Spatial Spread of Rabies Revisited: Influence of Age-Dependent Diffusion on Nonlinear Dynamics

We consider the spatio-temporal patterns of disease spread involving structured populations. We start with a general model framework in population biology and spatial ecology where the individual's spatial movement behaviors depend on its maturation status, and we show how delayed reaction diffusion equations with nonlocal interactions arise naturally. We then consider the impact of this delayed nonlocal interaction on the disease spread by revisiting the spatial spread of rabies in continental Europe during the period between 1945 and 1985. We show how the distinction of territorial patterns between juvenile and adult foxes, the main carriers of the rabies under consideration, yields a class of partial differential equations involving delayed and nonlocal terms that are implicitly defined by a hyperbolic-parabolic equation, and we show how incorporating this distinction into the model leads to a formula describing the relation of the minimal wave speed and the maturation time of foxes. We show how the homotopy argument developed by Chow, Lin, and Mallet-Paret can be applied to obtain the existence of a heteroclinic orbit between a disease-free equilibrium and an endemic state for the spatially averaged system of delay differential equations, and we illustrate how the technique developed by Faria, Huang, and Wu can be used to establish the existence of a family of traveling wavefronts in the neighborhood of the heteroclinic orbit for the corresponding spatial model. 1. Introduction. Spatial movement and reaction time lag are certainly two intrinsic features in biological systems; their interaction seems to be one of the many factors for possible complicated spatio-temporal patterns in a single species population without an external time-dependent forcing term. Modeling this interaction is nevertheless a highly nontrivial task, and recent progress indicates diffusive (partial or lattice) systems with nonlocal and delayed reaction nonlinearities arise very naturally. Such systems were investigated in the earlier work of Yamada [34], Pozio [24, 25], Redlinger [26, 27], and the modeling and analysis effort in the groundbreaking work by Britton [3], Gourley and Britton [9], Smith and Thieme [28] marked the beginning of the systematic study of a new class of nonlinear dynamical systems directly motivated by consideration of biological realities [10, 11]. This new class of nonlinear dynamical systems can be derived from the classical structured population model involving maturation-dependent spatial diffusion rates and nonlinear birth and natural maturation processes. More specifically, if we use u(t, x) to denote the total number of matured individuals in …