On epidemic models with nonlinear cross-diffusion

Modelling and simulation of infectious diseases help to predict the likely outcome of an epidemic. A well-known generic model type for the quantitative description of the epidemic evolution dynamics by an ordinary differential equation is provided by so-called SIR models. These models classify a population into “suscepti-ble” (S), “infected” (I) and “recovered” (R) subgroups. One very early and simple prototype of an SIR-model is due to Kermack and McKendrick (1927). It describes the population evolution by the system of ordinary differential equations dS dt = −βSI, dI dt = βSI − γI, dR dt = γI, with transmission rate β > 0 and recovery rate γ > 0. This equation can be written in general form as ∂S ∂t = f(S, I), ∂I ∂t = g(S, I), (1) where the epidemic interaction dynamics are modeled in the functions f and g. The variable “R” can be dropped since its evolution is already implicit in the equations. This basic SIR model can be extended by introducing a spatial distribution of both populations. In this contribution, it is shown how the spatial extension can be done by either a continuous or a discrete spatial distribution. For the continuous distribution, the model is typically formulated as a system of reaction-diffusion equations, where the reaction terms describe the local dynamics of susceptible and infected species, and the diffusion terms account for the spatial-distribution dynamics. For a discrete local distribution, the population is typically arranged in a series of patches. Each combination of population-type and patch corresponds to a particular variable. To describe the dynamics between different patches of the same population-type, the ordinary differential equations are extended by diffusion terms, which have an effect that is similar to the heat diffusion in Newton’s law of cooling. The dynamics are the same for continuous and discrete spatial distributions. At each location, both populations are present and interact locally according to an ordinary differential equation that governs the local epidemic interaction-dynamics. In addition, the spatial distribution allows us to model a spatial diffusion, which can be either “self-diffusion” or “cross-diffusion”. For self-diffusion, the diffusion rate of each population depends on its particular local variation, whereas for cross-diffusion, the rate also depends on the local variation of the other population. For an SIR model, cross-diffusion is consistent with the phenomenon that the susceptible population avoids areas with an elevated percentage of the infected population. It is known that diffusion in general and self-diffusion in particular aim towards an equilibrium solution, where the populations are homogeneously distributed. On the other hand, the coupling of diffusion with ordinary differential equations might provoke a nonequilibrium behavior, which is known as Turing instability. The extended SIR model provides a spatial pattern formation that is essentially driven by cross-diffusion. In this contribution, nonlinear constitutive equations are suggested and analyzed for both self-diffusion and cross-diffusion. In particular, the modelling of cross-diffusion is reconsidered by starting from a basic SIR model, which is extended by either a discrete or continuous spatial distribution. Comparing the discrete model with patches to the continuous PDE model prepares the way for a micro-macro transition, where the continuous model can be deduced from the discrete. This deduction allows us to advance in the development of epidemic models with cross-diffusion. This is further elaborated in the complete version of the paper.