Skip to main content

Advertisement

SpringerLink
Log in

Mathematical analysis of the role of repeated exposure on malaria transmission dynamics

  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

This paper presents a deterministic model for assessing the role of repeated exposure on the transmission dynamics of malaria in a human population. Rigorous qualitative analysis of the model, which incorporates three immunity stages, reveals the presence of the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction threshold is less than unity. This phenomenon persists regardless of whether the standard or mass action incidence is used to model the transmission dynamics. It is further shown that the region for backward bifurcation increases with decreasing average life span of mosquitoes. Numerical simulations suggest that this region increases with increasing rate of re-infection of first-time infected individuals. In the absence of repeated exposure (re-infection) and loss of infection-acquired immunity, it is shown, using a non-linear Lyapunov function, that the resulting model with mass action incidence has a globally-asymptotically stable endemic equilibrium when the reproduction threshold exceeds unity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S.J. Aneke, Mathematical modelling of drug resistant malaria parasites and vector population, Mathematical Methods in the Applied Sciences, 90(2002), 385–396.

    MathSciNet  Google Scholar 

  2. J.L. Aron, Acquired immunity dependent upon exposure in an SIRS epidemic model, Math. Biosci., 88(1988), 37–47.

    Article  MATH  MathSciNet  Google Scholar 

  3. J.L. Aron, Mathematical modelling of immunity to malaria, Math. Biosci., 90(1988), 385–396.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Baird, Resurgent malaria at the millennium: control strategies in crisis, Drugs, 59(2000), 719 743.

    Article  Google Scholar 

  5. C. Bowman, A.B. Gumel, P. van den Driessche, J. Wu and H. Zhu, A mathematical model for assessing control strategies against West Nile virus, Bull. of Math. Bio., 67(2005), 1107–1133.

    Article  Google Scholar 

  6. J. Carr, “Applications Centre Manifold Theory”, Springer-Verlag, New York, 1981.

    MATH  Google Scholar 

  7. C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 2(2004), 361–404.

    MathSciNet  Google Scholar 

  8. C. Castillo-Chavez, K. Cooke, W. Huang and S.A. Levin, Results on the dynamics for models for the sexual transmission of the human immunodeficiency virus, Appl. Math. Letters, 2(1989), 327–331.

    Article  MATH  Google Scholar 

  9. C. Castillo-Chavez, K. Cooke, W. Huang and S.A. Levin, The role of long incubation periods in the dynamics of HIV/AIDS.part 2: Multiple group models, In Carlos Castillo-Chavez, ed., Mathematical and Statistical Approaches to AIDS Epidemiology, Lecture Notes in Biomathematics, Springer-Verlag. 83(1989), 200–217.

  10. N. Chitnis, J.M. Cushing and J.M. Hyman, Bifurcation analysis of a mathematical model for malaria transmission, SIAM J. of Appl. Math., 67(2006), 24–45.

    Article  MATH  MathSciNet  Google Scholar 

  11. F.A.B. Coutinho, M.N. Burattini, L.F. Lopez and E. Massad, An approximate threshold condition for non-autonomous system: an application to a vector-borne infection, Mathematics and Computers in Simulation, 70(2005), 149–158.

    Article  MATH  MathSciNet  Google Scholar 

  12. K. Dietz, L. Molineaux and A. Thomas, A malaria model tested in the African Savannah, Bull. World Health Association, 50(1974), 347–357.

    Google Scholar 

  13. E.H. Elbasha and A.B. Gumel, Theoretical assessment of public health impact of imperfect Prophylactic HIV-1 vaccines with therapeutic benefits, Bull. Math. Biol., 68(2006), 577–614.

    Article  Google Scholar 

  14. Z. Feng, C. Castillo-Chavez and F. Capurro, A model for tuberculosis with exogeneous reinfection, Theor. Pop. Biol., 57(2000), 235–247.

    Article  MATH  Google Scholar 

  15. S.M. Garba, A.B. Gumel and M.R. Abu Bakar, Backward bifurcation in dengue transmission dynamics, Math. Biosci., 215(2008), 11–25.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Gemperli, P. Vounatsou, N. Sogoba and T. Smith, Malaria mapping using transmission models: application to survey data, American Journal of Epidemiology, 163(2006), 289–297.

    Article  Google Scholar 

  17. A.K. Ghosh, J. Chattppadhyay and P.K. Tapaswi, Immunity boosted by low exposure to infection in an SIRS model, Ecological Modelling, 87(1996), 227–233.

    Article  Google Scholar 

  18. K.P. Hadeler and C. Castillo-Chavez, A core group model for disease transmission, Math. Biosciences, 128(1995), 41–55.

    Article  MATH  Google Scholar 

  19. J.K. Hale, “Ordinary Differential Equations”, John Wiley and Sons, New York, 1969.

    MATH  Google Scholar 

  20. H.M. Hethcote, The Mathematics of infectious diseases, SIAM Rev., 42(2000), 599–963.

    Article  MATH  MathSciNet  Google Scholar 

  21. P. Hviid, Natural acquired immunity to Plasmodium falciparum in Africa, Acta Tropica, 95(2005), 265–269.

    Article  Google Scholar 

  22. H. Ishikawa, A. Ishi, N. Nagai, H. Ohmae, H. Masakazu, S. Suguri and J. Leafasia, A mathematical model for the transmission of Plasmodium vivax malaria, Parasitology International, 52(2003), 81–93.

    Article  Google Scholar 

  23. J.C. Koella, and R. Antia, Epidemiological models for the spread of anti-malarial resistance, Malaria Journal, 2(2003), 3.

    Article  Google Scholar 

  24. J.C. Koella, On the use of mathematical models of malaria transmission, Acta Tropica, 49(1991), 1–25.

    Article  Google Scholar 

  25. C. Kribs-zaleta, and J. Valesco-Hernandez, A simple vaccination model with multiple endemic state, Math. Biosci., 164(2000), 183–201.

    Article  MATH  Google Scholar 

  26. V. Lakshmikantham, S. Leela and A.A. Martynyuk, “Stability Analysis of Nonlinear Systems”, Marcel Dekker Inc., New York and Basel, 1989.

    MATH  Google Scholar 

  27. M. Lipsitch and M.B. Murray, Multiple equilibria: tuberculosis transmission require unrealistic assumptions, Theor. Popul. Biol., 63(2003), 169–170.

    Article  Google Scholar 

  28. G. Macdonald, “The Epidemiology and Control of Malaria”, Oxford University Press, London, 1957.

    Google Scholar 

  29. Z. Mukandavire, A.B. Gumel, W. Garira and J.M. Tchuenche, Mathematical analysis of a model for HIV-malaria co-infection, Mathematical Biosciences and Engineering, (2008).

  30. J. Nedelman, Inoculation and recovery rates in the malaria model of Dietz, Molineaux and Thomas, Mathematical Biosciences, 69(1984), 209–233.

    Article  MATH  MathSciNet  Google Scholar 

  31. J. Nedelman, Some new thoughts about some old malaria modelsintroductory review, Mathematical Biosciences, 73(1985), 159–182.

    Article  MATH  MathSciNet  Google Scholar 

  32. Pan American Health Organization. Annual malaria cases and deaths in the Americas 1998–2006, http://www.paho.org/English/AD/DPC/CD/mal-cases-deaths-1998–2006.pdf (accessed August, 2008).

  33. A.K. Rowe and R.W. Steketee, Prediction of the impact of malaria control efforts on all-cause child mortality in Sub-Saharan Africa, Am. J. Trop. Med. Hyg., 77(2007), 48–55.

    Google Scholar 

  34. K.S. Sharma, R. Chattopadhyay, K. Chakrabarti, S. Pati and C.E. Chitnis, Epidemiology of malaria transmission and development of natural immunity in a malaria endemic village, San Dulakudar, in Orissa state, India, American Journal of Tropical Medicine and Hygiene, 71(2004), 457–465.

    Google Scholar 

  35. O. Sharomi, C.N. Podder, A.B. Gumel, E.H. Elbasha, and J. Watmough, Role of incidence function in vaccine-induced backward bifurcation in some HIV models, Math. Biosci., 210(2007), 436–463.

    Article  MATH  MathSciNet  Google Scholar 

  36. O. Sharomi, C.N. Podder and A.B. Gumel, Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment, Math. Biosci and Engineering, 6(2008), 145–174.

    Article  MathSciNet  Google Scholar 

  37. C.P. Simon and J.A. Jacquez, Reproduction numbers and the stability of equilibrium of SI models for heterogeneous populations, SIAM J. Appl. Math., 52(1992), 541–576.

    Article  MATH  MathSciNet  Google Scholar 

  38. J. Tumwiine, J.Y.T. Mugisha and L.S. Luboobi, On oscillatory pattern of malaria dynamics in a population with temporary immunity, Comp. and Math. Methods of Medicine, 8(2007), 191–203.

    Article  MATH  MathSciNet  Google Scholar 

  39. P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180(2002), 29–48.

    Article  MATH  MathSciNet  Google Scholar 

  40. World Health Organization, World Malaria Report, (2002). http://www.rbm.who.int. (accessed July, 2008).

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada

    Ashrafi M. Niger & Abba B. Gumel

Authors
  1. Ashrafi M. Niger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abba B. Gumel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abba B. Gumel.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Niger, A.M., Gumel, A.B. Mathematical analysis of the role of repeated exposure on malaria transmission dynamics. Differ Equ Dyn Syst 16, 251–287 (2008). https://doi.org/10.1007/s12591-008-0015-1

Download citation

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-008-0015-1

MSC

Keywords

Search

Navigation