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.
A general mathematical model for population dispersal featuring long range taxis is presented and exemplified by the dispersal episode of the Africanized honey bees (Apis mellifera adansonii) throughout the American Continent. The mathematical model is a discrete-time and nonlocal model represented by an integrodifference recursion. A new taxis concept is defined and introduced into the mathematical model by an appropriate modification of the redistribution kernel. The model is capable of predicting the natural barrier for the expansion of the Africanized honey bees in the southern part of the Continent due to low winter temperatures. It also describes a sensitive expansion velocity with respect to the quality of resources, which can explain the AHB's astounding spread rate, by using two different kinds of population dynamics strategies, one for a resourceful environment and the other for poor regions.
AJI epidemiological population model is proposed to assess the impact of protection and/or treatment strategies applied to HIV infection. Sex-education campaigns are the available protection strategy, and drug (or association of drugs) administration is the treatment strategy considered. In this model we assumed recruitment and differential mortality rates for the homosexual population. In addition to the classical threshold contact rate related to the establishment of the disease, we obtained a threshold input rate.
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