A renewal equation approach to behavioural epidemic models: analytical and numerical issues

Epidemic models structured by the age of infection can be formulated in terms of a system of renewal equations and represent a very general mathematical framework for the analysis of infectious diseases ([1, 2]). Here, we propose a formulation of renawal equations that takes into account of the behavioral response of individuals to infection. We use the so called "information index", which is a distributed delay that summarizes the information available on current and past disease trend, and extend some results regarding compartmental behavioral models [3, 4, 5]. For the numerical solution of the equations we propose a non-standard approach [6] based on a non local discretization of the integral term characterizing the mathematical equations. We discuss classical problems related to the behaviour of this scheme and we prove the positivity invariance and the unconditional preservation of the stability nature of equilibria, with respect to the discretization parameter. These properties, together with the fact that the method can be put into an explicit form, actually make it a computationally attractive tool and, at the same time, a stand-alone discrete model describing the evolution of an epidemic. This is a joint work with Bruno Buonomo and Claudia Panico from University of Naples "Federico II", and Antonia Vecchio from IAC-CNR, Naples.