On the discretization and control of an SEIR epidemic model with a periodic impulsive vaccination

Abstract

This paper deals with the discretization and control of an SEIR epidemic model. Such a model describes the transmission of an infectious disease among a time-varying host population. The model assumes mortality from causes related to the disease. Our study proposes a discretization method including a free-design parameter to be adjusted for guaranteeing the positivity of the resulting discretetime model. Such a method provides a discrete-time model close to the continuous-time one without the need for the sampling period to be as small as other commonly used discretization methods require. This fact makes possible the design of impulsive vaccination control strategies with less burden of measurements and related computations if one uses the proposed instead of other discretization methods. The proposed discretization method and the impulsive vaccination strategy designed on the resulting discretized model are the main novelties of the paper. The paper includes (i) the analysis of the positivity of the obtained discrete-time SEIR model, (ii) the study of stability of the disease-free equilibrium point of a normalized version of such a discrete-time model and (iii) the existence and the attractivity of a globally asymptotically stable disease-free periodic solution under a periodic impulsive vaccination. Concretely, the exposed and infectious subpopulations asymptotically converge to zero as time tends to infinity while the normalized subpopulations of susceptible and recovered by immunization individuals oscillate in the context of such a solution. Finally, a numerical example illustrates the theoretic results.