Complexity
Volume 2020 (2020), Article ID 9614670, 12 pages
https://doi.org/10.1155/2020/9614670

Research Article

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

Yanmei Wang1,2 and Guirong Liu¹

¹School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China
²School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, Shanxi 030006, China

Correspondence should be addressed to Guirong Liu; lgr5791@sxu.edu.cn

Received 3 April 2020; Revised 16 May 2020; Accepted 19 May 2020; Published 31 May 2020

Guest Editor: Carlos Arturo Loredo Villalobos

Copyright © 2020 Yanmei Wang and Guirong Liu. 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.

Abstract.

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. Then, 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.

1. Introduction

Mathematical models have become a crucial tool in understanding dynamics of population growth [1–3]. In recent decades, some realistic mathematical models have been established to investigate dynamics of epidemic [4–10]. In order to simulate epidemic transmission process, many dynamic models have been established, such as SIS, SEIR, and SIRS models [11–13]. 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, , , and denote numbers of susceptible, infectious, and recovered individuals, respectively. is the recruitment rate of susceptible; denotes the disease propagation coefficient; and denote, respectively, the natural death rate and mortality caused by the disease; denotes the immunity loss rate; represents the transfer rate from infectious to susceptible; denotes the recovery rate of infectious individuals. In addition, , , , , , and .

From [15], (1) has disease-free equilibrium which is globally asymptotic stable in if . If , there exists a globally asymptotic stable endemic equilibrium .

However, dynamics of epidemic is often disturbed by some random factors. Hence, stochastic epidemic models are more realistic and have attracted much attention [16–19]. 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 . Here, represents Brownian motion on which is a complete probability space. denotes the intensity of . 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.

Motivated by the above work, we consider (2) and obtain stochastic stability of disease-free equilibrium and asymptotic behavior around endemic equilibrium of corresponding deterministic model (1).

Throughout this paper, we give the following hypotheses:

  is locally Lipschitz on for ;   and is nonincreasing on

From , (2) has disease-free equilibrium . By , if , then

2. Preliminaries

We will give some definitions and lemmas. Consider

Here, are, respectively, valued and valued functions defined on and . denotes -dimensional Brownian motion on . Assume that existence-and-uniqueness theorem is fulfilled. For , and . Denote and . Set .

Definition 1. ([[28], p.108]).

(i)Assume that is continuous on and . If there is such that, for ,

 then is positive-definite. In addition, is negative-definite if is positive-definite.(ii)Assume that is nonnegative and continuous on . If there is , satisfying for ,

then is decrescent.

Definition 2. ([[28], p.110]).

(i)If for any and , there is , satisfying for any with ,

 then trivial solution to (4) is stochastically stable.(ii)The trivial solution to (4) is stochastically asymptotically stable if it is stochastically stable, and for any , there is satisfying

 whenever .(iii)If for any , a.s., then trivial solution to (4) is almost surely exponentially stable in .

Lemma 1. ([[28], p.112]). If is positive-definite and decrescent, and is negative-definite, then trivial solution to (4) is stochastically asymptotically stable.
By Theorem 1 and Remark 1 in [26], the following result holds.

Lemma 2. (see [26]). For , there is a unique global positive solution to (2). Moreover,

is positively invariant.

3. Stability of Disease-Free Equilibrium

In epidemiology, stability has important practical significance.

Theorem 1. If , , then disease-free equilibrium to (2) is stochastically asymptotically stable in .

Proof. Denote . Define Lyapunov function

for , where is to be chosen later. Clearly, is positive-definite. Note that . From Definition 1 (ii), it follows that is decrescent. Now, we show that is negative-definite.
From It formula, for any ,

Obviously, we have

Substituting (12) and (13) into (11) yields

Note . Then, . Take

This yields that is negative-definite. From Lemma 1, is stochastically asymptotically stable in .

Lemma 3. For any , solution of (2) satisfies the following:

(i)If , then

(ii)If , then

Proof. Obviously, for . Define

Then,

Let and . From and (3),

Let . Then,

which yields

From the strong law of large numbers,

Then,

Obviously, if , then ; if , then . Lemma 3 holds.
By Lemma 3, the following result holds.

Theorem 2. Assume that

(i) or(ii).

Then, disease-free equilibrium of (2) is almost surely exponentially stable in .

Remark 1.

(i)If and , then holds. From Theorem 1, if , then disease-free equilibrium of (1) is asymptotically stable in . Hence, Theorem 1 extends Theorem 2.1 in [15].(ii)From Theorem 2, if , then disease-free equilibrium of (1) is exponentially stable in . Hence, Theorem 2 partially improves Theorem 2.1 in [15].(iii)From Theorem 1, if , then disease-free equilibrium of (2) is stochastically asymptotically stable in .

Remark 2.

(i)Assume that and . From condition in Theorem 2, if , then disease-free equilibrium of (2) is almost surely exponentially stable in . However, Theorem 2 in [26] implies that disease of (2) will become extinct if of Theorem 2 in [26] holds, i.e., .(ii)Assume that and . From Theorem 2, is almost surely exponentially stable in if , whereas disease will become extinct with probability one if in [26].

Obviously, condition of Theorem 2 is weaker than condition of Theorem 2 in [26].

Remark 3. Let . By Theorem 2, is almost surely exponentially stable in if condition (ii) holds. However, disease will become extinct if in [26]. Thus, condition (ii) of Theorem 2 is weaker than condition of Theorem 2 in [26].

Remark 4. From Remarks 2 and 3, Theorem 2 partially improves Theorem 2 in [26].

4. Asymptotic Properties around Endemic Equilibrium

In studying epidemic dynamics, we have interest in persistence of epidemic. We consider the behavior of solutions to (2) around endemic equilibrium of corresponding deterministic model (1). Denote

Theorem 3. If and , then

where be the solution of (2) with and

Proof. Define by

where

From It formula, (3), and ,

From (25)–(33),

Hence, we have

It follows from (35) that

From (36),

Consequently,

Remark 5. Theorem 3 shows that if , is small enough and then solution to (2) fluctuates around ; that is, disease will persist. Furthermore, if , then (34) becomes

which yields that for (1), is globally asymptotically stable in . This is consistent with Corollary 2.3 in [15]. Hence, Theorem 3 generalizes Corollary 2.3 in [15].

5. Numerical Simulations

By numerical simulation, we analyze the asymptotic behavior of model (2) so that readers can better understand our results. Let . Then, for . Let

Example 1. Take , and . By a simple computation, we obtain , , , and . Hence, the conditions of Theorem 1 hold. Furthermore, for (2), is stochastically asymptotically stable. Figure 1 supports the result.

(a)

(b)

(c)

Figure 1: Trajectories of deterministic model (1) and stochastic model (2) with (a) , (b) , and (c) .

Example 2. Take , and . Hence, and . Then, according to conclusion (i) in Theorem 2, solutions of (2) will tend almost surely exponentially to . However, from Corollary 2.3 in [15], the solution of deterministic model (1) will converge to . This demonstrates that noises can result in extinction of disease. Figure 2 clearly supports these results.

(a)

(b)

(c)

Figure 2: Trajectories of deterministic SIRS model (1) and stochastic SIRS model (2) with (a) , (b) , and (c) .

Example 3. Let , and such that and . Then, according to conclusion (ii) in Theorem 2, solutions of (2) will tend almost surely exponentially to . However, from Corollary 2.3 in [15], endemic equilibrium of (1) is globally asymptotically stable in . This represents the extinction of disease due to noise. Figure 3 clearly supports these results.

(a)

(b)

(c)

Figure 3: Trajectories of deterministic SIRS model (1) and stochastic SIRS model (2) with (a) , (b) , and (c) .

Example 4. Take . Then, and . By Theorem 3, solutions of (2) fluctuate around endemic equilibrium of deterministic model (1) in time average, which can be verified by using Figure 4. In addition, Figure 4 shows that the fluctuation increases with increase in .

(a)

(b)

(c)

Figure 4: Trajectories of deterministic SIRS model (1) and stochastic SIRS model (2) for different with .

Example 5. Take . Figure 5 plots the average in time of infected for different in (a) and (b), respectively. From Figure 5, the smaller the is, the smaller the number of infected cases is. In addition, when tends to 0, the number of infected cases will tend to 0. This result can also be derived from Theorem 2.

(a)

(b)

Figure 5: Trajectories of the average in time of infected for stochastic SIRS model (2) for different with . (a) . (b) .

Example 6. Take . Figure 6 plots the average in time of infected for different in (a) and (b), respectively. Figure 6 shows that the larger the is, the smaller the number of infected cases is. Furthermore, when is sufficiently large, the number of infected cases tends to 0. This result can be derived from Remark 1 (iii).

(a)

(b)

Figure 6: Trajectories of the average in time of infected for stochastic SIRS model (2) for different with . (a) . (b) .

Example 7. Take . Figure 7 plots the number of infected cases for different in (a) and (b), where in (a) and in (b). From Corollary 2.3 in [15], endemic equilibrium of deterministic model (1) is globally asymptotically stable in . Figure 7 shows that has a significant effect on both extinction and persistence of disease.

(a)

(b)

Figure 7: Trajectories of infected of stochastic SIRS model (2) for different with . (a) . (b) .

6. 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 Theorem 2.1 in [15]. Moreover, if the transfer rate from infectious to susceptible is sufficiently large, disease goes extinct. Then, exponential stability of disease-free equilibrium is obtained. This result partially improves Theorem 2.1 in [15] and Theorem 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. This 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 and , 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 is, the smaller the number of infected individuals will be. In addition, when tends to 0, the number of infected individuals will tend to 0. This result can also be derived from Theorem 2. Figure 6 shows that the larger the is, the smaller the number of infected will be. Furthermore, when is sufficiently large, the number of infected tends to 0. This 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), is the basic reproduction number. However, for stochastic SIRS model (2), is not a threshold parameter. From Theorem 2, no matter what the value of is, the disease could go extinct. This 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 . In addition, our numerical simulation results show that the disease goes extinct as long as . Regrettably, our theoretical results do not lead to this conclusion.

Data Availability

No data were used to support this study.

Conflicts of Interest

The 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.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 11971279).

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