A mathematical model of infectious disease contagion that accounts for population stratification based on immunity criteria is proposed. Our goal is to demonstrate the effectiveness of this idea in preventing different epidemics and to lessen the significant financial and human costs these diseases cause. We determined the fundamental reproduction rate, and with the help of this rate, we were able to examine the stability of the free equilibrium point and then proposed two control measures. The Pontryagin's maximum principle is used to describe the optimal controls, and an iterative approach is used to solve the optimality system. Finally, numerical simulations are carried out in MATLAB to verify the theoretical analysis.

In this paper, we propose a mathematical model of COVID-19 infection, taking into account the division of the population according to vaccination criteria. Our goal is to demonstrate the positive effect of receiving the third dose of the Corona vaccine. We proposed two strategies to limit the spread of the COVID-19 pandemic respectively awareness programs on the importance of the third dose of the vaccine and the delivery of treatment to infected individuals who have health problems. Pontryagin's maximum principle is applied in order to characterize the optimal controls, and the optimality system is resolved using an iterative approach. At last, numerical simulations are executed to verify the theoretical analysis using MATLAB.