Abstract
In this paper, we present a regime-switching SIR epidemic model with a ratio-dependent incidence rate and degenerate diffusion. We utilize the Markov semigroup theory to obtain the existence of a unique stable stationary distribution. We prove that the densities of the distributions of the solutions can converge in L1 to an invariant density under certain condition. Moreover, the sufficient conditions for the extinction of the disease, which means the disease will die out with probability one, are given in two cases. Meanwhile, we obtain a threshold parameter which can be utilized in identifying the stochastic extinction and persistence of the disease. Some numerical simulations are given to illustrate the analytical results.
Highlights
Based on the pioneering research[1], mathematical model provides effective control measures for infectious diseases and is an significant tool for analyzing the epidemiological characteristics of infectious diseases[2]
The long-time behavior of a regime-switching SIR epidemic model with a ratio-dependent incidence rate and degenerate diffusion are observed in this paper
The existence of a unique stable stationary distribution is obtained by using markov semigroup theory
Summary
In the study of SIR deterministic model, disease eradication and stability are two of the most concerns. Theorem 2.1 show that if 0S > 1, there exists a unique stationary distribution μ(⋅,⋅) of model (1.4) which is ergodic. The results mean that the model is stochastically asymptotic stability. Our result means that if 0S > 1, the disease I is stochastic persistent. We will investigate the stochastic extinction of the disease in model (1.4). To this end, we establish the following theorem. Be , of model (1.4) which is ergodic It means that the disease I is persistent stochastically. From the condition (i) in Theorem 2.2, we can derive that large noises can inhibit the outbreak of disease
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