The influence of prevention and isolation measures to control the infections of the fractional Chickenpox disease model

Abstract

In this paper, we propose a mathematical model using the Caputo fractional derivative (CFD) and two control signals to study the transmission dynamics and control of Chickenpox (Varicella) outbreak. The model consists of six compartments representing susceptible, vaccinated, exposed, infected with complications, infected without complications, and recovered individuals. We analyze the theoretical properties of the model, including existence, uniqueness, and boundedness of solutions, and calculate the basic reproduction number $\left(R_0\right)$. We identify equilibrium points and establish conditions for their stability. Sensitivity analysis helps identify the most influential parameters on $R_0$. We formulate a fractional optimal control problem (FOCP) by incorporating time-dependent prevention and isolation measures. The necessary optimality conditions are derived using Pontryagin’s maximum principle. Numerical simulations based on the Adams–Bashforth–Moulton (ABM) method illustrate the impact of control measures and fractional order on disease propagation. The results highlight the effectiveness of optimal controls and fractional order in understanding and managing epidemics, enhancing stability conditions. The study contributes to a better understanding of Chickenpox transmission dynamics and provides insights for disease control and management, aiding decision-makers and governments in taking preventive measures.