On Vaccination Strategies for a SISV Epidemic Model Guaranteeing the Nonexistence of Endemic Solutions

Abstract

A vaccination strategy based on the state feedback control theory is proposed. The objective is to fight against the propagation of an infectious disease within a host population. This propagation is modelled by means of a SISV (susceptible-infectious-susceptible-vaccinated) epidemic model with a time-varying whole population and with a mortality directly associated with the disease. The vaccination strategy adds four free-design parameters, with three of them being the feedback gains of the vaccination control law. The other one is used to switch off the vaccination if the proportion of susceptible individuals is smaller than a prescribed threshold. The paper analyses the positivity of such a model under the proposed vaccination strategy as well as the conditions for the existence of the different equilibrium points of its normalized model. The fact that an appropriate adjustment of the control gains avoids the existence of endemic equilibrium points in the normalized SISV model while guaranteeing the existence of a unique disease-free equilibrium point being globally exponentially stable is proved. This is a relevant novelty dealt with in this paper. The persistence of the infectious disease within a host population irrespective of the growing properties of the whole population can be avoided in this way. Such theoretical results are mathematically proved and, also, they are illustrated by means of simulation examples. Moreover, the performance of the proposed vaccination strategy in several real situations is studied in some simulation examples. One of them deals with the presence of uncertainties, which affects the synthesis of the vaccination control law, in the measures of the subpopulations of the model.