Biological invasion is an important area of research in mathematical biology and more so if it concerns species which are vectors for diseases threatening the public health of large populations. That is certainly the case for Aedes aegypti and the dengue epidemics in South America. Without the prospect of an effective and cheap vaccine in the near future, any feasible public policy for controlling the dengue epidemics in tropical climates must necessarily include appropriate strategies for minimizing the mosquito population factor. The present paper discusses some mathematical models designed to describe A. aegypti's vital and dispersal dynamics, aiming to highlight practical procedures for the minimization of its impact as a dengue vector. A continuous model including diffusion and advection shows the existence of a stable travelling wave in many situations and a numerical study relates the wavefront speed to a few crucial parameters. Strategies for invasion containment and its prediction based on measurable parameters are analysed.

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In this paper we propose some mathematical models for the transmission of dengue using a system of reaction-diffusion equations. The mosquitoes are divided into infected, uninfected and aquatic subpopulations, while the humans, which are divided into susceptible, infected and recovered, are considered homogeneously distributed in space and with a constant total population. We find Lie point symmetries of the models and we study theirs temporal dynamics, which provides us the regions of stability and instability, depending on the values of the basic offspring and the basic reproduction numbers. Also, we calculate the possible values of the wave speed for the mosquitoes invasion and dengue spread and compare them with those found in the literature. 2010 AMS Mathematics Classification numbers: 34D20, 35B35, 76M60, 92Bxx