A comprehensive spatial-temporal infection model

Abstract

Motivated by analogies between the spreading of human-to-human infections and of chemical processes, we develop a comprehensive model that accounts both for infection and for transport. In this analogy, the three different populations of infection models correspond to three chemical species. Areal densities emerge as the key variables, thus capturing the effect of spatial density. We derive expressions for the kinetics of the infection rates and for the important parameter R0, that include areal density and its spatial distribution. Coupled with mobility the model allows the study of various effects. We first present results for a batch reactor, the chemical process equivalent of the SIR model. Because density makes R0 a decreasing function of the process extent, the infection curves are different and smaller than for the standard SIR model. We show that the effect of the initial conditions is limited to the onset of the epidemic. We derive effective infection curves for a number of cases, including a back-and-forth commute between regions of low and high R0 environments. We then consider spatially distributed systems. We show that diffusion leads to traveling waves, which in 1-D geometries propagate at a constant speed and with a constant shape, both of which are sole functions of R0. The infection curves are slightly different than for the batch problem, as diffusion mitigates the infection intensity, thus leading to an effective lower R0. The dimensional wave speed is found to be proportional to the product of the square root of the diffusivity and of an increasing function of R0, confirming the importance of restricting mobility in arresting the propagation of infection. We examine the interaction of infection waves under various conditions and scenarios, and extend the wave propagation analysis to 2-D heterogeneous systems.