A sufficient condition for extinction and stability of a stochastic SIS
model with random perturbation
MOURAD EL IDRISSI, BILAL HARCHAOUI, ABDELADIM NAIT BRAHIM,
IBRAHIM BOUZALMAT, ADEL SETTATI, AADIL LAHROUZ
Department of Mathematics and Applications
Abdelmalek Essaadi University
Laboratory of mathematics and applications, FSTT, Abdelmalek Essaadi University, Tetouan, Morocco
MOROCCO
Abstract: The system dynamics of the randomly perturbed SIS depend on a certain threshold RS. If RS<1,
the disease is removed from our community, whereas an epidemic will occur if RS>1. However, what happens
when RS= 1? In this paper, we give a solution to this problem. Furthermore, we make some improvements to
the free disease equilibrium state E0when RS<1. Last, we give some computational simulations to explain our
results.
Key-Words: Stochastic epidemic models, SIS models, Stability of disease, Extinction of disease, Threshold,
Lyapunov function
1 Introduction
The standard SIS epidemic model is defined as the
following system
dS = (µ−µS −βSI +γI)dt,
dI = (−(µ+γ)I+βSI)dt, (1)
where Sand Iare the numbers of susceptible and
infected individuals, respectively. This model as-
sumes a vital dynamic with a mortality rate that cor-
responds to the birth rate, implying that S+I=
1. Besides, βis the rate of infection, and γis the
rate of recovery. A deterministic form of system (1)
given by the threshold R0=β
µ+γ[3]. In other
words, if R0≤1, then the free disease equilib-
rium state E0(1,0) is globally asymptotically stable.
While if R0>1,E0will become unstable, there is
an endemic state of equilibrium E∗1
R0
,R0−1
R0
that is globally asymptotically stable. During the
past few years, several mathematical programs for
transmission dynamics of infectious diseases have
been suggested [1, 2] such as (Susceptible-Infectious-
Susceptible), SEIR (Susceptible-Exposed-Infectious-
Recovered), SIRS (Susceptible-Infectious-Reduced-
Susceptible). The purpose of building these models
is to gain knowledge of the phenomenon of infec-
tious diseases and forecast the consequences of ap-
plying public health actions to reduce the propaga-
tion of diseases, which helps us plan successful strate-
gies for reducing the impact of infectious diseases.
The SIS models provide adequate classifications of
human population dynamics for particular bacterial
diseases such as malaria, some protozoan diseases
such as meningitis, and some sexually acquired dis-
eases such as tuberculosis (”gonorrhea”), where in-
dividuals usually build up their immunity to the dis-
ease over 24 hours and do not develop any resis-
tance to the disease when infected. There are various
forms of model SIS diseases in continuous determin-
istic and stochastic settings in the literature (see, e.g.,
[3, 4, 5, 6, 7, 8, 9, 10, 11]). In [4], CAI. has considered
the following stochastic form of model (1)
dS = (µ−µS −βSI +γI)dt
−σSIdB,
dI = (−(µ+γ)I+βSI)dt
+σSIdB.
(2)
According to the following initial conditions (S0, I0)
in the set ∆ = x∈R2
+;x1+x2= 1. Here,
Bis a Brownian motion on the probability space
(Ω,F,{Ft}t≥0,P)and σ > 0denotes the white
noise intensity. CAI. [4] has shown that the set ∆
is almost certainly positively invariant by the system
(2). Next, the authors studied the dynamic behavior
of I(t)as a function of the new stochastic thresh-
old RS=β
µ+γ+1
2σ2. They proved that if ei-
ther RS<1and β≥σ2or σ2> β ∨β2
2(µ+γ),
the disease will vanish. However, if RS>1, then
the disease will continue, and the model (2) will take
a unique stationary distribution. CAI. [4] also sug-
gested that if RS<1and β < σ2≤β2
2(µ+γ), then
the disease disappears with the probability of 1. Now,
Received: April 15, 2022. Revised: November 24, 2022. Accepted: December 19, 2022. Published: December 31, 2022.
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.40
Mourad El Idrissi, Bilal Harchaoui,
Abdeladim Nait Brahim, Ibrahim Bouzalmat,
Adel Settati, Aadil Lahrouz
E-ISSN: 2224-2678
367
Volume 21, 2022
in our work, we consider the following deterministic
system
dS =µ−µS −βS2I+γIdt,
dI =−(µ+γ)I+βS2Idt. (3)
Using the technique of perturbation on the parameter
β, we get the following stochastic form of the deter-
ministic model (3)
dS =µ−µS −βS2I+γIdt
−σS2IdB,
dI =−(µ+γ)I+βS2Idt
+σS2IdB.
(4)
Therefore, it is enough to study the SDE for I(t)
dI =−(µ+γ)I+β(1 −I)2Idt
+σ(1 −I)2IdB, (5)
≜f1(I)dt +f2(I)dB(t).
For any twice continuously differentiable V(.), the
formula of Itô associated with (4) is defined by
dV (X) = LV(X)dt +f2(X)∂V (X)
∂X dB(t),
where
LV(X) = f1(X)∂V (X)
∂X +1
2f2
2(X)∂2V(X)
∂X2,
is the generator of the process X∈(0,1).
In this article, we assume that β≥σ2is not exactly a
limitation because it indicates that the estimation error
σ2is smaller than the estimated value β. We inves-
tigated the case where RS≤1. More precisely, we
show that if RS<1, the equilibrium state without
disease E0is κ-th exponentially stable moment. If
RS= 1,E0is exponentially stable. Furthermore, the
disease will be extinct on average.
2 Stability of disease
In this section, we will investigate the stability of the
disease in the SDE system (4) to give the stochastic
threshold condition for disease control or elimination.
Theorem 2.1 Let (S0, I0)∈∆. If RS<1, then for
every κsuch that
0< κ < 2β
σ21
RS
−1,(6)
the solution I(t)satisfies
E(Iκ(t)) ≤Iκ
0e−ξt,
where
ξ=−κβ1−1
RS+1
2κσ2>0.(7)
Thus, the disease-free equilibrium state E0is κ-th mo-
ment exponentially stable.
Proof 1 Let the function of Lyapunov V(I) = Iκ,
where κ > 0is real constants check the condition
(6). By the formula of Itô, we obtain
dV (I) = LV(I)dt +κσ(1 −I)2IκdB, (8)
or
LV=κIκ−(µ+γ) + β(1 −I)2
+1
2σ2(κ−1) (1 −I)4,
≤κIκsup
0<x≤1−(µ+γ) + βx2−1
2σ2x4
+κ
2σ2,
≜κIκh(x) + κ
2σ2.(9)
We can show clearly that if β≥σ2and RS<1, then
h(x) = sup
0<x≤1−(µ+γ) + βx2−1
2σ2x4,
=β1−1
RS.(10)
Combining this with (9), we get
LIκ(t)≤ −ξIκ(t),
where ξis given in (7). Injecting it into (8), then inte-
grating the result and taking the expectations on both
sides, we get
E(Iκ(t)) ≤Iκ
0−ξt
0
E(Iκ(u)) du,
which implies with the Gronwall inequality that
E(Iκ(t)) ≤Iκ
0e−ξt.
The proof is finished.
Now, we will study the extinction of disease.
3 Extinction of disease
The following theorems discuss the situation where
the stochastic threshold RS= 1.
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DOI: 10.37394/23202.2022.21.40
Mourad El Idrissi, Bilal Harchaoui,
Abdeladim Nait Brahim, Ibrahim Bouzalmat,
Adel Settati, Aadil Lahrouz
E-ISSN: 2224-2678
368
Volume 21, 2022
Theorem 3.1 For any initial value (S0, I0)∈∆, if
RS= 1, then the solution of equation (5) follows
lim sup
t→∞
1
tt
0
I(s)ds = 0.(11)
Proof 2 Let (S0, I0)∈∆and the Lyapunov function
V(I) = log(I).
Using the formula of Itô, the equation (5), I≤1, the
binomial formula of Newton, and RS= 1, we have
dV =−(µ+γ) + β(1 −I)2−1
2σ2(1 −I)4dt
+σ(1 −I)2dB,
≤ − β−4σ2Idt +σ(1 −I)2dB. (12)
By integrating (12) from 0to t, we obtain
log I(t)≤log I(0) −β−4σ2t
0
I(s)ds
+σt
0
(1 −I(s))2dBs.(13)
According to the theorem of large numbers for mar-
tingales, there is a Ω1⊂Ωwith P(Ω1) = 1, so that
for every ω∈Ω1and ϵ > 0, there is T(w, ϵ)such
that for all t≥T, we obtain
log I(0) + σt
0
(1 −I(s))2dBs≤ϵt,
which implies with (13) that
eϵt ≥I(t)e
(β−4σ2)t
0
I(s)ds
,(14)
≜1
(β−4σ2)
d
dt
e
(β−4σ2)t
0
I(s)ds
.
By integrating (14) from Tto tand multiplying both
sides by 1
t, one obtains
1
tt
0
I(s)ds ≤1
(β−4σ2)tlog e(β−4σ2)∫T
0I(s)ds
+β−4σ2
ϵeϵT −eϵt.(15)
Hence, applying the rule of Hospital on (15), we get
lim sup
t→∞
1
tt
0
Is(ω)ds ≤ϵ
β−4σ2.
Letting ε→0, we obtain the requested result (11).
Theorem 3.2 Let (S0, I0)∈∆. If RS= 1, then for
all n > 0and ε > 0, we obtain
lim
I0→0
Psup
0≤t≤n
I(t)> ε= 0,(16)
that is, the disease-free steady state E0is stable in
probability.
Proof 3 Let κ≤1,(S0, I0)∈∆and the Lyapunov
function
V(I(t)) = Iκ(t).
Using the formula of Itô, (8), (9), (10), and RS= 1,
we obtain
dV (I(t)) ≤κ2
2σ2Iκ+κσ(1 −I)2IκdB.
Integrating this inequality between (0, t), and using
I≤1, we can have easily for κ≤1
Iκ(t)−Iκ(0) ≤κ2
2σ2t
+κσ t
0
(1 −I(s))2Iκ(s)dBs,
thus
sup
0≤t≤n
Iκ(t)≤Iκ(0) + κ2
2σ2n
+κσ sup
0≤t≤nt
0
(1 −I(s))2IκdBs.
By I < 1, we obtain
Psup
0≤t≤n
I(t)> ε≤IIκ
0≥ε
3+Iκ2
2σ2n≥ε
3
+Pκσ sup
0≤t≤n
Mt>ε
3,
where IAdenotes the characteristic function of A and
Mt=t
0
(1 −I(s))2Iκ(s)dBs,
which implies that
l=lim
I0→0
Psup
0≤t≤n
I(t)> ε,
≤Iκ2
2σ2n≥ε
3
+lim
I0→0
Pκσ sup
0≤t≤n
Mt>ε
3.(17)
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DOI: 10.37394/23202.2022.21.40
Mourad El Idrissi, Bilal Harchaoui,
Abdeladim Nait Brahim, Ibrahim Bouzalmat,
Adel Settati, Aadil Lahrouz
E-ISSN: 2224-2678
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Or, Mtis a continuous real-valued martingale, hence
by the inequality of Doob, we obtain
P=Pκσ sup
0≤t≤n
Mt>ε
3,
≤9κ2σ2
ε2Eη
0
(1 −I(s))2Iκ(s)dBs2,
≜9κ2σ2
ε2Eη
0(1 −I(s))2Iκ(s)2ds,
≤9κ2σ2
ε2η.
Using it in combination with (17), we get
lim
I0→0
Psup
0≤t≤n
I(t)> ε≤Iκ2σ2n
2≥ε
3
+9κ2σ2η
ε2.
By letting κ→0, we get the required statement (16).
4 Simulation
The following simulation illustrates that if RS= 1,
the stochastic disease will die when the deterministic
illness occurs.
Figure 1: Single path computer simulation of I(t)
for the SDE model (5) with initial condition I0= 0.4
and its related deterministic model for the parameters:
µ= 0.5,β= 0.902,γ= 0.4,σ= 0.2, then R0>1
and RS= 1.
5 Conclusion
This article studied a stochastic SIS epidemiological
model with a constant population size under white
noise control. We discussed the behavior of the
stochastic epidemiological SIS model over the long
term. We show sufficient conditions for the extinction
and stability of the disease. The stochastic popula-
tion model provides one of several possible stochastic
forms of the deterministic model. This model is gen-
eralizable. The argument is that the population can
experience sudden changes in its parameters [5].
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DOI: 10.37394/23202.2022.21.40
Mourad El Idrissi, Bilal Harchaoui,
Abdeladim Nait Brahim, Ibrahim Bouzalmat,
Adel Settati, Aadil Lahrouz
E-ISSN: 2224-2678
370
Volume 21, 2022
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Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
Mourad El idrissi and Adel Settati: carried out the
conceptualization, validation, formal analysis, writ-
ing - original draft, methodology, writing - review &
editing.
Bilal Harchaoui and Aadil Lahrouz: have imple-
mented the software, formal analysis, writing - origi-
nal draft, writing - review & editing.
Abdeladim Nait Brahim and Ibrahim Bouzalmat:
were responsible for validation, investigation, con-
ceptualization, writing - review & editing.
Sources of funding for research
presented in a scientific article or
scientific article itself
First, we are invited by your mathematical journals to
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International , CC BY 4.0)
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DOI: 10.37394/23202.2022.21.40
Mourad El Idrissi, Bilal Harchaoui,
Abdeladim Nait Brahim, Ibrahim Bouzalmat,
Adel Settati, Aadil Lahrouz
E-ISSN: 2224-2678
371
Volume 21, 2022
