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

)e 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. )ereafter, stochastic stability of the disease-free equilibrium in almost sure exponential and pth moment exponential is investigated. Finally, numerical examples are presented.


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][2][3][4][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][7][8][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: β 1 S(t)I 1 (t) 1 + α 1 S(t) + c 1 I 1 (t) + μ 1 S(t)I 1 (t) − β 2 S(t)I 2 (t) 1 + α 2 S(t) + c 2 I 2 (t) + μ 2 S(t)I 2 (t) + r 1 I 1 (t) + r 2 I 2 (t), _ I 1 (t) � β 1 S(t)I 1 (t) 1 + α 1 S(t) + c 1 I 1 (t) + μ 1 S(t)I 1 (t) − μ + a 1 + r 1 I 1 (t), _ I 2 (t) � β 2 S(t)I 2 (t) 1 + α 2 S(t) + c 2 I 2 (t) + μ 2 S(t)I 2 (t) − μ + a 2 + r 2 I 2 (t), where S(t) represents the number of susceptible at time t, I 1 (t) and I 2 (t) are the total population of the infected with virus V 1 and V 2 at time t, respectively, A represents the recruitment rate of the population, μ is the natural death rate of the population, r i is the treatment cure rate of the disease caused by virus V i , a i is the disease-related death rate, and β i is the infection coefficient, i � 1, 2. e incidence rate of disease is modeled by the specific functional response are saturation factors measuring the psychological or inhibitory effect. is specific functional response was introduced by Hattaf et al. [11], and here, it becomes to be a bilinear incidence rate if [12,13] if μ i � 0, and a Crowley-Martin functional response [14] if In the reality, epidemic systems are inevitably affected by environmental white noise. erefore, it is necessary to study how the noise influences the epidemic models. Consequently, many authors have studied stochastic epidemic models, see, e.g., [15][16][17]. For this, we consider the case in which the rates β i (i � 1, 2) are subject to random fluctuations, namely, β i dt is replaced by where B i (t) (i � 1, 2) are independent standard Brownian motions, and σ i > 0 represents the intensity of B i (t) for i � 1, 2. erefore, the corresponding stochastic system to (1) can be described by the following Itô equations: e 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 pth 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.
By summing all the equations of system (1), we find that the total population size N(t) � S(t) + I 1 (t) + I 2 (t) satisfies the inequality _  [20] can be used to deduce that us, the feasible solution set of the system equation of model (1) enters and remains in the region erefore, model (1) is well posed epidemiologically and mathematically [21]. Hence, it is sufficient to study the dynamics of model (1) in Γ.
It is easy to see that system (1) has a disease-free equilibrium state E 0 � (A/μ, 0, 0). erefore, the basic reproduction number is where We mention that the expressions of R 01 and R 02 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 E 0 . e Jacobian matrix of system (1) at the equilibrium E 0 is as follows: e three eigenvalues of Hence, the equilibrium E 0 will be locally asymptotically stable if R 0 < 1 and unstable when R 0 > 1. e following theorem discusses the global stability of the disease-free equilibrium E 0 .
Proof. Let U be the Lyapunov function defined as Differentiating U with respect to t along the positive solutions of system (1), we get We have Since S ≤ A/μ and the functions f i : is a solution of (1) contained entirely in the set Δ � (S, I 1 , I 2 ) ∈ Γ: _ U(t) � 0 . en, _ I 1 + _ I 2 � 0. We discuss four cases: From the second and third equations of (1), we have X 1 I 1 + X 2 I 2 � 0, which implies, according to (15), that I 1 � I 2 � 0. On the other hand, solutions of (1) contained in the plane Discrete Dynamics in Nature and Society 3 Case 2. If R 01 < 1 and R 02 � 1, then X 1 < 0 and en, X 1 I 1 + X 2 I 2 � 0 implies that I 1 � 0 and consequently e case R 01 � 1 and R 02 < 1 is analogue to the previous case.
Hence, X 1 I 1 � X 2 I 2 � 0, and by the same analysis in Case 2, we obtain that I 1 � I 2 � 0.
Proof. It is analogue to the previous proof.
Next, we investigate the local stability of system (1) at both-endemic equilibrium E * � (S * , I * 1 , I * 2 ). To obtain conditions for the existence of the equilibrium E * , system (1) is rearranged to get I * 1 and I * 2 which gives We have In addition, S * is given by the following cubic equation: where C 0 � μμ 1 μ 2 ϖ 1 ϖ 2 > 0, With the help of Descartes' rule of signs [24], equation (26) has a unique positive real root S * if any one of the following holds: (i) C 1 > 0 and C 2 > 0 (ii) C 1 > 0 and C 2 < 0 (iii) C 1 < 0 and C 2 < 0 Hence, system (1) has a unique positive equilibrium E * if β i − α i ϖ i > 0 for i � 1, 2, one of the conditions (i), (ii), and (iii) hold true, and S * > max i�1, e Jacobian matrix of system (1) at the equilibrium E * is determined by where (29)

Theorem 4. e endemic equilibrium E * is locally asymptotically stable if it exists.
Proof.
e characteristic equation of Jacobian matrix J E * can be written as where Note that en, it is easy to show that Q 2 > 0, Q 1 > 0, Q 0 > 0, and Q 2 Q 1 > Q 0 . us, by the Routh-Hurwitz criterion, all roots λ i (i � 1, 2, 3) of (30) have negative real part. erefore, the equilibrium E * of system (1) is asymptotically stable. □ Discrete Dynamics in Nature and Society

Stochastic SIS Epidemic Model
Let (Ω, F, F t t≥0 , P) be a complete probability space with filtration F t t ≥ 0 satisfying the usual conditions (i.e., it is increasing and right continuous while F 0 contains all P-null sets). We consider the following stochastic differential system: where x(t) ∈ R n , x(0) � x 0 represents the initial value, and f: R n × [0, +∞)↦R n and g: R n × [0, +∞)↦R n×m are locally Lipschitz functions in x. B(t) { } t≥0 is an m-dimensional standard Wiener process defined on the above probability space.
Let us suppose that f(0, t) � g(0, t) � 0 for all t ≥ 0 so that zero of R n is an equilibrium point of system (33).
Definition 1 (see [25]). e trivial solution x � 0 of system (33) is said to be almost surely exponentially stable if for all x 0 ∈ R n , we have lim sup Denote by C 2,1 (R n × [0, +∞); R + ) the family of all nonnegative functions V(x, t) defined on R n × [0, +∞) such that they are continuously twice differentiable in x and once in t. Denote by E(X) the mathematical expectation of a random variable X. If L acts on a function V ∈ C 2,1 (R n × [0, +∞); R + ), then where V t (x, t) � zV/zt, V x (x, t) � (zV/zx 1 , . . . , zV/zx n ), and V xx (x, t) � (z 2 V/zx i zx j ). By Itô's formula, we have Lemma 1 (see [26]). Suppose there exists a function V ∈ C 2,1 (R n × [0, +∞); R + ) satisfying the inequalities where p > 0 and K i (i � 1, 2, 3) are positive constants. en, the equilibrium of system (33) is pth moment exponentially stable. When p � 2, it is usually said to be exponentially stable in mean square, and the equilibrium x � 0 is globally asymptotically stable.

Existence and Uniqueness of the Global Positive
Solution. e following theorem shows that the solution of our system (2) is global and positive.

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 diseasefree 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 pth moment exponential stability of the disease-free equilibrium of model (2). It is shown that the magnitude of the intensity of noise σ i (i � 1, 2) will have an effective impact on stochastic stability of E 0 .

Data Availability
No data were used to support this study.

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