Advertisement

Bulletin of Mathematical Biology

, Volume 80, Issue 7, pp 1962–1987 | Cite as

Threshold Dynamics of a Temperature-Dependent Stage-Structured Mosquito Population Model with Nested Delays

Original Article
  • 43 Downloads

Abstract

Mosquito-borne diseases remain a significant threat to public health and economics. Since mosquitoes are quite sensitive to temperature, global warming may not only worsen the disease transmission case in current endemic areas but also facilitate mosquito population together with pathogens to establish in new regions. Therefore, understanding mosquito population dynamics under the impact of temperature is considerably important for making disease control policies. In this paper, we develop a stage-structured mosquito population model in the environment of a temperature-controlled experiment. The model turns out to be a system of periodic delay differential equations with periodic delays. We show that the basic reproduction number is a threshold parameter which determines whether the mosquito population goes to extinction or remains persistent. We then estimate the parameter values for Aedes aegypti, the mosquito that transmits dengue virus. We verify the analytic result by numerical simulations with the temperature data of Colombo, Sri Lanka where a dengue outbreak occurred in 2017.

Keywords

Mosquito Climate change Periodic delay Dengue Basic reproduction ratio Population dynamics 

References

  1. Abdelrazec A, Gumel AB (2017) Mathematical assessment of the role of temperature and rainfall on mosquito population dynamics. J Math Biol 74:1351–1395MathSciNetCrossRefMATHGoogle Scholar
  2. Alto BW, Lounibos LP, Mores CN, Reiskind MH (2008) Larval competition alters susceptibility of adult Aedes mosquitoes to dengue infection. Proc R Soc B 275:463–471CrossRefGoogle Scholar
  3. Bacaër N, Ait Dads EH (2012) On the biological interpretation of a definition for the parameter \(R_0\) in periodic population models. J Math Biol 65:601–621MathSciNetCrossRefMATHGoogle Scholar
  4. Bacaër N, Guernaoui S (2006) The epidemic threshold of vector-borne diseases with seasonality. J Math Biol 53:421–436MathSciNetCrossRefMATHGoogle Scholar
  5. Beck-Johnson LM, Nelson WA, Paaijmans KP, Read AF, Thomas MB, Bjornstad ON (2013) The effect of temperature on Anopheles mosquito population dynamics and the potential for malaria transmission. PLoS ONE 8(11):e79276CrossRefGoogle Scholar
  6. Christophers R (1960) Aedes aegypti (L.): the yellow fever mosquito. Cambridge University Press, CambridgeGoogle Scholar
  7. Costa EAPA, Santos EMM, Correia JF, Albuquerque CMR (2010) Impact of small variations in temperature and humidity on the reproductive activity and survival of Aedes aegypti (Diptera: Culicidae). Rev Bras Entomol 54:488–493CrossRefGoogle Scholar
  8. Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and the computation of the basic reproduction ratio \(R_0\) in the models for infectious disease in heterogeneous populations. J Math Biol 28:365–382MathSciNetCrossRefMATHGoogle Scholar
  9. Ewing DA, Cobbold CA, Purse BV, Nunn MA, White SM (2016) Modelling the effect of temperature on the seasonal population dynamics of temperate mosquitoes. J Theor Biol 400:65–79MathSciNetCrossRefMATHGoogle Scholar
  10. Fang J, Gourley S, Lou Y (2016) Stage-structured models of intra- and inter-specific competition within age classes. J Differ Equ 260:1918–1953MathSciNetCrossRefMATHGoogle Scholar
  11. Hale JK, Verduyn Lunel SM (1993) Introduction to functional differential equations. Springer, New YorkCrossRefMATHGoogle Scholar
  12. Hopp MJ, Foley JA (2001) Global-scale relationships between climate and the dengue fever vector Aedes aegypti. Clim Change 48:441–463CrossRefGoogle Scholar
  13. Inaba H (2012) On a new perspective of the basic reproduction number in heterogeneous environments. J Math Biol 22:113–128MathSciNetMATHGoogle Scholar
  14. Kot M (2001) Elements of mathematical ecology. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  15. Legros M, Lloyd AL, Huang Y, Gould F (2009) Density-dependent intraspecific competition in the larval stage of Aedes aegypti (Diptera: Culicidae): revisiting the current paradigm. J Med Entomol 46(3):409–419CrossRefGoogle Scholar
  16. Liang X, Zhao X-Q (2007) Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Commun Pure Appl Math 60:1–40MathSciNetCrossRefMATHGoogle Scholar
  17. Liang X, Zhang L, Zhao X-Q (2017) Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for Lyme disease). J Dyn Differ Equ.  https://doi.org/10.1007/s10884-017-9601-7 Google Scholar
  18. Liu K, Lou Y, Wu J (2017) Analysis of an age structured model for tick populations subject to seasonal effects. J Differ Equ 263:2078–2112MathSciNetCrossRefMATHGoogle Scholar
  19. Liu-Helmersson J, Stenlund H, Wilder-Smith A, Rocklov J (2014) Vectorial capacity of aedes aegypti: effects of temperature and implications for global dengue epidemic potential. PLoS ONE 9(3):e89783.  https://doi.org/10.1371/journal.pone.0089783 CrossRefGoogle Scholar
  20. Lou Y, Zhao X-Q (2017) A theoretical approach to understanding population dynamics with seasonal developmental durations. J Nonlinear Sci 27(2):573–603MathSciNetCrossRefMATHGoogle Scholar
  21. Marinho RA, Beserra EB, Bezerra-Gusmão MA, de Porto VS, Olinda RA, dos Santos CAC (2016) Effects of temperature on the life cycle, expansion, and dispersion of Aedes aegypti (Diptera: Culicidae) in three cities in Paraiba, Brazil. J Vector Ecol 41(1):1–10CrossRefGoogle Scholar
  22. McCauley E, Nisbet RM, De Roos AM, Murdoch WW, Gurney WSC (1996) Structured population models of herbivorous zooplankton. Ecol Monogr 66:479–501CrossRefGoogle Scholar
  23. Molnár PK, Kutz SJ, Hoar BM, Dobson AP (2013) Metabolic approaches to understanding climate change impacts on seasonal host-macroparasite dynamics. Ecol Lett 16:9–21CrossRefGoogle Scholar
  24. Ngarakana-Gwasira ET, Bhunu CP, Mashonjowa E (2014) Assessing the impact of temperature on malaria transmission dynamics. Afr Mat 25:1095–1112MathSciNetCrossRefMATHGoogle Scholar
  25. Nisbet RM, Gurney WS (1982) Modelling fluctuating populations. The Blackburn Press, New JerseyMATHGoogle Scholar
  26. Nisbet RM, Gurney WS (1983) The systematic formulation of population models for insects with dynamically varying instar duration. Theor Popul Biol 23:114–135MathSciNetCrossRefMATHGoogle Scholar
  27. Omori R, Adams B (2011) Disrupting seasonality to control disease outbreaks: the case of koi herpes virus. J Theor Biol 271:159–165MathSciNetCrossRefGoogle Scholar
  28. Rittenhouse MA, Revie CW, Hurford A (2016) A model for sea lice (Lepeophtheirus salmonis) dynamics in a seasonally changing environment. Epidemics 16:8–16CrossRefGoogle Scholar
  29. Shapiro LLM, Whitehead SA, Thomas MB (2017) Quantifying the effects of temperature on mosquito and parasite traits that determine the transmission potential of human malaria. PLoS Biol 15(10):e2003489.  https://doi.org/10.1371/journal.pbio.2003489 CrossRefGoogle Scholar
  30. Simoy MI, Simoy MV, Canziani GA (2015) The effect of temperature on the population dynamics of Aedes aegypti. Ecol Model 314:100–110CrossRefMATHGoogle Scholar
  31. Smith HL (1995) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. Mathematical surveys and monographs, vol 41. American Mathematical Society, ProvidenceGoogle Scholar
  32. Tatem AJ, Rogers DJ, Hay SI (2006) Global transport networks and infectious disease spread. Adv Parasitol 62:293–343CrossRefGoogle Scholar
  33. Thieme HR (2009) Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J Appl Math 70:188–211MathSciNetCrossRefMATHGoogle Scholar
  34. Thomé RCA, Yang HM, Esteva L (2010) Optimal control of Aedes aegypti mosquitoes by the sterile insect technique and insecticide. Math Biosci 223:12–23MathSciNetCrossRefMATHGoogle Scholar
  35. van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29–48MathSciNetCrossRefMATHGoogle Scholar
  36. Walter W (1997) On strongly monotone flows. Ann Pol Math 66:269–274MathSciNetCrossRefMATHGoogle Scholar
  37. Wang W, Zhao X-Q (2008) Threshold dynamics for compartmental epidemic models in periodic environments. J Dyn Differ Equ 20:699–717MathSciNetCrossRefMATHGoogle Scholar
  38. Wang X, Zhao X-Q (2017a) A malaria transmission model with temperature-dependent incubation period. Bull Math Biol 79:1155–1182MathSciNetCrossRefMATHGoogle Scholar
  39. Wang X, Zhao X-Q (2017b) Dynamics of a time-delayed Lyme disease model with seasonality. SIAM J Appl Dyn Syst 16(2):853–881MathSciNetCrossRefMATHGoogle Scholar
  40. Wang X, Tang S, Cheke RA (2016) A stage structured mosquito model incorporating effects of precipitation and daily temperature fluctuations. J Theor Biol 411:27–36CrossRefMATHGoogle Scholar
  41. Wang Y, Pons W, Fang J, Zhu H (2017) The impact of weather and storm water management ponds on the transmission of West Nile virus. R Soc Open Sci 4:170017.  https://doi.org/10.1098/rsos.170017 CrossRefGoogle Scholar
  42. Wilder-Smith A, Gubler DJ (2008) Geographic expansion of dengue: the impact of international travel. Med Clin North Am 92:1377–1390CrossRefGoogle Scholar
  43. Wu X, Magpantay FMG, Wu J, Zou X (2015) Stage-structured population systems with temporally periodic delay. Math Methods Appl Sci 38:3464–3481MathSciNetCrossRefMATHGoogle Scholar
  44. Yang HM, Macoris MLG, Galvani KC, Andrighetti MTM, Wanderley DMV (2009) Assessing the effects of temperature on the population of Aedes aegypti, the vector of dengue. Epidemiol Infect 137:1188–1202CrossRefGoogle Scholar
  45. Zhao X-Q (2017a) Basic reproduction ratios for periodic compartmental models with time delay. J Dyn Differ Equ 29:67–82MathSciNetCrossRefMATHGoogle Scholar
  46. Zhao X-Q (2017b) Dynamical systems in population biology, 2nd edn. Springer, New York, pp 285–315Google Scholar

Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of Western OntarioLondonCanada

Personalised recommendations