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Modelling and analysis of impulsive releases of sterile mosquitoes. (English) Zbl 1448.92296
Summary: To study the impact of releasing sterile mosquitoes on mosquito-borne disease transmissions, we propose two mathematical models with impulsive releases of sterile mosquitoes. We consider periodic impulsive releases in the first model and obtain the existence, uniqueness, and globally stability of a wild-mosquito-eradication periodic solution. We also establish thresholds for the control of the wild mosquito population by selecting the release rate and the release period. In the second model, the impulsive releases are determined by the closely monitored wild mosquito density, or the state feedback. We prove the existence of an order one periodic solution and find a relatively small attraction region, which ensures the wild mosquito population is under control. We provide numerical analysis which shows that a smaller release rate and more frequent releases are more efficient in controlling the wild mosquito population for the periodic releases, but an early release of sterile mosquitoes is more effective for the state feedback releases.

MSC:
92D30 Epidemiology
92D25 Population dynamics (general)
34D23 Global stability of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
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[1] L. Alphey, M. Benedict, R. Bellini, G.G. Clark, D.A. Dame, M.W. Service, and S.L. Dobson, Sterile-insect methods for control of mosquito-borne diseases: An analysis, Vector-Borne Zoonotic Dis. 10 (2010), pp. 295-311. doi: 10.1089/vbz.2009.0014[Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[2] R. Anguelov, Y. Dumont, and J.M.S. Lubuma, Mathematical modeling of sterile insect technology for control of anopheles mosquito, Comput. Math. Appl. 64 (2012), pp. 374-389. doi: 10.1016/j.camwa.2012.02.068[Crossref], [Web of Science ®], [Google Scholar] · Zbl 1252.92044
[3] D.D. Bainov and P.S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics, Pitman, London, 1993. [Google Scholar]
[4] D.D. Bainov and P.S. Simeonov, System with Impulse Effect Theory and Applications, Ellis Harwood series in Mathematics and its Applications, Ellis Harwood, Chichester, 1993. [Google Scholar]
[5] H.J. Barclay, Pest population stability under sterile releases, Res. Popul. Ecol. 24 (1982), pp. 405-416. doi: 10.1007/BF02515585[Crossref], [Google Scholar]
[6] H.J. Barclay and M. Mackuer, The sterile insect release method for pest control: A density-dependent model, Environ. Entomol. 9 (1980), pp. 810-817. doi: 10.1093/ee/9.6.810[Crossref], [Web of Science ®], [Google Scholar]
[7] A.C. Bartlett and R.T. Staten, The sterile insect release method and other genetic control strategies, in Radcliffe’s IPM World Textbook, 1996. Available at http://ipmworld.umn.edu/chapters/bartlett.htm. [Google Scholar]
[8] L.M. Cai, S.B. Ai, and J. Li, Dynamics of mosquitoes populations with different strategies for releasing sterile mosquitoes, SIAM J. Appl. Math. 74 (2014), pp. 1786-1809. doi: 10.1137/13094102X[Crossref], [Web of Science ®], [Google Scholar] · Zbl 1320.34071
[9] L.S. Chen, Pest control and geometric theory of Semi-continuous dynamical system, J. Beihua Univ. 12 (2011), pp. 1-9. [Google Scholar]
[10] H. Diaz, A.A. Ramirez, A. Olarte, and C. Clavijo, A model for the control of malaria using genetically modified vectors, J. Theoret. Biol. 276 (2011), pp. 57-66. doi: 10.1016/j.jtbi.2011.01.053[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1405.92144
[11] C. Dufourd and Y. Dumont, Impact of environmental factors on mosquito dispersal in the prospect of Sterile Insect Technique control, Comput. Math. Appl. 66 (2013), pp. 1695-1715. doi: 10.1016/j.camwa.2013.03.024[Crossref], [Web of Science ®], [Google Scholar] · Zbl 1345.34105
[12] Y. Dumont and J. M. Tchuenche, Mathematical studies on the sterile insect technique for the Chikungunya disease and Aedes albopictus, J. Math. Biol. 65 (2012), pp. 809-854. doi: 10.1007/s00285-011-0477-6[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1311.92175
[13] V.A. Dyck, J. Hendrichs, and A.S. Robinson, Sterile Insect Technique: Principles and Practice in Area-Wide Integrated Pest Managemen, Springer, Dordrecht, 2005. ISBN: 1-4020-4050-4. [Crossref], [Google Scholar]
[14] L. Esteva and H.M. Yang, Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique, Math. Biosci. 198 (2005), pp. 132-147. doi: 10.1016/j.mbs.2005.06.004[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1090.92048
[15] K.R. Fister, M.L. McCarthy, S.F. Oppenheimer, and C. Collins, Optimal control of insects through sterile insect release and habitat modification, Math. Biosci. 244 (2013), pp. 201-212. doi: 10.1016/j.mbs.2013.05.008[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1280.92025
[16] J.C. Flores, A mathematical model for wild and sterile species in competition: Immigration, Phys. A. 328 (2003), pp. 214-224. doi: 10.1016/S0378-4371(03)00545-4[Crossref], [Web of Science ®], [Google Scholar] · Zbl 1026.92051
[17] M.Z. Huang, J.X. Li, X.Y. Song, and H.J. Guo, Modeling Impulsive Injections of Insulin: Towards Artificial Pancreas, SIAM J. Appl. Math. 72 (2012), pp. 1524-1548. doi: 10.1137/110860306[Crossref], [Web of Science ®], [Google Scholar] · Zbl 1325.92045
[18] V. Lakshmikantham, D. Bainov, and P. Simeonov, Theory of Impulsive Differential Equations, Vol. 6, World Scientific, Singapore, 1989. [Crossref], [Google Scholar]
[19] J. Li, Simple mathematical models for interacting wild and transgenic mosquito populations, Math. Biosci. 189 (2004), pp. 39-59. doi: 10.1016/j.mbs.2004.01.001[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1072.92053
[20] J. Li, Differential equations models for interacting wild and transgenic mosquito populations, J. Biol. Dyn. 2 (2008), pp. 241-258. doi: 10.1080/17513750701779633[Taylor & Francis Online], [Google Scholar] · Zbl 1165.34027
[21] J. Li, Modeling of mosquitoes with dominant or recessive transgenes and Allee effects, Math. Biosci. Eng. 7 (2010), pp. 101-123. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1184.92031
[22] J. Li, Modelling of transgenic mosquitoes and impact on malaria transmission, J. Biol. Dyn. 5 (2011), pp. 474-494. doi: 10.1080/17513758.2010.523122[Taylor & Francis Online], [Google Scholar] · Zbl 1225.92033
[23] J. Li, Discrete-time models with mosquitoes carrying genetically-modified bacteria, Math. Biosci. 240 (2012), pp. 35-44. doi: 10.1016/j.mbs.2012.05.012[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1319.92043
[24] J. Li and Z.L. Yuan, Modelling releases of sterile mosquitoes with different strategies, J. Biol. Dyn. 9 (2015), pp. 1-14. doi: 10.1080/17513758.2014.977971[Taylor & Francis Online], [Google Scholar]
[25] A. Maiti, B. Patra, and G.P. Samanta, Sterile insect release method as a control measure of insect pests: A mathematical model, J. Appl. Math. Comput. 22 (2006), pp. 71-86. doi: 10.1007/BF02832038[Crossref], [Google Scholar] · Zbl 1149.92026
[26] M. Rafikov, L. Bevilacqua, and A.P.P. Wyse, Optimal control strategy of malaria vector using genetically modified mosquitoes, J. Theoret. Biol. 258 (2009), pp. 418-425. doi: 10.1016/j.jtbi.2008.08.006[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1405.92271
[27] M. Rafikov, A.P. Wyse, and L. Bevilacqua, Controlling the interaction between wild and transgenic mosquitoes, J. Nonlinear Syst. Appl. 1 (2010), pp. 21-31. [Google Scholar]
[28] S. S. Seirin Lee, R.E. Baker, E.A. Gaffney, and S.M. White, Optimal barrier zones for stopping the invasion of Aedes aegypti mosquitoes via transgenic or sterile insect techniques, Theoret. Ecol. 6 (2013), pp. 427-442. doi: 10.1007/s12080-013-0178-4[Crossref], [Web of Science ®], [Google Scholar]
[29] S.F. Shuai, Y.J. Li, and X.G. Chen, Summarize of common monitoring methods of mosquito vectors, J. Trop. Med. 13 (2013), pp. 1292-1296. [Google Scholar]
[30] R.C.A. Thome, H.M. Yang, and L. Esteva, Optimal control of Aedes aegypti mosquitoes by the sterile insect technique and insecticide, Math. Biosci. 223 (2010), pp. 12-23. doi: 10.1016/j.mbs.2009.08.009[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1180.92058
[31] S.M. White, P. Rohani, and S.M. Sait, Modelling pulsed releases for sterile insect techniques: Fitness costs of sterile and transgenic males and the effects on mosquito dynamics, J. Appl. Ecol. 47 (2010), pp. 1329-1339. doi: 10.1111/j.1365-2664.2010.01880.x[Crossref], [Web of Science ®], [Google Scholar]
[32] Wikipedia: Sterile Insect Technique (2013). Available at http://en.wikipedia.org/wiki/Sterileinsecttechnique. [Google Scholar]
[33] A.P.P. Wyse, L. Bevilacqua, and M. Rafikov, Simulating malaria model for different treatment intensities in a variable environment, Ecol. Model. 206 (2007), pp. 322-330. doi: 10.1016/j.ecolmodel.2007.03.038[Crossref], [Web of Science ®], [Google Scholar]
[34] http://news.163.com/15/0802/06/B008SANU00014Q4P.html. [Google Scholar]
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