Stationary distribution, extinction and density function for a stochastic HIV model with a Hill-type infection rate and distributed delay

Wenjie Zuo; Mingguang Shao; Wenjie Zuo; Mingguang Shao

doi:10.3934/era.2022206

Electronic Research Archive

2022, Volume 30, Issue 11: 4066-4085. doi: 10.3934/era.2022206

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Stationary distribution, extinction and density function for a stochastic HIV model with a Hill-type infection rate and distributed delay

Wenjie Zuo ^,,
Mingguang Shao

College of Science, China University of Petroleum (East China), Qingdao 266580, China

Received: 20 July 2022 Revised: 22 August 2022 Accepted: 24 August 2022 Published: 16 September 2022

In this article, we investigate the dynamics of a stochastic HIV model with a Hill-type infection rate and distributed delay, which are better choices for mass action laws. First, we transform a stochastic system with weak kernels into a degenerate high-dimensional system. Then the existence of a stationary distribution is obtained by constructing a suitable Lyapunov function, which determines a sharp critical value $ R_0^s $ corresponding to the basic reproduction number for the determined system. Moreover, the sufficient condition for the extinction of diseases is derived. More importantly, the exact expression of the probability density function near the quasi-equilibrium is obtained by solving the Fokker-Planck equation. Finally, numerical simulations are illustrated to verify the theoretical results.
- stochastic HIV model,
- Hill function,
- stationary distribution,
- extinction,
- density function
Citation: Wenjie Zuo, Mingguang Shao. Stationary distribution, extinction and density function for a stochastic HIV model with a Hill-type infection rate and distributed delay[J]. Electronic Research Archive, 2022, 30(11): 4066-4085. doi: 10.3934/era.2022206

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Abstract

In this article, we investigate the dynamics of a stochastic HIV model with a Hill-type infection rate and distributed delay, which are better choices for mass action laws. First, we transform a stochastic system with weak kernels into a degenerate high-dimensional system. Then the existence of a stationary distribution is obtained by constructing a suitable Lyapunov function, which determines a sharp critical value $ R_0^s $ corresponding to the basic reproduction number for the determined system. Moreover, the sufficient condition for the extinction of diseases is derived. More importantly, the exact expression of the probability density function near the quasi-equilibrium is obtained by solving the Fokker-Planck equation. Finally, numerical simulations are illustrated to verify the theoretical results.

References

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