Effects of vaccination on measles dynamics under fractional conformable derivative with Liouville–Caputo operator

Abstract

Deterministic models in mathematical epidemiology are important tools to comprehend dynamics of infectious diseases. Theory of ordinary differential equations with first-order derivatives is successfully being used for analysis of such diseases. However, most of the infectious diseases have non-Markovian characteristics and thereby require more sophisticated mathematical tools for modeling purposes. This research study investigates dynamics of measles infection with the help of a mathematical operator called conformable derivative of order $\alpha$ (the local derivative index) in the sense of Liouville–Caputo operator of order $\beta$ (the iterated or fractionalizing index). A new measles infection system is proposed that contains a population divided into five different compartments. Bounded solutions of the system exist within positively invariant region and steady-state analysis shows that the disease-free state is locally asymptotically stable when a threshold positive quantity does not exceed 1. The basic reproduction number ${\mathcal {R}}_0$ of the system is computed to be $2.2674e-02$ under the use of fractional conformable derivative whereas the least and the most sensitive parameters towards ${\mathcal {R}}_0$ have been computed via normalized forward sensitivity indices. Influence of various biological parameters on the system is observed with numerical simulations carried out using Adams–Moulton technique wherein the measles infection is found to be vanishing when $\alpha \ge 1$ under the fractional conformable derivative which helps to reduce the infection’s burden if either contact rate of infectious individuals with susceptible ones is duly reduced or the newly recruited individuals are given vaccination.