This study proposes a new fractional mathematical model to study the impact of vaccination on COVID-19 outbreaks by categorizing infected people into non-vaccinated, first dose-vaccinated, and second dose-vaccinated groups and exploring the transmission dynamics of the disease outbreaks. We present a non-linear integer order mathematical model of COVID-19 dynamics and modify it by introducing Caputo fractional derivative operator. We start by proving the good state of the model and then calculating its reproduction number. The Caputo fractional-order model is discretized by applying a reliable numerical technique. The model is proven to be stable. The classical model is fitted to the corresponding cumulative number of daily reported cases during the vaccination regime in India between 01 August 2021 and 21 July 2022. We explore the sensitivities of the reproduction number with respect to the model parameters. It is shown that the effective transmission rate and the recovery rate of unvaccinated infected individuals are the most sensitive parameters that drive the transmission dynamics of the pandemic in the population. Numerical simulations are used to demonstrate the applicability of the proposed fractional mathematical model via the memory index at different values of $0.7,0.8,0.9$ and 1. We discuss the epidemiological significance of the findings and provide perspectives on future health policy tendencies. For instance, efforts targeting a decrease in the transmission rate and an increase in the recovery rate of non-vaccinated infected individuals are required to ensure virus-free population. This can be achieved if the population strictly adhere to precautionary measures, and prompt and adequate treatment is provided for non-vaccinated infectious individuals. Also, given the ongoing community spread of COVID-19 in India and almost the pandemic-affected countries worldwide, the need to scale up the effort of mass vaccination policy cannot be overemphasized in order to reduce the number of unvaccinated infections with a view to halting the transmission dynamics of the disease in the population.

Ethiopia is one of the countries highly affected by the COVID-19 pandemic in Africa. A new study has looked at the transmission dynamics of the outbreak in Ethiopia based on the age categories of the infected individuals. Infected individuals are divided into three age categories (less than 14 years (I ${}_1$), 15–54 years (I ${}_2$), and above 54 years I ${}_3$). Chebyshev polynomial of the fourth kind is used to convert a fractional system into an algebraic system of equations, which is then solved using iterative methods. The fundamental reproduction number $({\mathcal{R}}_0)$ is calculated using the next-generation matrix technique, and the disease-free and endemic equilibrium points, as well as their stability, are carefully studied. We calculated the parameters and starting conditions provided for our model using data from the Ethiopian Public Health Institute (EPHI) and the Ethiopian Ministry of Health from 29 February 2021, to 7 June 2021.