Propagation Dynamics in a Reaction-Diffusion System on Zika Virus Transmission

This article studies the propagation dynamics in a reaction-diffusion system modeling the Zika virus transmission, during which the infected vectors and hosts are regarded as invaders in spatial domain R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document}. The system does not satisfy the mixed quasimonotone condition. A constant threshold is defined by the corresponding eigenvalue problem at the disease free steady state. For the traveling wave solutions connecting the disease free steady state to the endemic steady state, the threshold is the minimal wave speed determining the existence or nonexistence of monotone traveling wave solutions in the sense of components. For the initial value problem with fast decaying initial condition, the threshold describes the spatial expansion ability of the infected vectors and hosts. It is also proved that for the wave speed larger than the threshold, the corresponding traveling wave solution is unique in the sense of phase shift and exponentially asymptotic stable in the proper weighted functional space. The stability also shows the spreading speed in the corresponding Cauchy problem with special decaying initial condition.