Mathematical modeling and optimal control for COVID-19 with population behavior

To describe the spread of COVID-19, we develop a simple susceptible-vaccinated-infectious-recovery (SVIR) model. The transmission rate in this model incorporates the impact of caution and sense of safety on the transmission rate, making it biologically valid. The biological validity of the model is demonstrated by proving that a positive bound solution exists for this model. We find a formula for the basic reproduction function and use it to discuss the local asymptotic stability of the disease-free equilibrium (DFE). The existence of the endemic equilibrium (EE) is shown to be dependent not only on the basic reproduction number value but also on the level of caution. By fitting the model to COVID-19 data from several countries, we estimate parameter values with 95% confidence intervals. Sensitivity analysis reveals that the time-series solution is affected by both the level of caution and the sense of safety, indicating the importance of population behavior in disease spread. We introduce an optimal control problem for vaccination and show that the optimal strategy exists and is unique. Our simulations demonstrate that optimal vaccination is most effective for low sense of safety, leading to fewer infections and lower costs. For high levels of sense of safety, vaccines may have a negative impact, highlighting the importance of combining vaccination efforts with educational initiatives.