A Graph-Theoretic Framework for Zika Virus through Vertical Transmission and Stability Analysis.

The Zika virus, a mosquito-borne pathogen, poses a serious global health threat due to its ability to transmit vertically from infected mothers to newborns, leading to congenital abnormalities and adverse neurological outcomes. A rigorous understanding of vertical transmission dynamics is therefore essential for effective disease prevention and control. In this study, a graph-theoretic framework is proposed to model the human vertical transmission of Zika virus, the birth of babies with microcephaly and asymptomatic infected individuals. The model explicitly incorporates maternal infection states and vertically infected newborn compartments, capturing the mother-to-child transmission pathway. The resulting system is represented as a weighted directed graph, where vertices correspond to epidemiological compartments and directed edges denote transition mechanisms between states. The basic reproduction number $R_0$ is derived using an energy-based approach applied to the characteristic polynomial associated with the digraph structure, providing an analytical threshold for disease invasion. Furthermore, the Jacobian matrix obtained directly from the digraph at the disease-free equilibrium (DFE) is analyzed to investigate local stability using the Routh--Hurwitz and Gershgorin criteria. The analytical results highlight the critical role of vertical transmission pathways in shaping the overall dynamics of Zika virus spread and underscore the importance of maternal-focused intervention strategies in mitigating congenital Zika outcomes.