(\omega,t)}{dt}$
$\displaystyle\geq\mu-\varepsilon\beta({\color[rgb]{0,0,0}r})S(\omega,t)-(\mu+\nu+\gamma)S(\omega,t)+\gamma(1-2\varepsilon)$
$\displaystyle\geq\mu-\varepsilon\widecheck{\beta}S(\omega,t)-(\mu+\nu+\gamma)S(\omega,t)+\gamma(1-2\varepsilon).$

Following the same arguments in the proof of Theorem 4, we can assert that

$\liminf_{t\to\infty}S(\omega,t)\geq S_{0},\qquad\omega\in\bar{\Omega}.$

Recalling that $P(\bar{\Omega})=1$, we have

$\liminf_{t\to\infty}S(t)\geq S_{0}\qquad\text{a.s.}$

that combined with (11) gives us that

$\lim_{t\to\infty}S(t)=S_{0}\qquad\text{a.s.}$

Thus, under the condition of Theorem 8, we can say that any positive solution
of system (3) converges exponentially to the disease-free state
$x_{0}=(S_{0},0,0)$ almost surely.

Based on the definition in [2, 15], the _basic reproduction number_ , for our
model (3) with $\beta_{A}(r)=\beta_{I}(r):=\beta(r)$,
$\delta_{A}=\delta_{I}:=\delta$, in the semi-Markov random environment, can be
written from Theorem 8 as

$\mathcal{R}_{0}=\frac{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}\beta(r)S_{0}}{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\delta+\mu)}.$
(13)

We notice that we would have arrived to the same result if we had followed the
same arguments as in [15], that are based on the theory of basic reproduction
in random environment in [2].

######  Remark 2.

It easy to see that in the case of Markov-switching, that is for the
exponential holding time distribution in each regime, the basic reproduction
number for our model (3) with $\beta_{A}(r)=\beta_{I}(r):=\beta$,
$\delta_{A}=\delta_{I}:=\delta$ is

$\mathcal{R}_{0}=\frac{\sum_{r\in\mathcal{M}}\pi_{r}\beta(r)S_{0}}{\sum_{r\in\mathcal{M}}\pi_{r}(\delta+\mu)}=\frac{\sum_{r\in\mathcal{M}}\pi_{r}\beta(r)S_{0}}{(\delta+\mu)}.$

######  Proposition 9.

From (13), the following alternative conditions are valid

  * (i)

$\mathcal{R}_{0}<1$ if and only if
$\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\beta(r)S_{0}-(\delta+\mu))<0$,

  * (ii)

$\mathcal{R}_{0}>1$ if and only if
$\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\beta(r)S_{0}-(\delta+\mu))>0$.

The proof is immediate, so it is omitted.

###  3.2. $\beta_{A}(r)\neq\beta_{I}(r)$, or $r=1,\ldots,M$,
$\delta_{A}\neq\delta_{I}$

Let us define

$\widecheck{\beta}(r)=\max\\{\beta_{A}(r),\beta_{I}(r)\\},\qquad\hat{\delta}=\min\\{\delta_{A},\delta_{I}\\}.$

######  Theorem 10.

Let $\beta_{A}(r)\neq\beta_{I}(r)$, or $\delta_{A}\neq\delta_{I}$ in each
subsystem $r=1,\ldots,M$. If

$\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\widecheck{\beta}(r)S_{0}-(\hat{\delta}+\mu))<0,$
(14)

then the solution of system (3) with any initial value
$(x(0),\eta(0),r(0))\in\Gamma\times\mathbb{R}^{+}\times\mathcal{M}$ satisfies

$\displaystyle\lim_{t\to+\infty}S(t)$
$\displaystyle=\frac{\gamma+\mu}{\nu+\gamma+\mu}=:S_{0}\qquad\text{a.s.},$
(15) $\displaystyle\lim_{t\to+\infty}A(t)$ $\displaystyle=0\qquad\text{a.s.},$
(16) $\displaystyle\lim_{t\to+\infty}I(t)$ $\displaystyle=0\qquad\text{a.s}.$
(17)

###### Proof.

Let us prove conditions (16) and (17). By using equation (11), we have that
for all $\omega\in\Omega,$ and $t>T$

$\displaystyle\frac{d\ln(I(t)+A(t))}{dt}$
$\displaystyle\leq\widecheck{\beta}(r(t))(S_{0}+\varepsilon)-(\hat{\delta}-\mu).$

By the same arguments as in Theorem 8, we obtain that

$\displaystyle\limsup_{t\to\infty}\frac{\ln(I(t)+A(t))}{t}$
$\displaystyle\leq\frac{1}{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}}\bigg{[}\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\widecheck{\beta}(r)(S_{0}+\varepsilon)-(\hat{\delta}+\mu))\bigg{]}\qquad\text{a.s.}$

Thus, if
$\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\widecheck{\beta}(r)S_{0}-(\delta+\mu))<0$,
then for sufficiently small $\varepsilon>0$, we have
$\sum_{r\in\mathcal{M}}\pi_{r}(\widecheck{\beta}(r)(S_{0}+\varepsilon)-(\delta+\mu))<0$,
and consequently

$\lim_{t\to+\infty}A(t)=0,\qquad\text{and}\qquad\lim_{t\to+\infty}I(t)=0\qquad\text{a.s}.$

To prove assertion (15), we follow the same steps as in Theorem (8), by
considering that $\beta_{A}(r)$ and $\beta_{I}(r)$ are less than or equal to
$\widecheck{\beta}$.

With a different proof we can find another sufficient condition for the
extinction of the disease.

######  Theorem 11.

Let $\beta_{A}(r)\neq\beta_{I}(r)$ or $\delta_{A}\neq\delta_{I}$ in each
subsystem $r=1,\ldots,M$, and let $B(r)=(F-V)(r)$ as in (6). If

$\sum_{r\in\mathcal{M}}\pi_{r}m_{r}\lambda_{1}(B(r)+B(r)^{T})<0,$ (18)

where $\lambda_{1}$ is the maximum eigenvalue, then the solution of system (3)
with any initial value
$(x(0),\eta(0),r(0))\in\Gamma\times\mathbb{R}^{+}\times\mathcal{M}$ satisfies

$\displaystyle\lim_{t\to+\infty}S(t)$
$\displaystyle=\frac{\gamma+\mu}{\nu+\gamma+\mu}\qquad\text{a.s.},$ (19)
$\displaystyle\lim_{t\to+\infty}A(t)$ $\displaystyle=0\qquad\text{a.s.},$ (20)
$\displaystyle\lim_{t\to+\infty}I(t)$ $\displaystyle=0\qquad\text{a.s.}$ (21)

###### Proof.

By following the same arguments as in the proof of Theorem 8, we know that
(11) holds. Thus, we have that for any $\omega\in\Omega$ and any constant
$\varepsilon>0$, there exists $T(\omega,\varepsilon)>0$, such that for all
$t>T$

$\displaystyle\frac{dA(t)}{dt}$
$\displaystyle\leq\bigg{(}\beta_{A}(r(t))A(t)+\beta_{I}(r(t))I(t)\bigg{)}(S_{0}+\varepsilon)-(\alpha+\delta_{A}+\mu)A(t),$
$\displaystyle\frac{dI(t)}{dt}$ $\displaystyle=\alpha
A(t)-(\delta_{I}+\mu)I(t).$

We shall now prove assertions (20) and (21), by considering the comparison
system

$\displaystyle\frac{dw_{1}(t)}{dt}$
$\displaystyle=\bigg{(}\beta_{A}(r(t))w_{1}(t)+\beta_{I}(r(t))w_{2}(t)\bigg{)}(S_{0}+\varepsilon)-(\alpha+\delta_{A}+\mu)w_{1}(t),$
$\displaystyle\frac{dw_{2}(t)}{dt}$ $\displaystyle=\alpha
w_{1}(t)-(\delta_{I}+\mu)w_{2}(t),\qquad w_{1}(\bar{T})=A(T),\quad
w_{2}(T)=I(\bar{T})$

Let $w(t)=(w_{1}(t),w_{2}(t))^{T}$ and consider the function
$V(w(t))=\ln||w(t)||_{2}$. Then, let
$B_{\varepsilon}(r)=(F_{\varepsilon}-V_{\varepsilon})(r)$ the matrix in (6),
computed in $x_{0}(\varepsilon)=(S_{0}+\varepsilon,0,0)$. Then, we have

$\displaystyle\frac{dV(w(t))}{dt}$
$\displaystyle=\frac{1}{||w(t)||^{2}_{2}}\langle
w(t),\dot{w}(t)\rangle=\frac{w(t)^{T}}{||w(t)||_{2}}B_{\varepsilon}(r)\frac{w(t)}{||w(t)||_{2}}$
$\displaystyle=\frac{w(t)^{T}}{||w(t)||_{2}}\frac{\big{(}(B_{\varepsilon})(r)+(B_{\varepsilon})^{T}(r)\big{)}}{2}\frac{w(t)}{||w(t)||_{2}}$
$\displaystyle\leq\frac{\lambda_{1}\big{(}(B_{\varepsilon})(r)+(B_{\varepsilon})^{T}(r)\big{)}}{2}.$

By the same arguments in Theorem 8, invoking (7), assertions (20) and (21)
follows, and consequently (19).

######  Remark 3.

In the case of Markov-switching, the condition (18) becomes

$\sum_{r\in\mathcal{M}}\pi_{r}\lambda_{1}(B(r)+B(r)^{T})<0.$

######  Remark 4.

Let us fix $r\in\mathcal{M}$. Let us consider condition (14) and
$\mathcal{R}_{0}(r)$ in (2). It is easy to see that it holds

$\widecheck{\beta}(r)S_{0}-(\hat{\delta}+\mu)<0\Rightarrow\mathcal{R}_{0}(r)<1$

Now, let us consider condition (18). We have that

$\lambda_{1}(B(r)+B(r)^{T})=\beta_{A}(r)S_{0}-(\alpha+\delta_{A}+\mu)-(\delta_{I}+\mu)+\sqrt{(\beta_{A}(r)S_{0}-(\alpha+\delta_{A})+\delta_{I})^{2}+(\beta_{I}(r)S_{0}+\alpha)^{2}}.$

From this and from (5), it is easy to see that

$\lambda_{1}(B(r)+B(r)^{T})<0\Rightarrow\mathcal{R}_{0}(r)<1,$

and that if $\beta_{I}(r)S_{0}=\alpha$, it holds

$\lambda_{1}(B(r)+B(r)^{T})<0\Leftrightarrow\mathcal{R}_{0}(r)<1.$

In Section 7, we will compare numerically the two conditions (14) and (18), by
showing a case in which condition (14) is satisfied but (18) does not, and the
other case in which the vice versa occurs. Thus, it is sufficient that one of
the two conditions is verified to ensure the almost sure extinction.

##  4\. Persistence

In this section, we investigate the persistence in time mean of the disease.

######  Definition 12.

We say that system (3) is _almost surely persistent in the time mean_ , if

$\liminf_{t\to\infty}\frac{1}{t}\int_{0}^{t}S(u)du>0\qquad\liminf_{t\to\infty}\frac{1}{t}\int_{0}^{t}(I(u)+A(u))du>0,$

with probability one.

Let us remark that $I+A$ denote the fraction of individuals that may infect
the susceptible population.

######  Theorem 13.

Let us assume $\beta_{A}(r)=\beta_{I}(r):=\beta(r)$,
$\delta_{A}=\delta_{I}:=\delta$ in each subsystem $r=1,\ldots,M$. If
$\mathcal{R}_{0}>1$, then for any initial value
$(x(0),\eta(0),r(0))\in\mathring{\Gamma}\times\mathbb{R}^{+}\times\mathcal{M}$,
the following statement is valid with probability 1:

$\liminf_{t\to\infty}\frac{1}{t}\int_{0}^{t}S(u)du\geq\frac{\mu}{\mu+\nu+\widecheck{\beta}},$
(22)

$\liminf_{t\to\infty}\frac{1}{t}\int_{0}^{t}(I(u)+A(u))du\geq\frac{\mu+\nu+\gamma}{\widecheck{\beta}(\widecheck{\beta}+\gamma)}\frac{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\beta(r)S_{0}-(\delta+\mu))}{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}}.$
(23)

###### Proof.

For ease of notation, we will omit the dependence on $\omega$ and that on $t$
if not necessary. Since $A+I\leq 1$, from the first equation of system (2), we
have

$\frac{dS(t)}{dt}\geq\mu-(\mu+\nu)S-\beta(r)(A+I)S\geq\mu-[(\mu+\nu)+\beta(r)]S,$

integrating the above inequality and dividing both sides by $t$, we obtain

$\frac{1}{t}(\mu+\nu)\int_{0}^{t}S(u)du+\frac{1}{t}\int_{0}^{t}\beta(r(u))S(u)du\geq\mu-\frac{S(t)-S(0)}{t}.$
(24)

Then, for all $\omega\in\Omega$ it holds

$\lim_{t\to+\infty}\frac{S(t)-S(0)}{t}=0,\qquad\text{a.s.}$ (25)

From (24), it follows

$\liminf_{t\to+\infty}\int_{0}^{t}S(u)ds\geq\frac{\mu}{\mu+\nu+\widecheck{\beta}},\qquad\text{a.s.}$

and assertion (22) is proved.
Next, we will prove assertion (23). By summing the second and third equation
of (3), we have that

$\frac{d\ln(I(t)+A(t))}{dt}=\beta(r)S-\delta-\mu,$

from which, integrating both sides

$\begin{split}\ln(I(t)+A(t))&=\ln(I(0)+A(0))+\int_{0}^{t}\beta(r(u))S(u)du-\int_{0}^{t}(\delta+\mu)du\\\
&=\ln(I(0)+A(0))+\int_{0}^{t}\beta(r(u))\frac{\gamma+\mu}{\nu+\gamma+\mu}du-\int_{0}^{t}\beta(r(u))\left(\frac{\gamma+\mu}{\nu+\gamma+\mu}-S(u)\right)du-\int_{0}^{t}(\delta+\mu)du\\\
&\geq\ln(I(0)+A(0))+\int_{0}^{t}\beta(r(u))\frac{\gamma+\mu}{\nu+\gamma+\mu}du-\widecheck{\beta}\int_{0}^{t}\left(\frac{\gamma+\mu}{\nu+\gamma+\mu}-S(u)\right)du-\int_{0}^{t}(\delta+\mu)du.\end{split}$
(26)

Now, we have that

$\begin{split}\frac{dS(t)}{dt}&=\mu-(\mu+\nu)S-\beta(r)(A+I)S+\gamma-\gamma
S-\gamma(I+A)\\\
&\geq(\nu+\gamma+\mu)\left(\frac{\gamma+\mu}{\nu+\gamma+\mu}-S\right)-(\beta(r)+\gamma)(I+A)\\\
&\geq(\nu+\gamma+\mu)\left(\frac{\gamma+\mu}{\nu+\gamma+\mu}-S\right)-(\widecheck{\beta}+\gamma)(I+A),\end{split}$
(27)

from which, integrating both sides,

$\begin{split}(\nu+\gamma+\mu)\int_{0}^{t}\left(\frac{\gamma+\mu}{\nu+\gamma+\mu}-S(u)\right)du\leq
S(t)-S(0)+(\widecheck{\beta}+\gamma)\int_{0}^{t}(I(u)+A(u))du.\end{split}$
(28)

Combining (26) with (28), we obtain

$\begin{split}\ln(I(t)+A(t))&\geq\ln(I(0)+A(0))+\int_{0}^{t}\left(\beta(r(u))\frac{\gamma+\mu}{\nu+\gamma+\mu}-(\delta+\mu)\right)du\\\
&-\frac{\widecheck{\beta}}{\nu+\gamma+\mu}\left[S(t)-S(0)+(\widecheck{\beta}+\gamma)\int_{0}^{t}(I(u)+A(u))du\right].\end{split}$
(29)

For all $\omega\in\Omega$, (25) holds, moreover it is easy to see that

$\limsup_{t\to+\infty}\frac{\ln(I(t)+A(t))}{t}\leq 0,$

thus from (29), and the ergodic result (7), we get

$\liminf_{t\to+\infty}\frac{1}{t}\int_{0}^{t}(I(u)+A(u))du\geq\frac{\nu+\gamma+\mu}{\widecheck{\beta}(\widecheck{\beta}+\gamma)}\frac{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\beta(r)S_{0}-(\delta+\mu))}{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}},\qquad\text{a.s.}$

that is assertion (23).

Thus, by (ii) of Proposition 9, we conclude that the disease is persistent in
the time mean with probability 1.

######  Corollary 14.

Let us assume $\beta_{A}(r)=\beta_{I}(r):=\beta(r)$,
$\delta_{A}=\delta_{I}:=\delta$ in each subsystem $r=1,\ldots,M$. If
$\mathcal{R}_{0}>1$, then for any initial value
$(x(0),\eta(0),r(0))\in\mathring{\Gamma}\times\mathbb{R}^{+}\times\mathcal{M}$,
the following statements hold with probability 1:

$\liminf_{t\to\infty}\frac{1}{t}\int_{0}^{t}I(u)du\geq\frac{\alpha}{\alpha+\delta+\mu}\frac{\mu+\nu+\gamma}{\widecheck{\beta}(\widecheck{\beta}+\gamma)}\frac{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\beta(r)S_{0}-(\delta+\mu))}{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}},$
(30)

and

$\liminf_{t\to\infty}\frac{1}{t}\int_{0}^{t}A(u)du\geq\frac{\delta+\mu}{\alpha+\delta+\mu}\frac{\mu+\nu+\gamma}{\widecheck{\beta}(\widecheck{\beta}+\gamma)}\frac{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\beta(r)S_{0}-(\delta+\mu))}{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}}.$
(31)

###### Proof.

Integrating the third equation of system (3) and dividing both sides by $t$,
we have

$(\alpha+\delta+\mu)\frac{1}{t}\int_{0}^{t}I(u)du\geq\frac{\alpha}{t}\int_{0}^{t}(I(u)+A(u))-\frac{I(t)-I(0)}{t},$

Thus, from (23) it holds (30).

Now, as before, by integrating the third equation of system (3) and dividing
both sides by $t$, it is easy to see the (31) holds.

For the next result, we need to define

$\hat{\beta}(r)=\min\\{\beta_{A}(r),\beta_{I}(r)\\},\qquad\widecheck{\hat{\beta}}=\max_{r\in\mathcal{M}}\hat{\beta}(r),\qquad\text{and}\qquad\widecheck{\delta}=\max\\{\delta_{A},\delta_{I}\\}.$

######  Theorem 15.

Let us assume $\beta_{A}(r)\neq\beta_{I}(r)$ or $\delta_{A}\neq\delta_{I}$ in
each subsystem $r=1,\ldots,M$. If

$\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\hat{\beta}(r)S_{0}-(\widecheck{\delta}+\mu))>0$
(32)

then for any initial value
$(x(0),\eta(0),r(0))\in\mathring{\Gamma}\times\mathbb{R}^{+}\times\mathcal{M}$,
the following statement is valid with probability 1:

$\liminf_{t\to\infty}\frac{1}{t}\int_{0}^{t}S(u)du\geq\frac{\mu}{\mu+\nu+\widecheck{\beta}},$
(33)

$\liminf_{t\to\infty}\frac{1}{t}\int_{0}^{t}(I(u)+A(u))du\geq\frac{\mu+\nu+\gamma}{\widecheck{\hat{\beta}}(\widecheck{\beta}+\gamma)}\frac{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\hat{\beta}(r)S_{0}-(\widecheck{\delta}+\mu))}{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}}.$
(34)

###### Proof.

Assertion (33) can be proved in the same way as assertion (22) in Theorem 13,
by considering that
$-(\beta_{A}(r)A+\beta_{I}(r)I)S\geq-\widecheck{\beta}(A+I)S$. Let us prove
assertion (34). By summing the second and third equations of (3), we have that

$\displaystyle\frac{d\ln(I(t)+A(t))}{dt}$
$\displaystyle=\frac{1}{I+A}\left[(\beta_{A}(r)A+\beta_{I}(r)I)S-(\delta_{A}-\mu)A-(\delta_{I}-\mu)I\right]$
$\displaystyle\geq\hat{\beta}(r)S-(\widecheck{\delta}+\mu).$

Following similar arguments as in (26), we obtain that

$\begin{split}\ln(I(t)+A(t))&\geq\ln(I(0)+A(0))+\int_{0}^{t}\hat{\beta}(r(u))\frac{\gamma+\mu}{\nu+\gamma+\mu}du-\widecheck{\hat{\beta}}\int_{0}^{t}\left(\frac{\gamma+\mu}{\nu+\gamma+\mu}-S(u)\right)du-\int_{0}^{t}(\widecheck{\delta}+\mu)du.\end{split}$
(35)

Now, by the same steps as in (27), we obtain

$\begin{split}(\nu+\gamma+\mu)\int_{0}^{t}\left(\frac{\gamma+\mu}{\nu+\gamma+\mu}-S(u)\right)du\leq
S(t)-S(0)+(\widecheck{\beta}+\gamma)\int_{0}^{t}(I(u)+A(u))du.\end{split}$
(36)

By combining (35) and (36), we have

$\begin{split}\ln(I(t)+A(t))&\geq\ln(I(0)+A(0))+\int_{0}^{t}\left(\hat{\beta}(r(u))\frac{\gamma+\mu}{\nu+\gamma+\mu}-(\widecheck{\delta}+\mu)\right)du\\\
&-\frac{\widecheck{\hat{\beta}}}{\nu+\gamma+\mu}\left[S(t)-S(0)+(\widecheck{\beta}+\gamma)\int_{0}^{t}(I(u)+A(u))du\right].\end{split}$

Finally, with the same arguments as in Theorem 13, we obtain (34).

######  Remark 5.

Let us fix $r\in\mathcal{M}$. Let us consider condition (32) and
$\mathcal{R}_{0}(r)$ in (5). It is easy to see that it holds

$(\hat{\beta}(r)S_{0}-(\widecheck{\delta}+\mu))>0\Rightarrow\mathcal{R}_{0}(r)>1$

######  Corollary 16.

Let us assume $\beta_{A}(r)\neq\beta_{I}(r)$ or $\delta_{A}\neq\delta_{I}$ in
each subsystem $r=1,\ldots,M$. If

$\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\hat{\beta}(r)S_{0}-(\widecheck{\delta}+\mu))>0$

then for any initial value
$(x(0),\eta(0),r(0))\in\mathring{\Gamma}\times\mathbb{R}^{+}\times\mathcal{M}$,
the following statements hold with probability 1:

$\liminf_{t\to\infty}\frac{1}{t}\int_{0}^{t}I(u)du\geq\frac{\alpha}{\alpha+\delta_{I}+\mu}\frac{\mu+\nu+\gamma}{\widecheck{\hat{\beta}}(\widecheck{\beta}+\gamma)}\frac{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\hat{\beta}(r)S_{0}-(\widecheck{\delta}+\mu))}{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}},$

and

$\liminf_{t\to\infty}\frac{1}{t}\int_{0}^{t}A(u)du\geq\frac{\delta_{I}+\mu}{\alpha+\delta_{I}+\mu}\frac{\mu+\nu+\gamma}{\widecheck{\hat{\beta}}(\widecheck{\beta}+\gamma)}\frac{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\hat{\beta}(r)S_{0}-(\widecheck{\delta}+\mu))}{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}}.$

The proof is analogous to that of Corollary 14, by invoking (34).

######  Remark 6.

Let $\beta_{A}(r)=\beta_{I}(r):=\beta(r)$, $\delta_{A}=\delta_{I}:=\delta$,
for all $r=1,\ldots,M$. From the almost surely persistence in time mean proved
in Theorem 13, the following weak persistence follows: if $\mathcal{R}_{0}>1$,
then for any initial value
$(x(0),\eta(0),r(0))\in\mathring{\Gamma}\times\mathbb{R}^{+}\times\mathcal{M}$,
it holds

$\limsup_{t\to\infty}(I(t)+A(t))\geq\frac{\mu+\nu+\gamma}{\widecheck{\beta}(\widecheck{\beta}+\gamma)}\frac{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\beta(r)S_{0}-(\delta+\mu))}{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}}\qquad\text{a.s.}$

Let $\beta_{A}(r)\neq\beta_{I}(r)$ or $\delta_{A}\neq\delta_{I}$, for all
$r=1,\ldots,M$. From Theorem 15 follows: if
$\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\hat{\beta}(r)S_{0}-(\widecheck{\delta}+\mu))>0$
then for any initial value
$(x(0),\eta(0),r(0))\in\mathring{\Gamma}\times\mathbb{R}^{+}\times\mathcal{M}$,
it holds

$\limsup_{t\to\infty}(I(t)+A(t))\geq\frac{\mu+\nu+\gamma}{\widecheck{\hat{\beta}}(\widecheck{\beta}+\gamma)}\frac{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\hat{\beta}(r)S_{0}-(\widecheck{\delta}+\mu))}{\sum_{r\in\mathcal{M}}\pi_{r}m_{r}},\qquad\text{a.s}.$

In the case $\beta_{A}(r)=\beta_{I}(r):=\beta(r)$,
$\delta_{A}=\delta_{I}:=\delta$, the position of the value $\mathcal{R}_{0}$
(13) with respect to one determines the extinction or the persistence of the
disease, that is $\mathcal{R}_{0}$ is a threshold value. Thus, from Theorems
(8) and (13), we obtain the following corollary:

######  Corollary 17.

Let us assume $\beta_{A}(r)=\beta_{I}(r):=\beta(r)$,
$\delta_{A}=\delta_{I}:=\delta$, and consider $\mathcal{R}_{0}$ in (13).
Then,the solution of system (3) has the property that

  * (i)

If $\mathcal{R}_{0}<1$, for any initial value
$(x(0),\eta(0),r(0))\in\Gamma\times\mathbb{R}^{+}\times\mathcal{M}$, the
fraction of asymptomatic and infected individuals $A(t)$ and $I(t)$,
respectively, tends to zero exponentially almost surely, that is the disease
dies out with probability one;

  * (ii)

If $\mathcal{R}_{0}>1$, for any initial value
$(x(0),\eta(0),r(0))\in\mathring{\Gamma}\times\mathbb{R}^{+}\times\mathcal{M}$,
the disease will be almost surely persistent in time mean.

######  Remark 7.

Let us assume $\beta_{A}(r)\neq\beta_{I}(r)$ or $\delta_{A}\neq\delta_{I}$.
Let us fix $r\in\mathcal{M}$. Let us consider condition (32) and
$\mathcal{R}_{0}(r)$ in (5). It is easy to see that it holds

$(\hat{\beta}(r)S_{0}-(\widecheck{\delta}+\mu))>0\Rightarrow\mathcal{R}_{0}(r)>1$

In the case $\beta_{A}(r)\neq\beta_{I}(r)$ or $\delta_{A}\neq\delta_{I}$, we
find two regions, one where the system goes to extinction almost surely, and
the other where it is stochastic persistent in time mean. These two regions
are not adjacent, as there is a gap between them; thus we do not have a
threshold value separating these regions.

In the following Section 5, we investigate the omega-limit set of the system.
The introduction of the backward recurrence time process allows the considered
stochastic system to be a piecewise deterministic Markov process [15]. Thus,
in Section (6), we prove the existence of a unique invariant probability
measure for this process. Let us note that in the two subsequent sections, to
obtain our results, we follow mainly the approaches in [15] and, like them,
for simplicity, we restrict the analysis to a semi-Markov process with state
space $\mathcal{M}=\\{1,2\\}$. Hence, the external environmental conditions
can switch randomly between two states, for example favorable and adverse
weather conditions for the disease spread, or lockdown and less stringent
distance measures, considering that the disease transmission rate is also a
function of the contact rate.

##  5\. Omega-limit set

Let us assume in this section and in the subsequent one that
$\mathcal{M}=\\{1,2\\}$.
Let us define the omega-limit set of the trajectory starting from an initial
value $x(0)\in\Gamma$ as

$\tilde{\Omega}(x(0),\omega)=\bigcap_{T>0}\overline{\bigcup_{t>T}x(t,\omega,x(0))}$
(37)

We use the notation $\tilde{\Omega}$ for the limit set (37) in place of the
usual one $\omega$ in the deterministic dynamical systems for avoiding
conflict with the element $\omega$ in the probability sample space. In this
section, it will be shown that under some appropriate condition
$\tilde{\Omega}(x(0),\omega)$ is deterministic, i.e., it is constant almost
surely and it is independent of the initial value $x(0)$. Let us consider the
following assumption:

(H2) For some $r\in\mathcal{M}$, there exists a unique and globally
asymptotically stable endemic equilibrium
$x^{*}_{r}=(S^{*}_{r},A^{*}_{r},I^{*}_{r})$ for the corresponding system of
(3) in the state $r$.

Let us note that when $\beta_{A}(r)=\beta_{I}(r):=\beta(r)$ and
$\delta_{A}=\delta_{I}:=\delta$, in the case of persistence in time mean, by
(ii) of Proposition 9 the condition $\mathcal{R}_{0}>1$ implies that there
exists at least one state $r$ such that
$\pi_{r}m_{r}(\beta(r)S_{0}-(\delta+\mu))>0$, i.e. $\mathcal{R}_{0}(r)>1$. By
Theorem 6, $x_{r}^{*}$ is globally asymptotically stable in
$\mathring{\Gamma}$. Thus, if $\mathcal{R}_{0}>1$ condition (H2) is satisfied.
When $\beta_{A}(r)\neq\beta_{I}(r)$ or $\delta_{A}\neq\delta_{I}$, by Theorem
15, if equation (32) holds, then there exists at least one state $r$ such that
$(\hat{\beta}(r)S_{0}-(\widecheck{\delta}+\mu))>0$; from Remark 7 this implies
that $\mathcal{R}_{0}(r)>1$. By Theorem (7), we have the global asymptotic
stability of $x_{r}^{*}$ if $\mathcal{R}_{0}(r)>1$ under the additional
condition $\beta_{A}(r)<\delta_{I}$. However, it is easy to see that if this
last condition is verified, $(\hat{\beta}(r)S_{0}-(\widecheck{\delta}+\mu))>0$
cannot be valid. Thus, if we need (H2), we can suppose it holds for a state
for which $(\hat{\beta}(r)S_{0}-(\widecheck{\delta}+\mu))>0$ is not verified.
Indeed, we remember that this last condition is only sufficient to have
$\mathcal{R}_{0}(r)>1$ but not necessary.

Now, let us recall some concepts on the Lie algebra of vector fields [3, 11]
that we need for the next results. Let $w(y)$ and $z(y)$ be two vector fields
on $\mathbb{R}^{3}$. The Lie bracket $[w,z]$ is also a vector field given by

$[w,z]_{j}(y)=\sum_{k=1}^{3}\left(w_{k}\frac{\partial z_{j}}{\partial
y_{k}}(y)-z_{k}\frac{\partial w_{k}}{\partial y_{k}}(y)\right),\qquad j=1,2,3$

Assumption (H3):
A point $x=(S,A,I)\in\mathbb{R}^{3}_{+}$ is said to satisfy the _Lie bracket
condition_ , if vectors $u_{1}(x)$, $u_{2}(x)$,
$[u_{i},u_{j}](x)_{i,j\in\mathcal{M}}$,
$[u_{i},[u_{j},u_{k}]](x)_{i,j,k\in\mathcal{M}},\ldots$, span the space
$\mathbb{R}^{3}$, where for each $r\in\mathcal{M}$,

$u_{r}(x)=\left(\begin{matrix}\mu-\bigg{(}\beta_{A}(r)A+\beta_{I}(r)I\bigg{)}S-(\mu+\nu+\gamma)S+\gamma(1-A-I)\\\
\bigg{(}\beta_{A}(r)A+\beta_{I}(r)I(t)\bigg{)}S-(\alpha+\delta_{A}+\mu)A\\\
\alpha A-(\delta_{I}+\mu)I\end{matrix}\right).$ (38)

Without loss of generality, we can assume that condition (H2) holds for $r=1$.

######  Theorem 18.

Suppose that system (3) is persistent in time mean and the hypothesis (H2)
holds. Let us denote by $\xi^{r}_{t}(x(0))$ the solution of system (3) in the
state $r$ with initial value $x(0)\in\mathring{\Gamma}$, and let

$\Psi=\bigg{\\{}(S,A,I)=\xi^{e_{k}}_{t_{k}}\circ\ldots\circ\xi^{e_{1}}_{t_{1}}(x_{1}^{*}):t_{1},\ldots,t_{k}\geq
0\quad\text{and}\quad
e_{1},\ldots,e_{k}\in\mathcal{M},k\in\mathbb{N}\bigg{\\}}.$

Then, the following statements are valid:

  * (a)

The closure $\bar{\Psi}$ is a subset of the omega-limit set
$\tilde{\Omega}(x(0),\omega)$ with probability one.

  * (b)

If there exists a point $x_{*}:=(S_{*},A_{*},I_{*})\in\Psi$ satisfying the
condition (H3), then $\Psi$ absorbs all positive solutions, that is for any
initial value $x(0)\in\mathring{\Gamma}$, the value

$\hat{T}(\omega)=\inf\left\\{t>0:x(s,\omega,x(0))\in\Psi,\forall s>t\right\\}$

is finite outside a $\mathbb{P}$-null set. Consequently, $\bar{\Psi}$ is the
omega-limit set $\tilde{\Omega}(x(0),\omega)$ for any
$x(0)\in\mathring{\Gamma}$ with probability one.

The proof of the Theorem 18 follows by similar arguments to that of [15, Thm
9], thus we omit it.

##  6\. Invariant probability measure

In this section, we will prove the existence of an invariant probability
measure for the homogeneous Markov process $\\{(x(t),\eta(t),r(t)),t\geq 0\\}$
on the state space

$\mathcal{H}=\mathring{\Gamma}\times\mathbb{R}_{+}\times\mathcal{M}.$

Following [15], we introduce a metric $\hbar(\cdot,\cdot)$ on the state space
$\mathcal{H}$:

$\hbar\left((x_{1},s_{1},i),(x_{2},s_{2},j)\right)=\sqrt{|x_{1}-x_{2}|^{2}+|s_{1}-s_{2}|^{2}}+o(i,j),$
(39)

where

$o(i,j)=\begin{cases}0,&\text{if}\quad i=j,\\\ 1,&\text{if}\quad i\neq
j.\end{cases}$

Hence, $(X,\hbar(\cdot,\cdot),\mathcal{B}(\mathcal{H}))$ is a complete
separable metric space, where $\mathcal{B}(\mathcal{H})$ is the Borel
$\sigma$-algebra on $\mathcal{H}$.

To ensure the existence of an invariant probability measure on $\mathcal{H}$,
we use the following exclusion principle.

######  Lemma 19 (see [26]).

Let $\Phi=\left\\{\Phi_{t},t\geq 0\right\\}$ be a Feller process with state
space $(X,\mathcal{B}(X))$. Then either

  * a)

there exists an invariant probability measure on $X$, or

  * b)

for any compact set $C\subset X$,

$\lim_{t\to\infty}\sup_{\kappa}\frac{1}{t}\int_{0}^{t}\left(\int_{X}\mathbb{P}(u,x,C)\kappa({\mathrm{d}}x)\right){\mathrm{d}}u=0,$

where the supremum is taken over all initial distributions $\kappa$ on the
state space $X$, $x\in X$ is the initial condition for the process $\Phi_{t}$,
and $\mathbb{P}(t,x,C)=\mathbb{P}_{x}(\Phi_{t}\in C)$ is the transition
probability function.

To prove the Feller property of the process $\\{(x(t),\eta(t),r(t)),t\geq
0\\}$, we need the following lemma.

######  Lemma 20.

Let $\hbar(\cdot,\cdot)$ be the metric defined in (39). Then, for any $T>0$
and $\varepsilon>0$, we have

$\mathbb{P}\left\\{\max_{0\leq t\leq
T}\hbar\left((x(t,x_{1}),\eta(t),r(t)),(x(t,x_{2}),\eta(t),r(t))\right)\geq\varepsilon\right\\}\to
0$ (40)

as $|x_{1}-x_{2}|\to 0$, where $(x_{1},\eta(0),r(0)),$
$(x_{2},\eta(0),r(0))\in\mathcal{H}$ denote any two given initial values of
the process $\\{(x(t),\eta(t),r(t)),t\geq 0\\}$.

###### Proof.

By considering (38), it is easy to see that

$d\left(x(t,x_{1})-x(t,x_{2})\right)=\left(u_{r(t)}(x(t,x_{1})-u_{r(t)}(x(t,x_{2}))\right)dt.$
(41)

Applying the Itô formula to the function $|x(t,x_{1})-x(t,x_{2})|^{2}$, we
have

$\mathbb{E}|x(t,x_{1})-x(t,x_{2})|^{2}=|x_{1}-x_{2}|^{2}+2\mathbb{E}\left[\int_{0}^{t}\langle
x(s,x_{1})-x(s,x_{2}),u_{r(s)}(x(s,x_{1}))-u_{r(s)}(x(s,x_{2})\rangle\;ds\right]$
(42)

For ease of notation, let us define

$G(s,x_{k})=(\beta_{A}(r(s))A(s,x_{k})+\beta_{I}(r(s))I(s,x_{k})),\qquad
k=1,2.$

Now, we have that

$\displaystyle\langle
x(s,x_{1})-x(s,x_{2}),u_{r(s)}(x(s,x_{1}))-u_{r(s)}(x(s,x_{2})\rangle$
$\displaystyle=(S(s,x_{1})-S(s,x_{2}))(G(s,x_{1})S(s,x_{1})-G(s,x_{2})S(s,x_{2}))-(\mu+\nu+\gamma)(S(s,x_{1})-S(s,x_{2}))^{2}$
$\displaystyle-\gamma(S(s,x_{1})-S(s,x_{2}))(A(s,x_{1})-A(s,x_{2}))-\gamma(S(s,x_{1})-S(s,x_{2}))(I(s,x_{1})-I(s,x_{2}))$
$\displaystyle+(A(s,x_{1})-A(s,x_{2}))(G(s,x_{1})S(s,x_{1})-G(s,x_{2})S(s,x_{2}))-(\alpha+\delta_{A}+\mu)(A(s,x_{1})-A(s,x_{2}))^{2}$
$\displaystyle+\alpha(A(s,x_{1})-A(s,x_{2}))(I(s,x_{1})-I(s,x_{2}))-(\delta_{I}+\mu)(I(s,x_{1})-I(s,x_{2}))^{2}.$

Since

$\gamma(S(s,x_{1})-S(s,x_{2}))(A(s,x_{1})-A(s,x_{2}))\leq\frac{\gamma}{2}(S(s,x_{1})-S(s,x_{2}))^{2}+\frac{\gamma}{2}(A(s,x_{1})-A(s,x_{2}))^{2},$
$\gamma(S(s,x_{1})-S(s,x_{2}))(I(s,x_{1})-I(s,x_{2}))\leq\frac{\gamma}{2}(S(s,x_{1})-S(s,x_{2}))^{2}+\frac{\gamma}{2}(I(s,x_{1})-I(s,x_{2}))^{2},$

and

$\alpha(A(s,x_{1})-A(s,x_{2}))(I(s,x_{1})-I(s,x_{2}))\leq\frac{\alpha}{2}(A(s,x_{1})-A(s,x_{2}))^{2}+\frac{\alpha}{2}(I(s,x_{1})-I(s,x_{2}))^{2},$

we can write

$\displaystyle\begin{split}&\langle
x(s,x_{1})-x(s,x_{2}),u_{r(s)}(x(s,x_{1}))-u_{r(s)}(x(s,x_{2})\rangle\\\
&\leq(\mu+\nu+2\gamma)(S(s,x_{1})-S(s,x_{2}))^{2}+\left(\frac{\gamma}{2}+\frac{3\alpha}{2}+\delta_{A}+\mu\right)(A(s,x_{1})-A(s,x_{2}))^{2}\\\
&+\left(\delta_{I}+\mu+\frac{\gamma}{2}+\frac{\alpha}{2}\right)(I(s,x_{1})-I(s,x_{2}))^{2}\\\
&+|(S(s,x_{1})-S(s,x_{2}))(G(s,x_{1})S(s,x_{1})-G(s,x_{2})S(s,x_{2}))|\\\
&+|(A(s,x_{1})-A(s,x_{2}))(G(s,x_{1})S(s,x_{1})-G(s,x_{2})S(s,x_{2}))|\end{split}$
(43)

Now,

$\displaystyle\begin{split}&|(S(s,x_{1})-S(s,x_{2}))(G(s,x_{1})S(s,x_{1})-G(s,x_{2})S(s,x_{2}))|\\\
&\leq\widecheck{\beta}(S(s,x_{1})-S(s,x_{2}))\left[(S(s,x_{1})-S(s,x_{2}))(A(s,x_{1})+I(s,x_{1}))\right.\\\
&\left.+S(s,x_{2})(A(s,x_{1})+I(s,x_{1})-(A(s,x_{2})+I(s,x_{2})))\right]\\\
&\leq\widecheck{\beta}(S(s,x_{1})-S(s,x_{2}))^{2}+\widecheck{\beta}(S(s,x_{1})-S(s,x_{2}))(A(s,x_{1})+I(s,x_{1})-(A(s,x_{2})+I(s,x_{2})))\\\
&\leq\widecheck{\beta}(S(s,x_{1})-S(s,x_{2}))^{2}+\frac{\widecheck{\beta}}{2}\left[(S(s,x_{1})-S(s,x_{2}))^{2}+(A(s,x_{1})-A(s,x_{2}))^{2}\right]\\\
&+\frac{\widecheck{\beta}}{2}\left[(S(s,x_{1})-S(s,x_{2}))^{2}+(I(s,x_{1})-I(s,x_{2}))^{2}\right],\end{split}$
(44)

and, similarly

$\displaystyle\begin{split}&|(A(s,x_{1})-A(s,x_{2}))(G(s,x_{1})S(s,x_{1})-G(s,x_{2})S(s,x_{2}))|\\\
&\leq\widecheck{\beta}(A(s,x_{1})-A(s,x_{2}))\left[(S(s,x_{1})-S(s,x_{2}))(A(s,x_{1})+I(s,x_{1}))\right.\\\
&\left.+S(s,x_{2})(A(s,x_{1})+I(s,x_{1})-(A(s,x_{2})+I(s,x_{2})))\right]\\\
&\leq\frac{\widecheck{\beta}}{2}((A(s,x_{1})-A(s,x_{2}))^{2}+(S(s,x_{1})-S(s,x_{2}))^{2})\\\
&+\widecheck{\beta}((A(s,x_{1})-A(s,x_{2}))S(s,x_{2})(A(s,x_{1})+I(s,x_{1})-(A(s,x_{2})+I(s,x_{2})))\\\
&\leq\frac{\widecheck{\beta}}{2}((A(s,x_{1})-A(s,x_{2}))^{2}+(S(s,x_{1})-S(s,x_{2}))^{2})+\widecheck{\beta}((A(s,x_{1})-A(s,x_{2}))^{2})\\\
&+\frac{\widecheck{\beta}}{2}((A(s,x_{1})-A(s,x_{2}))^{2}+(I(s,x_{1})-I(s,x_{2}))^{2}).\end{split}$
(45)

Now, substituting (44) and (45) into (43) yields

$\displaystyle\begin{split}&\langle
x(s,x_{1})-x(s,x_{2}),u_{r(s)}(x(s,x_{1}))-u_{r(s)}(x(s,x_{2})\rangle\\\ &\leq
K|x(s,x_{1})-x(s,x_{2})|^{2},\end{split}$ (46)

where $K=\max\\{K_{1},K_{2},K_{3}\\}$, with

$K_{1}=\frac{5\widecheck{\beta}}{2}+(\mu+\nu+2\gamma),$
$K_{2}=\frac{5\widecheck{\beta}}{2}+\left(\frac{\gamma}{2}+\frac{3\alpha}{2}+\delta_{A}+\mu\right),$

and

$K_{3}=\widecheck{\beta}+\delta_{I}+\mu+\frac{\gamma}{2}+\frac{\alpha}{2}.$

From (42) and (46), following similar steps as in [15, Lemma 14], we have that

$\int_{0}^{T}\mathbb{E}\left|x(s,x_{1})-x(s,x_{2})\right|^{2}ds\leq|x(s,x_{1})-x(s,x_{2})|^{2}\int_{0}^{T}\exp(2Ks)ds\to
0$ (47)

as $|x(s,x_{1})-x(s,x_{2})|\to 0$. Moreover, from (41), it follows that

$\max_{0\leq t\leq
T}\left|x(s,x_{1})-x(s,x_{2})\right|\leq\left|x(s,x_{1})-x(s,x_{2})\right|+\int_{0}^{T}\left|u_{r(s)}(x(s,x_{1}))-u_{r(s)}(x(s,x_{2})\right|ds.$
(48)

For any $\varepsilon>0$, from Markov inequality and similar steps as in [15,
Lemma 14], we obtain

$\displaystyle\begin{split}&\mathbb{P}\left\\{\int_{0}^{T}\left|u_{r(s)}(x(s,x_{1}))-u_{r(s)}(x(s,x_{2})\right|ds\geq\varepsilon\right\\}\\\
&\leq\frac{T}{\varepsilon^{2}}\mathbb{E}\left[\int_{0}^{T}|u_{r(s)}(x(s,x_{1}))-u_{r(s)}(x(s,x_{2})|^{2}ds\right]\end{split}$
(49)

By similar arguments to (43), (44) and (45), one can find a positive number
$\bar{K}$ such that

$\left|u_{r(s)}(x(s,x_{1}))-u_{r(s)}(x(s,x_{2})\right|^{2}\leq\bar{K}\left|x(s,x_{1})-x(s,x_{2})\right|^{2}.$

Substituting this inequality into (49), by Fubini’s theorem we have

$\mathbb{P}\left\\{\int_{0}^{T}\left|u_{r(s)}(x(s,x_{1}))-u_{r(s)}(x(s,x_{2})\right|ds\geq\varepsilon\right\\}\leq\frac{\bar{K}T}{\varepsilon^{2}}\int_{0}^{T}\mathbb{E}\left|x(s,x_{1})-x(s,x_{2})\right|^{2}ds.$

By (47), then we get

$\mathbb{P}\left\\{\int_{0}^{T}\left|u_{r(s)}(x(s,x_{1}))-u_{r(s)}(x(s,x_{2})\right|ds\geq\varepsilon\right\\}\to
0$ (50)

as $|x(s,x_{1})-x(s,x_{2})|\to 0$. Definitively, by combining (48) with (50),
we obtain the claim (40).

######  Lemma 21.

The Markov process $\\{(x(t),\eta(t),r(t),t\geq 0\\}$ is Feller.

The proof requires the claim of Lemma 20 and it is analogous to that of [15,
Lemma 15], thus we omit it.

At this point, we can prove the existence of an invariant probability measure
by using Lemma (19).

######  Theorem 22.

Suppose that system (3) is persistent in time mean, then the Markov process
$\\{(x(t),\eta(t),r(t),t\geq 0\\}$ has an invariant probability measure
$\kappa^{*}$ on the state space $\mathcal{H}$.

###### Proof.

Let us consider the process $\\{(x(t),\eta(t),r(t)),t\geq 0\\}$ on a larger
state space

$\tilde{\mathcal{H}}=\tilde{\Gamma}\times\mathbb{R}_{+}\times\mathcal{M},$

where $\tilde{\Gamma}=\Gamma\setminus\\{(S,A,I):A+I=0\\}$. By Lemma 19 and 21,
we can prove the existence of an invariant probability measure $\kappa^{*}$
for the Markov process $h(t)=\\{(x(t),\eta(t),r(t)),t\geq 0\\}$ on
$\tilde{\mathcal{H}}$, provided that a compact subset
$C\subset\tilde{\mathcal{H}}$ exists such that

$\liminf_{t\to\infty}\frac{1}{t}\int_{0}^{t}\left(\int_{\tilde{\mathcal{H}}}\mathbb{P}(u,h,C)\kappa(dh)\right)du=\liminf_{t\to\infty}\frac{1}{t}\int_{0}^{t}\mathbb{P}(u,h_{0},C)du>0,$
(51)

for some initial distribution $\kappa=\delta_{h_{0}}$ with
$h_{0}\in\tilde{\mathcal{H}}$, where $\delta.$ is the Dirac function. Once the
existence of $\kappa^{*}$ is proved, we can easily see that $\kappa^{*}$ is
also an invariant probability measure of $\\{(x(t),\eta(t),r(t)),t\geq 0\\}$
on $\mathcal{H}$. Indeed, it is easy to prove that for any initial value
$(x(0),\eta(0),r(0))\in\mathring{\Gamma}\times\mathbb{R}^{+}\times\mathcal{M}$,
the solution $x(t)$ of system (3) does not reach the boundary
$\partial\tilde{\Gamma}$ when the system is persistent in time mean.
Consequently,
$k^{*}(\partial\tilde{\Gamma}\times\mathbb{R}_{+}\times\mathcal{M})=0$, which
implies therefore that $k^{*}$ is also an invariant probability measure on
$\mathcal{H}$. Thus, to complete the proof we just have to find a compact
subset $C\subset\tilde{\mathcal{H}}$ satisfying (51). Hereafter, in the proof,
we assume, without loss of generality that $\eta(0)=0$. Since system (3) is
persistent in time mean, there exists a constant $\iota>0$ such that

$\liminf_{t\to\infty}\frac{1}{t}\int_{0}^{t}(I(u)+A(u))du\geq\iota\qquad\text{a.s.}$

Following similar steps as in [15, Theorem 16], we obtain

$\liminf_{t\to\infty}\int_{0}^{t}\mathbb{P}\left\\{I(u)+A(u)\geq\frac{\iota}{2}\right\\}du\geq\frac{\iota}{2}.$
(52)

Now, we have to verify that $\mathbb{P}\\{\eta(u)>K\\}<\varepsilon$ holds for
any $u\in\mathbb{R}_{+}$, where $\varepsilon$ is a positive conctant such that
$\varepsilon<\frac{\iota}{4}$ and $K>0$ is a sufficiently large constant such
that $\max_{i\in\mathcal{M}}\left\\{1-F_{i}(K)\right\\}<\varepsilon/2$. To do
this, we follow similar steps as in [15, Theorem 16]. This implies that

$\liminf_{t\to\infty}\int_{0}^{t}\mathbb{P}\left\\{I(u)+A(u)\geq\frac{\iota}{2}\right\\}du\leq\varepsilon+\frac{1}{t}\int_{0}^{t}\mathbb{P}\left\\{I(u)+A(u)\geq\frac{\iota}{2},\eta(u)\leq
K\right\\}du.$

Combining this with (52), we have

$\liminf_{t\to\infty}\frac{1}{t}\int_{0}^{t}\mathbb{P}\left\\{I(u)+A(u)\geq\frac{\iota}{2},\eta(u)\leq
K\right\\}du\geq\frac{\iota}{4}.$

Let us consider the compact set
$C=\mathcal{D}\times[0,K]\times\mathcal{M}\in\tilde{\mathcal{H}},$ where the
set $\mathcal{D}$ is

$\left\\{(S,A,I)\in\Gamma:0\leq S+A+I\leq 1,I+A\geq\frac{\iota}{2}\right\\}.$

Then, it follows that

$\displaystyle\liminf_{t\to\infty}\frac{1}{t}\int_{0}^{t}\mathbb{P}(u,h_{0},C)du$
$\displaystyle=\liminf_{t\to\infty}\frac{1}{t}\int_{0}^{t}\mathbb{P}\left\\{x(u,\omega,x(0))\in\mathcal{D},\eta(u)\leq
K\right\\}du$
$\displaystyle=\liminf_{t\to\infty}\frac{1}{t}\int_{0}^{t}\mathbb{P}\left\\{(I(u)+A(u))\geq\frac{\iota}{2},\eta(u)\leq
K\right\\}$ $\displaystyle\geq\frac{\iota}{4}.$

Since $\iota>0$, we arrive to our claim.

Now, we show that the invariant probability measure $\kappa^{*}$ is unique by
using the property of Harris recurrence and positive Harris recurrence (for
standard definitions see, e.g., [18, 15]).

######  Proposition 23.

Suppose that system (3) is persistent in time mean and that assumptions
(H2)-(H3) hold, then the Markov process $\\{(x(t),\eta(t),r(t)),t\geq 0\\}$ is
positive Harris recurrent. Thus, the invariant probability measure
$\kappa^{*}$ of $\\{(x(t),\eta(t),r(t)),t\geq 0\\}$ on the state space
$\mathcal{H}$ is unique.

###### Proof.

The Harris recurrence of the process $\\{(x(t),\eta(t),r(t)),t\geq 0\\}$
follows by similar steps as in [15, Lemma 20]. Thus, from Theorem 22, we can
conclude that the process is positive Harris recurrent. Therefore, the
invariant probability measure $\kappa^{*}$ of $\\{(x(t),\eta(t),r(t)),t\geq
0\\}$ on $\mathcal{H}$ is unique [5, 18].

##  7\. Numerical experiments

In this section, we provide some numerical investigations to verify the
theoretical results. We assume that the conditional holding time distribution
$F_{i}(\cdot)$ of the semi-Markov process in state $i$ is a gamma distribution
$\Gamma(k_{i},{\theta_{i}})$, $i=1,\ldots,M$, whose probability density
function is

$f_{i}(t;k_{i},\theta_{i})=\frac{\theta_{i}^{k_{i}}t^{k_{i}-1}e^{-\theta_{i}t}}{\Gamma(k_{i})},\qquad\text{for}\quad
t>0,$

and the cumulative distribution function is given by

$F(t;k_{i},\theta_{i})=\int_{0}^{t}f(u;k_{i},\theta_{i})du,$

where $k_{i}>0$, $\theta_{i}>0$, and $\Gamma(\cdot)$ is the complete gamma
function. Let us note that if $k_{i}=1$, the gamma distribution becomes the
exponential distribution with parameter $\theta_{i}$.

We discuss the almost sure extinction and persistence in time mean of the
system in the following cases.

a)

b)

Figure 1. Dynamical behaviour of system (2) and the semi-Markov process
$r(t)$. The parameter values are: $\beta_{A}(1)=0.004$, $\beta_{I}(1)=0.008$,
$\beta_{A}(2)=0.97$, $\beta_{I}(2)=0.99$, $\delta_{A}=0.105$,
$\delta_{I}=0.1$, $\mu=1/(60*365)$, $\alpha=0.3$, $\gamma=0.03$, and
$\nu=0.01$. The parameters of $F_{1}$ and $F_{2}$ are respectively: a)
$k_{1}=6$, $\theta_{1}=0.8$ and $k_{2}=12$, $\theta_{2}=0.8$, b) $k_{1}=15$,
$\theta_{1}=0.8$ and $k_{2}=2$, $\theta_{2}=0.8$. Figure 2. Dynamical
behaviour of system (2) and the semi-Markov process $r(t)$. The parameter
values are: $\beta_{A}(1)=0.55$, $\beta_{I}(1)=0.5$, $\beta_{A}(2)=0.68$,
$\beta_{I}(2)=0.58$, $\delta_{A}=0.3$, $\delta_{I}=0.4$, $\mu=1/(60*365)$,
$\nu=0.01$, $\alpha=0.5$, $\gamma=0.01$. The parameters of $F_{1}$ and $F_{2}$
are $k_{1}=0.9$, $\theta_{1}=0.8$ and $k_{2}=2.5$, $\theta_{2}=0.8$,
respectively

_Case 1: $\beta_{A}(r)\neq\beta_{I}(r)$, $\delta_{A}(r)\neq\delta_{I}(r)$._ We
consider two states: $r_{1}$, in which the epidemic dies out, and $r_{2}$ in
which it persists. In Fig.1 a), we consider $F_{1}$, $F_{2}$ with parameters
$k_{1}=6$, $\theta_{1}=0.8$ and $k_{2}=12$, $\theta_{2}=0.8$, respectively.
Consequently, the respective mean sojourn times are $m_{1}=7.5$ and
$m_{2}=15$. The other parameters are: $\beta_{A}(1)=0.004$,
$\beta_{I}(1)=0.008$, $\beta_{A}(2)=0.97$, $\beta_{I}(2)=0.99$,
$\delta_{A}=0.105$, $\delta_{I}=0.1$, $\mu=1/(60*365)$, $\alpha=0.3$,
$\gamma=0.03$, and $\nu=0.01$. We have that

$\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\hat{\beta}(r)S_{0}-(\widecheck{\delta}+\mu))=4.283>0,$

thus the whole system is stochastically persistent in time mean. In Fig.1 b),
we consider $F_{1}$, $F_{2}$ with parameters $k_{1}=15$, $\theta_{1}=0.8$ and
$k_{2}=2$, $\theta_{2}=0.8$, respectively; we have $m_{1}=18.75$, $m_{2}=2.5$.
The other parameters are the same as in a). In this case,

$\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\widecheck{\beta}(r)S_{0}-(\hat{\delta}+\mu))=-0.0782<0,$

and the epidemic will go extinct almost surely, in the long run. Thus, we can
see the relevant role played by the mean sojourn times, indeed in the two
figures the parameters are the same, what changes is that in Fig.1 a) the mean
sojourn time in the persistent state is higher than that in the state of
extinction, while in Fig.1 b) the vice versa occurs.

_Case 2: $\beta_{A}(r)\neq\beta_{I}(r)$, $\delta_{A}(r)\neq\delta_{I}(r)$._
Here, we want to compare the two sufficient conditions for the almost sure
extinction (14) and (18). In Fig.2, we consider again two states $r_{1}$, in
which the epidemic dies out, and $r_{2}$ in which it persists. The parameters
are: $\beta_{A}(1)=0.55$, $\beta_{I}(1)=0.5$, $\beta_{A}(2)=0.68$,
$\beta_{I}(2)=0.58$, $\delta_{A}=0.3$, $\delta_{I}=0.4$, $\mu=1/(60*365)$,
$\nu=0.01$, $\alpha=0.5$, $\gamma=0.01$. Let us consider $F_{1}$ and $F_{2}$
with $k_{1}=0.9$, $\theta_{1}=0.8$ and $k_{2}=2.5$, $\theta_{2}=0.8$,
respectively. Hence, $m_{1}=1.125$ and $m_{2}=3.125$. We have

$\sum_{r\in\mathcal{M}}\pi_{r}m_{r}(\widecheck{\beta}(r)S_{0}-(\hat{\delta}+\mu))=0.05>0\qquad\text{and}\qquad\sum_{r\in\mathcal{M}}\pi_{r}m_{r}\lambda_{1}(B(r)+B(r)^{T})=-0.1959<0$

Thus, in this case condition (14) is not satisfied but (18) does. Vice versa,
in Fig.1 b), we have the opposite case, that is condition (14) is less than
zero, while (18) is equal to $1.0084$, hence greater than zero. By Theorems 10
and Theorem 11, we have the almost sure extinction in both cases, as we can
also see from Figs. 2 and 1 b).

a)

b)

c)

Figure 3. Dynamical behaviour of system (2) and the semi-Markov process
$r(t)$. The parameter values are: $\beta_{A}(1)=\beta_{I}(1)=0.05$,
$\beta_{A}(2)=\beta_{I}(2)=0.9$, $\delta_{A}=\delta_{I}=0.07$,
$\mu=1/(60*365)$, $\nu=0.01$, $\alpha=0.5$, $\gamma=0.02$. The parameters of
$F_{1}$ and $F_{2}$ are respectively: a) $k_{1}=4$, $\theta_{1}=0.8$ and
$k_{2}=15$, $\theta_{2}=0.8$, b) $k_{1}=15$, $\theta_{1}=0.8$ and $k_{2}=3$,
$\theta_{2}=0.8$, c) $k_{1}=18$, $\theta_{1}=0.8$ and $k_{2}=1$,
$\theta_{2}=0.8$

_Case 3: $\beta_{A}(r)=\beta_{I}(r):=\beta(r)$,
$\delta_{A}(r)=\delta_{I}(r):=\delta(r)$_. We consider two states: $r_{1}$, in
which the epidemic dies out, and $r_{2}$ in which it persists. The parameters
are: $\beta_{A}(1)=\beta_{I}(1)=0.05$, $\beta_{A}(2)=\beta_{I}(2)=0.9$,
$\delta_{A}=\delta_{I}=0.07$, $\mu=1/(60*365)$, $\nu=0.01$, $\alpha=0.5$,
$\gamma=0.02$. In Fig.3 a), we have $F_{1}$, $F_{2}$ with parameters
$k_{1}=4$, $\theta_{1}=0.8$ and $k_{2}=15$, $\theta_{2}=0.8$, respectively.
Consequently, $m_{1}=5$ and $m_{2}=18.75$, and

$\sum_{r\in\mathcal{M}}\pi_{r}m_{r}\left(\beta(r)\frac{\gamma+\mu}{\nu+\gamma+\mu}-(\delta+\mu)\right)=4.8809>0.$

Thus, the whole system is persistent in time mean. In Fig.3 b), $F_{1}$ and
$F_{2}$ have parameters $k_{1}=15$, $\theta_{1}=0.8$ and $k_{2}=3$,
$\theta_{2}=0.8$. Thus, $m_{1}=18.75$ and $m_{2}=3.75$, and

$\sum_{r\in\mathcal{M}}\pi_{r}m_{r}\left(\beta(r)\frac{\gamma+\mu}{\nu+\gamma+\mu}-(\delta+\mu)\right)=0.6506>0.$

Here, we stay on average longer in the state where the epidemic will go
extinct, with respect to the case a). However, although the system is
persistent in time mean, the threshold value lowers a lot and the fraction of
infectious symptomatic and asymptomatic individuals have the time to decays in
some time windows, and the susceptible to increases. In Fig.3 c) $F_{1}$ and
$F_{2}$ have parameters $k_{1}=18$, $\theta_{1}=0.8$ and $k_{2}=1$,
$\theta_{2}=0.8$, respectively. Thus, $m_{1}=22.5$ and $m_{2}=1.25$, and

$\sum_{r\in\mathcal{M}}\pi_{r}m_{r}\left(\beta(r)\frac{\gamma+\mu}{\nu+\gamma+\mu}-(\delta+\mu)\right)=-0.0812<0.$

Thus, in this case with the same parameters of the cases a) and b), we have
that the disease will go extinct almost surely, in the long run, stressing
again the relevance of the mean sojourn times to stem the epidemic.

##  8\. Conclusion

We have analyzed a SAIRS-type epidemic model with vaccination under semi-
Markov switching. In this model, the role of the asymptomatic individuals in
the epidemic dynamics and the random environment that possibly influences the
disease transmission parameters are considered. Under the assumption that both
asymptomatic and symptomatic infectious have the same transmission and
recovery rates, we have found the value of the basic reproduction number
$\mathcal{R}_{0}$ for our stochastic model. We have showed that if
$\mathcal{R}_{0}<1$ the disease will go extinct almost surely, while if
$\mathcal{R}_{0}>1$ the system is persistent in time mean. Then, we have
analyzed the model without restrictions, that is the transmission and recovery
rates of the asymptomatic and symptomatic individuals ca be possible
different. In this case, we have found two different sufficient conditions for
the almost sure extinction of the disease and a sufficient condition for the
persistence in time mean. However, the two regions of extinction and
persistence are not adjacent but there is a gap between them, thus we do not
have a threshold value dividing them.
In the case of disease persistence, by restricting the analysis to two
environmental states, under the Lie bracket conditions, we have investigated
the omega-limit set of the system. Moreover, we have proved the existence of a
unique invariant probability measure for the Markov process obtained by
introducing the backward recurrence process that keeps track of the time
elapsed since the latest switch. Finally, we have provided numerical
simulations to validate our analytical result and investigate the role of the
mean sojourn time in the random environments.

## Acknowledgments

This research was supported by the University of Trento in the frame “SBI-
COVID - Squashing the business interruption curve while flattening pandemic
curve (grant 40900013)”.

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