Browsing by Subject "Deterministic chaos -- Mathematical models"
Now showing 1 - 3 of 3
Results Per Page
Sort Options
Item Deterministic and stochastic discrete-time epidemic models with applications to amphibians(Texas Tech University, 2004-08) Emmert, Keith EricA discrete-time model is formulated for spread of disease in a structured host population. The host population is sub-divided into three developmental stages, larval, juvenile, and adult, and each stage can be infected by the pathogen. Recovery from the disease is possible with this model. We investigate conditions on the parameters where either the host population does not survive or the host population survives and is free from disease. The analysis assumes parameters of the model are constants. Several different submodels of the full structured epidemic model are studied and conditions are derived for global stability of the extinction equilibrium and local stability of the disease-free equilibrium. Numerical examples are presented to illustrate the dynamics of the model when the disease-free equilibrium is not stable. The motivation for this model is the spread of a fungal pathogen in an amphibian population. A second discrete-time deterministic and stochastic epidemic model is formulated for spread of disease in a structured host population. This model differs from the previous model because the parameters of this model are periodic. The host population is again subdivided, but this time into two developmental stages, juvenile and adult. Each stage can be infected by the pathogen, but there is no recovery from the disease. Several submodels of the full model are studied and conditions for global extinction as well as local stability of the disease-free solutions are given. Stochastic and deterministic examples illustrating the dynamics of the model are presented. The motivation for this model is the spread of a fungal pathogen in amphibian populations which are explosive breeders.Item Deterministic and stochastic models of virus dynamics(Texas Tech University, 2003-12) Perera, Niranjala CA variety of mathematical models ranging from very simple ones to complicated ones have been developed and analyzed in order to capture different phenomena associated with the spread of diseases. Even though none of these models behave exactly according to the observed clinical data, major features of disease dynamics can be captured merely by means of a simple model. The model introduced by Nowak and May [12] is such simple deterministic model of which a stability analysis has not been done. Our objectives in this endeavor are two-fold. The first objective of this thesis is to carry out a thorough analysis of the aforementioned deterministic model of virus dynamics while obtaining the related system of Ito stochastic differential equations which has not been obtained to date. The motivation for obtaining the related stochastic model is also two-fold. The first reason is the capability of stochastic models to capture the randomness associated with the disease dynamics. The second reason is while a deterministic model predicts a single outcome for a given set of parameter values, a stochastic model predicts an infinite set of possible outcomes weighed by their likelihoods and probabilities. Any mathematical model which describes virus dynamics, is not complete until it describes the immune response. With analogy to a predator-prey model, immune cells play the role of the predator while the virus plays the role of the prey. The immune response is triggered by encountering a foreign antigen. The role of the immune system is to fight off invasion by foreign pathogens. In this endeavor, our interest is a special kind of T cell, namely cytotoxic T lymphocyte (CTL) which can also identify and eliminate infected cells. Then the immune response is incorporated with the aforementioned simple model of virus dynamics. This is done under three different assumptions on the CTL proliferation rate. This evidently results in three different models. The second objective of this thesis is to carry out a thorough stability analysis of the three deterministic models of virus dynamics with the CTL response while obtaining respective systems of Ito stochastic differential equations.Item Mathematical models for the evolution of host-pathogen systems(Texas Tech University, 2004-08) McCormack, Robert KennethA continuous-time deterministic model of a host-pathogen system is developed and analyzed. The model includes the genetics of the pathogen. The host population is divided into susceptibles and infectives and the infective host population governs the growth of the pathogen population. It is shown that under certain conditions, the pathogen population obeys the Hardy-Weinberg law. A stochastic model is developed based on the deterministic formulation. Numerical simulations are used to compare the dynamics of the deterministic and stochastic models.