ABSTRACT
This contribution is a condensed version of Berres & Ruiz-Baier 2011. The goal is, on the one hand, to generate pattern formation in an epidemic model by a cross-diffusion term, and, on the other hand, to prevent blow-up by a nonlinear limitation of the cross-diffusion. These assumptions are designed to qualitatively reflect psychological behavior. The cross-diffusion term has the interpretation that the susceptible population moves away from increasing gradients of the
1 INTRODUCTION
The knowledge of spreading dynamics of infectious diseases helps to design prevention measures. A generic model category for the quantitative description of the epidemic evolution dynamics by an ordinary differential equation are the so-called SIR models, which classify a population into ‘susceptible’ (S), ‘infected’ (I) and ‘recovered’ (R) subgroups and balance the changes between these. One very early and simple prototype of a SIRmodel is due to Kermack and McKendrick 1927. It describes the population evolution by the system of ordinary differential equations
dS dt
SI dI dt
SI dR dt
I= − −α α= β βI =, ,
where α > 0 is the infection rate and β > 0 the recovery rate. There are several suggestions for improving the specification of these ODE-dynamics (Kim et al. 2010, Li et al. 2010), and structural modifications like SIR-models in networks (Liu & Zhang 2010). A key issue in epidemic modeling is the formation of spatial patterns. Based on a general setting in the two-dimensional reaction-diffusion framework for epidemic processes (Webb 1981), there are several suggestions for the combination of the system of ordinary differential equations of the SIR-model with a spatially two-dimensional diffusion equation of the involved variables (He & Stone 2003, Milner & Zhao 2008, Li & Zou 2009, Sun et al. 2009). Moreover, several contributions
infected population. In addition, it is assumed that the cross-diffusion effect depends on the local population density. For the nonlinear cross-diffusion it is assumed that there exists carelessness at a small and fatalism at a high total population number. At carelessness and fatalism the susceptible population decreases its tendency to avoid agents of the infected population. Such an avoidance is most effective for intermediate (neither too small nor too large) population numbers.
