Stability of a Nonlinear Stochastic Epidemic Model with Transfer from Infectious to Susceptible

We investigate a stochastic SIRS model with transfer from infectious to susceptible and nonlinear incidence rate. First, using stochastic stability theory, we discuss stochastic asymptotic stability of disease-free equilibrium of this model. Moreover, if the transfer rate from infectious to susceptible is sufficiently large, disease goes extinct. ,en, we obtain almost surely exponential stability of disease-free equilibrium, which implies that noises can lead to extinction of disease. By the Lyapunov method, we give conditions to ensure that the solution of this model fluctuates around endemic equilibrium of the corresponding deterministic model in average time. Furthermore, numerical simulations show that the fluctuation increases with increase in noise intensity. Finally, these theoretical results are verified by numerical simulations. Hence, noises play a vital role in epidemic transmission. Our results improve and extend previous related results.


Introduction
Mathematical models have become a crucial tool in understanding dynamics of population growth [1][2][3]. In recent decades, some realistic mathematical models have been established to investigate dynamics of epidemic [4][5][6][7][8][9][10]. In order to simulate epidemic transmission process, many dynamic models have been established, such as SIS, SEIR, and SIRS models [11][12][13]. In these models, the incidence rate is crucial. Classical disease transmission models adopt the standard or bilinear incidence rate. However, in the course of epidemic propagation, nonlinear incidence may be more realistic than other incidence rates [14]. In addition, infected individuals may recover after a period of treatment or become susceptible individuals directly due to transient antibody. In [15], a deterministic SIRS model with transfer from infectious to susceptible and nonlinear incidence can be modeled as follows: Here, S, I, and R denote numbers of susceptible, infectious, and recovered individuals, respectively. Λ is the recruitment rate of susceptible; β denotes the disease propagation coefficient; μ and α denote, respectively, the natural death rate and mortality caused by the disease; δ denotes the immunity loss rate; c 1 represents the transfer rate from infectious to susceptible; c 2 denotes the recovery rate of infectious individuals. In addition, Λ > 0, μ > 0, c 1 ≥ 0, c 2 ≥ 0, δ ≥ 0, and α ≥ 0.
However, dynamics of epidemic is often disturbed by some random factors. Hence, stochastic epidemic models are more realistic and have attracted much attention [16][17][18][19].
In [20], the authors discussed threshold behavior for a stochastic SIS model. In [21], asymptotic properties of a stochastic SIR model were considered. In [22,23], the authors investigated persistence and extinction for a stochastic SIRS model. In [24], the authors studied stability of a stochastic SIRS model. Fatini et al. [25] considered stochastic stability and instability for a stochastic SIR model. Recently, Wang et al. [26] established a stochastic SIRS epidemic model: with initial values S 0 > 0, I 0 > 0, andR 0 > 0. Here, B(t) represents Brownian motion on (Ω, F, P) which is a complete probability space. σ 2 denotes the intensity of B(t). Other parameters are defined as (1). Model (2) covers many stochastic models as particular cases (see, for example, [15,22,27]). In [26], extinction and persistence are obtained.
As is well known, stability of the dynamic system means that solutions are insensitive to small changes of initial value. Hence, stability is one of the important topics encountered in applications. However, because of the complexity of stochastic dynamics, there are not many results on stability of stochastic differential equations.

Preliminaries
We will give some definitions and lemmas. Consider Here, F and G are, respectively, R d − valued and Definition 1 ([ [28], p.108]) then V is decrescent.
By eorem 1 and Remark 1 in [26], the following result holds.

Stability of Disease-Free Equilibrium
In epidemiology, stability has important practical significance. (2) is stochastically asymptotically stable in D.
is demonstrates that noises 8 Complexity can result in extinction of disease. Figure 2 clearly supports these results.
Example 6. Take β � 0.5. Figure 6 plots the average in time of infected (1/t) t 0 I(s)ds for different c 1 in (a) and (b), respectively. Figure 6 shows that the larger the c 1 is, the  Figure 7 shows that σ 2 has a significant effect on both extinction and persistence of disease.

Conclusions
Stability is one of the important topics encountered in applications. However, because of the complexity of stochastic dynamics, there are not many results on stability analysis of stochastic differential equations.
Based on this, we investigate stochastic stability of a stochastic SIRS model. To begin with, using stochastic stability theory, we study stochastic asymptotic stability of disease-free equilibrium of (2), which generalizes eorem 2.1 in [15]. Moreover, if the transfer rate from infectious to susceptible is sufficiently large, disease goes extinct.
en, exponential stability of disease-free equilibrium is obtained. is result partially improves eorem 2.1 in [15] and eorem 2 in [26] and demonstrates that noises can result in extinction of the disease. Furthermore, by the Lyapunov method, we give conditions to ensure that solution of (2) fluctuates around endemic equilibrium of (1) in time average. is generalizes Corollary 2.3 in [15]. At last, numerical simulations are presented to confirm theoretical results and find new properties. Figure 4 shows that if R 0 > 1 and σ 2 < μ(a 3 + 1)/(a 2 I * ), then the solution of (2) fluctuates around endemic equilibrium of (1). Moreover, Figure 4 also shows that the fluctuation increases with increase in noise intensity. From Figure 5, the smaller the β is, the smaller the number of infected individuals will be. In addition, when β tends to 0, the number of infected individuals will tend to 0. is result can also be derived from eorem 2. Figure 6 shows that the larger the c 1 is, the smaller the number of infected will be. Furthermore, when c 1 is sufficiently large, the number of infected tends to 0. is result can be derived from Remark 1 (iii). Figure 7 shows that noise intensity has a significant effect on both extinction and persistence of the disease. Hence, noises play a vital role in epidemic transmission. For deterministic SIRS model (1), R 0 is the basic reproduction number. However, for stochastic SIRS model (2), R 0 is not a threshold parameter. From eorem 2, no matter what the value of R 0 is, the disease could go extinct. is can also be verified by the examples in this paper.
Although there are important findings revealed by the above investigation, the results still have some limitations. One may consider stochastic asymptotic stability in R 3 + . In addition, our numerical simulation results show that the disease goes extinct as long as R 0 < 1. Regrettably, our theoretical results do not lead to this conclusion.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
All the authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.