Mathematical Modeling of COVID-19 with Vaccination Using Fractional Derivative: A Case Study

Vaccination against any infectious disease is considered to be an effective way of controlling it. This paper studies a fractional order model with vaccine efficacy and waning immunity. We present the model's dynamics under vaccine efficacy, the impact of immunization, and the waning of the vaccine on coronavirus infection disease. We analyze the model under their equilibrium points. The model under the equilibrium points is discussed and proven that it is locally asymptotically stable if R-v is lesser than unity. We present the backward bifurcation hypothesis of the model and show that there is a parameter beta(2) that causes the backward bifurcation in the imperfect vaccine model. We show certain assumptions when psi = 1 for the imperfect vaccine case; the model is both stable globally asymptotically at the disease-free (R-0 <= 1) and endemic cases (R-0 > 1). By using infected cases from the recent wave throughout Pakistan, we shall estimate the model parameters and calculate the numerical value of the basic reproductive number R-0 approximate to 1.2591. We present the comprehensive graphical results for the realistic parameter values and show many useful suggestions regarding the elimination of the infection from society. The vaccination efficacy that provides an important role in disease elimination is discussed in detail.