STABILITY ANALYSIS OF SEXUAL AND SYMPTOMATIC VECTOR-BORNE ZIKA VIRUS TRANSMISSION USING GRAPH THEORY

Abstract

Zika virus (ZIKV) is primarily transmitted through infected Aedes mosquitoes, while sexual transmission provides an additional pathway for disease spread. In this study, a deterministic ZIKV transmission model is formulated using a weighted directed graph to incorporate both vector-borne and sexual transmission pathways, with particular consideration of symptomatic and asymptomatic human infections. The human and mosquito epidemiological compartments are represented as vertices, while transmission, progression, recovery, and removal processes are assigned as biologically meaningful edge weights. The resulting graph structure provides a systematic representation of the interactions among the different epidemiological states. The disease-free equilibrium ????0 is identified by setting all exposed and infected compartments to zero. To investigate its local stability, the infected subsystem is linearised at ????0 and represented through the Jacobian decomposition ???? = ???? − ????, where ???? describes the appearance of new infections and ???? represents transitions and removal processes. The characteristic equation of the Jacobian is derived, and the Routh–Hurwitz criterion is applied to establish the stability conditions. In particular, ℛ0 < 1 provides the threshold for local asymptotic stability of the disease-free equilibrium, whereas ℛ0 > 1 indicates the potential for disease invasion. The proposed framework links the graph-based representation of ZIKV transmission with equilibrium stability analysis