Threshold dynamics and density function of a stochastic cholera transmission model

Ying He; Bo Bi; Ying He; Bo Bi

doi:10.3934/math.20241065

AIMS Mathematics

2024, Volume 9, Issue 8: 21918-21939. doi: 10.3934/math.20241065

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Research article

Threshold dynamics and density function of a stochastic cholera transmission model

Ying He ¹,
Bo Bi ^{2
,
,}

1.
School of Mathematics and Statistics, Northeast Petroleum University, Daqing 163318, China
2.
School of Public Health, Hainan Medical University, Haikou 571199, China

Received: 20 March 2024 Revised: 17 June 2024 Accepted: 02 July 2024 Published: 11 July 2024
MSC : 37H05, 37H30, 60H10

Cholera, as an endemic disease around the world, has imposed great harmful effects on human health. In addition, from a microscopic viewpoint, the interference of random factors exists in the process of virus replication. However, there are few theoretical studies of viral infection models with biologically reasonable stochastic effects. This paper studied a stochastic cholera model used to describe transmission dynamics in China. In this paper, we adopted a special method to simulate the effect of environmental perturbations to the system instead of using linear functions of white noise, i.e., the transmission rate of environment to human was satisfied Ornstein–Uhlenbeck processes, which is a more practical and interesting. First, it was theoretically proved that the solution to the stochastic model is unique and global, with an ergodic stationary distribution. Moreover, by solving the corresponding Fokker–Planck equation and using our developed algebraic equation theory, we obtain the exact expression of probability density function around the quasi-equilibrium of the stochastic model. Finally, several numerical simulations are provided to confirm our analytical results.
- Cholera transmission model,
- ergodic stationary distribution,
- Fokker–Planck equation,
- density function,
- extinction
Citation: Ying He, Bo Bi. Threshold dynamics and density function of a stochastic cholera transmission model[J]. AIMS Mathematics, 2024, 9(8): 21918-21939. doi: 10.3934/math.20241065

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Abstract

Cholera, as an endemic disease around the world, has imposed great harmful effects on human health. In addition, from a microscopic viewpoint, the interference of random factors exists in the process of virus replication. However, there are few theoretical studies of viral infection models with biologically reasonable stochastic effects. This paper studied a stochastic cholera model used to describe transmission dynamics in China. In this paper, we adopted a special method to simulate the effect of environmental perturbations to the system instead of using linear functions of white noise, i.e., the transmission rate of environment to human was satisfied Ornstein–Uhlenbeck processes, which is a more practical and interesting. First, it was theoretically proved that the solution to the stochastic model is unique and global, with an ergodic stationary distribution. Moreover, by solving the corresponding Fokker–Planck equation and using our developed algebraic equation theory, we obtain the exact expression of probability density function around the quasi-equilibrium of the stochastic model. Finally, several numerical simulations are provided to confirm our analytical results.

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