In this article, we propose a mathematical model for the transmission of coronavirus-19 disease (COVID-19) to understand under which conditions it will be eradicated or persisted. The dynamics of COVID-19 for this study is segregated into seven compartments: susceptible, vaccinated, exposed, symptomatically infected, asymptomatically infected, quarantined, and recovered. The model has both disease-free and endemic equilibria. The effective reproduction number (R 0 (ν)) is computed using the next-generation matrix method. It has been proven that the disease-free equilibrium and endemic equilibrium are globally asymptotically stable. A backward bifurcation analysis is implemented and in the form of a theorem, it is stated under what condition our model undergoes a backward bifurcation. Then the model is fitted with COVID-19 infected cases reported from March 31, 2020, to July 31, 2021, in Iran. In this time interval, Iran has experienced 4 waves of this disease. An optimal control model has been formulated and solved to realize the positive effects of performance of vaccination and treatment of quarantined individuals to prevent the spread of COVID-19. Finally, sensitivity analysis and numerical simulations approved that the implementation of social distancing, quarantine, vaccination, and putting on face masks will help to minimize the spread of the COVID-19 virus.
Mathematics Subject Classification (2000) 92B05, 92D25, 92D30, 37N25, 34D23.