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Mathematical modeling of sterile insect technology for control of anopheles mosquito. (English) Zbl 1252.92044
Summary: The Sterile Insect Technology (SIT) is a nonpolluting method of control of the invading insects that transmit disease. The method relies on the release of sterile or treated males in order to reduce the wild population of anopheles mosquitos. We propose two mathematical models. The first model governs the dynamics of the anopheles mosquito. The second model, the SIT model, deals with the interaction between treated males and wild female anopheles. Using the theory of monotone operators, we obtain dynamical properties of a global nature that can be summarized as follows. Both models are dissipative dynamical systems on the positive cone \(\mathbb R^4_+\). The value \(R=1\) of the basic offspring number \(R\) is a forward bifurcation for the model of the anopheles mosquitos, with the trivial equilibrium 0 being globally asymptotically stable (GAS) when \(R \leq 1\), whereas 0 becomes unstable and one stable equilibrium is born with well determined basins of attraction when \(R>1\). For the SIT model, we obtain a threshold number \(\hat{\lambda}\) of treated male mosquitoes above which the control of wild female mosquitoes is effective. That is, for \(\lambda>\hat{\lambda}\) the equilibrium 0 is GAS. When \(0<\lambda \leq \hat{\lambda}\), the number of equilibria and their stability are described together with their precise basins of attraction. These theoretical results are rephrased in terms of possible strategies for the control of the anopheles mosquitos and are illustrated by numerical simulations.

MSC:
92C60 Medical epidemiology
92D40 Ecology
93C95 Application models in control theory
65C20 Probabilistic models, generic numerical methods in probability and statistics
93A30 Mathematical modelling of systems (MSC2010)
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