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Stability and backward bifurcation in a malaria transmission model with applications to the control of malaria in China. (English) Zbl 1356.92081
Summary: In this paper, we consider a deterministic malaria transmission model with standard incidence rate and treatment. Human population is divided into susceptible, infectious and recovered subclasses, and mosquito population is split into susceptible and infectious classes. It is assumed that, among individuals with malaria who are treated or recovered spontaneously, a proportion moves to the recovered class with temporary immunity and the other proportion returns to the susceptible class. Firstly, it is shown that two endemic equilibria may exist when the basic reproduction number \(\mathcal{R}_0 < 1\) and a unique endemic equilibrium exists if \(\mathcal{R}_0 > 1 .\) The presence of a backward bifurcation implies that it is possible for malaria to persist even if \(\mathcal{R}_0 < 1\). Secondly, using geometric method, some sufficient conditions for global stability of the unique endemic equilibrium are obtained when \(\mathcal{R}_0 > 1\). To deal with this problem, the estimate of the Lozinskiĭ measure of a \(6 \times 6\) matrix is discussed. Finally, numerical simulations are provided to support our theoretical results. The model is also used to simulate the human malaria data reported by the Chinese Ministry of Health from 2002 to 2013. It is estimated that the basic reproduction number \(\mathcal{R}_0 \approx 0.0161\) for the malaria transmission in China and it is found that the plan of eliminating malaria in China is practical under the current control strategies.

MSC:
92D30 Epidemiology
34C23 Bifurcation theory for ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
Software:
MATCONT
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