Article

# Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique.

Departamento de Matemáticas, Facultad de Ciencias, UNAM 04510 México, D.F., Mexico.

Mathematical Biosciences (Impact Factor: 1.45). 01/2006; 198(2):132-47. DOI: 10.1016/j.mbs.2005.06.004 Source: PubMed

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**ABSTRACT:**We propose and analyze a new compartmental model of dengue transmission with memory between human-to-mosquito and mosquito-to-human. The memory is incorporated in the model by using a fractional differential operator. A threshold quantity R0, similar to the basic reproduction number, is worked out. We determine the stability condition of the disease-free equilibrium (DFE) E0 with respect to the order of the fractional derivative $\alpha$ and R0. We determine $\alpha$ dependent threshold values for R0, below which DFE (E0) is always stable, above which DFE is always unstable, and at which the system exhibits a Hopf-type bifurcation. It is shown that even though R0 is less than unity, the DFE may not be always stable, and the system exhibits a Hopf-type bifurcation. Thus, making R0 < 1 for controlling the disease is no longer a sufficient condition. This result is synergistic with the concept of backward bifurcation in dengue ODE models. It is also shown that R0 > 1 may not be a sufficient condition for the persistence of the disease. For a special case, when $\alpha$ = 1/2 , we analytically verify our findings and determine the critical value of R0 in terms of some important model parameters. Finally, we discuss about some dengue control strategies in light of the threshold quantity R0.Communications in Nonlinear Science and Numerical Simulation 08/2013; · 2.77 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Insect pests pose a major threat to a balanced ecology as it can threaten local species as well as spread human diseases; thus, making the study of pest control extremely important. In practice, the sterile insect release method (SIRM), where a sterile population is introduced into the wild population with the aim of significantly reducing the growth of the population, has been a popular technique used to control pest invasions. In this work we introduce an integro-differential equation to model the propagation of pests in a heterogeneous environment, where this environment is divided into three regions. In one region SIRM is not used making this environment conducive to propagation of the insects. A second region is the eradication zone where there is an intense release of sterile insects, leading to decay of the population in this region. In the final region we explore two scenarios. In the first case, there is a small release of sterile insects and we prove that if the eradication zone is sufficiently large the pests will not invade. In the second case, when SIRM is not used at all in this region we show that invasions always occur regardless of the size of the eradication zone. Finally, we consider the limiting equation of the integro-differential equation and prove that in this case there is a critical length of the eradication zone which separates propagation from obstruction. Moreover, we provide some upper and lower bound for the critical length.Journal of Mathematical Biology 05/2014; · 2.37 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The relationship between epidemiology, mathematical modeling and computational tools allows to build and test theories on the development and battling of a disease. This PhD thesis is motivated by the study of epidemiological models applied to infectious diseases in an Optimal Control perspective, giving particular relevance to Dengue. Dengue is a subtropical and tropical disease transmitted by mosquitoes, that affects about 100 million people per year and is considered by the World Health Organization a major concern for public health. The mathematical models developed and tested in this work, are based on ordinary differential equations that describe the dynamics underlying the disease, including the interaction between humans and mosquitoes. An analytical study is made related to equilibrium points, their stability and basic reproduction number. The spreading of Dengue can be attenuated through measures to control the transmission vector, such as the use of specific insecticides and educational campaigns. Since the development of a potential vaccine has been a recent global bet, models based on the simulation of a hypothetical vaccination process in a population are proposed. Based on Optimal Control theory, we have analyzed the optimal strategies for using these controls, and respective impact on the reduction/eradication of the disease during an outbreak in the population, considering a bioeconomic approach. The formulated problems are numerically solved using direct and indirect methods. The first discretize the problem turning it into a nonlinear optimization problem. Indirect methods use the Pontryagin Maximum Principle as a necessary condition to find the optimal curve for the respective control. In these two strategies several numerical software packages are used.01/2014;

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