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We present a simple SVIR (susceptible, vaccinated, infected, recovered) model to analyze the spread of COVID‐19, incorporating the influence of the population's caution on the transmission rate, which is considered nonlinear in current number of infected. Demonstrating a positive bound solution confirms the model's biological relevance. Through a formula for the basic reproduction number, we explore the local asymptotic stability of the disease‐free equilibrium (DFE) and endemic equilibrium (EE), showing that the existence of the EE relies on the basic reproduction number. Furthermore, we establish the global stability of the DFE by constructing a Lyapunov function. We present an optimal control problem for vaccination, demonstrating the existence and uniqueness of the optimal strategy. Our simulations indicate that optimal vaccination is effective in reducing infections and costs. We also investigate the effect of integrating education into the model to underscore its importance in decreasing disease transmission rates and reducing the necessity for vaccine uptake.
Forrest et al. (Tue,) studied this question.