International Journal of Systems Science and Applied Mathematics

| Peer-Reviewed |

Stability and Time-Scale Analysis of Malaria Transmission in Human-Mosquito Population

Received: 09 September 2016    Accepted: 04 November 2016    Published: 02 December 2016
Views:       Downloads:

Share This Article

Abstract

More realistic human-mosquito mathematical model in which re-infected asymptomatic humans are considered is presented. The Next Generation Matrix technique is used to construct epidemiological threshold known as the reproduction number. Locally and globally asymptotically stable disease-free equilibrium conditions for the model are established. Possible time-scale of events for model transition from non-endemic to endemic is analyzed. Results show that the buildup of the latent asymptomatic humans at steady state is the main dynamics of malaria in the endemic region.

DOI 10.11648/j.ijssam.20170201.11
Published in International Journal of Systems Science and Applied Mathematics (Volume 2, Issue 1, January 2017)
Page(s) 1-9
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Malaria Transmission, Stability Analysis, Mathematical Modeling

References
[1] Malaria." WHO. N. p., n.d. Web. 06 Sept. 2015.
[2] K. A. Cullen and P. M Arguin. "Malaria Surveillance--United States, 2012." Morbidity And Mortality Weekly Report. Surveillance Summaries (Washington, D. C.: 2002) 63. 12 (2014): 1-22. MEDLINE with Full Text. Web. 6 Sept. 2015.
[3] M. Dako-Gyeke and H. M. Kofie. "Factors Influencing Prevention And Control Of Malaria Among Pregnant Women Resident In Urban Slums, Southern Ghana." African Journal Of Reproductive Health 19.1 (2015): 44-53. MEDLINE with Full Text. Web. 7 Sept. 2015.
[4] G. H. Bledsoe, Malaria primer for clinicians in the United States, South. Med. J., 12(1998): pp. 1197-1204.
[5] J. D. Charlwood, T. Smith, P. F. Billingsley, W. Takken, E. Lyimo and J. Meuwissen, Survival and infection probabilities of anthropophagic anophelines from an area of high prevalence of Plasmodium falciparum in humans, Bull. Entomol. Res., 87(1997): pp. 445-453.
[6] http://www.niaid.nih.gov/topics/malaria/pages/lifecycle.aspx accessed in September 2016.
[7] G. Killeen, U. Fillinger, I. Kiche, L. Gouagna and B. Knols. Eradication of Anopheles gambiae from Brazil: lessons for malaria control in Africa?, Lancet Infect Dis., 10(2002): pp. 618-627.
[8] C. R. Newton, T. E. Taylor, R. O. Whitten. Pathophysiology of fatal falciparum malaria in African children. Am J Trop Med Hyg 58 (1998): 673-683.
[9] J. Sachs and P. Malaney. 2002. The economic and social burden of malaria. Nature 415: 680-685.
[10] P. Brown, Trials and tribulations of a malaria vaccine, New Scientist (1991) 18-19.
[11] H. M. Yang, Malaria transmission model for different levels of acquired immunity and temperature dependent parameters (vector). J. Public Health, 34 (2000), 223-231.
[12] B. Ogutu, A. B. Tiono, M. Makanga, Z. Premji, A. D. Gbado, D. Ubben, A. C. Marrast, O. Gaye. Treatment of asymptomatic carriers with artemether-lumefantrine: an opportunity to reduce the burden of malaria. Malaria Journal, [online]. [viewed 09/11/2016].
[13] Population Reference Bureau, World population data, 2015.
[14] R. Aguas, L. J. White, R. W. Snow and M. G. Gomes. Prospects for malaria eradication in Subsaharan Africa”. PLoS ONE3 (3) 2008.
[15] L. Molineaux, G. R. Shidrawi, J. L. Clarke, J. R. Boulzaguet, and T. S. Ashkar, Assessment of insecticidal impact on the malaria mosquitoes vectorial capacity, from data on the man-biting rate and age-composition. Bulletin of the World Health Organisation, 57 (1979), 265-274.
[16] J. Nedelman. “Inoculation and recovery rates in the malaria model of Dietz”. Molineaux and Thomas Mathematical Biosciences, 69(1984), 209-233.
[17] N. Maire, T. Smith, A. Ross, S. Owusu-Agyei, K. Dietz. A model for natural immunity to asexual blood stages of Plasmodium Falciparum malaria in endemic areas. Am. J Trop. Med. Hyg., 75 (2006), 19-31.
[18] J. A. Filipe, E. M. Riley, C. J. Drakelgy, C. J. Sutherland, A. C. Cthani. “Determination of the processes driving the acquisition of immunity to malaria using a mathematical transmission model. PLoS Comput. Biol., 3(12) (2007), 2567-2579.
[19] T. Boysema and C. Drakeley. Epidemiology and Infectivity of Plasmodium Falciparum and Plasmodium Vivax gametocytes in relation to malaria control and elimination. Clin. Microbiol. Rev., 24 (2011), 377-410.
[20] N. Chitnis, 2002. Using Mathematical Models in Controlling the Spread of Malaria. Unpublished thesis (PhD), University of Arizona, Tucson, USA.
[21] R. M. Anderson and R. M. Mag. Infectious diseases of humans: Dynamics and control, Oxford University Press, Oxford, 1991
[22] C. Emmanuel and U. Odo, Current trend in malarial Chemotherapy. Academic journal, 7(4) (2008), 350-355.
[23] K. Annan and M. Fisher. Stability Conditions of Chagas-HIV Co-infection Disease Model Using the Next Generation Method. Applied Mathematical Sciences, Vol. 7, No. 57(2013), 2815-2832.
[24] J. P. LaSalle, Stability theory for ordinary differential equations. J. Di_erential Equations 4 (1968), 57-65.
[25] L. J. S. Allen, An Introduction to Mathematical Biology, Prentice Hall, Upper Saddle River, NJ, 2007.
Author Information
  • School of Science and Technology, Georgia Gwinnett College, Lawrenceville, Georgia, USA

  • School of Science and Technology, Georgia Gwinnett College, Lawrenceville, Georgia, USA

Cite This Article
  • APA Style

    Kodwo Annan, Cedrick Dizala Mukinay. (2016). Stability and Time-Scale Analysis of Malaria Transmission in Human-Mosquito Population. International Journal of Systems Science and Applied Mathematics, 2(1), 1-9. https://doi.org/10.11648/j.ijssam.20170201.11

    Copy | Download

    ACS Style

    Kodwo Annan; Cedrick Dizala Mukinay. Stability and Time-Scale Analysis of Malaria Transmission in Human-Mosquito Population. Int. J. Syst. Sci. Appl. Math. 2016, 2(1), 1-9. doi: 10.11648/j.ijssam.20170201.11

    Copy | Download

    AMA Style

    Kodwo Annan, Cedrick Dizala Mukinay. Stability and Time-Scale Analysis of Malaria Transmission in Human-Mosquito Population. Int J Syst Sci Appl Math. 2016;2(1):1-9. doi: 10.11648/j.ijssam.20170201.11

    Copy | Download

  • @article{10.11648/j.ijssam.20170201.11,
      author = {Kodwo Annan and Cedrick Dizala Mukinay},
      title = {Stability and Time-Scale Analysis of Malaria Transmission in Human-Mosquito Population},
      journal = {International Journal of Systems Science and Applied Mathematics},
      volume = {2},
      number = {1},
      pages = {1-9},
      doi = {10.11648/j.ijssam.20170201.11},
      url = {https://doi.org/10.11648/j.ijssam.20170201.11},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ijssam.20170201.11},
      abstract = {More realistic human-mosquito mathematical model in which re-infected asymptomatic humans are considered is presented. The Next Generation Matrix technique is used to construct epidemiological threshold known as the reproduction number. Locally and globally asymptotically stable disease-free equilibrium conditions for the model are established. Possible time-scale of events for model transition from non-endemic to endemic is analyzed. Results show that the buildup of the latent asymptomatic humans at steady state is the main dynamics of malaria in the endemic region.},
     year = {2016}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Stability and Time-Scale Analysis of Malaria Transmission in Human-Mosquito Population
    AU  - Kodwo Annan
    AU  - Cedrick Dizala Mukinay
    Y1  - 2016/12/02
    PY  - 2016
    N1  - https://doi.org/10.11648/j.ijssam.20170201.11
    DO  - 10.11648/j.ijssam.20170201.11
    T2  - International Journal of Systems Science and Applied Mathematics
    JF  - International Journal of Systems Science and Applied Mathematics
    JO  - International Journal of Systems Science and Applied Mathematics
    SP  - 1
    EP  - 9
    PB  - Science Publishing Group
    SN  - 2575-5803
    UR  - https://doi.org/10.11648/j.ijssam.20170201.11
    AB  - More realistic human-mosquito mathematical model in which re-infected asymptomatic humans are considered is presented. The Next Generation Matrix technique is used to construct epidemiological threshold known as the reproduction number. Locally and globally asymptotically stable disease-free equilibrium conditions for the model are established. Possible time-scale of events for model transition from non-endemic to endemic is analyzed. Results show that the buildup of the latent asymptomatic humans at steady state is the main dynamics of malaria in the endemic region.
    VL  - 2
    IS  - 1
    ER  - 

    Copy | Download

  • Sections