Persistence or extinction of disease in stochastic epidemic models and dynamically consistent discrete Lotka-Volterra competition models

Two distinct topics are considered. The first topic concerns nonstandard finite-difference (NSFD) schemes for the Lotka-Volterra competition model, and the second topic concerns the persistence or extinction of disease in stochastic epidemic models.

A problem of interest in numerical analysis is to derive a discrete-time model which can be used to approximate the solution of an ordinary differential equation (ODE) or system of ODEs. These discrete models can be constructed by applying finite difference schemes to a given ODE. The goal is to derive a discrete model which preserves the properties of the corresponding continuous model. In Chapter 2, the Lotka-Volterra competition model is introduced and some well-known properties of the model are stated. A general class of discrete-time competition models constructed from a NSFD scheme is considered. Sufficient conditions are derived such that the elements of this class of difference equations are dynamically consistent with the Lotka-Volterra competition model. The discrete models are shown to preserve the positivity of solutions, existence and stability conditions of the equilibrium points, boundedness of solutions, and monotonicity of the Lotka-Volterra system.

In deterministic epidemic theory, the basic reproduction number and type reproduction numbers, are well-known thresholds used to determine whether a disease will persist or become extinct. For stochastic epidemic models, there are similar thresholds which are used to estimate the probability of disease persistence or extinction. Typically, the deterministic and stochastic thresholds are discussed separately. In Chapter 3, some well-known deterministic (ODE) epidemic models from the literature are considered. The basic reproduction number and type reproduction numbers are calculated for each of these models and a corresponding continuous-time Markov chain (CTMC) model is derived. For each of the CTMC models, a stochastic threshold is computed as well as the probability of disease persistence or extinction. In addition, a new relationship is illustrated between the deterministic and stochastic thresholds.

Factors such as spatial heterogeneity, connectivity, and dispersal of individuals through worldwide travel can significantly affect disease dynamics. The effects of these factors have been studied for several infectious diseases including influenza, severe acute respiratory syndrome (SARS), and tuberculosis. In these studies, the population is split into several groups or patches and dispersal is allowed between these patches. In Chapter 4, deterministic and stochastic multi-patch epidemic models with and without demographics are derived and analyzed. As in Chapter 3, the basic reproduction number is calculated for the deterministic models. Two types of stochastic multi-patch models are explored: CTMC models and stochastic differential equation (SDE) models. For the CTMC models, the stochastic threshold is computed as well as the probability of disease persistence or extinction. Numerical examples illustrate the differences between the deterministic and stochastic patch models.