Stability Analysis and Optimal Control of Zika Virus Transmission Using the Q − Homotopy Analysis Transform Method

Abstract

The Zika virus (ZIKV) poses significant public health challenges due to its rapid transmission and severe effects on pregnant women and new-borns. In this paper, we develop a compartment mathematical model to study the transmission dynamics of the Zika virus between human and mosquito populations. The human population is divided into susceptible, exposed, infected, pregnant, infected infant, and recovered classes, while the mosquito population includes susceptible, exposed, and infected compartments. The basic reproduction number  is derived using the next generation matrix method to determine the threshold condition for disease persistence. By examining the existence and uniqueness, local and global stability analyses of both the disease free and endemic equilibrium are performed using linearization and Lyapunov function techniques. We incorporated optimal control to identify effective prevention, treatment, and vector management strategies in the model. To obtain approximate analytical solutions, we employ the q − homotopy analysis transform method (q − HATM), which provides a rapidly convergent series solution without restrictive assumptions. The results demonstrate that reducing the contact rate and enhancing recovery significantly decrease  and prevent disease outbreak.