Minimal epidemic models with information index: from compartmental to integral formulation

Recent advances in epidemiological modelling have increasingly emphasised the role of sociological factors in describing epidemic dynamics. In this paper, we first provide an overview of the main results regarding the formulation of minimal ordinary differential equations (ODEs) models with information-dependent contact patterns. We then discuss how the minimal ODE models may be extended to integral models through the formulation of specific constitutive equations. We recall the results concerning the integral model that generalises the ODE models with prevalence-dependent contact patterns. Furthermore, we provide a follow-up by also considering the integral model with incidence-dependent contact patterns. For this new model, we study the asymptotic properties of the solutions and obtain sufficient conditions for the stability of the steady states, expressed in terms of the memory kernel and the infectivity function. We numerically show how the memory kernel affects the dynamical outcomes of the model when specific infectivity functions are considered.