Stability analysis of epidemiological models incorporating heterogeneous infectivity

In this paper we analyze general deterministic epidemiological models described by autonomous ordinary differential equations taking into account heterogeneity related to the infectivity and vital dynamics, in which the flow into the compartment of the susceptible individuals is given by a generic function. Our goal is to provide a new tool that facilitates the qualitative analysis of equilibrium points, which represent the disease free population, generalizing the result presented by Leite et al. (Math Med Biol J IMA 17:15–31, 2000) , and population extinction. The epidemiological models exposed are the type SEIRS (Susceptible-Exposed-Infectious-Recovered-Susceptible) and SEIR (Susceptible-Exposed-Infectious-Recovered) with vaccination. Moreover, we computed the basic reproduction number from the models by van den Driessche and Watmough (Math Biosci 180:29–48, 2002) and correlate this threshold parameter with the stability of the equilibrium point representing the disease free population.