The effect of time scales on SIS epidemic model

Abstract

The distribution of diseases is one of the most interesting real-world phenomena which can be systematically studied through a mathematical model. A well-known simple epidemic model with surprising dynamics is the SIS model. Usually, the time domains that are widely used in mathematical models are limited to real numbers for the case of continuous time or to integers for the case of discrete time. However, a disease pandemic such as an influenza pandemic regularly disappears from a population and then recurs after a period of time. Additionally, collecting actual data continuously is time-consuming, relatively expensive, and really impractical. It seems that using a continuous-time model to describe observed data may not always be possible due to time domain conflict. The purpose of this paper is, therefore, to study the qualitative behavior of SIS models on continuous, discrete, and mixed continuous-discrete time scales. We investigate their dynamic behavior and examine how this behavior changes in the different time scale domains. We show that the dynamic behavior can change in a systematic manner from simple stable steady-state solutions for the continuous time domain to complicated chaotic solutions for the discrete-time domain.