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.

This study introduces an SIRS compartmental mathematical model encompassing vaccination and variable immunity periods for infectious diseases. I derive a basic reproduction number formula and assess the local and global stability of disease-free and the local stability of the endemic equilibria. I demonstrate that the basic reproduction number in the presence of a vaccine is highly sensitive to the rate of immunity loss, and even a slight reduction in this rate can significantly contribute to disease control. Additionally, I have derived a formula to calculate the critical efficacy period required for a vaccine to effectively manage and control the disease.The analysis conducted for the model suggests that increasing the vaccine’s immunity duration (efficacy) decelerates disease dynamics, leading to reduced rates of reinfection and less severe disease outcomes. Furthermore, this delay contributes to a decrease in the basic reproduction number (\(R_0\)), thus facilitating more rapid disease control efforts.

We formulate a simple susceptible‐infectious‐recovery (SIR) model to describe the spread of the coronavirus under strict social restrictions. The transmission rate in this model is exponentially decreasing with time. We find a formula for basic reproduction function and estimate the maximum number of daily infected individuals. We fit the model to induced death data in Italy, United States, Germany, France, India, Spain, and China over the period from the first reported death to August 7, 2020. We notice that the model has excellent fit to the disease death data in these countries. We estimate the model's parameters in each of these countries with 95% confidence intervals. We order the strength of social restrictions in these countries using the exponential rate. We estimate the time needed to reduce the basic reproduction function to one unit and use it to order the quality of social restrictions in these countries. The social restriction in China was the strictest and the most effective and in India was the weakest and the least effective. Policy‐makers may apply the Chinese successful social restriction experiment and avoid the Indian unsuccessful one.

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.