Dynamics of a Diffusive SIR Epidemic Model with Time Delay

This paper is devoted to a reaction-diffusion system for a SIR epidemic model with time delay and incidence rate. Firstly, the nonnegativity and boundedness of solutions determined by nonnegative initial values are obtained. Secondly, the existence and local stability of the disease-free equilibrium as well as the endemic equilibrium are investigated by analyzing the characteristic equations. Finally, the global asymptotical stability are obtained via Lyapunov functionals.     

Qualitative behavior of numerical solutions to an s-i-s epidemic model

A finite difference scheme along the characteristics is used to approximate the solution of an age-dependent s-i-s epidemic model. The global behavior of the discrete solution resulting from the algorithm is investigated. It is shown that a nontrivial discrete periodic solution is generated by a periodic force of infection. Sufficient (and explicit) threshold conditions for the existence and stability of a unique nontrivial periodic solution are given. Results from numerical experiments are presented. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 317–337, 1998     

Epidemic dynamics and spatial segregation driven by cognitive diffusion and nonlinear incidence

We propose susceptible-infected-susceptible epidemic reaction–diffusion models with cognitive movement and nonlinear incidence $S^q{I}^p$ $(p,q>0)$ in a spatially heterogeneous environment. The cognitive dispersal term takes either random diffusion or symmetric diffusion. Building upon the $L^{\infty }$-estimates of positive solutions under $p,q>0$, we state the asymptotic dynamics for , $q>0$. The numerical results reveal spatial segregation of susceptible and infected populations: (a) the heterogeneous random diffusion can segregate the population and reduce the infection fraction significantly; (b) the segregation phenomenon disappears as the ratio $p/q$ approaches one from below; (c) the disease-free region strengthens the segregation induced by heterogeneous random diffusion; (d) the segregation governed by random diffusion is more sensitive to the incidence mechanism; (e) the distribution of steady states driven by symmetric diffusion is always similar to that by homogeneous diffusion.     

Polynomial ergodicity of an SIRS epidemic model with density-dependent demographics

In this paper, a stochastic susceptible-infective-recovered-susceptible (SIRS) model with density-dependent demographics is proposed to study the dynamics of transmission of infectious diseases under stochastic environmental fluctuations. We demonstrate that the position of the basic reproduction number $R_0^s$ with respect to 1 is the threshold between extinction and persistence of the disease under mild extra conditions. That is, under mild extra conditions, when , the disease is eradicated with probability 1; when $R_0^s>1$, the disease is persistent almost surely and the Markov process has a unique stationary distribution and is polynomial ergodic. As an application, we use the 2017 influenza A data from Western Asia to estimate the parameter values of the model and based on that investigate the effect of random noises on the dynamics of the model. Our study reveals that the basic reproduction number $R_0^s$ is negatively correlated with the noise intensity for the infected but positively correlated with that for the susceptible population, which are different from the findings obtained in the existing literature.