On the dynamics of SVIR model with Caputo and modified Mittag–Leffler fractional derivatives

Abstract

This study develops a memory-sensitive $\s$ epidemic model incorporating fractional-order dynamics to capture the nuanced progression and control of infectious diseases. The population is stratified into four compartments—susceptible ($\S)$, vaccinated ($\V$), infectious ($I$), and recovered ($\R$)--while the model integrates two distinct fractional operators: the classical Caputo derivative ($\c)$ and the modified Mittag-Leffler kernel ($\m$). These operators allow for a richer representation of temporal memory and hereditary effects in disease transmission and intervention outcomes. Rigorous mathematical analysis confirms the system’s non-negativity, boundedness, and the existence and uniqueness of solutions, ensuring biological and theoretical consistency. The basic reproduction number $R_0$ is derived using the next-generation matrix approach, and its threshold behavior is linked to the stability of the disease-free equilibrium. Numerical simulations, executed via a fractional Runge–Kutta scheme, reveal that the $\m$ formulation yields smoother compartmental transitions and prolonged epidemic tails compared to the $\c$ model, despite both ensuring disease eradication when $R_0 < 1$.