A computational model of varicella dynamics with age-dependent waning immunity: stability, parameter estimation, and optimal control via finite difference schemes

Varicella (chickenpox), caused by the varicella–zoster virus, remains a public health concern due to its high transmissibility and the gradual waning of vaccine-induced immunity. In this work, we develop an age-structured mathematical model describing varicella transmission while incorporating vaccination, waning immunity, treatment, and isolation. The stability of the disease-free and endemic equilibrium is analyzed using Lyapunov functional techniques, providing conditions for disease eradication or persistence. Model parameters are estimated from epidemiological data using a weighted least squares approach. Numerical simulations based on finite difference schemes illustrate the epidemic dynamics and support the analytical results. In addition, the model is extended to an optimal control framework to minimize the combined costs of vaccination and treatment. Simulation results show that integrated control strategies significantly reduce infection prevalence and overall healthcare costs. This study provides a computational framework combining stability analysis, parameter estimation, and optimal control for infectious diseases with age-dependent waning immunity.