Analysis of a Deterministic and a Stochastic SIS Epidemic Model with Double Epidemic Hypothesis and Specific Functional Response

Naim, Mouhcine; Lahmidi, Fouad

doi:https://doi.org/10.1155/2020/5362716

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2020 | Article ID 5362716 | https://doi.org/10.1155/2020/5362716

Analysis of a Deterministic and a Stochastic SIS Epidemic Model with Double Epidemic Hypothesis and Specific Functional Response

Mouhcine Naim¹and Fouad Lahmidi¹

Academic Editor: Garyfalos Papashinopoulos

Received17 Mar 2020

Accepted11 May 2020

Published26 May 2020

Abstract

The purpose of this paper is to investigate the stability of a deterministic and stochastic SIS epidemic model with double epidemic hypothesis and specific nonlinear incidence rate. We prove the local asymptotic stability of the equilibria of the deterministic model. Moreover, by constructing a suitable Lyapunov function, we obtain a sufficient condition for the global stability of the disease-free equilibrium. For the stochastic model, we establish global existence and positivity of the solution. Thereafter, stochastic stability of the disease-free equilibrium in almost sure exponential and th moment exponential is investigated. Finally, numerical examples are presented.

1. Introduction

Epidemiology is the study of the spread of infectious diseases with the objective to trace factors that are responsible for or contribute to their occurrence. Mathematical modeling has become an important tool in analyzing the epidemiological characteristics of infectious diseases and can provide useful control measures (see, for example, [1–5]).

In classical epidemic models, the susceptible individuals can be infected with only a disease. In the real world, the susceptible individuals can be infected by two or more kinds of diseases at the same time such as HBV coinfection with HCV and HDV and HIV coinfection with HBV, HCV, and TB. Recently, the authors of [6–9] investigated the epidemic model SIS (where infection with the disease does not confer permanent immunity against reinfection so that those who survived the infection revert to the class of wholly susceptible individuals [10]) with double epidemic hypothesis which has two epidemic diseases caused by two different viruses. In this paper, we consider a deterministic SIS model with double epidemic hypothesis described by the following differential system:where represents the number of susceptible at time , and are the total population of the infected with virus and at time , respectively, represents the recruitment rate of the population, is the natural death rate of the population, is the treatment cure rate of the disease caused by virus , is the disease-related death rate, and is the infection coefficient, . The incidence rate of disease is modeled by the specific functional response , where are saturation factors measuring the psychological or inhibitory effect. This specific functional response was introduced by Hattaf et al. [11], and here, it becomes to be a bilinear incidence rate if , a saturated incidence rate if or , a Beddington–DeAngelis functional response [12, 13] if , and a Crowley–Martin functional response [14] if , .

In the reality, epidemic systems are inevitably affected by environmental white noise. Therefore, it is necessary to study how the noise influences the epidemic models. Consequently, many authors have studied stochastic epidemic models, see, e.g., [15–17]. For this, we consider the case in which the rates () are subject to random fluctuations, namely, is replaced by , where () are independent standard Brownian motions, and represents the intensity of for . Therefore, the corresponding stochastic system to (1) can be described by the following Itô equations:with , .

The rest of this paper is organized in the following manner. In Section 2, we present a local stability analysis of the equilibria and a global stability analysis of the disease-free equilibrium for the deterministic model (1). In Section 3, we prove that the stochastic model (2) has a unique global positive solution, and we give sufficient conditions for the almost sure exponential stability and the th moment exponential stability of the disease-free equilibrium. Numerical examples will be presented in Section 4. Finally, we close the paper with a brief conclusion.

2. Deterministic SIS Epidemic Model

For biological reasons, we assume that the initial conditions of system (1) satisfy

Thus, system (1) is positive [18], that is, , and for all . In fact, by Proposition 2.1 in [19], we have

By summing all the equations of system (1), we find that the total population size satisfies the inequalitywhich ensures that if . The standard comparison theorem [20] can be used to deduce that

Thus, the feasible solution set of the system equation of model (1) enters and remains in the region

Therefore, model (1) is well posed epidemiologically and mathematically [21]. Hence, it is sufficient to study the dynamics of model (1) in .

It is easy to see that system (1) has a disease-free equilibrium state . Therefore, the basic reproduction number iswhere

We mention that the expressions of and can also be obtained by applying the next generation matrix method provided by van den Driessche and Watmough [22].

Now, we investigate the local stability of the disease-free equilibrium . The Jacobian matrix of system (1) at the equilibrium is as follows:

The three eigenvalues of are , , and . Hence, the equilibrium will be locally asymptotically stable if and unstable when .

The following theorem discusses the global stability of the disease-free equilibrium .

Theorem 1. If , then the disease-free equilibrium of (1) is globally asymptotically stable in .

Proof. Let be the Lyapunov function defined asDifferentiating with respect to along the positive solutions of system (1), we getWe haveSince and the functions are increasing, then . Thus,Therefore, ensures that . Suppose that is a solution of (1) contained entirely in the set . Then, . We discuss four cases:

Case 1. If and , thenFrom the second and third equations of (1), we have , which implies, according to (15), that . On the other hand, solutions of (1) contained in the plane satisfy , which implies that as .

Case 2. If and , then andThen, implies that and consequently . Suppose that ; then, . Hence, ; then, which is a contradiction. Then, .

Case 3. The case and is analogue to the previous case.

Case 4. If and , then such that , . Hence, , and by the same analysis in Case 2, we obtain that .

Hence, by LaSalle’s invariance principle [23], every solution to equations of system (1), with initial conditions in , approaches as . Thus, is globally asymptotically stable.

Now, if , then system (1) has the disease-free equilibrium for , , wherewith and

Theorem 2. If and , then the equilibrium is locally asymptotically stable.

Proof. The Jacobian matrix of system (1) at the equilibrium is determined bywhereClearly, is an eigenvalue of . Since because and the function is increasing, then . Hence, if . The other two eigenvalues of are determined by the following equation:whereSince , then and . Thus, by the Routh–Hurwitz criterion, the eigenvalues of have negative real part. Therefore, the equilibrium of system (1) is asymptotically stable if and .
Furthermore, if , then system (1) has the disease-free equilibrium for , , wherewith and

Theorem 3. If and , then the equilibrium is locally asymptotically stable.

Proof. It is analogue to the previous proof.
Next, we investigate the local stability of system (1) at both-endemic equilibrium . To obtain conditions for the existence of the equilibrium , system (1) is rearranged to get and which givesWe have if for , and . In addition, is given by the following cubic equation:whereWith the help of Descartes’ rule of signs [24], equation (26) has a unique positive real root if any one of the following holds:(i) and (ii) and (iii) and Hence, system (1) has a unique positive equilibrium if for , one of the conditions , , and hold true, and .
The Jacobian matrix of system (1) at the equilibrium is determined bywhere

Theorem 4. The endemic equilibrium is locally asymptotically stable if it exists.

Proof. The characteristic equation of Jacobian matrix can be written aswhereNote thatThen, it is easy to show that , , , and . Thus, by the Routh–Hurwitz criterion, all roots of (30) have negative real part. Therefore, the equilibrium of system (1) is asymptotically stable.

3. Stochastic SIS Epidemic Model

Let be a complete probability space with filtration satisfying the usual conditions (i.e., it is increasing and right continuous while contains all -null sets). We consider the following stochastic differential system:where , represents the initial value, and and are locally Lipschitz functions in . is an -dimensional standard Wiener process defined on the above probability space.

Let us suppose that for all so that zero of is an equilibrium point of system (33).

Definition 1. (see [25]). The trivial solution of system (33) is said to be almost surely exponentially stable if for all , we have

Denote by the family of all nonnegative functions defined on such that they are continuously twice differentiable in and once in . Denote by the mathematical expectation of a random variable . If acts on a function , thenwhere , , and .

By Itô’s formula, we have

Lemma 1. (see [26]). Suppose there exists a function satisfying the inequalitieswhere and are positive constants. Then, the equilibrium of system (33) is th moment exponentially stable. When , it is usually said to be exponentially stable in mean square, and the equilibrium is globally asymptotically stable.

3.1. Existence and Uniqueness of the Global Positive Solution

The following theorem shows that the solution of our system (2) is global and positive.

Theorem 5. For any initial value , there is a unique solution to (2) on , and this solution remains in with probability one.

Proof. Let . The total population in system (2) verifies the equationIf for all (a.s.), then we getHence, by integration, we haveThen, (a.s.), soSince the coefficients of system (2) are locally Lipschitz continuous, then by the work of Mao [25] for any initial value , there is a unique local positive solution on , where is the explosion time. To show that this solution is global, we only need to prove (a.s.).
Let such that . For , we define the stopping time:Then,Consider the function defined for byCalculating the differential of along the solution trajectories of system (2) and using Itô’s formula, for all and , we getwhereAccording to (41), we have for all (a.s.). Hence,Therefore,whereIntegrating both sides of (48) from 0 to and after taking the expectation on both sides, we obtain thatSince , thenwhere is the indicator function of . Note that there are some components of equal to . Therefore,Thus,By combining (50) and (53), we get that, for all ,Extending to 0, we obtain for all , . Hence, . As , then (a.s.) which completes the proof.

3.2. Almost Sure Exponential Stability

The goal of this section is to establish a sufficient condition for the almost sure exponential stability of the disease-free equilibrium in . For this, we consider

Proposition 1. almost surely converges exponentially to 0 if

Proof. By Itô’s formula, we haveIntegrating both sides from 0 to yields thatwhere , , are continuous local martingales with . Moreover, we havewhere are positive constants. Thus, the strong law of large numbers for local martingales [27] implies thatIt follows thatThe proposition is proved.

Then, we obtain the following theorem.

Theorem 6. If , then the disease-free equilibrium of stochastic system (2) is almost surely exponentially stable in .

Proof. It suffices to prove that converges to 0 exponentially (a.s.). Then, by Proposition 1, it suffices to prove thatBy Itô’s formula, we haveSincewe haveSet and . Then,Since , , hence,Therefore,This completes the proof.

3.3. Moment Exponential Stability

In this section, we investigate the th moment exponential stability of the disease-free equilibrium in of stochastic system (2).

We use Lemma 1 to prove the following theorem.

Theorem 7. Let . Ifthen the disease-free equilibrium of stochastic system (2) is th moment exponentially stable in .

Proof. Let and . We define the Lyapunov function as follows:where is a positive constant which will be determined later. By Itô’s formula, we haveFrom Young’s inequality, for , we haveThen,whereNow, we choose sufficiently small such that . In view of condition (69), we have for ; hence, we can choose positive such that for . According to Lemma 1, the proof is completed.

Remark 1. From Lemma 1, Theorem 7, and the case , we get that ifthen the disease-free equilibrium of stochastic system (2) is globally asymptotically stable in .

4. Numerical Examples

In this section, we give some numerical examples in order to illustrate our theoretical results in Theorem 1 and Theorem 6.

Example 1. We consider the deterministic SIS system with parameters , , , , , , , , , , , , , and . By calculation, we have . Hence, according to Theorem 1, the disease-free equilibrium is globally asymptotically stable, which means that the disease dies out.

Example 2. In this example, we consider the stochastic SIS system with parameters the same as in Example 1 and , , Then, we have . Thus, from Theorem 6, we can conclude that the disease-free equilibrium is almost surely exponentially stable.
Now, we choose and . Then, we have . Therefore, the condition of Theorem 6 is not satisfied. Consequently, if the magnitude of the intensity of noise is large, then the disease in the stochastic model will go extinct.

5. Conclusion

In this paper, we have proposed and analyzed a new stochastic SIS epidemic model with double epidemic hypothesis and specific functional response by introducing random perturbations of white noise. Firstly, in the absence of noise, we have derived sufficient conditions for local asymptotic stability of the equilibria; also, we have proved the global stability for disease-free equilibrium. Next, we have established global existence and positivity of the solution for our stochastic model. In addition, we have given a sufficient condition for the almost sure exponential stability and th moment exponential stability of the disease-free equilibrium of model (2). It is shown that the magnitude of the intensity of noise will have an effective impact on stochastic stability of .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2020 Mouhcine Naim and Fouad Lahmidi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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