Dynamics of an SLIR model with nonmonotone incidence rate and stochastic perturbation

Jinhui Zhang; Jingli Ren; Xinan Zhang; Jinhui Zhang; Jingli Ren; Xinan Zhang

doi:10.3934/mbe.2019274

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 5504-5530. doi: 10.3934/mbe.2019274

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Dynamics of an SLIR model with nonmonotone incidence rate and stochastic perturbation

1.
School of Mathematics and Statistics, Zhengzhou University, Henan Zhengzhou, 450001, China
2.
College of Science, Zhongyuan University of Technology, Henan Zhengzhou, 450007, China
3.
School of Mathematics and Statistics, Central China Normal University, Hubei Wuhan 430079, China

Received: 31 January 2019 Accepted: 09 April 2019 Published: 14 June 2019

In this paper we study an SLIR epidemic model with nonmonotonic incidence rate, which describes the psychological effect of certain serious diseases on the community when the number of infectives is getting larger. By carrying out a global analysis of the model and studying the stability of the disease-free equilibrium and the endemic equilibrium, we show that either the number of infective individuals tends to zero or the disease persists as time evolves. For the stochastic model, we prove the existence, uniqueness and positivity of the solution of the model. Then, we investigate the stability of the model and we prove that the infective tends asymptotically to zero exponentially almost surely as $R_0 < 1$. We also proved that the SLIR model has the ergodic property as the fluctuation is small, where the positive solution converges weakly to the unique stationary distribution.
- epidemic model,
- nonmonotone incidence rate,
- psychological effect,
- global stability,
- stochastic perturbation,
- ergodic property
Citation: Jinhui Zhang, Jingli Ren, Xinan Zhang. Dynamics of an SLIR model with nonmonotone incidence rate and stochastic perturbation[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5504-5530. doi: 10.3934/mbe.2019274

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Abstract

In this paper we study an SLIR epidemic model with nonmonotonic incidence rate, which describes the psychological effect of certain serious diseases on the community when the number of infectives is getting larger. By carrying out a global analysis of the model and studying the stability of the disease-free equilibrium and the endemic equilibrium, we show that either the number of infective individuals tends to zero or the disease persists as time evolves. For the stochastic model, we prove the existence, uniqueness and positivity of the solution of the model. Then, we investigate the stability of the model and we prove that the infective tends asymptotically to zero exponentially almost surely as $R_0 < 1$. We also proved that the SLIR model has the ergodic property as the fluctuation is small, where the positive solution converges weakly to the unique stationary distribution.

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