Unwillingness for vaccination in India: Real-life data application with queueing analysis of Covid-19 mode

Abstract

Vaccination is seen as a weapon that not only profits an individual to fight the deadly SARS-CoV-2 virus but also helps in building herd immunity for the general public. There are a lot of individuals who are keen and willing to get inoculated by the Covid-19 vaccine and return back to their normal life. However, there is some public who still are unwilling for the vaccination due to vaccine hesitancy. And even after vaccination(one dose), there may be cases of breakthrough infections with mild symptoms. Unwillingness for vaccination and breakthrough infections are now very important aspects of this ongoing epidemic amid the vaccination process that needs to be challenged by responsible stakeholders. Thus, we will first formulate an ordinary differential equation (ODE) model consisting of willing and unwilling (for vaccination) population along with breakthrough infection. We will then extend ODE model to a queueing model that accounts for the movement of willing and non-willing susceptible individuals as well. The governing equations are formed and the transition rates from one stage to another are taken to follow an exponential distribution. The transient state probabilities of the model are acquired by applying the matrix method and thereafter Runge-Kutta 4th order method is employed to attain various performance indices.  The parameters are estimated for our system based on the standard nonlinear least-squares method using the real data of the vaccinated (one dose) population for India between the time period of 5 March 2021 to 3 July 2021. We would first aim to study the impact of this system on the basic reproduction number $\rho^{0}$. The sensitivity analysis of $\rho^{0}$ is done to recognize the important estimated parameters which may help in controlling the epidemic. Then the impact of the unwillingness and breakthrough infections directly helping in transmission of the infection amid the vaccination process is shown by uncertainty analysis using the technique of PRCC and contour plots. Numerical analysis is also provided to certify the queueing model.