Existence and stability of solution for time-delayed nonlinear fractional differential equations

Abstract

The objective of this study is to analyze the existence and stability of solutions for a mathematical model of nonlinear systems of time-delayed fractional differential equations involving Atangana-Baleanu-Caputo ([Formula: see text]) fractional derivatives with vaccination. The epidemiological states of the total population are classified as susceptible individuals [Formula: see text], infected individuals [Formula: see text], quarantined individuals [Formula: see text], recovered individuals [Formula: see text], vaccinated individuals [Formula: see text] and insusceptible/protected persons [Formula: see text]. By considering coordinate transformation, we linearize our system and obtain a matrix coefficient for the proposed time-delayed fractional differential equation. We establish sufficient requirements and inequalities using the generalized Gronwall inequality and Krasnoselskii's fixed-point theorem to prove the existence of solution for our problem. Additionally, we construct sufficient requirements for stability analysis by employing the Laplace transform and matrix measure theory. We also investigate criteria for the local stability of the epidemic and disease-free equilibrium points. Finally, we provide a numerical illustration to validate the effectiveness of our results.