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Unformatted text preview: Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain ∗ Jing Li and Xingfu Zou Department of Applied Mathematics University of Western Ontario London, ON N6A 5B7, Canada E-mails: [email protected] and [email protected] Accepted by Bulletin of Mathematical Biology, March 2009 Abstract In this paper, with the assumptions that an infectious disease in a population has a fixed latent period and the latent individuals of the population may diffuse, we formulate an SIR model with a simple demographic structure for the population living in a spatially continuous environment. The model is given by a system of reaction-diffusion equations with a discrete delay accounting for the latency and a spatially non-local term caused by the mobility of the individuals during the latent period. We address the existence, uniqueness and positivity of solution to the initial-value problem for this type of system. Moreover, we investigate the traveling wave fronts of the system and obtain a critical value c ∗ which is a lower bound for the wave speed of the traveling wave fronts. Although we can not prove that this value is exactly the minimal wave speed, numeric simulations seem to suggest that it is. Furthermore, the simulations on the PDE model also suggest that the spread speed of the disease indeed coincides with c ∗ . We also discuss how the model parameters affect c ∗ . Keywords. Latent period, diffusion, traveling waves, wave speed, spread speed, non-local in- fection. * Research supported by NSERC, by NCE-MITACS of Canada and by PREA of Ontario. 1 1 Introduction Mathematical models have been extensively used to study the dynamics of infectious diseases in population level. Most continuous time models are in the form of ordinary differential equations (ODEs) (see, e.g., [7, 15]). Such ODEs models assume that the population are well mixed, and the transmission are instantaneous. In reality, the environment in which a population lives is often heterogeneous making it necessary to distinguish the locations. Also, some diseases have a latency: infected individuals do not infect other susceptible individuals until some time later. Taking the human tuberculosis or bovine tuberculosis as an example, it may take months for the infection to develop to the infectious stage (see, e.g., [1, 5, 14, 25] and references therein). In order for a model to be more realistic, the above factors should be incorporated into the model. For the latter, in their recent work , van den Driessche et al. set up an SEIR model with a general probability function p ( t ) to account for the probability that an exposed individual still remains in the exposed class t time units after entering the exposed class. When p ( t ) is subject to a negatively exponentially distribution, the model reduces to one still described by an ODEs system. The situation when p ( t ) is a step function is more interesting. Indeed, when p ( t ) =...
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This note was uploaded on 02/10/2012 for the course MTH 487 taught by Professor Jhonopera during the Spring '11 term at Cleveland State.
- Spring '11
- Applied Mathematics