Some discrete-time SI, SIR, and SIS epidemic models

Abstract

Discrete-time models, or difference equations, of some well-known SI, SIR, and SIS epidemic models are considered. The discrete-time SI and SIR models give rise to systems of nonlinear difference equations that are similar in behavior to their continuous analogues under the natural restriction that solutions to the discrete-time models be positive. It is important that the entire system be considered since the difference equation for infectives I in an SI model has a logistic form which can exhibit period-doubling and chaos for certain parameter values. Under the restriction that S and I be positive, these parameter values are excluded. In the case of a discrete SIS model, positivity of solutions is not enough to guarantee asymptotic convergence to an equilibrium value (as in the case of the continuous model). The positive feedback from the infective class to the susceptible class allows for more diverse behavior in the discrete model. Period-doubling and chaotic behavior is possible for some parameter values. In addition, if births and deaths are included in the SI and SIR models (positive feedback due to births) the discrete models exhibit periodicity and chaos for some parameter values. Single-population and multi-population, discrete-time epidemic models are analyzed.