June  2020, 40(6): 3467-3484. doi: 10.3934/dcds.2020042

A stage structured model of delay differential equations for Aedes mosquito population suppression

1. 

School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou 510320, China

2. 

Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

3. 

Center for Applied Mathematics

4. 

College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China

* Corresponding author: Moxun Tang

Dedicated to Prof. Wei-Ming Ni on the occasion of his 70th birthday

Received  February 2019 Published  October 2019

Tremendous efforts have been devoted to the development and analysis of mathematical models to assess the efficacy of the endosymbiotic bacterium Wolbachia in the control of infectious diseases such as dengue and Zika, and their transmission vector Aedes mosquitoes. However, the larval stage has not been included in most models, which causes an inconvenience in testing directly the density restriction on population growth. In this work, we introduce a system of delay differential equations, including both the adult and larval stages of wild mosquitoes, interfered by Wolbachia infected males that can cause complete female sterility. We clarify its global dynamics rather completely by using delicate analyses, including a construction of Liapunov-type functions, and determine the threshold level $ R_0 $ of infected male releasing. The wild population is suppressed completely if the releasing level exceeds $ R_0 $ uniformly. The dynamical complexity revealed by our analysis, such as bi-stability and semi-stability, is further exhibited through numerical examples. Our model generates a temporal profile that captures several critical features of Aedes albopictus population in Guangzhou from 2011 to 2016. Our estimate for optimal mosquito control suggests that the most cost-efficient releasing should be started no less than 7 weeks before the peak dengue season.

Citation: Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042
References:
[1]

P. AgnewM. HideC. Sidobre and Y. Michalakis, A minimalist approach to the effects of density dependent competition on insect life history traits, Ecol. Entomol., 27 (2002), 396-402.  doi: 10.1046/j.1365-2311.2002.00430.x.  Google Scholar

[2]

F. BaldacchinoF. CaputoA. DragoT. A. DellaF. Montarsi and A. Rizzoli, Control methods against invasive Aedes mosquitoes in Europe: A review, Pest. Manag. Sci., 71 (2015), 1471-1485.   Google Scholar

[3]

S. BhattP. W. GethingO. J. BradyJ. P. MessinaA. W. FarlowC. L. MoyesJ. M. DrakeJ. S. BrownsteinA. G. HoenO. SankohM. F. MyersD. B. GeorgeT. JaenischG. R. W. WintC. P. SimmonsT. W. ScottJ. J. Farrar and S. I. Hay, The global distribution and burden of dengue, Nature, 496 (2013), 504-507.   Google Scholar

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P. CaillyA. TranT. BalenghienG. L'AmbertC. Toty and P. Ezanno, A climate driven abundance model to assess mosquito control strategies, Ecol. Model., 227 (2012), 7-17.  doi: 10.1016/j.ecolmodel.2011.10.027.  Google Scholar

[5]

Q. Cheng, Q. L. Jing, R. C. Spear, J. M. Marshall, Z. C. Yang and P. Gong, Climate and the timing of imported cases as determinants of the dengue outbreak in Guangzhou, 2014: Evidence from a mathematical model, PLoS Negl. Trop. Dis., 10 (2016), e0004417. doi: 10.1371/journal.pntd.0004417.  Google Scholar

[6]

D. A. FocksD. G. HaileE. Daniels and G. A. Mount, Dynamic life table model for Aedes aegypti (Diptera: Culicidae): Simulation results and validation, J. Med. Entomol., 30 (1993), 1018-1028.  doi: 10.1093/jmedent/30.6.1018.  Google Scholar

[7]

H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Monographs and Textbooks in Pure and Applied Mathematics, 57. Marcel Dekker, Inc., New York, 1980.  Google Scholar

[8]

A. A. HoffmannB. L. MontgomeryJ. PopoviciI. Iturbe-OrmaetxeP. H. JohnsonF. MuzziM. GreenfieldM. DurkanY. S. LeongY. DongH. CookJ. AxfordA. G. CallahanN. KennyC. OmodeiE. A. McGrawP. A. RyanS. A. RitchieM. Turelli and S. L. O'Neill, Successful establishment of Wolbachia in Aedes populations to suppress dengue transmission, Nature, 476 (2011), 454-457.  doi: 10.1038/nature10356.  Google Scholar

[9]

L. C. HuM. G. HuangM. X. TangJ. S. Yu and B. Zheng, Wolbachia spread dynamics in stochastic environments, Theor. Popul. Biol., 106 (2015), 32-44.  doi: 10.1016/j.tpb.2015.09.003.  Google Scholar

[10]

L. C. HuM. G. HuangM. X. TangJ. S. Yu and B. Zheng, Wolbachia spread dynamics in multi-regimes of environmental conditions, J. Theor. Biol., 462 (2019), 247-258.  doi: 10.1016/j.jtbi.2018.11.009.  Google Scholar

[11]

L. C. HuM. X. TangZ. D. WuZ. Y. Xi and J. S. Yu, The threshold infection level for Wolbachia invasion in random environments, J. Diff. Equ., 266 (2019), 4377-4393.  doi: 10.1016/j.jde.2018.09.035.  Google Scholar

[12]

M. G. HuangL. C. Hu and B. Zheng, Comparing the efficiency of Wolbachia driven Aedes mosquito suppression strategies, J. Appl. Anal. Comput., 9 (2019), 211-230.   Google Scholar

[13]

M. G. HuangJ. W. LouL. C. HuB. Zheng and J. S. 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.  Google Scholar

[14]

M. G. HuangM. X. Tang and J. S. Yu, Wolbachia infection dynamics by reaction-diffusion equations, Sci. China Math., 58 (2015), 77-96.  doi: 10.1007/s11425-014-4934-8.  Google Scholar

[15]

M. G. HuangJ. S. YuL. C. Hu and B. Zheng, Qualitative analysis for a Wolbachia infection model with diffusion, Sci. China Math., 59 (2016), 1249-1266.  doi: 10.1007/s11425-016-5149-y.  Google Scholar

[16]

P. F. Jia, L. Lu, X. Chen, J. Chen, L. Guo, X. Yu and Q. Y. Liu, A climate-driven mechanistic population model of Aedes albopictus with diapause, Para. Vect., 9 (2016), 175 pp. doi: 10.1186/s13071-016-1448-y.  Google Scholar

[17]

R. M. Lana, M.M. Morais, T. F. M. de Lima, T. G. de Senna Carneiro, L. M. Stolerman, J. P. C. dos Santos, J. J. C. Cortés, A. E. Eiras and C. T. Codeço, Assessment of a trap based Aedes aegypti surveillance program using mathematical modeling, PLoS one, 13 (2018), e0190673. doi: 10.1371/journal.pone.0190673.  Google Scholar

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Y. J. Li, F. Kamara, G. F. Zhou, S. Puthiyakunnon, C. Y. Li, Y. X. Liu, Y. H. Zhou, L. J. Yao, G. Y. Yan and X.-G. Chen, Urbanization increases Aedes albopictus larval habitats and accelerates mosquito development and survivorship, PLoS Negl. Trop. Dis., 8 (2014), e0003301. doi: 10.1371/journal.pntd.0003301.  Google Scholar

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[20]

Z. LiuY. Zhang and Y. Yang, Population dynamics of Aedes (Stegomyia) albopictus (Skuse) under laboratory conditions, Acta Entomol. Sin., 28 (1985), 274-280.   Google Scholar

[21]

F. LiuC. Zhou and P. Lin, Studies on the population ecology of Aedes albopictus 5. The seasonal abundance of natural population of Aedes albopictus in Guangzhou, Acta Sci. Natur. Univ. Sunyatseni, 29 (1990), 118-122.   Google Scholar

[22]

F. LiuC. YaoP. Lin and C. Zhou, Studies on life table of the natural population of Aedes albopictus, Acta Sci. Natur. Univ. Sunyatseni, 31 (1992), 84-93.   Google Scholar

[23]

P. A. RossN. M. EndersbyH. L. Yeap and A. A. Hoffmann, Larval competition extends developmental time and decreases adult size of wMelPop Wolbachia infected Aedes aegypti, Am. J. Trop. Hyg., 91 (2014), 198-205.  doi: 10.4269/ajtmh.13-0576.  Google Scholar

[24]

H. Smith, An Introduction to Delay Differential Equations with Applications to Life Sciences, Texts in Applied Mathematics, 57. Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar

[25]

J. WaldockN. L. ChandraJ. LelieveldY. ProestosE. MichaelG. Christophides and P. E. Parham, The role of environment variables on Aedes albopictus biology and Chikungunya epidemiology, Pathog. Glob. Health., 107 (2013), 224-240.   Google Scholar

[26]

R. K. Walsh, C. Bradley, C. Apperson and F. Gould, An experimental field study of delayed density dependence in natural populations of Aedes albopictus, PLoS One, 7 (2012), e0035959. Google Scholar

[27]

R. K. WalshL. FacchinelliJ. M. RamseyJ. G. Bond and F. Gould, Assessing the impact of density dependence in field populations of Aedes aegypti, J. Vect. Ecol., 36 (2011), 300-307.  doi: 10.1111/j.1948-7134.2011.00170.x.  Google Scholar

[28]

T. WalkerP. H. JohnsonL. A. MoreikaI. Iturbe-OrmaetxeF. D. FrentiuC. J. McmenimanY. S. LeongY. DongJ. AxfordP. KriesnerA. L. LloydS. A. RitchieS. L. O'Neill and A. A. Hoffmann, The wMel Wolbachia strain blocks dengue and invades caged Aedes aegypti populations, Nature, 476 (2011), 450-453.  doi: 10.1038/nature10355.  Google Scholar

[29]

E. Waltz, US reviews plan to infect mosquitoes with bacteria to stop disease, Nature, 533 (2016), 450-451.  doi: 10.1038/533450a.  Google Scholar

[30]

WHO, Global Strategy for Dengue Prevention and Control 2012-2020, World Health Organization, Geneva, 2012. Google Scholar

[31]

Z. Y. XiC. C. H. Khoo and S. L. Dobson, Wolbachia establishment and invasion in an Aedes aegypti laboratory population, Science, 310 (2005), 326-328.  doi: 10.1126/science.1117607.  Google Scholar

[32]

J. S. Yu, Modeling mosquito population suppression based on delay differential equations, SIAM J. Appl. Math., 78 (2018), 3168-3187.  doi: 10.1137/18M1204917.  Google Scholar

[33]

J. S. Yu and B. Zheng, Modeling Wolbachia infection in mosquito population via discrete dynamical models, J. Differ. Equ. Appl., (2019).  doi: 10.1080/10236198.2019.1669578.  Google Scholar

[34]

B. ZhengW. L. GuoL. C. HuM. G. Huang and J. S. Yu, Complex Wolbachia infection dynamics in mosquitoes with imperfect maternal transmission, Math. Biosci. Eng., 15 (2018), 523-541.  doi: 10.3934/mbe.2018024.  Google Scholar

[35]

B. ZhengX. P. LiuM. X. TangZ. Y. Xi and J. S. Yu, Use of age-stage structural models to seek optimal Wolbachia-infected male mosquito releases for mosquito-borne disease control, J. Theor. Biol., 472 (2019), 95-109.  doi: 10.1016/j.jtbi.2019.04.010.  Google Scholar

[36]

B. ZhengM. X. Tang and J. S. Yu, Modeling Wolbachia spread in mosquitoes through delay differential equation, SIAM J. Appl. Math., 74 (2014), 743-770.  doi: 10.1137/13093354X.  Google Scholar

[37]

B. ZhengM. X. TangJ. S. Yu and J. X. Qiu, Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission, J. Math. Biol., 76 (2018), 235-263.  doi: 10.1007/s00285-017-1142-5.  Google Scholar

[38]

B. ZhengJ. S. YuZ. Y. Xi and M. X. Tang, The annual abundance of dengue and Zika vector Aedes albopictus and its stubbornness to suppression, Ecol. Model., 387 (2018), 38-48.  doi: 10.1016/j.ecolmodel.2018.09.004.  Google Scholar

[39]

Z. Zhong and G. He, The life table of laboratory Aedes albopictus under various temperatures, Academic J. Sun Yat-Sen Univ. of Med. Sci., 9 (1988), 35-39.   Google Scholar

[40]

Z. Zhong and G. He, The life and fertility table of Aedes albopictus under different temperatures, Acta Entom. Sinica, 33 (1990), 64-70.   Google Scholar

show all references

References:
[1]

P. AgnewM. HideC. Sidobre and Y. Michalakis, A minimalist approach to the effects of density dependent competition on insect life history traits, Ecol. Entomol., 27 (2002), 396-402.  doi: 10.1046/j.1365-2311.2002.00430.x.  Google Scholar

[2]

F. BaldacchinoF. CaputoA. DragoT. A. DellaF. Montarsi and A. Rizzoli, Control methods against invasive Aedes mosquitoes in Europe: A review, Pest. Manag. Sci., 71 (2015), 1471-1485.   Google Scholar

[3]

S. BhattP. W. GethingO. J. BradyJ. P. MessinaA. W. FarlowC. L. MoyesJ. M. DrakeJ. S. BrownsteinA. G. HoenO. SankohM. F. MyersD. B. GeorgeT. JaenischG. R. W. WintC. P. SimmonsT. W. ScottJ. J. Farrar and S. I. Hay, The global distribution and burden of dengue, Nature, 496 (2013), 504-507.   Google Scholar

[4]

P. CaillyA. TranT. BalenghienG. L'AmbertC. Toty and P. Ezanno, A climate driven abundance model to assess mosquito control strategies, Ecol. Model., 227 (2012), 7-17.  doi: 10.1016/j.ecolmodel.2011.10.027.  Google Scholar

[5]

Q. Cheng, Q. L. Jing, R. C. Spear, J. M. Marshall, Z. C. Yang and P. Gong, Climate and the timing of imported cases as determinants of the dengue outbreak in Guangzhou, 2014: Evidence from a mathematical model, PLoS Negl. Trop. Dis., 10 (2016), e0004417. doi: 10.1371/journal.pntd.0004417.  Google Scholar

[6]

D. A. FocksD. G. HaileE. Daniels and G. A. Mount, Dynamic life table model for Aedes aegypti (Diptera: Culicidae): Simulation results and validation, J. Med. Entomol., 30 (1993), 1018-1028.  doi: 10.1093/jmedent/30.6.1018.  Google Scholar

[7]

H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Monographs and Textbooks in Pure and Applied Mathematics, 57. Marcel Dekker, Inc., New York, 1980.  Google Scholar

[8]

A. A. HoffmannB. L. MontgomeryJ. PopoviciI. Iturbe-OrmaetxeP. H. JohnsonF. MuzziM. GreenfieldM. DurkanY. S. LeongY. DongH. CookJ. AxfordA. G. CallahanN. KennyC. OmodeiE. A. McGrawP. A. RyanS. A. RitchieM. Turelli and S. L. O'Neill, Successful establishment of Wolbachia in Aedes populations to suppress dengue transmission, Nature, 476 (2011), 454-457.  doi: 10.1038/nature10356.  Google Scholar

[9]

L. C. HuM. G. HuangM. X. TangJ. S. Yu and B. Zheng, Wolbachia spread dynamics in stochastic environments, Theor. Popul. Biol., 106 (2015), 32-44.  doi: 10.1016/j.tpb.2015.09.003.  Google Scholar

[10]

L. C. HuM. G. HuangM. X. TangJ. S. Yu and B. Zheng, Wolbachia spread dynamics in multi-regimes of environmental conditions, J. Theor. Biol., 462 (2019), 247-258.  doi: 10.1016/j.jtbi.2018.11.009.  Google Scholar

[11]

L. C. HuM. X. TangZ. D. WuZ. Y. Xi and J. S. Yu, The threshold infection level for Wolbachia invasion in random environments, J. Diff. Equ., 266 (2019), 4377-4393.  doi: 10.1016/j.jde.2018.09.035.  Google Scholar

[12]

M. G. HuangL. C. Hu and B. Zheng, Comparing the efficiency of Wolbachia driven Aedes mosquito suppression strategies, J. Appl. Anal. Comput., 9 (2019), 211-230.   Google Scholar

[13]

M. G. HuangJ. W. LouL. C. HuB. Zheng and J. S. 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.  Google Scholar

[14]

M. G. HuangM. X. Tang and J. S. Yu, Wolbachia infection dynamics by reaction-diffusion equations, Sci. China Math., 58 (2015), 77-96.  doi: 10.1007/s11425-014-4934-8.  Google Scholar

[15]

M. G. HuangJ. S. YuL. C. Hu and B. Zheng, Qualitative analysis for a Wolbachia infection model with diffusion, Sci. China Math., 59 (2016), 1249-1266.  doi: 10.1007/s11425-016-5149-y.  Google Scholar

[16]

P. F. Jia, L. Lu, X. Chen, J. Chen, L. Guo, X. Yu and Q. Y. Liu, A climate-driven mechanistic population model of Aedes albopictus with diapause, Para. Vect., 9 (2016), 175 pp. doi: 10.1186/s13071-016-1448-y.  Google Scholar

[17]

R. M. Lana, M.M. Morais, T. F. M. de Lima, T. G. de Senna Carneiro, L. M. Stolerman, J. P. C. dos Santos, J. J. C. Cortés, A. E. Eiras and C. T. Codeço, Assessment of a trap based Aedes aegypti surveillance program using mathematical modeling, PLoS one, 13 (2018), e0190673. doi: 10.1371/journal.pone.0190673.  Google Scholar

[18]

Y. J. Li, F. Kamara, G. F. Zhou, S. Puthiyakunnon, C. Y. Li, Y. X. Liu, Y. H. Zhou, L. J. Yao, G. Y. Yan and X.-G. Chen, Urbanization increases Aedes albopictus larval habitats and accelerates mosquito development and survivorship, PLoS Negl. Trop. Dis., 8 (2014), e0003301. doi: 10.1371/journal.pntd.0003301.  Google Scholar

[19]

H. L. Lin, T. Liu, T. Song, L. F. Lin, J. P. Xiao, J. Y. Lin, J. F. He, H. J. Zhong, W. B. Hu, A. P. Deng, Z. Q. Peng, W. J. Ma and Y. H. Zhang, Community involvement in dengue outbreak control: An integrated rigorous intervention strategy, PLoS Negl. Trop. Dis., 10 (2016), e0004919. doi: 10.1371/journal.pntd.0004919.  Google Scholar

[20]

Z. LiuY. Zhang and Y. Yang, Population dynamics of Aedes (Stegomyia) albopictus (Skuse) under laboratory conditions, Acta Entomol. Sin., 28 (1985), 274-280.   Google Scholar

[21]

F. LiuC. Zhou and P. Lin, Studies on the population ecology of Aedes albopictus 5. The seasonal abundance of natural population of Aedes albopictus in Guangzhou, Acta Sci. Natur. Univ. Sunyatseni, 29 (1990), 118-122.   Google Scholar

[22]

F. LiuC. YaoP. Lin and C. Zhou, Studies on life table of the natural population of Aedes albopictus, Acta Sci. Natur. Univ. Sunyatseni, 31 (1992), 84-93.   Google Scholar

[23]

P. A. RossN. M. EndersbyH. L. Yeap and A. A. Hoffmann, Larval competition extends developmental time and decreases adult size of wMelPop Wolbachia infected Aedes aegypti, Am. J. Trop. Hyg., 91 (2014), 198-205.  doi: 10.4269/ajtmh.13-0576.  Google Scholar

[24]

H. Smith, An Introduction to Delay Differential Equations with Applications to Life Sciences, Texts in Applied Mathematics, 57. Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar

[25]

J. WaldockN. L. ChandraJ. LelieveldY. ProestosE. MichaelG. Christophides and P. E. Parham, The role of environment variables on Aedes albopictus biology and Chikungunya epidemiology, Pathog. Glob. Health., 107 (2013), 224-240.   Google Scholar

[26]

R. K. Walsh, C. Bradley, C. Apperson and F. Gould, An experimental field study of delayed density dependence in natural populations of Aedes albopictus, PLoS One, 7 (2012), e0035959. Google Scholar

[27]

R. K. WalshL. FacchinelliJ. M. RamseyJ. G. Bond and F. Gould, Assessing the impact of density dependence in field populations of Aedes aegypti, J. Vect. Ecol., 36 (2011), 300-307.  doi: 10.1111/j.1948-7134.2011.00170.x.  Google Scholar

[28]

T. WalkerP. H. JohnsonL. A. MoreikaI. Iturbe-OrmaetxeF. D. FrentiuC. J. McmenimanY. S. LeongY. DongJ. AxfordP. KriesnerA. L. LloydS. A. RitchieS. L. O'Neill and A. A. Hoffmann, The wMel Wolbachia strain blocks dengue and invades caged Aedes aegypti populations, Nature, 476 (2011), 450-453.  doi: 10.1038/nature10355.  Google Scholar

[29]

E. Waltz, US reviews plan to infect mosquitoes with bacteria to stop disease, Nature, 533 (2016), 450-451.  doi: 10.1038/533450a.  Google Scholar

[30]

WHO, Global Strategy for Dengue Prevention and Control 2012-2020, World Health Organization, Geneva, 2012. Google Scholar

[31]

Z. Y. XiC. C. H. Khoo and S. L. Dobson, Wolbachia establishment and invasion in an Aedes aegypti laboratory population, Science, 310 (2005), 326-328.  doi: 10.1126/science.1117607.  Google Scholar

[32]

J. S. Yu, Modeling mosquito population suppression based on delay differential equations, SIAM J. Appl. Math., 78 (2018), 3168-3187.  doi: 10.1137/18M1204917.  Google Scholar

[33]

J. S. Yu and B. Zheng, Modeling Wolbachia infection in mosquito population via discrete dynamical models, J. Differ. Equ. Appl., (2019).  doi: 10.1080/10236198.2019.1669578.  Google Scholar

[34]

B. ZhengW. L. GuoL. C. HuM. G. Huang and J. S. Yu, Complex Wolbachia infection dynamics in mosquitoes with imperfect maternal transmission, Math. Biosci. Eng., 15 (2018), 523-541.  doi: 10.3934/mbe.2018024.  Google Scholar

[35]

B. ZhengX. P. LiuM. X. TangZ. Y. Xi and J. S. Yu, Use of age-stage structural models to seek optimal Wolbachia-infected male mosquito releases for mosquito-borne disease control, J. Theor. Biol., 472 (2019), 95-109.  doi: 10.1016/j.jtbi.2019.04.010.  Google Scholar

[36]

B. ZhengM. X. Tang and J. S. Yu, Modeling Wolbachia spread in mosquitoes through delay differential equation, SIAM J. Appl. Math., 74 (2014), 743-770.  doi: 10.1137/13093354X.  Google Scholar

[37]

B. ZhengM. X. TangJ. S. Yu and J. X. Qiu, Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission, J. Math. Biol., 76 (2018), 235-263.  doi: 10.1007/s00285-017-1142-5.  Google Scholar

[38]

B. ZhengJ. S. YuZ. Y. Xi and M. X. Tang, The annual abundance of dengue and Zika vector Aedes albopictus and its stubbornness to suppression, Ecol. Model., 387 (2018), 38-48.  doi: 10.1016/j.ecolmodel.2018.09.004.  Google Scholar

[39]

Z. Zhong and G. He, The life table of laboratory Aedes albopictus under various temperatures, Academic J. Sun Yat-Sen Univ. of Med. Sci., 9 (1988), 35-39.   Google Scholar

[40]

Z. Zhong and G. He, The life and fertility table of Aedes albopictus under different temperatures, Acta Entom. Sinica, 33 (1990), 64-70.   Google Scholar

Figure 1.  The dynamical complexity of (3) not covered by Theorems 2.4 and 2.5. The parameter values are specified in (25) and $ K_L = 5\times 10^5 $. $ \overline{R} = 10^5\in (0, R_0) $ in A-C, and $ \overline{R} = R_0 = 1.073\times 10^6 $ in D. A. The solution with $ \overline\phi = 1.1\times 10^5>L_1 $ and $ \overline\psi = 7.5\times 10^3<A_1 $ rotates around the unstable equilibrium point $ E_1 $ before moving towards $ E_0 $. B and C. Either $ E_0 $ or $ E_2 $ may attract solutions with $ \overline\phi>L_1 $ and $ \overline\psi<A_1 $, or $ \overline\phi<L_1 $ and $ \overline\psi>A_1 $; see (26) for the values of $ L_1 $ and $ A_1 $. D. Either $ E_0 $ or $ E^*_2 $ may attract solutions with $ \overline\phi>L_2^* $ and $ \overline\psi<A_2^* $, or solutions with $ \overline\phi<L_2^* $ and $ \overline\psi>A_2^* $
Figure 2.  The temporal profiles of wild Aedes albopictus population in Guangzhou from 2011 to 2016. The profiles were simulated by (3) with $ \overline{R}\equiv 0 $, supplemented by the temperature-dependent rates estimated from (27) and Table 2
Figure 3.  The temperature dependency of the threshold releasing level $ R_0 $. The figure was simulated by (24) with the daily rates estimated from (27) and Table 2. It exhibits a quasi-periodicity as temperature annually, and peaks from July to October during the high-risk season of dengue fever
Figure 4.  Suppression of the wild Aedes albopictus population during the high-incidence season of dengue fever in 2012. The curves were generated by substituting the same parameter values and the initial data described in Table 2 into (3). The mosquito population is reduced more than $ >95\% $ on September 12, 2012, and the same low level is kept in phase Ⅱ of the five weeks behind
Table 1.  The life table of Aedes albopictus. The parameters are adapted to Aedes albopictus population in subtropical monsoon climate as in Guangzhou
Para. Definition value Reference
$N$ Number of eggs laid by a female 200 [20,22,39]
$\mu_E$ Hatch rate of egg (day$^{-1}$) Dependent of T [6,17,25]
$\beta$ Mean larvae produced by a female (day$^{-1}$) $\beta=2N\mu_E/\tau_A$
$m$ Minimum larva mortality rate (day$^{-1}$) Dependent of T [4,5,25]
$\mu$ Pupation rate (day$^{-1}$) Dependent of T [6,17,25]
$\bar{\alpha}$ Pupa survival rate (day$^{-1}$) Dependent of T [6,17,25]
$\delta$ Adult female mortality rate (day$^{-1}$) Dependent of T [4,5,25]
$\tau_E$ Development period of egg (days) (3.7, 18.3) [16,18,22,38]
$\tau_L$ Development period of larva (days) (5.2, 27.7) [16,18,22,38]
$\tau_P$ Development period of pupa (days) (1.5, 8.6) [16,18,22,38]
$\tau_A$ Mean longevity of female (days) (4.8, 40.9) [16,18,22,38]
Para. Definition value Reference
$N$ Number of eggs laid by a female 200 [20,22,39]
$\mu_E$ Hatch rate of egg (day$^{-1}$) Dependent of T [6,17,25]
$\beta$ Mean larvae produced by a female (day$^{-1}$) $\beta=2N\mu_E/\tau_A$
$m$ Minimum larva mortality rate (day$^{-1}$) Dependent of T [4,5,25]
$\mu$ Pupation rate (day$^{-1}$) Dependent of T [6,17,25]
$\bar{\alpha}$ Pupa survival rate (day$^{-1}$) Dependent of T [6,17,25]
$\delta$ Adult female mortality rate (day$^{-1}$) Dependent of T [4,5,25]
$\tau_E$ Development period of egg (days) (3.7, 18.3) [16,18,22,38]
$\tau_L$ Development period of larva (days) (5.2, 27.7) [16,18,22,38]
$\tau_P$ Development period of pupa (days) (1.5, 8.6) [16,18,22,38]
$\tau_A$ Mean longevity of female (days) (4.8, 40.9) [16,18,22,38]
Table 2.  The parameter values of (27) and the temperature-dependent mortality rates
Para. Value Reference
$ \mu_E $ $ \rho_E = 0.24, a_E = 10798, b_E = 100000, c_E = 14184 $ [6,17,25]
$ \mu = \mu_L $ $ \rho_L = 0.2088, a_L = 26018, b_L = 55990, c_L = 304.6 $ [6,17,25]
$ \bar{\alpha} = \mu_P $ $ \rho_P = 0.384, a_P = 14931, b_P = -472379, c_P = 148 $ [6,17,25]
$ m $ $ \left\{ \begin{array}{ll} 0.0000866 T^2-0.00368 T+0.09, \, \, T\ge12.5 ^oC, \\0.5, \, \, {\rm else}. \end{array} \right. $ [4,5,25]
$ \delta $ $ \left\{ \begin{array}{ll} 0.000114 T^2-0.00427 T+0.1278, \, \, T\ge 15 ^oC, \\0.5, \, \, {\rm else}. \end{array} \right. $ [4,5,25]
Para. Value Reference
$ \mu_E $ $ \rho_E = 0.24, a_E = 10798, b_E = 100000, c_E = 14184 $ [6,17,25]
$ \mu = \mu_L $ $ \rho_L = 0.2088, a_L = 26018, b_L = 55990, c_L = 304.6 $ [6,17,25]
$ \bar{\alpha} = \mu_P $ $ \rho_P = 0.384, a_P = 14931, b_P = -472379, c_P = 148 $ [6,17,25]
$ m $ $ \left\{ \begin{array}{ll} 0.0000866 T^2-0.00368 T+0.09, \, \, T\ge12.5 ^oC, \\0.5, \, \, {\rm else}. \end{array} \right. $ [4,5,25]
$ \delta $ $ \left\{ \begin{array}{ll} 0.000114 T^2-0.00427 T+0.1278, \, \, T\ge 15 ^oC, \\0.5, \, \, {\rm else}. \end{array} \right. $ [4,5,25]
Table 3.  The least releasing levels $ r_1 $ in phase I, $ r_2 $ in phase II, and the total numbers in the two phase to reduce $ >95\% $ of wild Aedes albopictus on September 12, 2012, and keep the same low level in phase II of the five weeks behind. The same parameter values and the initial data in Figure 2, except $ \overline R(t) = r_1 $ in phase Ⅰ, and $ \overline R(t) = r_2 $ in phase Ⅱ, were inserted into (3) for simulations
Phase Ⅰ Phase Ⅱ (5 weeks) Total number
Suppression period, $ r_1 $, Release times $ r_2 $, Release times
5 weeks, $ 4.4 \times 10^9 $, 12 $ 1.24 \times 10^7 $, 12 $ 5.29 \times 10^{10} $
6 weeks, $ 5.11 \times 10^8 $, 14 $ 8.48 \times 10^6 $, 12 $ 7.26 \times 10^9 $
7 weeks, $ 1.48 \times 10^8 $, 16 $ 8.42 \times 10^6 $, 12 $ 2.47 \times 10^9 $
8 weeks, $ 7.65 \times 10^7 $, 19 $ 5.89 \times 10^6 $, 12 $ 1.52 \times 10^9 $
9 weeks, $ 4.06 \times 10^7 $, 21 $ 4.63 \times 10^6 $, 12 $ 9.08 \times 10^8 $
10 weeks, $ 2.1 \times 10^7 $, 23 $ 5.48 \times 10^6 $, 12 $ 5.49 \times 10^8 $
Phase Ⅰ Phase Ⅱ (5 weeks) Total number
Suppression period, $ r_1 $, Release times $ r_2 $, Release times
5 weeks, $ 4.4 \times 10^9 $, 12 $ 1.24 \times 10^7 $, 12 $ 5.29 \times 10^{10} $
6 weeks, $ 5.11 \times 10^8 $, 14 $ 8.48 \times 10^6 $, 12 $ 7.26 \times 10^9 $
7 weeks, $ 1.48 \times 10^8 $, 16 $ 8.42 \times 10^6 $, 12 $ 2.47 \times 10^9 $
8 weeks, $ 7.65 \times 10^7 $, 19 $ 5.89 \times 10^6 $, 12 $ 1.52 \times 10^9 $
9 weeks, $ 4.06 \times 10^7 $, 21 $ 4.63 \times 10^6 $, 12 $ 9.08 \times 10^8 $
10 weeks, $ 2.1 \times 10^7 $, 23 $ 5.48 \times 10^6 $, 12 $ 5.49 \times 10^8 $
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