A Mathematical Model for the Population Dynamics of Malaria with a Temperature Dependent Control

  • A. Nwankwo
  • D. OkuonghaeEmail author
Original Research


In this work, we develop a mathematical model for malaria transmission that investigates the impact of temperature on the efficacy of ITNs. The disease free state of the autonomous version of the formulated model is seen to be globally asymptotically stable in the absence of disease induced mortality when the associated reproduction number is less than unity. Also, when the associated reproduction number of the non-autonomous model is greater than unity, the disease persist in the population. Sensitivity and uncertainty analysis of the autonomous model, using the associated reproduction number, number of infected humans and vectors as response functions, show that the probability of transmission from human to vectors (\(\beta _{hv}\)), maximum biting rate of vectors (\(b_{max}\)), vector carrying capacity (\(k_v\)) as well as ITN efficacy and coverage (\(\varepsilon _b\) and \(b_o\)) have the most impact on all three response functions. In the non-autonomous model, for temperature values in the range \([18\pm 4-32\pm 4]\,^\circ \)C and using the number of infected humans and vectors as response functions, it can be seen that the bed-net coverage lost its significance for temperature values in the range \([22\pm 4-30\pm 4]\,^\circ \)C and \([22\pm 4-26\pm 4] \,^\circ \)C respectively. Numerical simulations of the time-averaged reproduction number for the case where the efficacy of the bed-net is temperature-dependent showed that temperature values in the range [22–27] \(\,^\circ \)C effects the efficacy of the bed-nets most. Numerical simulations of the non-autonomous model for the case where ITN efficacy is temperature-dependent, we determine temperature ranges that make for increase in the effectiveness of the bed-nets in the fight against malaria.


Malaria Non-autonomous model Efficacy (ITNs) Global stability Sensitivity analysis 



  1. 1.
    Agusto, F.B., Gumel, A.B., Parham, P.E.: Qualitative assessment of the role of temperature variation on malaria transmission dynamics. J. Biol. Syst. 24(4), 1–34 (2015)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Agusto, F.B., Del Valle, S.Y., Blayneh, K.W., Ngonghala, C.N., Goncalvese, M.J., Li, N., Zhao, R., Gong, H.: The impact of bed-net use on malaria prevalence. J. Theor. Biol. 320, 58–65 (2013)CrossRefzbMATHGoogle Scholar
  3. 3.
    Beck-Johnson, L.M., Nelson, W.A., Paaijmans, K.P., Read, A.F., Thomas, M.B., Bjrnstad, O.N.: The effect of temperature on anopheles mosquito population dynamics and the potential for malaria transmission. PLoS One 8(11), e79276 (2013). CrossRefGoogle Scholar
  4. 4.
    Birget, P.L.G., Koella, J.C.: An epidemiological model of the effects of insecticide-treated bed nets on malaria transmission. PLoS One 10(12), e0144173 (2015). CrossRefGoogle Scholar
  5. 5.
    Blower, S.M., Dowlatabadi, H.: Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example. Int. Stat. Rev. 2, 229–243 (1994)CrossRefzbMATHGoogle Scholar
  6. 6.
    Castillo-Chavez, C., Song, B.: Dynamical model of tuberclosis and their applications. Math. Biosci. Eng. 1(2), 361–404 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chitnis, N., Hyman, J.M., Cushing, J.M.: Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bull. Math. Biol. 70, 1272–1296 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8., climate: Kisumu. Accessed May 2017
  9. 9.
    Davis, G.J.: Assessing the impact of temperature change on the effectiveness of insecticide-treated nets. Int. J. Agents Technol. 3(3), 35–48 (2011)CrossRefGoogle Scholar
  10. 10.
    Diekmann, O., Heesterbeek, J., Metz, J.: On the definition and the computation of the basic reproduction ratio r0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28, 365–382 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dumont, Y., Chiroleu, F.: Vector control for the chikungunya disease. Math. Biosci. Eng. 7, 105–111 (2010)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Garba, S.M., Gumel, A.B., Abu Bakar, M.R.: Backward bifurcations in dengue transmission dynamics. Math. Biosci. 1, 11–25 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gimba, B., Bala, S.I.: Modeling the impact of bed-net use and treatment on malaria transmission dynamics. Int. Sch. Res. Not. (2017). Google Scholar
  14. 14.
    Glunt, K.D., Oliver, S.V., Hunt, R.H., Paaijmans, K.P.: The impact of temperature on insecticide toxicity against the malaria vectors Anopheles arabiensis and Anopheles funetus. Malar. J. 13, 350 (2018)CrossRefGoogle Scholar
  15. 15.
    Glunt, K.D., Paaijmans, K.P., Read, A.F., Thomas, M.B.: Environmental temperatures significantly change the impact of insecticides measured using WHOPES protocols. Malar. J. 17, 131 (2014)CrossRefGoogle Scholar
  16. 16.
    Glunt, K.D., Blanford, J.I., Paaijmans, K.P.: Chemicals, climate, and control: increasing the effectiveness of malaria vector control tools by considering relevant temperatures. PLoS Pathog. 9(10), e1003602 (2013). CrossRefGoogle Scholar
  17. 17.
    Gumel, A.B.: Causes of backward bifurcation in some epidemiological models. J. Math. Anal. Appl. 395, 355–365 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hodjati, M.H., Curtis, C.F.: Effects of permethrin at different temperatures on pyrethroid-resistant and susceptible strains of Anopheles. Med. Vet. Entomol. 13, 415–422 (1999)CrossRefGoogle Scholar
  19. 19.
    Kenya National Bureau of Statistics (SCDIC) Accessed May 2017Google Scholar
  20. 20.
    Khan, H.A.A., Akram, W.: The Effect of Temperature on the Toxicity of Insecticides against Musca domestica L.: Implications for the Effective Management of Diarrhea. PLoS ONE 9(4): e95636. (2014)
  21. 21.
    Lakshmikantham, V., Leela, S.: Differential and Integral Inequalities: Theory and Applications. Academic Press, NewYork (1969)zbMATHGoogle Scholar
  22. 22.
    Lou, Y., Zhao, X.Q.: A climate-based malaria transmission model with structured vector population. SIAM J. Appl. Math 70, 2023–2044 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Magal, P., Zhao, X.Q.: Global attractors and steady states for uniformly persistent dynamical systems. SIAM JSIAM J. Appl. Math. 37, 251–275 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    McLeod, R.G., Brewster, J.F., Gumel, A.B., Slonowsky, D.A.: Sensitivity and uncertainty analyses for a sars model with time-varying inputs and outputs. Math. Biosci. Eng. 3(3), 527–544 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Mordecai, E.A., Paaijmans, K.P., Johnson, L.R., Balzer, C., Ben-Horin, T., Moor, E., McNally, A., Pawar, S., Ryan, S.J., Smith, T.C., Lafferty, K.D.: Optimal temperature for malaria transmission is dramatically lower than previously predicted. Ecol. Lett. 16, 22–30 (2013)CrossRefGoogle Scholar
  26. 26.
    Ngonghala, C.N., Del Valle, S.Y., Zhao, R., Mohammed-Awel, J.: Quantifying the impact of decay in bed-net efficacy on malaria transmission. J. Theor. Biol. 363, 247–261 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ngonghala, C.N., Zhao, R., Mohammed-Awel, J., Prosper, O.: Interplay between insecticide-treated bed-nets and mosquito demography: implications for malaria control. J. Theor. Biol. 397, 179–192 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Okuneye, K., Gumel, A.B.: Analysis of a temperature and rainfall dependent model for malaria transmission Dynamics. Math. Bios. (2015). zbMATHGoogle Scholar
  29. 29.
    Okuonghae, D., Omosigho, S.E.: Analysis of a mathematical model for tuberculosis: What could be done to increase case detection. J. Theor. Biol. 269, 31–45 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Okuonghae, D., Ikhimwin, B.O.: Dynamics of a mathematical model for tuberculosis with variability in susceptibility and disease progressions due to difference in awareness level. Front. Microbiol. 6, 1530 (2016). CrossRefGoogle Scholar
  31. 31.
    Russell, T.L., Govella, N.J., Azizi, S., Drakeley, C.J., Kachur, S.P., Killeen, G.F.: Increased proportions of outdoor feeding among residual malaria vector populations following increased use of insecticide-treated nets in rural Tanzania. Malar. J. 10, 80 (2011)CrossRefGoogle Scholar
  32. 32.
    La Salle, J.P.: The stability of dynamical systems, Regional Conference Series in Applied Mathematics. SIAM Philadephia (1976)Google Scholar
  33. 33.
    Smith, H.L.: Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, p. 41. Soc, Am. Math (1995)Google Scholar
  34. 34.
    Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Walker, P.G.T., Griffin, J.T., Ferguson, N.M., Ghani, A.C.: Estimating the most efficient allocation of interventions to achieve reductions in Plasmodium falciparum malaria burden and transmission in Africa: a modelling study. Lancet Global Health. 4(7), e474–484 (2016)CrossRefGoogle Scholar
  36. 36.
    Wang, W., Zhao, X.-Q.: Threshold dynamics for compartmental epidemic models in periodic environments. Journal of Dyn. Differ. Equ. 20, 699–717 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Whiten, S.R., Peterson, R.K.D.: The Influence of Ambient Temperature on the Susceptibility of Aedes aegypti (Diptera: Culicidae) to the Pyrethroid Insecticide Permethrin. Journal of Medical Entomology 53, 1 (2016)CrossRefGoogle Scholar
  38. 38.
    World Health Organisation WHO: WHO recommended long-lasting insecticidal mosquito nets. World Health Organization, Geneva (2012)Google Scholar
  39. 39.
    World Health Organisation (WHO), 2015. Global Malaria Programme, World Malaria ReportGoogle Scholar
  40. 40.
    Zhang, F., Zhao, X.Q.: A periodic epidemic model in a patchy environment. Journal of Mathematical Analysis and Application 325, 496–516 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Zhao, X.Q.: Dynamical systems in population biology. Springer, NewYork (2003)CrossRefzbMATHGoogle Scholar
  42. 42.
    Zhao, X.Q.: Uniform persistence and periodic co-existence states in infinite-dimensional periodic semiflows with applications. Can. Appl. Math. Q. 3, 473–495 (1995)zbMATHGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BeninBenin CityNigeria

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