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The Threshold Infection Level for \({{Wolbachia }}\) Invasion in a Two-Sex Mosquito Population Model

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Abstract

In this paper, we formulate a new \({{Wolbachia }}\) infection model in a two-sex mosquito population with stage structure. Some key factors of \({{Wolbachia }}\) infection, including cytoplasmic incompatibility (CI), male killing (MK) effect, maternal transmission, fecundity cost due to fitness effect and different mortality rates for infected individuals, are captured. Dynamical analysis has been carried out, and the basic reproduction number \(R_0\) for \({{Wolbachia }}\) infection has been calculated. Our analysis shows that \({{Wolbachia }}\) can establish in a mosquito population if \(R_0\) is greater than unity. If \(R_0\) is less than unity, \({{Wolbachia }}\) establishment still can be achieved if backward bifurcation occurs. Under this circumstance, the initial values lying in the basin of attraction of the stable \({{Wolbachia }}\)-established equilibrium are essential to guarantee \({{Wolbachia }}\) establishment. In particular, the method to find the basin of attraction and evaluate the threshold initial values is given. Besides, according to a comparison of different releasing strategies, it is shown that, from the perspective of economy and disease control, keeping the number of infected female mosquitoes to a necessary minimum by relying on higher number of male mosquitoes released is a desirable strategy. Moreover, global and local sensitivity analysis and numerical simulation have been performed to explore the impact of model parameters to the success of population establishment. Our results suggest that low levels of MK effect and fitness costs as well as high levels of CI and maternal inheritance are in favor of \({{Wolbachia }}\) establishment. Moreover, not considering MK effect and incomplete CI effect may result in the underestimation of the number of infected mosquitoes needed to be released.

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Acknowledgements

Dan Li is supported by the National Natural Science Foundation of China (No. 11801209) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 18KJB110004). Hui Wan is supported by the Jiangsu Overseas Research and Training Program for University Prominent Young and Middle-aged Teachers and Presidents, the NSF of the Jiangsu Higher Education Committee of China (No. 17KJA110002) and a project funded by PAPD of Jiangsu Higher Education Institutions. The authors thank two reviewers and Prof. Guihong Fan for very helpful suggestions and comments.

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  1. School of Mathematical Science, Huaiyin Normal University, Huaian, 223300, People’s Republic of China

    Dan Li

  2. Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, People’s Republic of China

    Hui Wan

  3. Department of Mathematics, Purdue University, West Lafayette, IN, 47906, USA

    Hui Wan

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  1. Dan Li
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Li, D., Wan, H. The Threshold Infection Level for \({{Wolbachia }}\) Invasion in a Two-Sex Mosquito Population Model. Bull Math Biol 81, 2596–2624 (2019). https://doi.org/10.1007/s11538-019-00620-1

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