Mathematical Models of Epidemics with Infection Time

After an infectious contact, there is a time lapse before the individual actually becomes infected. In this paper, we present different ways of incorporating this incubation or infection time into epidemic mathematical models. For simplicity, we consider the Susceptible–Infective–Susceptible and the Susceptible–Infective–Recovered–Susceptible models with no demographic effects, so we can concentrate on the infection process. We study the different methods from a modeling point of view to determine their biological validity and find that not all methods presented in the literature are valid. Specifically, the infection part of the model should only move individuals from one compartment to another but should not change the total population. We consider models with no delay, discrete delay, distributed delay, exposed populations, and fractional derivatives. We analyze the methods that are realistic and find their steady solutions, stability, and bifurcations. We also investigate the effect of the duration of the infection time on the solutions. Numerical simulations are conducted and guidelines on how to chose a method are presented.