A SIR Model with Spatially Distributed Multiple Populations Interactions for Disease Dissemination

Abstract

In this contribution we analyze a discretized SIR (Susceptible, Infectious, and Removed) compartmental model, to investigate the role of individual interactions in the spread of diseases.  The compartments S_{i, j}, I_{i,j} and R_{i, j} (i, j, = 1,2, ..., n) are spatially distributed in a two-dimensional n x n network. We assume that the dynamics follow the well-known SIR-like iteration within the population in each (i,j) site. Moreover, the dynamics are enriched by considering a multi-population interaction following a Gaussian spatial distribution. Therefore, the mobility of individuals between distinct networks is measured from the width alpha of the Gaussian distribution. The interaction of individuals between distinct sites, responsible for the contagion between different populations, is assumed to occur in a time interval smaller than a fixed interval $h$ so that, the total population in each site (i, j), given by N_{i, j} =S_{i, j} + I_{i, j} + R_{i, j}, remained constant (for example, individual leaves his home site (i, j) to work at a neighboring site and returns to his home in a time interval less than h ). We numerically explore some scenarios of population interaction, based on distinct choices of width alpha, that include a hypothetically rapidly closedness and reopening of the economy. The results found show interesting dynamics in the infected population due to the interaction parameter alpha(t) between the populations. Finally, the model can be applied to evaluate the spread of diseases such as COVID-19 enabling decision making in different contexts.