Author(s)

Jie Xu, Tiansi Zhang

College of Science, University of Shanghai for Science and Technology, Shanghai, China.

The article investigates a SIQR epidemic model with specific nonlinear incidence rate and stochastic model based on the former, respectively. For deterministic model, we study the existence and stability of the equilibrium points by controlling threshold parameter R₀ which determines whether the disease disappears or prevails. Then by using Routh-Hurwitz criteria and constructing suitable Lyapunov function, we get that the disease-free equilibrium is globally asymptotically stable if R₀<1 or unstable if R₀>1. In addition, the endemic equilibrium point is globally asymptotically stable in certain region when R₀>1. For the corresponding stochastic model, the existence and uniqueness of the global positive solution are discussed and some sufficient conditions for the extinction of the disease and the persistence in the mean are established by defining its related stochastic threshold R₀^s. Moreover, our analytical results show that the introduction of random fluctuations can suppress disease outbreak. And numerical simulations are used to confirm the theoretical results.

Epidemic Model, Specific Nonlinear Incidence Rate, Lyapunov Function, Stability, Existence, Persistence

Xu, J. and Zhang, T. (2019) Dynamic Analysis for a SIQR Epidemic Model with Specific Nonlinear Incidence Rate. Journal of Applied Mathematics and Physics, 7, 1840-1860. doi: 10.4236/jamp.2019.78126.