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# The persistence and extinction of a stochastic SIS epidemic model with Logistic growth

*Advances in Difference Equations*
**volume 2018**, Article number: 68 (2018)

## Abstract

The dynamical properties of a stochastic susceptible-infected epidemic model with Logistic growth are investigated in this paper. We show that the stochastic model admits a nonnegative solution by using the Lyapunov function method. We then obtain that the infected individuals are persistent under some simple conditions. As a consequence, a simple sufficient condition that guarantees the extinction of the infected individuals is presented with a couple of illustrative examples.

## Introduction

Some mathematical models, for instance, see [1–5], have been employed to describe and understand epidemic transmission dynamics since the work of Kermack and McKendrick [6] was proposed. The classical compartment models were proposed and investigated on the ground of some restrictive assumptions including a constant total population size and a constant recruitment rate for the susceptible individuals. This assumption is relatively reasonable for a short-lasting disease. While in reality, the population sizes of human beings and other creatures are generally variable, instead of keeping constant for a long run. As an example of this phenomenon, Ngonghala *et al.* pointed out that malaria in developing countries took place with growth of local population size. When it concerns the variable population size, some recent literature works, such as Ngonghala *et al.* [7], Busenberg and Driessche [8], Wang *et al.* [9], Zhao *et al.* [10], Zhu and Hu [11], Li *et al.* [12], had considered the effect of population size on the epidemic dynamics. We would like to mention the work by Wang *et al.* [9], in which they constructed an SIS epidemic model under the assumption that the susceptible individuals followed the Logistic growth:

where \(S(t)\) and \(I(t)\) denote the numbers of the susceptible and the infected individuals at time *t*, respectively; *r* is the intrinsic growth rate of the susceptible individuals; *a* is the carrying capacity of the community in the absence of infection; *d* is the natural death rate; *γ* represents the recovery rate of the infected individuals; *ε* is the disease-induced death rate; \(\beta(I)\) is the transmission rate and is given in the following form:

All the parameters are assumed to be nonnegative. When \(p=0\), \(\beta(I)\) is equal to the constant transmission rate *β*. In this paper, we shall consider the following deterministic SIS endemic model:

where \(b=r/a\). We set \(N(t)\) is the total population at time *t*, then

Wang *et al.* [9] showed that the domain

is a positively invariant set with respect to model (1). Moreover, the disease-free equilibrium \(E_{0}(a,0)\) of model (1) always exists, and if the basic reproductive number

\(E_{0}(a,0)\) is globally asymptotically stable. If \(R_{0}>1\), then \(E_{0}(a,0)\) is unstable, and there is a unique endemic equilibrium \(E^{*}(S^{*},I^{*})\) which is globally asymptotically stable. Here

The compartment models are inevitably affected by the environmental noise. We assume that the transmission coefficient *β* is subject to the environmental white noise, that is,

where \(B(t)\) is a standard Brownian motion, *σ* is the intensity of environmental white noise. In order to explore the stochastic effect, when the constant transmission rate is replaced by a random variable, we consider the corresponding stochastic SIS epidemic model:

The sum of the two equations for the population size \(N(t)\) of models (3) and (9) is

Throughout this paper, we will work on the complete probability space \((\Omega, \{\mathcal{F}_{t}\}_{t\geq 0},P)\) with its filtration \(\{\mathcal{F}_{t}\}_{t\geq 0}\) satisfying the usual conditions (i.e., it is right continuous and \(\mathcal{F}_{0}\) contains all P-null sets). We will investigate the dynamical properties of stochastic SIS model from several aspects: the result that stochastic model (9) admits a unique positive solution will be studied in the next section. The sufficient conditions of the persistence for the infected individuals would be derived. Further, we still find a simple condition to reach the extinction for the infected individuals. As a consequence, several illustrative examples are carried out to support the main results of this paper.

## Existence and uniqueness of positive solution

In this section, we first show that the solution of system (9) is positive and global. Our proof is motivated by the work of Mao *et al.* [13].

### Theorem 1

*There exists a unique solution*
\((S(t),I(t))\)
*of system* (9) *on*
\(t\geq 0\)
*for any initial value*
\((S(0),I(0))\in \mathbb{R}^{2}_{+}\), *and the solution will remain in*
\(\mathbb{R}^{2}_{+}\)
*with probability* 1, *namely*
\((S(t),I(t))\in \mathbb{R}^{2}_{+} \)
*for all*
\(t\geq 0\)
*almost surely*.

### Proof

Since the coefficients of model (9) satisfy local Lipschitz conditions for any initial value \((S(0),I(0))\in \mathbb{R}^{2}_{+}\), there exists a unique local solution on \(t\in[0,\tau_{e})\), where \(\tau_{e}\) is the explosion time. Next, we will show that the solution of model (9) is global. To this end, we need to show that \(\tau_{e}=\infty\) holds almost surely. Let \(k_{0}>0\) be sufficiently large such that \(S(0)\) and \(I(0)\) all lie within the interval \([\frac{1}{k_{0}}, k_{0}]\). For all \(k\geq k_{0}\), we define the stopping time

Throughout this paper, we set \(\inf\emptyset=\infty\). Clearly, \(\tau_{k}\) is an increasing function as \(k\rightarrow\infty\). We set \(\tau_{\infty}=\lim_{k\to \infty} \tau_{k}\), according to the definition of stopping time, we get that \(\tau_{\infty}\leq\tau_{e}\) a.s. If we can show that \(\tau_{\infty}=\infty \) a.s., then \(\tau_{e}=\infty\) a.s. From now on, our proof will go by contradiction. If this statement is false, then there exists a pair of constants \(T>0\) and \(\varepsilon\in(0,1)\) such that \(P\{\tau_{\infty}\leq T\}>\varepsilon\), hence there exists an integer \(k_{1}>k_{0}\) such that

We define a \(C^{2}\)-function \(V:\mathbb{R}^{2}_{+}\rightarrow\mathbb{R}_{+}\) as follows:

where *k* is a constant determined later. Generalized Itô's formula gives that

Choose the constant

which implies that

The remainder of the proof follows that in Zhao *et al.* [10]. □

## Persistence in the mean

In this section, we shall investigate the persistence property of model (9). The solution of model (9) is said to be persistent in the mean if

For convenience, we define the following notation:

### Lemma 1

([13], Strong law of large numbers)

*Let*
\(M=\{M_{t}\}_{t\geq0}\)
*be a real*-*valued continuous local martingale vanishing at*
\(t=0\). *Then*

*and also*

### Theorem 2

*Let*
\((S(t), I(t))\)
*be a solution of system* (9) *with any initial value*
\((S(0),I(0))\in \Pi\). *If*

*then the density of the infected individuals obeys the following expression*:

### Proof

Integrating both sides of the second equation of model (9) gives that

Then generalized Itô's formula acting on model (9) leads to

Integrating both sides of (24) and (25) from 0 to *t* and dividing by *t*, we have that

We combine (23), (26), and (27) and derive that

then

We denote

by strong law of large numbers for martingales, together with the facts \(0< S(t)\), \(I(t)< K_{0}\), which yields that

therefore,

The proof is complete. □

### Example 1

Let the parameters of model (9) be

and the initial value be \((S(0),I(0))=(5,2)\), then the threshold of model (9) is computed as

which is consistent with the result of Theorem 2 (see Fig. 1).

## Extinction

In the previous section, we have investigated the persistence of the solution to model (9). In this section, we shall prove that the density of the infected individuals will be driven to extinction with a negative exponential power under some simple assumptions.

### Theorem 3

*Let*
\((S(t), I(t))\)
*be the solution of model* (9) *with the initial value*
\((S(0),I(0))\in \Pi\). *If*

*or*

*holds*, *then the density of the infected individuals will decline to zero exponentially with probability one*. *That is to say*,

*or*

### Proof

From the second equation of model (9), we have

The fact \(S\leq K_{0}\) leads to the following result:

We denote

by the strong law of large numbers for martingales, we then have

Condition (35) of Theorem 3 gives that

On the other hand, expression (39) can be computed as follows:

By similar discussion, together with condition (36), we take superior limit on both sides of (44) and derive that

The proof is complete. □

### Example 2

We set the parameters of model (9) are

and the initial value is \((S(0),I(0))=(35,50)\). It is easy to check that

Theorem 3 is satisfied, and the infected individuals will decline to zero according to Fig. 2. If we choose another group of parameters

and the initial value \((S(0),I(0))=(1,8)\) in order to meet condition (36), after substitution, we then get that

which also means that the infected individuals definitely tend to zero with the rate of a negative exponential power as shown in Fig. 3.

## Conclusion

The dynamical properties of the stochastic SIS model with Logistic growth are paid more attention to in this paper. According to the approach shown in many recent literature works, we still construct a \(C^{2}\)-function to show that the stochastic SIS epidemic model admits a unique positive global solution. Based on the general assumption of this paper, the total population is separated into two compartments: one is the susceptible, another is the infected. We also assume that the transmission rate *β* is perturbed by a white noise. The two indicators \(\tilde{R}_{0}\) and \(\breve{R}_{0}\) are kind of thresholds of this paper: when \(\tilde{R}_{0}> 1\), under some extra conditions, the density of the infected individuals keeps persistent; when \(\breve{R}_{0}<1\) holds or (36) is valid, the density of the infected individuals declines to zero in a long run. Several illustrative examples support the main results of this paper.

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## Acknowledgements

The authors would like to thank Zhanshuai Miao for his discussion and the anonymous referees for their good comments. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11201075, 11601085), Natural Science Foundation of Fujian Province of China (Grant No. 2016J01015, 2017J01400).

## Author information

### Affiliations

### Contributions

The main idea of this paper was proposed by JL. JL prepared the manuscript initially, LC and FW performed all the steps of the proofs in this research. All authors read and approved the final manuscript.

### Corresponding author

Correspondence to Fengying Wei.

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We claim that none of the authors have any competing interests in the manuscript.

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## About this article

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### MSC

- 60H10
- 37H10

### Keywords

- Extinction
- Persistence
- Stochastic SIS model
- Logistic growth