Date of Award


Degree Type


Degree Name

Doctor of Philosophy


Applied Mathematics


Dr. Xingfu Zou

Second Advisor

Dr. L. Wahl

Third Advisor

Dr. P. Yu


Assuming that an infectious disease has a fixed latent period and latent individuals in the population may disperse in a spatially heterogeneous environment, we derive three new models of SIR type, which are more realistic than the existing related ones. The first one considers a 2-patch environment and ignores the demographic structure. The model is given by a system of delay differential equations (DDEs). It is a generalization of the classical Kermack-McKendrick SIR model, and it preserves some properties that the Kermack-McKendrick model processes. We show that the ratio of final sizes in two patches is fully determined by the ratio of dispersion rates of susceptible individuals between the two patches. We also numerically explore the patterns by which the disease dies out and have observed multiple outbreaks of the disease before it goes to extinction. The second model considers a general n-patch environment but incorporates a simple demographic structure, also resulting in a system of DDEs. Assuming the irreducibility of dispersal rates matrices of the infected classes, an expression of the basic reproduction number Ro is obtained. It is shown that disease free equilibrium is globally asymptotically stable if Ro < 1, and unstable if Ro > 1. In the latter case, there is at least one interior equilibrium and the disease is uniformly persistent. For n = 2, two special cases are considered to obtain more detailed results on how the disease latency and the population dispersal jointly affect the disease dynamics. The third model deals with a spatially continuous environment, and is given by a delayed system of reaction-diffusion equations with a spatially non-local term. We address the well-posedness of the model but the main concern is traveling wave fronts iii of the model. We obtain a critical value c* which is shown to be a lower bound for the wave speed of 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 model equations also suggest that the disease spread speed coincides with c*. We also discuss how the model parameters affect c*.



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