Epidemic Models with Pulse Vaccination and Time Delay

Abstract

In this thesis we discuss deterministic compartmental epidemic models. We study the asymp- totic stability of the disease-free solution of models with pulse vaccination campaigns.

The main contributions of this thesis are to extend the literature of pulse vaccination models with delay. We take results for ordinary differential equation models and extend them to models with delay differential equations. Model generalizations include the use of a general incidence term as an upper bound for the actual incidence, and the use of switch parameters to approximate time-varying parameters.

In particular, we look at contact rate parameters which are piecewise constant or time-varying. We extend literature results for non-delay general incidence models to find uniform asymptotic stability of the disease-free solution which helps us to add delay. We find an upper bound for the susceptible population under pulse vaccination and use this bound to tighten results for eradication thresholds: that is, we use this upper bound to find sufficient conditions for the uniform asymptotic stability of the disease-free solution of delayed pulse vaccination models. We extend literature results for constant contact rate bilinear incidence delay models to models with periodic time-varying contact rate, and determine conditions under which the disease-free solution is uniformly asymptotically stable for small delay. We also find conditions for disease permanence in the corresponding non-delay, time-varying-parameter pulse vaccination model. For piecewise- constant contact rate bilinear incidence models we again find thresholds which guarantee uniform asymptotic stability under small delay.

We additionally discuss the effects of time-varying total population on our results, through a change of variables to population fractions. The total population is commonly held constant in the literature, for analytical simplicity, so we survey the methods for time-varying total population and the effects of such variation on the pulse vaccination schemes. We retain thresholds for eradication by considering the compartment populations as fractions of the total, instead of population numbers. The result is also applied to constant-population delay systems. When changing from standard incidence to bilinear incidence in delay systems, we discuss a way to estimate the effect of time-varying N.

We support our theory with simulation results.