The Basic Reproduction Number for Cellular SIR Networks

Abstract

The basic reproduction number $R_0$ is the average number of new infections produced by a typical infective individual in the early stage of an infectious disease, following the introduction of few infective individuals in a completely susceptible population. If $R_0<1$, then the disease dies, whereas for $R_0>1$ the infection can invade the host population and persist. This threshold quantity is well studied for SIR compartmental or mean field models based on ordinary differential equations, and a general method for its computation has been proposed by van den Driessche and Watmough. We concentrate here on SIR epidemiological models that take into account the contact network N underlying the transmission of the disease. In this context, it is generally admitted that $R_{0}$ can be approximated by the average number $R_{2,3}$ of infective individuals of generation three produced by an infective of generation two. We give here a simple analytic formula of $R_{2,3}$ for SIR cellular networks. Simulations on two-dimensional cellular networks with von Neumann and Moore neighborhoods show that $R_{2,3}$ can be used to capture a threshold phenomenon related the dynamics of SIR cellular network and confirm the good quality of the simple approach proposed recently by Aparicio and Pascual for the particular case of Moore neighborhood.