Some mathematical models and scientific computations in epidemiology and immunology
Abstract
It has been observed in several settings that schistosomiasis is less prevalent in segments of a river with a fast current than in those with a slow current. Some believe that this could be attributed to the flush-away of snails by the fast flow and low density of residents in regions with fast flow. In the first part of this thesis, we describe and analyze a mathematical model for schistosomiasis in which castration of infected snails, effective chemoprophylaxis, and flush-away of parasite larvae are incorporated. The model can be applied to different aquatic environments —lake regions and river regions— by varying the velocity of the "river flow." We determine a reproduction number for the disease directly related to its persistence and extinction. We show that the reproduction number is a decreasing function of the river velocity and the prevention/treatment rate and numerical simulations suggest that the mean parasite load is low when either the velocity of river or the prevention/treatment rate is high. Finally, we obtain a critical value for the chemoprophylaxis rate that indicates the minimum effort needed in order to eradicate the disease. In the second part of this thesis, we study a general framework in epidemics —an S-I-R model with directed spatial diffusion— in which susceptible individuals move away from the foci of infection and all individuals move away from overcrowded regions. The demographic changes are neglected as we assume that the duration of the epidemics is short compared with the mean life span of a human being. The model consists of three hyperbolic partial differential equations, the sum of these equations being parabolic. Positivity of solutions is established and regularity of solutions is discussed. The system is a mixed-type degenerate system in which three eigenvalues suggest a heat-like, a hyperbolic, or a backward heat-like equation. A robust first-order numerical scheme is designed and finite time blow-up of some solutions is illustrated through numerical simulations. A numerical test of the finite time blow-up of solution is proposed. The model has a wide range of applications, such as the spreading of bubonic plague in Europe in the XIVth century and refugee displacement in modern society.
Degree
Ph.D.
Advisors
Milner, Purdue University.
Subject Area
Mathematics
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