The Discrete-time SIR, SIS Epidemical Model

In this paper,dynamical analysis of some discrete-time epidemical models are studied. We analyze the effect of general susceptible probability and dispersal between patches on disease spreading and discuss the relationship between continuous and discrete model with discrete method.In the first part,a discrete S-I-R model with general function of remaining susceptible probability is analysed.The basic reproduction number is established, existence and stability of equilibria are studied with conditions only related to this number.Some sufficient conditions for the permanence and global stability of the system are also given.Mathematical results of the model suggest that overcrowded population may be the reason for disease spreading and improving the medical technology is an effective way for disease control.In the second part,we change the general function of remaining susceptible probability for the S-I-R model,existence and local stability of equilibria are studied.A bistability bifurcation occurs when we assumed that infections are modeled as Possion processes in single Patch discrete-time S-I-R models.Two patches discrete-time S-I-R models with disease-enhanced and disease-suppressed dispersal between patches may give rise to two stable equilibria.Here,we focus on the joint impact of dispersal and disease on the population with potentially complex population dynamics.Disease-induced and disease-suppressed dispersal appear to play a critical role on the generation and support of multiple attractors,they increase the likelihood of disease persistence.In other words, dispersal enhances persistence. In the third part,a basic continuous SIS model without disease deaths is proposed. For the continuous model there is always a unique global asymptotically stable equilibrium. We define two different discrete SIS systems that approximate to the continuous system in two ways and study the stability of the models.Period-doubling and chaetic behaviors are excluded.The results of global stability show that the behaviors of new discrete-time models are similar to the continuous SIS model,but the basic reproduction number and endemic equlibira are different between the continuous model and the second discrete-time model if h is not sufficiently small.