10.30495/ijm2c.2022.1936520.1225

Stability Analysis of a Malaria Transmission Model for the Effect of Infected Immigrants with Temperature and Rainfall Dependent Parameters

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  • Victor Yiga*1,
  • Hasifa Nampala2,
  • Julius Tumwiine3,
  1. Department of Mathematics, Faculty of science, Kyambogo University
  2. Department of Mathematics, Faculty of Science, Kyambogo University, Kyambogo, Uganda
  3. Department of Mathematics, Mbarara University of Science and Technology, Mbarara, Uganda

Received: 27-07-2021

Accepted: 24-04-2022

Published in Issue 30-06-2022

Copyright (c) 2024 International Journal of Mathematical Modeling & Computations

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Yiga, V., Nampala, H., & Tumwiine, J. (2022). Stability Analysis of a Malaria Transmission Model for the Effect of Infected Immigrants with Temperature and Rainfall Dependent Parameters. International Journal of Mathematical Modelling & Computations, 12(2), 115-130. https://doi.org/10.30495/ijm2c.2022.1936520.1225
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Abstract

A human host-mosquito vector model for transmission of malaria with infow of infected immigrants is formulated. The mosquito population includes aquatic stages (eggs, larvae, and pupae) and mature stages which have highly temperature and rainfall dependent life cycles. Model analysis reveals that the model only attains two (2) endemic equilibria; one in absence of the vector population and the other in presence of the vector population. The endemic equilibrium without the mosquito vector population is unstable. The endemic equilibrium with the vector population is locally stable and globally unstable. Numerical simulations of the model reveal that the proportion of infected humans introduced into the community does not significantly change the pattern of malaria transmission.
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References

  1. A. Abdelrazec and A. B. Gumel, Mathematical assessment of the role of temperature and rainfall
  2. on mosquito population dynamics, J. Math. Biol., 74 (2017) 1351–1395.
  3. F. B. Agusto, A. B. Gumel and P. E. Parham, Qualitative assessment of the role of temperature
  4. variations on malaria transmission dynamics, J. Biol. Syst., 23 (4) (2015) 1550030.
  5. S. Altizer, A. Dobson, P. Hosseini, P. Hudson, M. Pascual and P. Rohani, Seasonality and the
  6. dynamics of infectious diseases, Ecol. Lett., 9 (2006) 467–484.
  7. M.N. Bayoh and S. W. Lindsay, Effect of temperature on the development of the aquatic stages of
  8. Anopheles gambiae sensu stricto (Diptera: Culicidae), Bull. Entomol. Res., 93 (2003) 375–381.
  9. L. M. Beck-Johnson, W. A. Nelson, K. P. Paaijmans, A. F. Read, M. B. Thomas and O. N. Bjornstad,
  10. The importance of temperature fluctuations in understanding mosquito population dynamics and
  11. malaria risk, R. Soc. Open Sci., 4 (2017) 160969.
  12. L. M. Beck-Johnson, W. A. Nelson, K. P. Paaijmans, A. F. Read, M. B. Thomas and O. N. Bjornstad,
  13. The effect of temperature on Anopheles mosquito population dynamics and the potential for malaria
  14. transmission, PLoS One, 8 (2013) e79276.
  15. K. Blayneh, Y. Cao and H. D. Kwon, Optimal control of vector-borne diseases: treatment and
  16. prevention, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009) 587–618.
  17. A. Bomblies, Modeling the role of rainfall patterns in seasonal malaria transmission, Clim. Change,
  18. (2012) 673–685.
  19. F. Brauer and P. van den Driessche, Models for transmission of disease with immigration of infectives,
  20. Math. Biosci., 171 (2001) 141–154.
  21. C. O.Buckee, A. J. Tatem and C. J. E. Metcalf, Seasonal population movements and the surveillance
  22. and control of infectious diseases, Trends Parasitol., 33 (2017) 10–20.
  23. C. Cosner, J. C. Beier, R. S. Cantrell, D. Impoinvil, L. Kapitanski, M. D. Potts, A.Troyo and S.
  24. Ruan, The effects of human movement on the persistence of vector-borne diseases, J. Theor. Biol.,
  25. (2009) 550–560.V. Yiga et al./ IJM2C, 12 - 02 (2022) 115-130. 129
  26. A. Egbendewe-Mondzozo, M. Musumba, B. A. McCarl and X. Wu, Climate change and vector-borne
  27. diseases: an economic impact analysis of malaria in Africa, Int. J. Environ. Res. Public Health, 8
  28. (2011) 913–930.
  29. A. F¨arnert, K. Wyss, S. Dashti and P. Naucler, Duration of residency in a non-endemic area and
  30. risk of severe malaria in African immigrants, Clin. Microbiol. Infect., 21 (2015) 494–501.
  31. J. T. Griffin, T. D. Hollingsworth, H. Reyburn, C. J. Drakeley E. M. Riley and A. C. Ghani, Gradual
  32. acquisition of immunity to severe malaria with increasing exposure, Proc. R. Soc. B., 282 (2015)
  33. T. Ikemoto, Tropical malaria does not mean hot environments, J. Med. Entomol., 45 (2008) 963–969.
  34. T. M. Lunde, K. Diriba, E. Loha, S. Asgeir and B. Lindtjrn. A dynamic model of some malariatransmitting Anopheline mosquitoes of the Afrotropical region. I. Model description and sensitivity
  35. analysis, Malar. J., 12 (2013) 28.
  36. T. M. Lunde, M. N. Bayoh and B. Lindtjrn, How malaria models relate temperature to malaria
  37. transmission, Parasit. Vectors, 6 (2013) 20.
  38. C. L. Lyons, M. Coetzee and S. L. Chown, Stable and fluctuating temperature effects on the development rate and survival of two malaria vectors, Anopheles arabiensis and Anopheles funestus,
  39. Parasit. Vectors, 6 (2013) 104.
  40. O.S. Makinde and G. J. Abiodun, The impact of rainfall and temperature on malaria dynamics in
  41. the KwaZulu-Natal province, South Africa, Commun. Stat. Case Stud. Data Anal. Appl., 6 (2020)
  42. –108.
  43. P. Martens and L. Hall, Malaria on the move: human population movement and malaria transmission,
  44. Emerging. Infect. Di., 6 (2000) 103.
  45. C. Mukandavire, G. Musuka, G. Magombedze and Z. Mukandavire, Malaria model with immigration
  46. of infectives and seasonal forcing in transmission, Int. J. of Appl. Math. Comput., 2 (2010) 1–16.
  47. A. Y. Mukhtar, J. B. Munyakazi and R. Ouifki, Assessing the role of climate factors on malaria
  48. transmission dynamics in South Sudan, Math. Biosci., 310 (2019) 13–23.
  49. E. A. Mordecai, Optimal temperature for malaria transmission is dramatically lower than previously
  50. predicted, Ecol. Lett., 16 (2013) 22–30.
  51. E. T. Ngarakana-Gwasira, C. P. Bhunu, M. Masocha and E. Mashonjowa, Assessing the role of
  52. climate change in malaria transmission in Africa, Malar. Res. Treat., 2016 (2016).
  53. B. A. Okech, L. C. Gouagna, E. W. Kabiru, E. Walczak, J. C. Beier, G. Yan and J. I. Githure,
  54. Resistance of early midgut stages of natural Plasmodium falciparum parasites to high temperatures
  55. in experimentally infected Anopheles gambiae (Diptera: Culicidae), J. Parasitol., 90 (2004) 764–768.
  56. K. Okuneye, A. Abdelrazec and A. B. Gumel, Mathematical analysis of a weather-driven model for
  57. the population ecology of mosquitoes, Math. Biosci. Eng., 15 (2018) 57–93.
  58. K. P. Paaijmans, A. F. Read and M. B. Thomas, Understanding the link between malaria risk and
  59. climate, Proc. Natl. Acad Sci U S A, 106 (2009) 13844–13849.
  60. K. P. Paaijmans, M. O. Wandago, A. K. Githeko and W. Takken, Unexpected high losses of Anopheles gambiae larvae due to rainfall, PLoS One, 2 (2007) e1146.
  61. P.E. Parham, D. Pople, C. Christiansen-Jucht, S. Lindsay, W. Hinsley and E. Michael, Modeling the
  62. role of environmental variables on the population dynamics of the malaria vector Anopheles gambiae
  63. sensu stricto, Malar. J., 11 (2012) 271.
  64. D. K. Pindolia, A.J. Garcia, A. Wesolowski, D.L. Smith, C. O. Buckee, A.M. Noor, R. W. Snow and
  65. A. J. Tatem, Human movement data for malaria control and elimination strategic planning, Malar.
  66. J., 11 (2012) 205.
  67. J. M. Reinhold, C. R. Lazzari and C. Lahond`ere, Effects of the environmental temperature on Aedes
  68. aegypti and Aedes albopictus mosquitoes: a review, Insects., 9 (2018) 158.
  69. J. Shaman and J. F. Day, Reproductive phase locking of mosquito populations in response to rainfall
  70. frequency, PLoS One, 2 (2007) e331.
  71. S. T. Stoddard, A. C. Morrison, G. M. Vazquez-Prokopec, V. P. Soldan, T. J. Kochel, U. Kitron, J. P.
  72. Elder and T. W. Scott, The role of human movement in the transmission of vector-borne pathogens,
  73. PLoS. Negl. Trop. Dis., 3 (2009) e481.
  74. D. Sunita and B. Nisha, Stability analysis of a human-mosquito model of malaria with infective
  75. immigrants, Int. J. Math. Comput. sci., 11 (2017) 85–89.
  76. A. Tabbabi, A. A. Alkishe, A. M. Samy, A. Rhim and A. T. Peterson, Malaria in North Africa: A
  77. Review of the Status of Vectors and Parasites, J. Entomol. Sci., 55 (2020) 25–37.
  78. A. Tran, G. l’Ambert, G. Lacour, R. Benot, M. Demarchi, M. Cros, P. Cailly, M. Aubry-Kientz, T.
  79. Balenghien and P. Ezanno, A rainfall and temperature driven abundance model for Aedes albopictus
  80. populations, Int. J. Env. Res. Pub. He., 10 (2013) 1698–1719.
  81. J. Tumwiine, J. Y. T. Mugisha and L. S. Luboobi, A host-vector model for malaria with infective
  82. immigrants, J. Math. Anal. Appl., 361 (2010) 139–149.
  83. X. N. Wang and X. Q. Zhao, A periodic vector-bias malaria model with incubation period, SIAM J.
  84. Appl. Math., 77 (2017) 181–201.
  85. A. Wesolowski, N. Eagle, A. J. Tatem, D. L. Smith, A. M. Noor, R. W. Snow and C. O. Buckee,
  86. Quantifying the impact of human mobility on malaria, Science, 338 (2012) 267–270.
  87. M. T. White, J. T. Griffin, T. S. Churcher, N. M. Ferguson, M. G. Basanez and A. C. Ghani,
  88. Modelling the impact of vector control interventions on Anopheles gambiae population dynamics,
  89. Parasit. Vectors, 4 (2011) 153.
  90. A. Y. N. Win, T. M. Maung, K. T. Wai, T. Oo, A. Thi, R. Tipmontree, N. Soonthornworasiri, M.
  91. Kengganpanich and J. Kaewkungwal, Understanding malaria treatment-seeking preferences within
  92. the public sector amongst mobile/migrant workers in a malaria elimination scenario: a mixedmethods study, Malar. J., 16 (2017) 462.
  93. World Health Organisation, World Malaria Report, World Health Organisation, Geneva, Switzerland, (2019).
  94. World Health Organisation and UNICEF, Achieving the malaria MDG target: reversing the incidence130 V. Yiga et al./ IJM2C, 12 - 02 (2022) 115-130.
  95. of malaria 2000-2015, World Health Organisation, Geneva, Switzerland, (2015).
  96. T. K. Yamana, A. Bomblies, I. M. Laminou, J. B. Duchemin and E. A. Eltahir, Linking environmental
  97. variability to village-scale malaria transmission using a simple immunity model, Parasit. Vectors, 6
  98. (2013) 226.
  99. V. Yiga, H. Nampala and J. Tumwiine, Analysis of the model on the effect of seasonal factors on
  100. malaria transmission dynamics, J. Appl. Math., 2020 (2020), doi:10.1155/2020/8885558.

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