A Pontryagin principle and optimal control of spreading COVID-19 with vaccination and quarantine subtype

A mathematical model is a beautiful and powerful way to depict the condition of epidemiological disease transmission. In this work, we used a nonlinear differential equation to construct a mathematical model of COVID-19. Nonlinear differential equation illustrates the spread of COVID-19 disease incorporating the vaccinated and quarantined subpopulations. A compartment of a model of COVID-19 disease was carried out involving several control variables and several biological assumptions. Applying the control variables to a mathematical model is the prevention of direct contact between infected and susceptible subpopulations, a vaccination control process, and an intensive handling of infected and quarantined populations. In the next section, an investigation of the positivity and boundedness of the solution COVID-19 disease, and an analysis of the existence and uniqueness of the solution was carried out. Then, the existence of the control variables involved in the mathematical model that has been designed is demonstrated. Furthermore, by applying the Pontryagin Principle to determine the optimal conditions and best values ​​for each control variable that holds on. On the other hand, in addition to the mathematical analysis result, provides numerical simulations using MATLAB software as one of the steps in describing the behavior of the dynamical solution or the phase portrait. Finally, the last section shows that the optimal control condition carried out is able to reduce the density of infected and quarantined subpopulations, respectively. Hence, it is in line with the functional objective that has been constructed.