Analysis of an epidemiological model driven by multiple noises: Ergodicity and convergence rate☆
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:
- 1.
How does the level of susceptible individuals change when infected individuals tend to become extinct?
- 2.
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 . If the probability distribution of the solution converges exponentially to a unique invariant distribution, indicating that the system is mixing [29]. If 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.
Section snippets
Model formulation and preliminaries
We assume that the total population is divided into three parts: the susceptible (with density St at time t), the infected (with density It at time t), and the recovered (with density Rt at time t). Assume that all individuals are born susceptible. The epidemic model to be studied is described as:
The biological meanings of parameters and functions appeared are listed in Table 1.
In the modeling of infectious diseases, the effective
Main results
Lemma 3.1 Consider the following system:For any given initial value the solution of the system (3.1) admits a unique invariant probability measure ψ. In addition, [31]
In the following, we aim to analyze the dynamics of the stochastic system (2.4). To proceed, we define Theorem 3.1 When It converges exponentially to 0 with probability one, i.e.,Moreover, the
Conclusions and future work
In the present study, we investigate the dynamics of an epidemiological model with multiple noise perturbations. The stochastic analysis method and the ergodic theory of Markov processes are employed to carry out the dynamic of the stochastic switched system. The results indicate that completely determines the stochastic persistence and extinction of the underlying system:
- (a)
If the probability distribution of the solution of the underlying system converges exponentially to a unique
Acknowledgments
The authors would like to thank the editor and anonymous referees for their valuable comments and suggestions. Also, the authors would extend their sincere thanks to Professor Nguyen Hai Dang for his valuable comments.
References (49)
- et al.
Global stability of general cholera models with nonlinear incidence and removal rates
J. Frankl. Inst.
(2015) - et al.
Global analysis of a vector-host epidemic model in stochastic environments
J. Frankl. Inst.
(2019) - et al.
Pulse vaccination strategy in the sir epidemic model
Bull. Math. Biol.
(1998) - et al.
Global dynamics of an infinite dimensional epidemic model with nonlocal state structures
J. Differ. Equ.
(2018) - et al.
An edge-based sir model for sexually transmitted diseases on the contact network
J. Theor. Biol.
(2018) - et al.
A stochastic model for internal HIV dynamics
J. Math. Anal. Appl.
(2008) - et al.
A stochastic vector-borne epidemic model: quasi-stationarity and extinction
Math. Biosci.
(2017) - et al.
Fluctuating periodic solutions and moment boundedness of a stochastic model for the bone remodeling process
Math. Biosci.
(2018) - et al.
The sis epidemic model with Markovian switching
J. Math. Anal. Appl.
(2012) - et al.
A stochastic vector-borne epidemic model: quasi-stationarity and extinction
Math. Biosci.
(2017)
A stochastic sirs epidemic model with infectious force under intervention strategies
J. Differ. Equ.
Global threshold dynamics of a stochastic differential equation sis model
J. Math. Anal. Appl.
Threshold dynamics and ergodicity of an sirs epidemic model with Markovian switching
J. Differ. Equ.
A new way of investigating the asymptotic behaviour of a stochastic sis system with multiplicative noise
Appl. Math. Lett.
Quasi-sure exponential stabilization of stochastic systems induced by G-Brown. Motion with discrete time feedback control
J. Math. Anal. Appl.
Stability analysis of inertial Cohen–Grossberg neural networks with Markovian jumping parameters
Neurocomputing
The threshold of a stochastic susceptible-infective epidemic model under regime switching
Nonlinear Anal.: Hybrid Syst.
Asymptotic profiles of steady states for a diffusive sis epidemic model with mass action infection mechanism
J. Differ. Equ.
The threshold of a stochastic sirs epidemic model with saturated incidence
Appl. Math. Lett.
Mathematical analysis of a virus dynamics model with general incidence rate and cure rate
Nonlinear Anal.: Real World Appl.
Exclusion and persistence in deterministic and stochastic chemostat models
J. Differ. Equ.
Certain properties related to well posedness of switching diffusions
Stoch. Process. Appl.
Portmanteau theorem for unbounded measures
Stat. Probab. Lett.
Ergodic property of the chemostat: a stochastic model under regime switching and with general response function
Nonlinear Anal.: Hybrid Syst.
Cited by (3)
A new idea on density function and covariance matrix analysis of a stochastic SEIS epidemic model with degenerate diffusion
2020, Applied Mathematics LettersCitation Excerpt :Moreover, it is very necessary to point out that the methods mentioned above can be also applied to study other stochastic epidemic models with non-degenerate diffusion (see e.g. [18,19]).
Dynamical behavior of a stochastic SIQS model via isolation with regime-switching
2023, Journal of Applied Mathematics and ComputingThreshold dynamics and sensitivity analysis of a stochastic semi-markov switched sirs epidemic model with nonlinear incidence and vaccination
2021, Discrete and Continuous Dynamical Systems - Series B
- ☆
Z. Qiu was supported by the National Natural Science Foundation of China (11671206, 11971232). T. Feng was supported by the Scholarship Foundation of China Scholarship Council (201806840120) and the Fundamental Research Funds for the Central Universities (30918011339).