A non standard finite difference model for a class of renewal equations in epidemiology

Mathematical models based on non-linear integral and integro-differential equations are gaining increasing attention in mathematical epidemiology due to their ability to incorporate the past infection dynamic into its current development. This property is particularly suitable to represent the evolution of diseases where the dependence of infectivity on the time since becoming infected plays a crucial role. These renewal equation models contain an integral term describing the contribution of the force of infection to the total infectivity and need, in general, numerical simulations for a complete understanding and quantitative description. For a general model which includes demographic effects [1, 2], we propose a non-standard approach [3] 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.