Developing a mathematical model for optimal cost-effective treatment strategies applied to a diphtheria outbreak

Abstract

In this study, we seek to apply a system of nonlinear ordinary differential equa- tions to analyze how the dynamics of primary infection affect the proliferation of Diphtheria. We prove existence, uniqueness, positivity, and boundedness of the solution. Also investigate the qualitative behavior of the models, and find a threshold parameter that guarantee the asymptotic stability of the equilibrium points, which is known as basic reproduction number. The parameters are intro- duced in different terms of equations of the model are determined by fitting to match daily cases data sets using nonlinear least-squares method. The aim of this work is to determine the optimal control of treatment and vaccination adminis- tration schemes useful in controlling the epidemic situation especially in poorly resourced settings. The optimal treatments represent the efficacy of vaccination and treatment inhibiting diphtheria infection and preventing new infections with an objective functional which minimizes the infected populations and minimizes the systematic cost based on the percentage effect of the treatment strategies. The existence and the uniqueness of the optimal pair are discussed. A charac- terization of the optimal controls via adjoint variables is established. We obtain an optimality system that we solve numerically by a iterative Forward-Backward Sweep method. We also discuss cost-effective treatment strategy to obtain the least cost-effective objectivefunction.