A fractional study on the vaccination effect to control the COVID-19 epidemic

In this chapter, we address the global challenge posed by the COVID-19 pandemic, a highly infectious disease that has claimed millions of lives worldwide. Fractional operators prove to be more effective in assessing infection spread and gaining deeper insights into epidemic dynamics. Hence, we employ two widely recognized fractional operators, namely the Caputo and Atangana–Baleanu derivatives, to investigate the impact of vaccination on COVID-19 spread. We first present the model under the Caputo operator and look at some fundamental mathematical aspects of this non-integer-order model, such as solution positivity and equilibria stability in the case where R0<1. We further extend the COVID-19 model with a well-known nonsingular Atangana–Baleanu (AB) fractional operator and study the existence of a unique solution of the model using the fixed-point theorem. The Ulam–Hyers stability is presented with the help of non-linear analysis. Sensitivity analysis of the model parameters is also discussed. To simulate fractional models, we derive the results using a novel Newton interpolation polynomial-based approach. Graphical representations at different fractional-order values are presented, providing a comprehensive understanding of the implications of fractional derivatives in modeling the vaccination effects on COVID-19 dynamics.