Introduction

Spatial spread of communicable diseases is closely related to the spatial heterogeneity of the environment and the spatial-temporal movement of the hosts. Mathematical modeling of disease spread normally starts with the consideration of the transmission dynamics within a population which is homogeneous in terms of host structures and environmental variation, and then follows by the examination of the impact on the transmission dynamics of the refined and detailed biological/epidemiological structures and patterns of spatial dispersal/diffusion of the hosts.

Epidemic theory for homogeneous populations has shown that the basic reproductive number, which may be considered as the fitness of a pathogen in a given population, must be greater than unity for the pathogen to invade a susceptible population (Anderson and May, 1991; Brauer and Castillo-Chavez, 2000; Diekmann and Heesterbeek, 2000; Edelstein-Keshet, 1988; Jones and Sleeman, 2003; Murray, 2003;

Thieme, 2003). It is natural to ask how spatial movement of the hosts affects the spatial-temporal spread pattern of the disease if the basic reproduction number for an otherwise homogeneous population exceeds unity.

Answers to the above question obviously depend on the manner in which hosts move into, out of, and within the considered geographical region. For example, adding an immigration term so that infective individuals enter the system at a constant rate clearly allows the persistence of the disease, because if it dies out in one region then the arrival of an infective from elsewhere can trigger another epidemic. Indeed, a constant immigration term has a mildly stabilizing effect on the dynamics and tends to increase the minimum number of infective individuals observed in the models (Bolker and Grenfell, 1995). Spread of diseases in a heterogeneous population has also been intensively studied using patchy or metapopulation models. These models are formulated under the assumption that the host population under consideration can be divided into multipatches so that the host population within a patch is considered as homogeneous, and the heterogeneity is associated with the rates with which individuals move from one patch to another (Arino and van den Dreissche, 2006).

Another popular way to incorporate the spatial movement of hosts into epidemic models is to assume some types of host random movements, leading to reaction-diffusion equations. See, for example, Busenberg and Travis (1983), Capasso (1978), Capasso and Wilson (1997), De Mottoni et al. (1979), Gudelj et al. (2004), Fitzgibbon et al. (2007), Webb (1981). This strand of theoretical developments built on the pioneering work of Fisher (1937), who used a logistic-based reaction-diffusion model to investigate the spread of an advantageous gene in a spatially extended population. With initial conditions corresponding to a spatially localized introduction, such models predict the eventual establishment of a well-defined invasion front which divides the invaded and uninvaded regions and moves into the uninvaded region with a constant velocity. The velocity at which an infection wave moves is set by the rate of divergence from the (unstable) disease-free state and can be determined by linear methods (Murray, 2003).

Most reaction-diffusion (or reduced/related space-dependent integral) epidemic models are space-dependent extensions of the classical Kermack-McKendrik (Kermack and McKendrik 1927) deterministic compartmental model for a directly transmitted viral or bacterial agent in a closed population consisting of susceptibles, infectives, and recovereds. Their model leads to a nonlinear integral equation which has been studied extensively. The deterministic model of Bartlett (1956) predicts a wave of infection moving out from the initial source of infection. Kendall (1957) generalized the Kermack-McKendrik model to a space-dependent integro-differential equation. Aronson (1977) argued that the three-component Kendall model can be reduced to a scalar one and extended the concept of asymptotic speed of propagation developed in Aronson and Weinberger (1975) to the scalar epidemic model. The Kendall model assumes that the infected individuals become immediately infectious and does not take into account the fact that most infectious diseases have an incubation period. This incubation period was considered by Diekmann (1978, 1979) and Thieme (1977a, 1977b, 1979), using a nonlinear (double) integral equation model. For further study on velocity of spatial spread, we refer to Mollison (1991), van den Bosch et al. (1990), the monograph of Rass and Radicliffe (2003), and references cited therein. Most of these studies concern the existence of traveling waves, and their relation to the disease propagation/spreadrate. For additional studies, see Ai and Huang (2005), Cruickshank et al. (1999), Hosono and Ilyas (1995), Kuperman and Wio (1999), Zhao and Wang (2004), etc.

Despite these studies on reaction-diffusion epidemic models, however, there are very few studies on modeling spatial spread of specific diseases using partial differential equation models. In this chapter, we review some previous studies on modeling spatial spread of specific communicable diseases using reaction-diffusion equations. Selected topics include the transmission of rabies in fox population (Källen et al., 1984; Källen et al., 1985; Murray et al., 1986), dengue (Takahashi et al., 2005), West Nile virus (Lewis et al., 2006; Ou and Wu, 2006), hantavirus spread in mouse populations (Abramson and Kenkre, 2002), Lyme disease (Caraco et al., 2002), and feline immunodeficiency virus (FIV) (Fitzgibbon et al., 1995; Hilker et al., 2007).

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