### Mathematical Biosciences and Engineering

2019, Issue 5: 5504-5530. doi: 10.3934/mbe.2019274
Research article Special Issues

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

• 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.

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

### Related Papers:

• 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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