Asymptotic Behavior and Stationary Distribution of a Nonlinear Stochastic Epidemic Model with Relapse and Cure

In this paper, by introducing environmental perturbation, we extend an epidemic model with graded cure, relapse, and nonlinear incidence rate from a deterministic framework to a stochastic differential one. *e existence and uniqueness of positive solution for the stochastic system is verified. Using the Lyapunov function method, we estimate the distance between stochastic solutions and the corresponding deterministic system in the time mean sense. Under some acceptable conditions, the solution of the stochastic system oscillates in the vicinity of the disease-free equilibrium if the basic reproductive number R0 ≤ 1, while the random solution oscillates near the endemic equilibrium, and the system has a unique stationary distribution if R0 > 1. Moreover, numerical simulation is conducted to support our theoretical results.


Introduction
Mathematical models can improve our understanding of the dynamics of infectious diseases, predict the transmission trend, and help us formulate preventive measures. e classical SIR and SIS epidemic models established by Kermack and McKendrick are one of the most important models in epidemiology [1,2]. From then on, a large number of researchers have proposed and investigated more accurate epidemic models, taking into account different forms of incidence rate, intervention strategies, random perturbation, and other factors. In particular, the human body has certain immune mechanisms to keep itself healthy, and individuals recovered from some diseases may relapse and become reinfected [3][4][5][6]. erefore, it is natural to consider immune effects in mathematical models. Recently, to explore infectious diseases in which infected individuals may be permanently rehabilitated or reinfected, Lahrouz et al. [7] proposed a nonlinear SIR epidemic model with relapse and graded cure as follows: ((I(t)) + εI(t) + cR(t), _ I(t) � βS(t)f(I(t)) − μ 2 + ε + λ I(t) + αR(t), _ R(t) � λI(t) − μ 3 + c + α R(t), where S(t), I(t), and R(t) denote the numbers of the population that are susceptible, infective, and recovered with temporary immunity, respectively. e parameter μ is the growth rate of S; μ i (i � 1, 2, 3) denotes the death rates of S, I, and R, respectively; λ is the recovery rate of I; α, ε, and c denote the relapse rate, the temporary immunity, and cure rate, respectively; β is the transmission rate from S to I. All the parameters are assumed to be positive, and it is biologically meaningful to suppose that μ 1 ≤ min μ 2 , μ 3 . e nonlinear incidence rate βS(t)f(I(t)) in model (1) reflects the heterogeneous mixing of susceptible and infective population, and the force of infection f(I) is a function of C ∞ on [0, ∞) such that f(0) � 0, 0 < f(I) < f′((0))I, ∀I > 0, (S(0), I(0), R(0)) ∈ R 3 + . (2) Lahrouz et al. [7] have studied the global dynamics of system (1). e basic reproduction number is computed as and the unique disease-free equilibrium E 0 � (μ/μ 1 , 0, 0) is globally asymptotically stable if R 0 < 1. Under some additional conditions, system (1) has a unique endemic equilibrium E * and it is globally asymptotically stable if R 0 > 1. A large amount of research studies have found that the spread of diseases is naturally subject to random environmental perturbation, such as unpredictable human exposure and meteorological factors [8,9]. Hence, an increasing number of stochastic epidemic models including environmental noise have been developed [10][11][12][13]. In the present paper, motivated by the approach in [14,15], we introduce system (1) environmental noise which is directly proportional to S, I, and R and establish the following stochastic system: roughout the paper, we let (Ω, F, F t t≥0 , P) be a complete probability space with a 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), B i (t)(i � 1, 2, 3) denotes a scalar Brownian motion defined on the complete probability space Ω, and R 3 e rest of the paper is organized as follows. In Section 2, we prove the existence and uniqueness of the positive solution for system (4), and some long time behavior of the solution is discussed. In Section 3, we analyze the asymptotic behavior of system (4) near the disease-free equilibrium and estimate the distance between stochastic solutions and the corresponding deterministic system in the time mean sense. In Section 4, asymptotic behavior near endemic equilibrium is analyzed, and we also obtain sufficient conditions for the existence of stationary distribution and persistence of diseases. In Section 5, numerical simulation is displayed to support our theoretical results. A brief conclusion is given in the last section.

Existence and Uniqueness of the Positive Solution
In this section, we present two main results. e first theorem guarantees the existence and uniqueness of the positive solution for system (4), and the second one shows some long time behavior of the solution.
Define a nonnegative C 2 -function V: where C > 0 is a positive constant determined later, and the nonnegativity can be obtained from u − 1 − ln u ≥ 0 for any u > 0.
Let n ≥ n 0 and T > 0 be arbitrary. Applying Itô's formula to V, we obtain that where 2 Journal of Mathematics Here, we choose C such that Cβf ′ (0) − μ 2 ≤ 0. en, substituting the inequality into (8) yields that Furthermore, where τ n ∧T � min τ n , T . Taking the expectation of the above inequality yields Set Ω n � τ n ≤ T for n ≥ n 1 , then P(Ω k ) ≥ ε due to (6). Note that, for every ω ∈ Ω n , at least one of S(τ n , ω), I(τ n , ω), R(τ n , ω) equals to either n or 1/n. Hence, Following from (12), we obtain where I Ω n (ω) is the indicator function of Ω n . Taking n ⟶ ∞, we have ∞ > + ∞, which is a contradiction. e conclusion is confirmed.

Asymptotic Behavior around the Disease-Free Equilibrium
For the deterministic system (1), the unique disease-free equilibrium E 0 � (μ/μ 1 , 0, 0) is globally asymptotically stable if the reproduction number R 0 < 1. e following theorem Journal of Mathematics 3 shows the asymptotic behavior of the stochastic system (4) near E 0 .
Define a nonnegative C 2 function V: R 3 + ⟶ R + as follows: Together with (19)-(22), we obtain Integrating both sides of (24) and taking expectation, we obtain Journal of Mathematics 5 Hence, e above theorem shows that the solution of system (4) oscillates near the disease-free equilibrium E 0 in the time mean sense if R 0 ≤ 1, and the magnitude of the oscillation is proportional to the intensity of noise. From the perspective of biology, the disease will be controlled in a small range if the intensity of noise is sufficiently small.

Stationary Distribution and Asymptotic Behavior around the Endemic Equilibrium
In this section, we turn to the case when the reproduction number R 0 > 1 and discuss sufficient conditions for the persistence of disease. We first recall some general results. Consider ℓ-dimension stochastic equation: where X(t) is a homogeneous Markov process in ℓ-dimension Euclidean space R ℓ . e diffusion matrix is defined as follows: Lemma 1 (see [17] for all x ∈ R ℓ , where f: R ℓ ⟶ R be a function integrable with respect to the measure μ. For the deterministic system (1), there exists at least one positive equilibrium E * � (S * , I * , R * ) if R 0 > 1. Moreover, assume the condition holds; then, the equilibrium E * is unique and globally stable according to eorem 5.1 in [7].

Journal of Mathematics
Denote an ellipsoid Σ � (S, I, R): m 1 (S − S * ) 2 + m 2 (I − I * ) 2 + m 3 (R − R * ) 2 ≤ δ}, then the ellipsoid Σ lies entirely in R 3 + if 0 < δ < min m 1 S 2 * , m 2 I 2 * , m 3 R 2 * . Take D to be any neighborhood of Σ with D⊆R 3 + , then there exists some C > 0 such that LV ≤ − C for any x ∈ R 3 + ∖ D. at is, condition (ii) holds. e diffusion matrix of system (4) is given by Choose M � min (S,I,R)∈D⊂R 3 en, condition ( i) in Lemma 1 is satisfied. erefore, according to Lemma 1, the stochastic system (4) has a unique stationary distribution and it is ergodic. Moreover, the dynamical behavior around the endemic equilibrium E * satisfies e proof is completed. eorem 4 shows that, under certain conditions, the solution of system (4) will oscillate around the endemic equilibrium E * for the long time. Furthermore, the following theorem indicates that the disease will be almost surely persistent in the time mean sense. □ Theorem 5. Suppose the conditions in eorem 4 hold; then, the solution of system (4) has the property that Proof. According to the proof process of eorem 4, we have Integrating the above inequality from 0 to t, we obtain where From the strong law of large numbers for local martingales, we have lim t⟶∞ (M(t)/t) � 0 a.s.  and the value of parameters as μ � 0.99, 002, c � 0.0007, λ � 0.007, and α � 0.0001. e selection of these parameters satisfies the condition of eorem 3, and numerical simulation is displayed in Figure 1. Moreover, we can see that when the noise intensity decreases, and the vibration of the solution for system (4) also decreases.
Similarly, take the value of parameters as μ � 0.99, μ 1 � 0.18, μ 2 � 0.1999, μ 3 � 0.5, β � 0.4, θ � 0.1, ε � 0.002, c � 0.007, λ � 0.07, and α � 0.0001. It is easy to compute that R 0 > 1. e numerical simulation is shown in Figure 2, from which one can see that the solution of stochastic system vibrates around the endemic equilibrium E * . Moreover, Figure 3 shows the histograms of S(t), I(t), and R(t), and it indicates the existence of a unique stationary distribution for system (4), and the disease will be almost surely persistent in the time mean sense.

Conclusion
Since most systems in the real world are disturbed by random and unpredictable perturbation, we introduce environmental noise of white noise type into the transmission of disease and study a stochastic version of a nonlinear SIRS epidemic model with relapse and cure. e reproduction number R 0 is a threshold parameter. If the conditions of eorems 3 and 4 hold, according to R 0 ≤ 1 or R 0 > 1, we prove that the solution of the stochastic model oscillates in the vicinity of the disease-free equilibrium and the endemic equilibrium, respectively, and the fluctuation intensity is proportional to the white noise intensity. roughout the paper, we use numerical simulations to illustrate our theoretical results. In particular, we also prove that the stochastic SIR model has a unique ergodic stationary distribution, and the disease will be prevalent in the sense of time mean under some conditions. However, based on our theoretical analysis and numerical simulations, it is not clear that the disease can be eradicated.

Data Availability
All data generated or analyzed during this work are included in this article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.