Mathematical Model of COVID-19 with Imperfect Vaccine and Virus Mutation

In this study, we investigated the effect of a partially protective vaccine on COVID-19 infection with the original and mutant viruses using a deterministic mathematical model. The model we developed consists of $S$ (susceptible), $V$ (vaccinated), $I_1$ (infected with the original virus), $I_2$ (infected with the mutant virus), and $R$ (recovered) subcompartments. In the model, we examined the effect of both artificial active immunity (vaccinated) and natural active immunity (infected). Since it is known that the recovery and mortality rates of the original virus and the mutant virus are different in COVID-19, we took this into account in the study. First of all, we obtained the basic reproduction number using the next-generation matrix method. We analyzed the local stability of the disease-free equilibrium point and the endemic equilibrium point of the model using the Routh-Hurwitz criterion and the global stability with the help of Lyapunov functions. Using the Castillo-Chavez and Song Bifurcation Theorem, we demonstrate the existence of a backward bifurcation that occurs when the vaccine is not effective enough, leading to the simultaneous existence of both disease-free and endemic equilibrium points, even when the basic reproduction number is below 1. We estimated the three model parameters by parameter estimation and identified the model-sensitive parameters by local sensitivity analysis. We found that the parameter representing vaccine efficacy is the most sensitive to the basic reproduction number, and that increasing vaccine efficacy will reduce the average number of secondary cases. The three different simulations we present to illustrate the basic mechanisms underlying the dynamics of our model and to support the analytical findings suggest that there is a strong relationship between vaccine efficacy and the course of the epidemic, and that it is necessary to produce vaccines with higher efficacy and increase the vaccination rate to reduce the average number of secondary cases and the likelihood that infected individuals will remain under the influence of the epidemic for a long time.