Optimal control of infectious disease: Information-induced vaccination and limited treatment

Abstract

In this study, a nonlinear compartmental model for an infectious disease is proposed. The spread of information is considered due to spread of disease in the population. The dynamics of the information is modeled by a separate rate equation and it is assumed that the growth of the information depends on the density of the infective population. The model accounts for the effect of information on vaccination coverage during the epidemic outbreak when medical resources for treatment are limited. Further, considering information-induced vaccination and treatment as controls, an optimal control problem is proposed which minimizes costs incurred due to the disease burden and applied controls. The total incurred cost is determined by taking a weighted sum of cost of productivity loss due to disease and costs incurred in applying control interventions. A higher nonlinearity in cost is considered for information induced vaccination efforts. With the help of Pontryagin’s Maximum Principle, the control system is analyzed and optimal control profiles for the applied controls are obtained. We further numerically explore the optimal control problem. A comparative study is made by choosing following control strategies: (A) execution of only information-induced vaccination, (B) implementation of only treatment and (C) execution of both the policies simultaneously. We observe that the comprehensive use of control interventions reduces the severity of the disease burden and also minimizes the economic burden incurred due to these interventions. Further, the effect of the basic reproduction number on the proposed control strategies as well as on the dynamics of infectious diseases is investigated. The numerical results infer that the treatment is more effective and economically feasible for a mild epidemic, while the information induced vaccination is more efficient for a serious epidemic.