Global dynamics of difference equations for SIR epidemic models with a class of nonlinear incidence rates

Abstract

In this paper, we propose a discrete SIR epidemic model whose discretization scheme preserves the global stability of equilibria for a class of continuous SIR epidemic models. From a biological motivation, the infection rate of the model is given by unspecified functions which incorporates a latency period with some distribution. By identifying the basic reproduction number R 0 of the model, the global asymptotic stability of the equilibria of the model is fully determined by applying discrete-time analogue of Lyapunov functionals when the infection rate has a suitable monotone property. Moreover, our result indicates that the latency period does not influence the global dynamics of the model.

Dynamic consistent NSFD scheme for a viral infection model with cellular infection and general nonlinear incidence

“…However, since a discrete model generally can exhibit more complicated dynamical behavior than continuous models such as bifurcations and chaos (see [25,28] and references therein). Fortunately, a nonstandard finite difference scheme (NSFD) has been proposed by Mickens [29] and received much attention (see [20,[30][31][32][33][34][35][36][37][38] and references therein). One important advantage of Mickens' scheme is that it performs well in preserving the major dynamical properties of the approximated original continuous models (see [20,[35][36][37][38]).…”

A non-standard finite difference method for a hepatitis B virus infection model with spatial diffusion

“…It is a natural requirement of an adequate numerical method that it possess the discrete equivalents of the qualitative properties the continuous system satisfies. Enatsu et al [8] pointed out that how to choose the discrete schemes which preserve the global asymptotic stability for equilibria of the corresponding continuous-time epidemic models was an open problem. Until now, there exist some excellent works on the preservation of the global stability of equilibria for systems of ordinary differential equations [5 -8,11,12,18,23].…”

A delayed SEIRS epidemic model with impulsive vaccination and nonlinear incidence rate

“…In addition to vaccination, the incidence rate of a disease is also an important factor to affect the model. Bilinear incidence rate βSI is commonly used in many models (see [1,3,4,6,9,20]). But when the number of susceptible individuals becomes large, it is unreasonable to assume that the infected population is proportion to the number of susceptible individuals, because the number of susceptible population with every infective contact within a certain time is limited.…”