Homogenous mixing and network approximations in discrete-time formulation of a SIRS model

A discrete-time deterministic epidemic model is proposed to better understand the contagious dynamics and the behaviour observed in the incidence of real infectious diseases. For this purpose, we analyse a SIRS model both in a random-mixing approach and in a small-world network formulation. The models include the basic parameters that characterize an epidemic: infection and recovery times, as well as mechanisms of contagion. Depending on the parameters, the random-mixing model has different types of behaviour of an epidemic: pathogen extinction; endemic infection; sustained oscillations and dynamic extinction. Spatial effects are included in our network-based approach, where each individual of a population is represented by a node of a small-world network. Our network-based approach includes rewiring connections to account for time-varying network structure, a consequence of the natural response to the emergence of an epidemic (e.g. avoiding contacts with infected individuals). Random and adaptive rewiring conditions are analysed and numerical simulation are made. A comparison of model predictions with the actual effects of COVID-19 infection on population that occurred in Italy and France is produced. Results of the time series of infected people show that our adaptive evolving networks represent effective strategies able to decrease the epidemic spreading.