# Dynamics of a stochastic SIS epidemic model with nonlinear incidence rates

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## Abstract

In this paper, considering the impact of stochastic environment noise on infection rate, a stochastic SIS epidemic model with nonlinear incidence rate is proposed and analyzed. Firstly, for the corresponding deterministic system, the threshold which determines the extinction or permanence of the disease is obtained by analyzing the stability of the equilibria. Then, for the stochastic system, the global dynamics is investigated by using the theory of stochastic differential equations; especially the threshold dynamics is explored when the stochastic environment noise is small. The results show that the condition for the epidemic disease to go to extinction in the stochastic system is weaker than that of the deterministic system, which implies that stochastic noise has a significant impact on the spread of infectious diseases and the larger stochastic noise is conducive to controlling the epidemic diseases. To illustrate this phenomenon, we give some computer simulations with different intensities of the stochastic noise.

## Keywords

Stochastic SIS epidemic model Nonlinear incidence rate Extinction Permanence in mean## MSC

37H10 60H10 92C60 92D30## 1 Introduction

Infectious diseases are the public enemy of mankind and have brought great catastrophe to mankind. Authors were committed to finding ways to control infectious diseases from pathology, epidemiology, culture and other aspects. The mathematical modeling method is considered as an effective method to understand the development and evolution of variables [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Mathematical models have been used to study the spread and evolution of infectious diseases in the human population. For example, Bernoulli in 1760 proposed the first mathematical model in epidemiology, for studying the spread and inoculation of smallpox. By classifying human populations into three separate categories: the susceptible *S*, the infected *I* and the removed *R*, Kermack and McKendrick [14] in 1927 proposed a well-known compartmental model. It is assumed in the model that the susceptible class can transform into the infective class through the contact with infected persons, and the infectives can be recovered through treatment so that have permanent immunity. Therefore, it is now well known as the SIR model, which has been widely studied by [15, 16, 17, 18, 19, 20]. However, some research showed that some diseases, such as influenza [21], viral diarrhea [22] and hand, foot and mouth disease [23], the immunity gained after an illness is temporary, then part of the recovered can transfer to the susceptible population again, this model is known as the SIS model [24].

Many researchers pay special attention to the incidence rate of infectious diseases. A nonlinear incidence rate plays an important role in the evolution of infectious diseases, because epidemic models described by nonlinear incidence rates may be more suitable and realistic, which also exhibit much richer dynamics. For example, the standard incidence rate \(\beta \frac{SI}{N}\) or the bilinear incidence rate \(\beta SI\) is proposed and used in reference [25, 26, 27, 28]. And saturation infection rate \(\frac{\beta SI}{1+\alpha I}\) is used in reference [29]. A special non-monotone with the form \(\beta S^{p} I^{q}\) is proposed and investigated by Severo [30], Liu et al. [31], Hethcote et al. [32], and Y. Li and Muldowney [33]. About more general forms of incidence functions, please see Pugliese [34], Thieme [35], Korobeinikov [36], Ruan and Wang [37] and Huang [38].

*Λ*is the recruitment rate of the population including the birth and migration.

*α*is mortality due to illness.

*p*is positive integer. The biological significance of other parameters please see Liu et al. [31].

Generally in the dynamic modeling of infectious diseases, we will first consider a deterministic model, however, considering the real world is filled with random and unpredictable, using stochastic model to model the dynamic of infectious diseases is more practical. Different stochastic disturbance approaches have been introduced into epidemic models. On the whole, there are four common random stochastic approaches. The first one is to introduce the parameters’ disturbance to a deterministic system (see, e.g., [39, 40, 41, 42, 43, 44, 45, 46, 47]), the second one is to investigate the stochasticity by using the method of time Markov chains (see, e.g., [48, 49, 50, 51, 52]). The third one is to consider Lévy jump noise (see, e.g., [53, 54, 55]). The fourth one is to study stochastic disturbance around the positive equilibria of a deterministic system (see, e.g., [56, 57]). Similar ideas have also been used in other modeling and analysis, for example [58, 59, 60, 61, 62].

Our main objective in the rest of present paper is to attempt to establish the threshold dynamics of system (3) similar to the deterministic system.

## 2 Preliminaries

Throughout this paper, we let \(\mathbb{R}^{d}\): the d-dimensional Euclidean space. \(\mathbb{R}^{d}_{+}:= \{x\in\mathbb{R}^{d}: x_{i} > 0, 1\leq i\leq d\}\), i.e. the positive cone.

*x*and once in

*t*. We set

### Lemma 2.1

(The one-dimensional Itô’s formula [63])

*Let*\(x(t)\)

*be an Itô’s process on*\(t\geq0\)

*with the stochastic differential*

*where*\(f\in\mathcal{L}^{1}(\mathbb{R}_{+};\mathbb{R})\)

*and*\(g\in\mathcal {L}^{2}(\mathbb{R}_{+};\mathbb{R})\).

*Let*\(V\in C^{2,1}(\mathbb{R}^{d} \times \mathbb{R}_{+};\mathbb{R})\).

*Then*\(V(x(t),t)\)

*is again an Itô’s process with the stochastic differential given by*

Let *f* be an integrable function on \([0,+\infty)\), define \(\langle f(t)\rangle=\frac{1}{t}\int_{0}^{t}f(\theta)\,d\theta\). Then we have the following definition [43].

### Definition 2.1

- (i)
the diseases \(I(t)\) is said to be extinctive if \(\lim_{t\rightarrow+\infty}I(t)=0\);

- (ii)
the diseases \(I(t)\) is said to be permanent in mean if there exists a positive constant

*λ*such that \(\liminf_{t\rightarrow+\infty}\langle I(t)\rangle\geq\lambda\).

By using the methods from Lahrouz and Omari [42], we can prove the following lemma.

### Lemma 2.2

*For any initial value*\((S_{0}, I_{0})\in R^{2}_{+}\), *there exists a unique solution*\((S(t), I(t))\)*to system* (3) *on*\(t\geq0\), *and the solution will remain in*\(R^{2}_{+}\)*with probability one*, *namely*, \((S(t), I(t))\in R^{2}_{+}\)*for all*\(t\geq0\)*almost surely*.

### Proof

*ϵ*, which satisfies \(\epsilon\leq\epsilon_{0}\), define the stopping time \(\tau _{\epsilon}\) by

*V*is positive definite. Using Itô’s formula, we get

*ϵ*, then

By using the methods from Ji et al. [64], we can prove the following lemma and remark.

### Lemma 2.3

*For any initial value*\((S_{0}, I_{0})\in\overline{R}^{2}_{+}\), *there exists a unique solution*\((S(t), I(t))\)*to system* (3) *on*\(t\geq0\), *and the solution will remain in*\(\overline{R}^{2}_{+}\)*with probability* 1, *namely*, \((S(t), I(t))\in\overline{R}^{2}_{+}\)*for all*\(t\geq0\)*a*.*s*.

### Remark 2.1

By using the methods from Meng et al. [43], we can prove the following lemma.

### Lemma 2.4

*Let*\((S(t), I(t))\)

*be a solution of system*(3)

*with initial value*\((S(0), I(0))\in R^{2}_{+}\).

*Then*

### Proof

*k*. Thus, there exists a positive \(k_{0}(\omega)\), for almost all \(\omega\in \varOmega\), for which (4) holds when \(k\geq k_{0}(\omega)\). Hence, if \(k\geq k_{0}(\omega)\) and \(k\delta\leq t\leq(k+1)\delta\), then, for almost all \(\omega\in\varOmega\),

*ε*, there exist a constant \(T(\omega)\) and a set \(\varOmega_{\epsilon}\), such that \(\mathbb{P}(\varOmega_{\epsilon })\geq1-\epsilon\) and for \(t\geq T(\omega)\), \(\omega\in\varOmega_{\epsilon}\),

## 3 Dynamics of the deterministic system

### Theorem 3.1

*Define*

*Then*,

*for system*(1),

*we have*

- (i)
*if*\(\mathcal{R}<1\),*it has a unique stable ‘diseases*-*extinction’ equilibrium point*\(E_{0}\),*which implies the extinction of the diseases*; - (ii)
*if*\(\mathcal{R}>1\),*it has a stable positive equilibrium*\(E^{*}\),*which implies the permanence of the disease*.

### Proof

## 4 Dynamics of the stochastic system

In this section, we try to explore the conditions leading to the extinction and persistence of the infectious disease.

### 4.1 Extinction

### Theorem 4.1

*For system*(3),

- (i)
*If*\(\sigma^{2}>\max\{\beta (\frac{\mu}{\varLambda} )^{p},\frac{\beta^{2}}{2(\mu+\alpha+\gamma)}\}\),*then the infectious disease of system*(3)*goes to extinction almost surely*. - (ii)
*If*\(\sigma^{2}<\beta (\frac{\mu}{\varLambda} )^{p}\),*then the infectious disease of system*(3)*goes to extinction almost surely for*\(\mathcal{R}^{*}<1\).

*Moreover*, \(\lim_{t\rightarrow+\infty}S(t)=\frac{\varLambda}{\mu}\),

*almost surely*.

### Proof

*t*gives

*Case I:*\(\sigma^{2}>\beta (\frac{\mu}{\varLambda} )^{p}\)

*.*In this case, we can easily see from (8) that

*Case II:*\(\sigma^{2}<\beta (\frac{\mu}{\varLambda} )^{p}\)

*.*In this case, we can similarly have

*t*large enough. By the first equation of system (3), we have

### Remark 4.1

Theorem 4.1 shows that when \(\mathcal{R}^{*}<1\), the infectious disease of system (3) dies out almost surely, that is to say, large white noise stochastic disturbance can lead to epidemic extinction.

### Remark 4.2

Note that \(\mathcal{R}^{*}=\mathcal{R}-\frac{\sigma^{2} (\frac{\varLambda }{\mu} )^{2p}}{2(\mu+\alpha+\gamma)}\). Obviously, \(\mathcal{R}<1\) leads to \(\mathcal{R}^{*}<1\), while the other side is not true. This implies that the condition for \(I(t)\) going to extinction in the deterministic system is stronger than its stochastic counterpart due to the effect of the white noise disturbance.

### 4.2 Permanence in mean

*t*and dividing by

*t*on both sides of system (3) yields

*t*and dividing by

*t*on both sides of (16) yields

*p*being odd, there are two cases we should discuss.

*p*is an even number, let \(p=2n, n\in N\). Then we have

*p*is an odd number, let \(p=2n-1,n\in N\). Then

### Theorem 4.2

*If*\(\mathcal{R}^{*}>1\),

*then the infectious disease*

*I*

*is permanent in mean*,

*moreover*,

*I satisfies*

### Remark 4.3

Theorem 4.1 and Theorem 4.2 show that the condition for the disease to go to extinction or permanence strongly depends on the intensity of white noise disturbances. And small white noise disturbances will be beneficial to long-term prevalence of the disease, conversely, large white noise disturbances may cause the epidemic disease to die out.

## 5 Numerical simulation

In the following, by employing the Euler Maruyama (EM) method [63, 65], we make some numerical simulations to illustrate the extinction and persistence of the diseases in stochastic system and corresponding deterministic system for comparison.

## 6 Conclusion

The aim of this paper is to make contributions to understand the dynamics of SIS epidemic models with nonlinear incidence rate. First, we expand a deterministic SIS epidemic model by introducing the extra mortality. For the modified system, by analyzing the stability of equilibria, we define a threshold which determines the extinction and permanence of the epidemic disease. Second, we establish a stochastic system by introducing the white noise disturbance into the deterministic system. For the stochastic system, we define a new threshold associated with its deterministic counterpart and analyze the dynamics of the system based on the new threshold by using the theory of stochastic differential equations. Our results show that there exists a significant difference of threshold of the stochastic system from its deterministic counterpart. The difference caused by the introduction of stochastic white noise makes the extinction conditions of the diseases in the stochastic system are weaker than that of the corresponding deterministic model. However, in the present model, the nonlinear incidence rate takes the form \(\beta S^{p}(t)I(t)\), which is a special case of the nonlinear incidence rate \(\beta S^{p}(t)I^{q}(t)\) with \(q=1, p\in N\), for the more general case \(p, q\in R^{+}\), we do not give an effective analysis method at present. The analysis of this scheme in such case is left to our further study.

## Notes

### Availability of data and materials

Data sharing not applicable to this article as all data sets are hypothetical during the current study.

### Authors’ contributions

NG and YS designed the study and carried out the analysis. NG, YS and XW contributed to writing the paper. JL performed numerical simulations. All authors read and approved the final manuscript.

### Funding

This work is supported by Shandong Provincial Natural Science Foundation of China (No. ZR2015AQ001), the National Natural Science Foundation of China (No. 11371230), SDUST Research Fund (2014TDJH102) and SDUST Innovation Fund for Graduate Students (SDKDYC180348).

### Competing interests

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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