SIRS dynamics on a diffusive agent system

Since its introduction in 1927 by Kermack and McKendrick, SIR compartmental models have been the basis of mathematical epidemiology. In this work we consider a SIRS epidemics on a system of mobile agents, which can interact during a finite period of time that depends on their dynamics. Thus, as the probability of disease transmission depends on this contact time, the spatial dynamics will strongly influence the disease evolution. By combining individual-based simulations and mean-field arguments, we study the dependency of the equilibrium populations on motility parameters, specifically the active speed and tumbling frequency. We find that the equilibrium epidemic size exhibits two very distinct, non-trivial scaling regimes with the motility parameters, depending on whether the system is in the ballistic or diffusive regime. Our mean-field estimates lead to an effective renormalization of the transition rates that allow building a phase-diagram that separates endemic and disease free phases. We find an excellent agreement between numerical simulations and mean-field estimates.