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

Kodwo Annan; Cedrick Dizala Mukinay

doi:doi:10.11648/j.ijssam.20170201.11

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

Kodwo Annan, Cedrick Dizala Mukinay

Published in International Journal of Systems Science and Applied Mathematics (Volume 2, Issue 1)

Received: 9 September 2016 Accepted: 4 November 2016 Published: 2 December 2016

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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.

Published in	International Journal of Systems Science and Applied Mathematics (Volume 2, Issue 1)
DOI	10.11648/j.ijssam.20170201.11
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), 2016. 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.

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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

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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

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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

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@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://article.sciencepublishinggroup.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}
}

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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  -

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Author Information

Kodwo Annan

School of Science and Technology, Georgia Gwinnett College, Lawrenceville, Georgia, USA
Cedrick Dizala Mukinay

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

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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

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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

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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

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@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://article.sciencepublishinggroup.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  -

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