Spatial effects of two-stage contagion: a Cellular Automata model

Abstract

This paper investigates the impact of spatial factors on epidemic dynamics using a two-stage contagion model with two possible outcomes, whose probability of occurrence depends of a parameter \(q \in [0,1]\). The model considers direct contagion (\(q=1\)), where contact with an ill individual causes a susceptible individual to become infected, and indirect contagion (\(q=0\)), where a susceptible individual, in contact with an infected individual, does not become ill but, enters an exposed intermediate stage, weakening their resistance to the disease. To incorporate spatial effects, we first define a rigorous integro-differential model expressed in terms of the densities of each class into which the population can be divided. The time evolution of these variables is determined by integrals that account for the range of influence of the contagion over the susceptible classes. Under certain conditions, this integro-differential system can be reduced to a system of ordinary differential equations for the average population in each class. Nonetheless, even for low-dimensional cases, the solution to this extended system appears to evade analytical methods. Alternatively, we represent the population using a cellular automata model on a two-dimensional square grid. We demonstrate that the outcome is influenced by the neighborhood size r; in the limit of large r, the model converges to the mean field approximation, where interactions follow a law of mass action. We specifically explore the impact of the initial spatial configuration on the population’s asymptotic behavior, highlighting its significance in cases of local bistability. Additionally, we establish that oscillatory spatial behavior can emerge when considering a recovered or death class. These findings lay the groundwork for future studies on multi-stage contagion in complex networks.