Examining the Relationship Between Infection Power Rate and the

Critical Threshold in Stochastic SIS Epidemic Modeling

B. HARCHAOUI, M. EL IDRISSI, A. EL HAITAMI, A. NAIT BRAHIM, A. SETTATI,

A. LAHROUZ, M. EL JARROUDI, M. ER-RIANI, T. AMTOUT

Department of Mathematics and Applications

Abdelmalek Essaadi University

Laboratory of Mathematics and Applications, FSTT, Abdelmalek Essaadi University, Tetouan, Morocco

MOROCCO

Abstract: The stochastic SIS epidemic model is well-known for its critical threshold Rs, indicating the transition

between disease eradication (Rs<1) and epidemic outbreaks (Rs>1). However, the scenario where Rs= 1

has been uncertain. We present a definitive resolution to this pivotal issue. Additionally, we introduce advance-

ments in analyzing the disease-free state of equilibrium when Rs<1to deepen our understanding of the system

dynamics. To validate our theoretical developments and provide visual evidence, extensive computer simulations

are conducted, enhancing the comprehensiveness and applicability of our findings to the broader field of epi-

demiology and infectious disease modeling. The implications of our results extend to public health policies and

interventions aimed at effectively managing and controlling infectious diseases in different communities where

Rshovers around the critical value.

Key-Words: SIS model, Stochastic epidemic models, Extinction of disease, Stability of disease, Lyapunov

function, Threshold.

Received: May 9, 2022. Revised: July 17, 2023. Accepted: August 14, 2023. Published: September 7, 2023.

1 Introduction

In the early 20th century, a transformative shift oc-

curred in epidemiology with the pioneering work of

renowned scientists, including Anderson Gray McK-

endrick and Janet-Leigh Claypon, who introduced

mathematical modeling. Since then, mathematical

modeling has become an indispensable tool, pro-

foundly impacting outbreak and epidemic manage-

ment and playing a crucial role in guiding evidence-

based public health interventions. Epidemiology has

evolved through the contributions of notable physi-

cians like Quinto Tiberio Angelerio, who skillfully

handled the plague outbreak in Alghero, Sardinia, for

1582. However, it was during the 19th century that

modern epidemiology as a scientific discipline began

to take shape. Often regarded as the father of mod-

ern epidemiology, John Snow made a pivotal discov-

ery when he traced a devastating cholera outbreak in

London to the contaminated water from the Broad

Street pump. This groundbreaking investigation can

be considered the seminal event that laid the foun-

dation for the science of epidemiology as we know

it today. Epidemiology is a scientific discipline that

delves into the study of epidemics, diseases in gen-

eral, and even health-related conditions unrelated to

diseases. The origins of this field can be traced back

to ancient Greece, where the renowned physician Hip-

pocrates of Kos made significant contributions by dis-

tinguishing between epidemic and endemic diseases.

Epidemiology extends beyond human health, exam-

ining diseases affecting plants and domestic and live-

stock animals. An epidemic is an abnormal and sub-

stantial upsurge of a particular disease within a pop-

ulation that occurs relatively quickly. The intricate

process of disease transmission involves many influ-

ential factors, encompassing both the characteristics

of the infectious agent and the complexities of the

host population. Regarding the infectious agent, its

inherent properties, such as the mode of transmission

(e.g., respiratory droplets, direct contact), the duration

of infectivity, and its response to medical interven-

tions like treatments and vaccines, are critical deter-

minants of its spread among individuals. Equally sig-

nificant are the host population factors that contribute

to the dynamics of an epidemic. Social interactions,

demographics (e.g., age, gender), cultural practices,

geographic distribution, and economic conditions all

play pivotal roles in shaping the vulnerability and re-

silience of a population to the disease. Throughout

the annals of recorded history, human civilization has

grappled with recurrent epidemics and pandemics.

These infectious disease outbreaks have inflicted pro-

found human suffering, societal turbulence, and eco-

nomic disruptions. In light of such formidable chal-

lenges, accurately predicting outbreak progression is

paramount to effectively mitigating their adverse im-

pacts. Central to this endeavor lies the field of epi-

demiologic modeling, which serves as a fundamental

WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE

DOI: 10.37394/23208.2023.20.8

B. Harchaoui, M. El Idrissi,

A. El Haitami, A. Nait Brahim, A. Settati,

A. Lahrouz, M. El Jarroudi, M. Er-Riani, T. Amtout

E-ISSN: 2224-2902

Volume 20, 2023

tool in understanding the dynamics of disease trans-

mission and devising informed strategies for contain-

ment and prevention. The impact of infectious dis-

eases on human life is enormous. Each year, mil-

lions of people suffer or die from different diseases.

There are many questions about the propagation of

diseases. For instance, how many people will be

touched together and thus need treatment? How long

will the disease outbreak last? What is the potential

for a vaccination strategy to reduce the seriousness

of the epidemic? In epidemiology, the importance of

mathematical modeling is significant because it can

provide information about the mechanisms underly-

ing the propagation of diseases and propose strate-

gies for their control. The first mathematical model

of epidemiology known was developed and solved

by Daniel Bernoulli in 1760. Using compartmental-

ized models, the basics of modern mathematical epi-

demiology were established in the early 20th century,

[1]. Mathematical epidemiology expanded exponen-

tially. Various mathematical models were articulated

and discussed, [2], [3], [4], [5], [6]. The classical SIS

model for a constant population is

dS = [µ(1 −S) + γI −βSI]dt,

dI = [βSI −(µ+γ)I]dt. (1)

Nonetheless, real-world scenarios are often charac-

terized by abundant stochastic elements and precar-

ious occurrences. This underscores the significance

of employing stochastic calculus as a robust method-

ology for elucidating these genuine random phenom-

ena manifesting in the natural world. The integration

of stochasticity into epidemic models and population

dynamics has yielded compelling findings of notable

consequence. The technique of parameter perturba-

tion has been widely adopted by various researchers,

[7], [8], [9], [10], [11], [12], to enhance their in-

vestigations. The studies, [6], [13], [14], specifi-

cally delved into color noise effects. Their studies

meticulously scrutinized the intricate dynamics of an

epidemic model under finite regime-switching condi-

tions. Motivated by the insights gained from these

preceding endeavors, our current research focuses on

the influence of environmental fluctuations, [4], [6],

[15], [16], [17], [18], [19]. These fluctuations are hy-

pothesized to manifest as variations in the parameter

βwithin the deterministic model (1) discussed earlier.

As such, this study aims to unravel the implications of

integrating these fluctuations. So that

β−→ β+σdB

dt .

We establish the ensuing stochastic model by assim-

ilating the described perturbations into the determin-

istic framework presented as equation (1).

dS = [µ(1 −S) + γI −βSI]dt −σSIdB,

dI = [βSI −(µ+γ)I]dt +σSIdB. (2)

In this work, we study the stochastic SIS epidemic

model with the non-linearity power function, that is











dS(t) = [µ(1 −S) + γI −βSpI]dt

−σSpIdB(t),

dI(t) = [βSpI−(µ+γ)I]dt

+σSpIdB(t),

(3)

where p≥1is the non-linearity power constant.

Therefore, it is enough to study the SDE for I(t)

dI(t)=[β(1 −I(t))pI(t)−(µ+γ)I(t)] dt

+σ(1 −I(t))pI(t)dB(t),(4)

where Sand Iare the numbers of susceptible and in-

fected individuals. This model assumes a vital dy-

namic with a mortality rate corresponding to the birth

rate, implying that S+I= 1. Besides, the param-

eter denoted as βpertains to the infection rate, rep-

resenting the rate at which new infections are con-

tracted. Conversely, γcorresponds to the recovery

rate, delineating the pace at which individuals recu-

perate from the disease. A deterministic form of (1)

is given R0=β

µ+γ. When the basic reproduction

number R0≤1, the globally asymptotically stable

condition is achieved at the disease-free equilibrium

state E0(1,0). Conversely, when R0>1, the stabil-

ity of E0is compromised, leading to the emergence

of an endemic equilibrium state E∗1

R0,R0−1

R0

which attains global asymptotic stability. In the

past few years, a number of mathematical programs

for the transmission dynamics of infectious diseases

have been proposed, [2], [4], [5], [6], [7], [8], [13],

[20], [21], [22], [23], as SIS (Susceptible-Infectious-

Susceptible), SIRS (Susceptible-Infectious-Reduced-

Susceptible). The SIS models provide proper catego-

rizations of human population dynamics for specific

bacterial diseases such as malaria, certain protozoan

diseases such as meningitis and some sexually dis-

eases such as tuberculosis (”gonorrhea”) in which in-

dividuals generally build up immunity to the disease

within 24 hours and do not develop resistance to the

disease when infected. Based on 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 > 0indicates the in-

tensity of the white noise. Next, the authors investi-

gated the dynamic behavior of I(t)as a function of

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DOI: 10.37394/23208.2023.20.8

B. Harchaoui, M. El Idrissi,

A. El Haitami, A. Nait Brahim, A. Settati,

A. Lahrouz, M. El Jarroudi, M. Er-Riani, T. Amtout

E-ISSN: 2224-2902

Volume 20, 2023

the new threshold

Rs=β

µ+γ+1

2σ2.

They proved that if either Rs<1and β≥σ2or

σ2> β ∨β2

2(µ+γ), the disease will disappear. How-

ever, if Rs>1, then the disease will persist. In

[24], the author also suggested that if Rs<1and

β < σ2≤β2

2(µ+γ), then the disease disappears with

the probability 1. In this paper, we will assume that

β≥1

2σ2is not exactly a limitation because it indi-

cates that the estimation error σ2is smaller than the

estimated value β. We investigated the case where

Rs≤1. More precisely, we prove if Rs= 1,E0is

exponentially stable. Furthermore, the disease is ex-

tinct in the mean. The manuscript follows a structured

approach, beginning with Section 2, which presents

exponentially stable for the system described in equa-

tion 3. In Section 3, we establish sufficient condi-

tions for disease extinction. Subsequently, Section 4

provides an in-depth discussion of our theoretical dis-

coveries and includes numerical simulations to illus-

trate them. Finally, a concise conclusion succinctly

outlines the primary contributions made in this study.

2 Exponentially stable

In this section, our objective is to analyze the stability

of the disease within the SDE system (3) and establish

the stochastic threshold condition for disease control

or eradication.

Theorem 2.1 Let (S0, I0)∈∆. If Rs<1, then for

any nsuch as

0< n < 2βσ−2R−1

S−1,(5)

the solution I(t)meets

E(In(t)) ≤In

0exp (−ct),

where

c=−nβ1− R−1

S+n

2σ2>0.(6)

Thus, the disease-free equilibrium state E0is n-th

moment exponentially stable.

Proof. Let V1(I) = In,n > 0be any real constant

verifying the condition (5). By the Itô formula we get

for p > 1

dV1(I) = LIndt +nσ(1 −I)pIndB, (7)

and

LV1(I) = nIn−(µ+γ) + β(1 −I)p

2σ2(n−1) (1 −I)2p,

≤nInsup

0<x≤1−(µ+γ) + βxp

−1

2σ2x2p+n

2σ2.(8)

We can show clearly that if β≥1

2σ2and Rs<1then

sup

0<x≤1−(µ+γ) + βxp−1

2σ2x2p

=β1−1

Rs,(9)

Combining this with (8) we get

LIn(t)≤ −cIn(t),

where ccan be found in (6). By injecting it into (7),

then integrating the result and taking the expectations

on both sides yields

EV1(I(t)) ≤V1(I(0)) −ct

EV1(I(u))du,

which implies with the Gronwall inequality that

E(In(t)) ≤In

0exp (−ct).

3 Extinction of the disease

The subsequent theorems address the scenario where

the stochastic threshold Rs= 1.

Theorem 3.1 For a given initial value (S0, I0)∈∆,

if Rs= 1, then the solution of the equation (3) follows

lim

t→∞

tt

I(s)ds = 0.(10)

Proof. Let V2(I) = log(I). With the equation (3),

I≤1,Rs= 1, and the formula of Itô we get

dV2(I) = −(µ+γ) + β(1 −I)p

−σ2

2(1 −I)2pdt +σ(1 −I)pdB,

≤ −pβ−σ2Idt +σ(1 −I)pdB.

(11)

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By integrating (11) and using Newton’s binomial for-

mula we obtain

log I(t)≤ −pβ−σ2t

I(s)ds +log I(0)

+σt

(1 −I(s))pdBs.(12)

By the large number theorem for martingales it exists

Ω1⊂Ωwith P(Ω1)=1, so that for all ω∈Ω1and

ϵ > 0, there exist T(w, ϵ)such that for any t≥Twe

get

log I(0) + σt

(1 −I(s))pdB(s)≤ϵt,

which implies with (12) that

p(β−σ2)

dt exp pβ−σ2t

I(s)ds

=I(t)exp pβ−σ2t

I(s)ds,

≤exp (ϵt).(13)

By integrating (13) from Tto tand multiplying the

two sides by 1

twe have

tt

I(s)ds

≤1

p(β−σ2)tlog exp pβ−σ2T

I(s)ds

ϵβ−σ2exp (ϵt)−exp (ϵT ).(14)

Then, by applying twice the rule of the Hospital on

(14) we get

lim sup

t→∞

tt

Is(ω)ds ≤ϵ

p(β−σ2).

By letting ε→0, the desired result (10) is obtained.

Theorem 3.2 Let (S0, I0)∈∆. If Rs= 1, then for

any η > 0and ε > 0we have

lim

I0→0

Psup

0≤t≤η

I(t)> ε= 0,(15)

that is the disease-free steady state E0is stable in

probability.

Proof. If Rs= 1, then using the formula of Itô, (7),

(8) and (9) we get

dV1(I(t)) ≤1

2n2σ2In+nσ(1 −I)pIndB,

Integrating between 0and t, it is easy to have for

n≤1

In(t)−In(0) ≤1

2n2σ2t

+nσ t

(1 −I(s))pIn(s)dBs,

then

sup

0≤t≤η

In(t)≤In(0) + 1

2n2σ2η

+nσ sup

0≤t≤ηt

(1 −I(s))pIn(s)dB(s).

Using I < 1we get

Psup

0≤t≤η

I(t)> ε≤IIn(0)≥ε

3+In2

2σ2η≥ε

+Pnσ sup

0≤t≤η

Mt>ε

3,

where IAis the characteristic function of A and

Mt=t

(1 −I(s))pIn(s)dB(s),

which implies that

lim

I0→0

Psup

0≤t≤η

I(t)> ε(16)

≤In2

2σ2η≥ε

+lim

I0→0

Pnσ sup

0≤t≤η

Mt>ε

3.

Moreover, Mtis a real-valued continuous martingale,

so by Doob inequality we get

Pnσ sup

0≤t≤η

Mt>ε

3

≤9n2σ2

ε2Eη

(1 −I(s))pIn(s)dBs2,

=9n2σ2

ε2Eη

((1 −I(s))pIn(s))2ds,

≤9n2σ2

ε2η.

Combining it with (16) we have

lim

I0→0

Psup

0≤t≤η

I(t)> ε≤In2

2σ2η≥ε

+9n2σ2

ε2η.

By letting n→0we obtain the requested formulation

(15).

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B. Harchaoui, M. El Idrissi,

A. El Haitami, A. Nait Brahim, A. Settati,

A. Lahrouz, M. El Jarroudi, M. Er-Riani, T. Amtout

E-ISSN: 2224-2902

Volume 20, 2023

4 Simulation

To validate the robustness of our discoveries, we

intend to carry out numerical simulations employ-

ing the Milstein scheme as our chosen computational

approach, as outlined in [25]. More precisely, we

plan to discretize equation (3) utilizing the subsequent

scheme:











Sk+1 =Sk+µk−µkSk−βkSp

kIk+γkIk∆t

−σkSp

kIk√∆tτk−σ2

2Sp

kIk(τ2

k−1)∆t,

Ik+1 =Ik+−(µk+γk)Ik+βkSp

kIk∆t

+σkSp

kIk√∆tτk+σ2

2Sp

kIk(τ2

k−1)∆t.

In this context, τk, (k= 1,2, ...) symbolizes an uncor-

related stochastic variable adhering to standard nor-

mal distributions, specifically represented as N(0,1).

Example 1 We choose µ= 0.5,β= 0.92,γ= 0.4,

σ= 0.2,p= 1,I0= 0.4, then

R0>1,Rs= 1.

Consequently, based on Theorems 3.2, 3.1, and Fig.

1, the stochastic disease will be removed from the pop-

ulation while the deterministic disease occurs.

Example 2 Set µ= 0.4,β= 0.9,γ= 0.45,σ= 0.5,

p= 1,I0= 0.4, then

R0>1,Rs<1.

Thus, based on Theorem 2.1, and Fig. 2, the disease-

free equilibrium state E0is n-th moment exponen-

tially stable.

0 500 1000 1500 2000 2500 3000 3500 4000

Time(t)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

I(t)

Fig. 1. Trajectories of I(t)for SDE model 3 by

using the parameters of Example 1.

0 500 1000 1500 2000 2500 3000 3500 4000

Time(t)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

I(t)

Fig. 2. Trajectories of I(t)for SDE model 3 by

using the parameters of Example 2.

5 Conclusion

This study rigorously examined a stochastic SIS epi-

demiological model within a constant-sized popula-

tion under white noise control. Through our anal-

ysis, we have comprehensively explored the long-

term behavior of the SIS stochastic epidemiological

model and drawn significant insights. Our findings

provide compelling evidence for the overall stabil-

ity of the disease dynamics and the eventual extinc-

tion of the disease within the population. This indi-

cates that the control measures implemented, repre-

sented by the white noise control in this context, have

effectively mitigated disease-sustained transmission,

leading to disease-free equilibrium over time. The

implications of our research extend to public health

and disease management strategies. By elucidating

the model stability and extinction properties, we of-

fer valuable guidance for designing effective control

interventions and optimizing resource allocation to

combat infectious diseases in similar settings. Nev-

ertheless, we acknowledge that the stochastic nature

of infectious disease dynamics poses inherent com-

plexities. Future research could explore further re-

finements to the model, considering additional factors

such as demographic heterogeneity, spatial interac-

tions, and temporal variations in control measures to

capture real-world epidemiological scenarios better.

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DOI: 10.37394/23208.2023.20.8

B. Harchaoui, M. El Idrissi,

A. El Haitami, A. Nait Brahim, A. Settati,

A. Lahrouz, M. El Jarroudi, M. Er-Riani, T. Amtout

E-ISSN: 2224-2902

Volume 20, 2023

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A. El Haitami, A. Nait Brahim, A. Settati,

A. Lahrouz, M. El Jarroudi, M. Er-Riani, T. Amtout

E-ISSN: 2224-2902

Volume 20, 2023

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Contribution of individual authors to

the creation of a scientific article

(ghostwriting policy)

Bilal Harchaoui, Mourad El Idrissi: were responsi-

ble for the conceptualization, validation, formal anal-

ysis, writing - original draft, methodology, writing

- review & editing. Adil El Haitami, Abdeladim

Nait Brahim: have implemented the software, for-

mal analysis, writing - original draft, writing - review

& editing. Adel Settati, Aadil Lahrouz, Mustapha

El Jarroudi: have implemented the software, formal

analysis, writing - original draft, writing - review &

editing. Mustapha Er-Riani, Tarik Amtout: Car-

ried out the validation, investigation, conceptualiza-

tion, writing - review & editing.

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To begin with, we have received an invitation from

your esteemed mathematical journals to submit our

work. Additionally, the authors declare that they have

no identifiable financial conflicts of interest or per-

sonal relationships that could influence the findings

and conclusions presented in this article.

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International , CC BY 4.0)

This article is published under the terms of the Cre-

ative Commons Attribution License 4.0

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE

DOI: 10.37394/23208.2023.20.8

B. Harchaoui, M. El Idrissi,

A. El Haitami, A. Nait Brahim, A. Settati,

A. Lahrouz, M. El Jarroudi, M. Er-Riani, T. Amtout

E-ISSN: 2224-2902

Volume 20, 2023