Global dynamics and density function in a class of stochastic SVI epidemic models with Lévy jumps and nonlinear incidence

Xiaodong Wang; Kai Wang; Zhidong Teng; Xiaodong Wang; Kai Wang; Zhidong Teng

doi:10.3934/math.2023148

AIMS Mathematics

2023, Volume 8, Issue 2: 2829-2855. doi: 10.3934/math.2023148

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Global dynamics and density function in a class of stochastic SVI epidemic models with Lévy jumps and nonlinear incidence

Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi 830017, Xinjiang, China

Received: 18 September 2022 Revised: 20 October 2022 Accepted: 27 October 2022 Published: 10 November 2022
MSC : 92D25, 92D30, 37C75, 37H30, 37L45

The paper studies the global dynamics and probability density function for a class of stochastic SVI epidemic models with white noise, Lévy jumps and nonlinear incidence. The stability of disease-free and endemic equilibria for the corresponding deterministic model is first obtained. The threshold criteria on the stochastic extinction, persistence and stationary distribution are established. That is, the disease is extinct with probability one if the threshold value $ R_{0}^{s} < 1 $, and the disease is persistent in the mean and any positive solution is ergodic and has a unique stationary distribution if $ R_{0}^{s} > 1 $. Furthermore, the approximate expression of the log-normal probability density function around the quasi-endemic equilibrium of the stochastic model is calculated. A new technique for the calculation of the probability density function is proposed. Lastly, the numerical examples and simulations are presented to verify the main results.
- stochastic SVI model,
- Lévy jumps,
- extinction and persistence in the mean,
- stationary distribution,
- probability density function
Citation: Xiaodong Wang, Kai Wang, Zhidong Teng. Global dynamics and density function in a class of stochastic SVI epidemic models with Lévy jumps and nonlinear incidence[J]. AIMS Mathematics, 2023, 8(2): 2829-2855. doi: 10.3934/math.2023148

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Abstract

The paper studies the global dynamics and probability density function for a class of stochastic SVI epidemic models with white noise, Lévy jumps and nonlinear incidence. The stability of disease-free and endemic equilibria for the corresponding deterministic model is first obtained. The threshold criteria on the stochastic extinction, persistence and stationary distribution are established. That is, the disease is extinct with probability one if the threshold value $ R_{0}^{s} < 1 $, and the disease is persistent in the mean and any positive solution is ergodic and has a unique stationary distribution if $ R_{0}^{s} > 1 $. Furthermore, the approximate expression of the log-normal probability density function around the quasi-endemic equilibrium of the stochastic model is calculated. A new technique for the calculation of the probability density function is proposed. Lastly, the numerical examples and simulations are presented to verify the main results.

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