Stochastic and Renewal Methods Applied to Epidemic Models

The thesis is made up of three chapters, which analyse the disease spreads within the population in three different settings. The first chapter analyses the effects of the adaptive vaccination strategy on the infectious disease dynamics in a closed population and a demographically open population. The analytical methods are used to show that the cumulative force of infection for the closed population and the endemic force of infection in the demographically open population can be significantly reduced by combining two factors: the vaccine effectiveness and the vaccination rate. The impact of these factors on the force of infection can transform an endemic steady state into a disease free state. The second chapter analyses the SIS Epidemic dynamics using the Birth-Death Markov processes. The Susceptible-Infected-Susceptible (SIS) model is defined with the population of constant size (M); the susceptible population (S) and the infected population (I) have the same rate of birth and death (mu); the disease spreads with the transmission rate (beta). Using a stochastic method, it is shown that the magnitude of the disease spread depends on the Reproductive Number (R = beta/mu). In the long run, the stochastic equilibrium and the deterministic equilibrium yield the same infected size equilibrium (1-(1/R)) in proportion. Finally, the asymptotic distribution of the infected size is shown to follow a normal distribution with mean (1-(1/R))M and variance (M/R). The third chapter studies the impact of the Gamma distribution of the individual lifetime on the SIS Epidemic Dynamic. The approaches are both numerical and analytical methods. The composite Newton-Cotes quadrature formulas are implemented in order to provide the best accurate estimation of the infected size equilibrium. The numerical solution of the infected size was computed and the results show that the infected size is an increasing function of the shape parameter (k) with a phase of acceleration and a phase of deceleration before reaching a stable value (1-(1/(2R))). The numerical solution was also compared with the analytical solution provided by the Extreme Value Theory. The results consolidate the numerical solutions of the infected size. However, the analytical approximation is not valid for shape parameters (k) less than 1.