Modeling of COVID-19 with vaccination and optimal control

COVID-19 is a respiratory disease caused by the virus SARS-CoV-2. As of now, millions of people have died as a result of the devastating COVID-19 outbreak. Vaccination can lessen the severity of COVID-19, even though there is no known cure for it. Here, we propose and develop a novel COVID-19 epidemiological vaccination model. Then, we find the disease-free and endemic equilibria for our proposed model and compute the basic reproduction number (R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document}). We discuss the sensitivity analysis to visualize the impact of different parameters mainly associated with vaccination. Through numerical simulation, it has been shown that increasing treatment and vaccination rates reduces the transmission and increases the recovery, respectively. By taking into account two control parameters-namely, those related to the disease's rate of diagnosis and transmission-the suggested model is extended to an optimal control problem. It has been observed that both control parameters have a significant impact on limiting the spread of COVID-19. The main goal of this study is to find out how vaccination, when compared to no vaccination, can reduce the spread of disease.