A mathematical model for spatially expanding infected area of epidemics transmitted through heterogeneously distributed susceptible units

Little is known about the effect of environmental heterogeneity on the spatial expansion of epidemics. In this work, to focus on the question of how the extent of epidemic damage depends on the spatial distribution of susceptible units, we develop a mathematical model with a simple stochastic process, and analyze it. We assume that the unit of infection is immobile, as town, plant, etc. and classify the units into three classes: susceptible, infective and recovered. We consider the range expanded by infected units, the infected range R, assuming a certain generalized relation between R and the total number of infected units k, making use of an index, a sort of fractal dimension, to characterize the spatial distribution of infected units. From the results of our modeling analysis, we show that the expected velocity of spatial expansion of infected range is significantly affected by the fractal nature of spatial distribution of immobile susceptible units, and is temporally variable. When the infection finally terminates at a moment, the infected range at the moment is closely related to the nature of spatial distribution of immobile susceptible units, which is explicitly demonstrated in our analysis.