Analysis of an epidemiological model driven by multiple noises: Ergodicity and convergence rate

Abstract

Environmental noise is unavoidable in the spread of infectious diseases. In this paper, we present a mathematical system to investigate the impact of environmental noise on disease transmission dynamics. The model incorporates Brownian noise, Markovian switching noise and nonlinear incidence. The results show that the long-term dynamics of the stochastic system is determined by a threshold parameter which is closely related to the stochastic noise. If the threshold is greater than zero all solutions converge exponentially to a unique invariant probability distribution, while if the threshold is less than zero the infectious diseases are extinct at an exponential rate and the level of susceptible individuals converges weakly to a unique invariant probability distribution. The threshold parameter also provides essential guidelines for accessing control of the diseases and implies that the environmental noise may be beneficial to contain the infectious diseases. The results extend and generalize previous work in understanding the dynamics of stochastic epidemic models with Markov switching. The theoretical approach can also be applied to the stochastic systems driven by white noise.

Introduction

The outbreak of infectious diseases poses a great danger to human health and may cause unpredictable economic losses and social problems. Research on infectious diseases has always been a hot topic in epidemiology, biomathematics, biostatistics and other related fields in recent years. Among the numerous research methods, mathematical modeling is a powerful tool for studying the law of infectious disease dynamics, providing a strong theoretical basis for the formulation of relevant control strategies [1], [2], [3], [4], [5], [6]. The basic frameworks of classical epidemiological models can be traced back to the SIR and SIS compartmental structures based on ordinary differential equations (ODEs). These ODEs-based epidemiological models describe how dependent variables (susceptible, infected, recovering, etc.) change over time, and have contributed considerably to our understanding of infectious diseases such as smallpox, influenza, and measles [7], [8], [9], [10].

The spread of infectious diseases may subject to some uncertainties and stochastic phenomena due to fluctuations in the natural environment. Recently, Dalal et al. [11] pointed out that stochastic differential equations (SDEs) can be applied to explore the dynamics of infectious diseases. Li [12] indicated that stochastic systems can be used to study the transmission dynamics of infectious diseases in small communities. By running a stochastic model several times, we can obtain the distribution of the predicted variables, while a deterministic system will give a single predicted value [13]. However, the current research on the dynamics of stochastic systems has many challenges due to the lack of mathematical techniques.

In epidemiology, there are two common ways to introduce stochastic factors into ODEs-based epidemiological models. One is to assume that the spread of diseases is subject to some small random fluctuations. Physically, these small random fluctuations can be described by white noise [14]. Another is to assume that the key factors, such as the birth rate and the effective contact rate, are affected by random switching of external environmental regimes. These random switching factors are usually described by telegraph noise [15]. So far, by introducing random perturbations in different ways, many scholars have studied the dynamics of SDEs-based models, including stochastic persistence and extinction, stationary distribution and ergodicity [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]. These papers have served a pivotal role in understanding the transmission dynamics of infectious diseases. However, most of these models have considered either white noise or telegraph noise alone. For instance, Cai et al. [18] studied a stochastic SIRS epidemic model with white noise under intervention strategies, and their results suggest that white noise can suppress disease outbreaks. Li et al. [22] proposed a stochastic SIRS epidemic model with telegraph noise, and they obtained a threshold that determines the dynamics of the stochastic system.

In this paper, by following the methods of [25], [28], [29], [30], we study a model that includes both white noise and telegraph noise explicitly. We extend the model of Liu [31] by including a nonlinear incidence rate. The new model allows us to examine the effects of both white noise and telegraph noise on the transmission dynamics of diseases. We remark that for the model proposed in Liu [31], the threshold for the existence of a unique ergodic stationary distribution and extinction was calculated. However, two critical questions were ignored:

• How does the level of susceptible individuals change when infected individuals tend to become extinct?

• How fast does the solution of the system converge to the stationary distribution when there is a stationary distribution?

In the following, we aim to answer the above two questions. In Section 2, we present the derivation of the stochastic system and some preliminary results. In Section 3, we theoretically investigate the mathematical properties of the stochastic system. The long-term dynamics of the system are shown to be determined by the criteria $\mathcal{R}$. If $\mathcal{R}>0,$ the probability distribution of the solution converges exponentially to a unique invariant distribution, indicating that the system is mixing [29]. If $\mathcal{R}<0,$ the diseases are extinct at an exponential rate, while the level of susceptible individuals converges weakly to a unique invariant probability distribution. Finally, the paper concludes with a brief discussion.