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June  2022, 27(6): 3039-3052. doi: 10.3934/dcdsb.2021172

## Stage-structured models for interactive wild and periodically and impulsively released sterile mosquitoes

 1 Department of Mathematical Sciences, The University of Alabama in Huntsville, Huntsville, AL 35899, USA 2 Center for Applied Mathematics, Guangzhou University, Guangzhou, 510006, China

* Corresponding author: Jianshe Yu

Received  February 2021 Revised  May 2021 Published  June 2022 Early access  June 2021

Fund Project: This work is supported in part by National Natural Science Foundation of China (12071095, 11971127, 11631005)

A two-dimensional stage-structured model for the interactive wild and sterile mosquitoes is derived where the wild mosquito population is composed of larvae and adult classes and only sexually active sterile mosquitoes are included as a function given in advance. The strategy of constant releases of sterile mosquitoes is considered but periodic and impulsive releases are more focused on. Local stability of the origin and the existence of a positive periodic solution are investigated. While mathematical analysis is more challenging, numerical examples demonstrate that the model dynamics, determined by thresholds of the release amount and the release waiting period, essentially match the dynamics of the alike one-dimensional models. It is also shown that richer dynamics are exhibited from the two-dimensional stage-structured model.

Citation: Shangbing Ai, Jia Li, Jianshe Yu, Bo Zheng. Stage-structured models for interactive wild and periodically and impulsively released sterile mosquitoes. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3039-3052. doi: 10.3934/dcdsb.2021172
##### References:
 [1] H. J. Barclay, Pest population stability under sterile releases, Res. Popul. Ecol., 24 (1982), 405-416.  doi: 10.1007/BF02515585. [2] H. J. Barclay, Mathematical models for the use of sterile insects, In: Sterile Insect Technique. Principles and Practice in Area-Wide Integrated Pest Management, (V.A. Dyck, J. Hendrichs, and A.S. Robinson, Eds.), Springer, Heidelberg, (2005), 147–174. doi: 10.1007/1-4020-4051-2_6. [3] H. J. Barclay and M. Mackuer, The sterile insect release method for pest control: A density dependent model, Environ. Entomol., 9 (1980), 810-817.  doi: 10.1093/ee/9.6.810. [4] N. Becker, Mosquitoes and Their Control, Kluwer Academic/Plenum, New York, 2003. [5] G. Briggs, The endosymbiotic bacterium Wolbachia induces resistance to dengue virus in Aedes aegypti, PLoS Pathog., 6 (2012), e1000833. [6] L. Cai, S. Ai and G. Fan, Dynamics of delayed mosquitoes populations models with two different strategies of releasing sterile mosquitoes, Math. Biosci. Engi., 15 (2018), 1181-1202.  doi: 10.3934/mbe.2018054. [7] L. Cai, S. Ai and J. Li, Dynamics of mosquitoes populations with different strategies for releasing sterile mosquitoes, SIAM J. Appl. Math., 74 (2014), 1786-1809.  doi: 10.1137/13094102X. [8] CDC, Life cycle: The mosquito, 2021, https://www.cdc.gov/dengue/resources/factsheets/mosquitolifecyclefinal.pdf. [9] C. Dye, Intraspecific competition amongst larval aedes aegypti: Food exploitation or chemical interference, Ecol. Entomol., 7 (1982), 39-46.  doi: 10.1111/j.1365-2311.1982.tb00642.x. [10] R. M. Gleiser, J. Urrutia and D. E. Gorla, Effects of crowding on populations of aedes albifasciatus larvae under laboratory conditions., Entomologia Experimentalis et Applicata, 95 (2000), 135-140.  doi: 10.1046/j.1570-7458.2000.00651.x. [11] L. Hu, M. Huang, M. Tang, J. Yu and B. Zheng, Wolbachiaspread dynamics in stochastic environments, Theor. Popul. Biol., 106 (2015), 32-44.  doi: 10.1016/j.tpb.2015.09.003. [12] M. Huang, X. Song and J. Li, Modeling and analysis of impulsive releases of sterile mosquitoes, J. Biol. Dyn., 11 (2017), 147-171.  doi: 10.1080/17513758.2016.1254286. [13] M. Huang, J. Luo, L. Hu, B. Zheng and J. Yu, Assessing the efficiency of Wolbachia driven aedes mosquito suppression by delay differential equations, J. Theor. Biol., 440 (2018), 1-11.  doi: 10.1016/j.jtbi.2017.12.012. [14] J. Li, Simple stage-structured models for wild and transgenic mosquito populations, J. Diff. Eqns. Appl., 15 (2009), 327-347.  doi: 10.1080/10236190802566491. [15] J. Li, Malaria model with stage-structured mosquitoes, Math. Biol. Eng., 8 (2011), 753-768.  doi: 10.3934/mbe.2011.8.753. [16] J. Li, New revised simple models for interactive wild and sterile mosquito populations and their dynamics, J. Biol. Dyn., 11 (2017), 316-333.  doi: 10.1080/17513758.2016.1216613. [17] J. Li and S. Ai, Impulsive releases of sterile mosquitoes and interactive dynamics with time delay, J. Biol. Dyn., 14 (2020), 289-307.  doi: 10.1080/17513758.2020.1748239. [18] J. Li, L. Cai and Y. Li, Stage-structured wild and sterile mosquito population models and their dynamics, J. Biol. Dyn., 11 (2017), 79-101.  doi: 10.1080/17513758.2016.1159740. [19] Y. Li and X. Liu, An impulsive model for Wolbachia infection control of mosquito-borne diseases with general birth and death rate functions, Nonl. Anal. Real World Appl., 37 (2017), 421-432.  doi: 10.1016/j.nonrwa.2017.03.003. [20] M. Otero, H. G. Solari and N. Schweigmann, A stochastic population dynamics model for Aedes aegypti: Formulation and application to a city with temperate climate, Bull. Math. Biol., 68 (2006), 1945-1974.  doi: 10.1007/s11538-006-9067-y. [21] T. Walker, P. H. Johnson and L. A. Moreika, et al., The wMel Wolbachia strain blocks dengue and invades caged Aedes aegypti populations, Nature, 476 (2011), 450–453. doi: 10.1038/nature10355. [22] Wikipedia, Sterile Insect Technique, 2021, https://en.wikipedia.org/wiki/Sterile_insect_technique. [23] Z. Xi, C. C. Khoo and S. I. Dobson, Wolbachiaestablishment and invasion in an Aedes aegypti laboratory population, Science, 310 (2005), 326-328.  doi: 10.1126/science.1117607. [24] J. Yu, Modelling mosquito population suppression based on delay differential equations, SIAM, J. Appl. Math., 78 (2018), 3168-3187.  doi: 10.1137/18M1204917. [25] J. Yu, Existence and stability of a unique and exact two periodic orbits for an interactive wild and sterile mosquito model, J. Diff. Eqns, 269 (2020), 10395-10415.  doi: 10.1016/j.jde.2020.07.019. [26] J. Yu and J. Li, Dynamics of interactive wild and sterile mosquitoes with time delay, J. Biol. Dyn., 13 (2019), 606-620.  doi: 10.1080/17513758.2019.1682201. [27] J. Yu and J. Li, Global asymptotic stability in an interactive wild and sterile mosquito model, J. Diff. Eqns., 269 (2020), 6193-6215.  doi: 10.1016/j.jde.2020.04.036. [28] X. Zhang, S. Tang and R. A. Cheke, Models to assess how best to replace dengue virus vectors with Wolbachia-infected mosquito populations, Math. Biosci., 269 (2015), 164-177.  doi: 10.1016/j.mbs.2015.09.004. [29] B. Zheng, M. Tang and J. Yu, Modeling Wolbachia spread in mosquitoes through delay differential equation, SIAM J. Appl. Math., 74 (2014), 743-770.  doi: 10.1137/13093354X. [30] B. Zheng, M. Tang, J. Yu and J. Qiu, Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission, J. Math. Biol., 76 (2018), 235-263.  doi: 10.1007/s00285-017-1142-5. [31] X. Zheng, D. Zhang, Y. Li, C. Yang, Y. Wu and X. Liang et al., Incompatible and sterile insect techniques combined eliminate mosquitoes, Nature, 572 (2019), 56–61. doi: 10.1038/s41586-019-1407-9.

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##### References:
 [1] H. J. Barclay, Pest population stability under sterile releases, Res. Popul. Ecol., 24 (1982), 405-416.  doi: 10.1007/BF02515585. [2] H. J. Barclay, Mathematical models for the use of sterile insects, In: Sterile Insect Technique. Principles and Practice in Area-Wide Integrated Pest Management, (V.A. Dyck, J. Hendrichs, and A.S. Robinson, Eds.), Springer, Heidelberg, (2005), 147–174. doi: 10.1007/1-4020-4051-2_6. [3] H. J. Barclay and M. Mackuer, The sterile insect release method for pest control: A density dependent model, Environ. Entomol., 9 (1980), 810-817.  doi: 10.1093/ee/9.6.810. [4] N. Becker, Mosquitoes and Their Control, Kluwer Academic/Plenum, New York, 2003. [5] G. Briggs, The endosymbiotic bacterium Wolbachia induces resistance to dengue virus in Aedes aegypti, PLoS Pathog., 6 (2012), e1000833. [6] L. Cai, S. Ai and G. Fan, Dynamics of delayed mosquitoes populations models with two different strategies of releasing sterile mosquitoes, Math. Biosci. Engi., 15 (2018), 1181-1202.  doi: 10.3934/mbe.2018054. [7] L. Cai, S. Ai and J. Li, Dynamics of mosquitoes populations with different strategies for releasing sterile mosquitoes, SIAM J. Appl. Math., 74 (2014), 1786-1809.  doi: 10.1137/13094102X. [8] CDC, Life cycle: The mosquito, 2021, https://www.cdc.gov/dengue/resources/factsheets/mosquitolifecyclefinal.pdf. [9] C. Dye, Intraspecific competition amongst larval aedes aegypti: Food exploitation or chemical interference, Ecol. Entomol., 7 (1982), 39-46.  doi: 10.1111/j.1365-2311.1982.tb00642.x. [10] R. M. Gleiser, J. Urrutia and D. E. Gorla, Effects of crowding on populations of aedes albifasciatus larvae under laboratory conditions., Entomologia Experimentalis et Applicata, 95 (2000), 135-140.  doi: 10.1046/j.1570-7458.2000.00651.x. [11] L. Hu, M. Huang, M. Tang, J. Yu and B. Zheng, Wolbachiaspread dynamics in stochastic environments, Theor. Popul. Biol., 106 (2015), 32-44.  doi: 10.1016/j.tpb.2015.09.003. [12] M. Huang, X. Song and J. Li, Modeling and analysis of impulsive releases of sterile mosquitoes, J. Biol. Dyn., 11 (2017), 147-171.  doi: 10.1080/17513758.2016.1254286. [13] M. Huang, J. Luo, L. Hu, B. Zheng and J. Yu, Assessing the efficiency of Wolbachia driven aedes mosquito suppression by delay differential equations, J. Theor. Biol., 440 (2018), 1-11.  doi: 10.1016/j.jtbi.2017.12.012. [14] J. Li, Simple stage-structured models for wild and transgenic mosquito populations, J. Diff. Eqns. Appl., 15 (2009), 327-347.  doi: 10.1080/10236190802566491. [15] J. Li, Malaria model with stage-structured mosquitoes, Math. Biol. Eng., 8 (2011), 753-768.  doi: 10.3934/mbe.2011.8.753. [16] J. Li, New revised simple models for interactive wild and sterile mosquito populations and their dynamics, J. Biol. Dyn., 11 (2017), 316-333.  doi: 10.1080/17513758.2016.1216613. [17] J. Li and S. Ai, Impulsive releases of sterile mosquitoes and interactive dynamics with time delay, J. Biol. Dyn., 14 (2020), 289-307.  doi: 10.1080/17513758.2020.1748239. [18] J. Li, L. Cai and Y. Li, Stage-structured wild and sterile mosquito population models and their dynamics, J. Biol. Dyn., 11 (2017), 79-101.  doi: 10.1080/17513758.2016.1159740. [19] Y. Li and X. Liu, An impulsive model for Wolbachia infection control of mosquito-borne diseases with general birth and death rate functions, Nonl. Anal. Real World Appl., 37 (2017), 421-432.  doi: 10.1016/j.nonrwa.2017.03.003. [20] M. Otero, H. G. Solari and N. Schweigmann, A stochastic population dynamics model for Aedes aegypti: Formulation and application to a city with temperate climate, Bull. Math. Biol., 68 (2006), 1945-1974.  doi: 10.1007/s11538-006-9067-y. [21] T. Walker, P. H. Johnson and L. A. Moreika, et al., The wMel Wolbachia strain blocks dengue and invades caged Aedes aegypti populations, Nature, 476 (2011), 450–453. doi: 10.1038/nature10355. [22] Wikipedia, Sterile Insect Technique, 2021, https://en.wikipedia.org/wiki/Sterile_insect_technique. [23] Z. Xi, C. C. Khoo and S. I. Dobson, Wolbachiaestablishment and invasion in an Aedes aegypti laboratory population, Science, 310 (2005), 326-328.  doi: 10.1126/science.1117607. [24] J. Yu, Modelling mosquito population suppression based on delay differential equations, SIAM, J. Appl. Math., 78 (2018), 3168-3187.  doi: 10.1137/18M1204917. [25] J. Yu, Existence and stability of a unique and exact two periodic orbits for an interactive wild and sterile mosquito model, J. Diff. Eqns, 269 (2020), 10395-10415.  doi: 10.1016/j.jde.2020.07.019. [26] J. Yu and J. Li, Dynamics of interactive wild and sterile mosquitoes with time delay, J. Biol. Dyn., 13 (2019), 606-620.  doi: 10.1080/17513758.2019.1682201. [27] J. Yu and J. Li, Global asymptotic stability in an interactive wild and sterile mosquito model, J. Diff. Eqns., 269 (2020), 6193-6215.  doi: 10.1016/j.jde.2020.04.036. [28] X. Zhang, S. Tang and R. A. Cheke, Models to assess how best to replace dengue virus vectors with Wolbachia-infected mosquito populations, Math. Biosci., 269 (2015), 164-177.  doi: 10.1016/j.mbs.2015.09.004. [29] B. Zheng, M. Tang and J. Yu, Modeling Wolbachia spread in mosquitoes through delay differential equation, SIAM J. Appl. Math., 74 (2014), 743-770.  doi: 10.1137/13093354X. [30] B. Zheng, M. Tang, J. Yu and J. Qiu, Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission, J. Math. Biol., 76 (2018), 235-263.  doi: 10.1007/s00285-017-1142-5. [31] X. Zheng, D. Zhang, Y. Li, C. Yang, Y. Wu and X. Liang et al., Incompatible and sterile insect techniques combined eliminate mosquitoes, Nature, 572 (2019), 56–61. doi: 10.1038/s41586-019-1407-9.
Schematic illustration of the phase-plane analysis in Theorem 3.3
Schematic illustration of the Poincare map in Theorem 3.3
Schematic illustration of the phase-plane analysis in Theorem 3.4
Parameters are given in (23) and the release amount threshold $c^* = 4.35979$. With $c = 3 < c^*$, the threshold waiting period threshold $T^*$ seems between 6 and 7. When $T = 6$, the origin is locally, not globally, asymptotically stable and there exists a locally asymptotically stable positive periodic solution as shown in the left figure. When $T = 7$, there exists a globally asymptotically stable positive periodic solution as shown in the right figure. Here only the solution curves for the larvae are presented
Parameters are still given in (23) with the same release amount threshold $c^* = 4.35979$, but now $c = 6 > c^*$. When $T = 5.5$, the origin is globally asymptotically stable as shown in the left figure, and when $T = 7$ there exists a globally asymptotically stable positive periodic solution as shown in the right figure. Here again only the solution curves for the larvae are presented
With the same parameters in (23), and also the same $c = 6 > c^* = 4.35979$ when $T = 6$, the origin is only locally asymptotically stable and there exists a locally asymptotically stable positive periodic solution
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