The epidemiology of childhood related diseases: Stability analysis and optimal control.

This work examines optimum vaccination strategies for childhood illnesses (such as chicken pox, mumps, German measles and hepatitis B). The theory involves the stability analysis of an epidemiological SIR Model. Contact rate and vaccination parameter were assumed as periodic. A stability analysis for the Disease Free State (trivial solution) was done. For periodically driven systems, the Floquet theory is the standard tool (leading to Orbital stability). Another approach to the analysis of the stability of the trivial solution was done by Singular Perturbations theory which uses the different time scales of disease infection and population dynamics. This leads to the definition of Instantaneous stability. The results lead in parallel, to two optimization problems from which we obtain optimum vaccination strategies. We define a set of solutions for the optimization problems and offer candidate solutions for every optimization problem which belongs to the set of optimal solutions. Furthermore both control scenarios (Orbital and Instantaneous stability) were simulated. The investigations indicate that it is mostly better to apply the Instantaneous stability criterion which leads to an almost optimum Floquet factor. The focusing exclusively on the Floquet factor (Orbital stability) can lead to situations in which the Instantaneous stability is not satisfied and for some illnesses, short-term epidemics may occur. The segment on Hepatitis B is a case study. An age-structured hepatitis B model was analyzed. Motivation for this investigation is an earlier published work on Bi-stability in the case of a model without age structure. We look in particular at the question of whether Bi-stability can occur in the case of an age structured model