The Spatial Spread of Infectious Diseases
Mathematical Biosciences Institute, The Ohio State University
(December 5, 2006 3:30 PM - 4:30 PM)
Spatial heterogeneity, habitat connectivity, and rates of movement can have large impacts on whether a disease, such as rabies, persists or becomes extinct. In this talk, we consider equilibrium properties for a pair of related frequency-dependent SIS epidemic models in which the movement of susceptible and infected individuals in discrete or continuous space is represented by a system of differential equations, or a coupled pair of reaction-diffusion equations, respectively.
In both models, local differences in disease transmission and recovery rates characterize whether regions are low-risk or high-risk, and these differences collectively determine whether the spatial domain (or habitat) is low-risk or high-risk. We relate the basic reproduction number (R_0) to the speed with which infected individuals move within the habitat. For low-risk habitats, the disease-free equilibrium (DFE) is stable provided that the rate at which infected individuals move lies above a threshold value. For high-risk habitats, the DFE is always unstable. When the DFE is unstable, a unique endemic equilibrium (EE) exists. This EE tends to a spatially inhomogeneous DFE as the rate at which susceptible individuals move becomes very small. The limiting DFE is positive on all low-risk regions and can also be positive on some high-risk regions. Sufficient conditions for the limiting DFE to be positive or zero on high-risk regions are given, and these conditions are illustrated using numerical examples.
This work is in collaboration with Linda Allen (Texas Tech), Ben Bolker (Florida), and Yuan Lou (Ohio State).