A Delayed Mathematical Model to break the life cycle of Anopheles Mosquito

Muhammad A. Yau, Bootan Rahman


In this paper, we propose a delayed mathematical model to break the life cycle of anopheles mosquito at the larva stage by incorporating a time delay τ at the larva stage that accounts for the period of growth or development to pupa. We prove local stability of the system’s equilibrium and find the critical values for Hopf bifurcation to occur. Also, we find that the system’s equilibrium undergoes stability switching from stable to periodic to unstable and vice versa when the time delay τ crosses the imaginary axis from the left half plane to the right half plane in the (Re,Im) plane. Finally, we perform some numerical simulations and the results agree well with the analytical analysis. This is the first time such a model is proposed.


Delayed model; Anopheles mosquito; Malaria Control; Hopf bifurcation; Larva; Stability analysis

Full Text:



J. L. Aron, Mathematical Modeling of Immunity to Malaria, Mathematical Biosciences 90, 385-396 (1988). 32

T. J. N. Bailey, The Mathematical Theory of Infectious Diseases and its Application, (Griffin, London, 1975), 2nd edition.

L. A. Baton and L. C. Ranford-Cartwright, Spreading the Seeds of Million-murdering Death: Metamorphoses of Malaria in the Mosquito, Trends in Parasitology 21, 573-580 (2005).

Bates, M., The natural history of mosquitoes, The Macmillan Company, New York. pp 379 (1949).

Beltrami, E, Mathematics for dynamic modeling, Academic Press. N.Y (1989).

C. C. Carlos, Malaria: In Infectious Diseases, (J. B. Lipincolt Company, Philadelphia, (1989).

Coetzee M, Distribution of the African malaria vectors of the Anopheles gambiae complex, American Journal of tropical medicine and hygiene, 70(2):103-104 (2004).

Davidson, G, Estimation of the population of Anopheline Mosquito, nature,174:792-793 (1954).

K. Dietz, L. Molineaux, and A. Thomas, A Malaria Model Tested in the Africa Savanna, Bull of World Hearlth Organization 50, 347-357 (1974). 32

C. Faraj, S. Ouahbi, E. Adlaoui, D. Boccolini, R. Romi, and R. E. Aou ad, Risque de Reemergence du Paludisme au Maroc. Etude de la Capacite Vectorielle d’Anopheles Labranchiae dans une Zone Rizicole au Nord du Pays, Parasite 15, 605-610 (2008).

K. L. Gage, T. R. Burkot, R. J. Eisen, and E. B. Hayes, Climate and Vectorborne Diseases, American Journal of Preventive Medicine 35 (5), 43-450 (2008).

J. L. Gallup and J. D. Sachs, The Economic Burden of Malaria, American Journal of Tropical Medicine and Hygiene 64, 85-96 (2001).

Garret-Jones, C, Prognosis for interruption of malaria transmission through assessment of the mosquitoes vectorial capacity, Nature. 204: 1173-1175 (1964).

H. M. Giles and D. A. Warrel, Bruce-Chwatt’s Essential Malariology, (Hodder Arnold Publication, London, 2002), 4th edition. 31, 33

Gillies M.T. Coetzee A, A supplement to the anophelinae of Africa south of Sahara, The South African Institute of Medical Research, Johannesburg. South Africa pp 55. http://www.who.int/topics/malaria/en/ (1987).

T. Hayakawa, R. Culleton, H. Otani, T. Horii, and K. Tanabe, Big Bang in the Evolution of Extant Malaria Parasites, Molecular Biology and Evolution 25(10), 2233-2239, (2008).

Z. Hawass, Y. Z. Gad, S. Ismail, R. Khairat, D. Fathalla, N. Hasan, A. Ahmed, H. Elleithy, M. Ball, F. Gaballah, S. Wasef, M. Fateen, H. Amer, P. Gostner, A. Selim, A. Zink, and C. M. Pusch, Ancestry and Pathology in King Tutankhamun’s Family, The Journal of the American Medical Association 303, 638-647 (2010).

G. Macdonald, The Epidemiology and Control of Malaria, (Oxford University Press, London, 1957). 30

G. Macdonald, The Analysis of Infection Rates in Diseases in Which Super-infection Occurs, Tropical Diseases Bulletin 47, 907-915 (1950). 32

P. Martens, R. S. Kovats, S. Nijhof, P. de Vries, M. J. T. Levermore, D. J. Bradley, J. Cox, and A. J. McMichael, Climate Change and Future Populations at Risk of Malaria, Global Environmental Change 9, S89-S107 (1999).

K. Marsh, Malaria Disaster in Africa, Lancet 352, 924-925. (1998).

Muhammad A. Yau, A Mathematical Model to Break the Life Cycle of Anopheles Mosquito, Shiraz E Medical Journal, Vol.12, No.3, 2011.

G.A.NgwaandW.S.Shu, A mathematical Model for Endemic Malariawith Variable Human and Mosquito Populations, Mathematical and Computer Modelling 32, 747-763 (2000).

S. Nourridine, M. I. Teboh-Ewungkem, and G. A. Ngwa, A Mathematical Model of the Population Dynamics of Disease Transmitting Vectors with Spatial Consideration, Journal of Biological Dynamics 1751-3758 (2011).

S. M. Rich, F. H. Leendertz, G. Xu, M. LeBreton, C. F. Djoko, M. N. Aminake, E. E. Takang, J. L. D. Diffo, B. L. Pike, B. M. Rosenthal, P. Formenty, C. Boesch, F. J. Ayala, and N. D. Wolfe, The Origin of Malignant Malaria, Proceedings of the National Academy of Sciences 106 (35), 14902-14907 (2009).

R. Ross, The Prevention of Malaria, (John Murray, London, 1911). 31

S. G. Staedke, E. W. Nottingham, J. Cox, M. R. Kamya, P. J. Rosenthal, and G. Dorsey, Proximity to Mosquito Breeding Sites as a Risk Factor for Clinical Malaria Episodes in an Urban Cohort of Ugandan Children, American Journal of Tropical Medicine and Hygiene 69 (3), 244-246 (2003).

I. W. Sherman, Malaria: Parasite Biology, Pathogenesis, and Protection, (ASM Press,1998) 30.

S. Sainz-Elipe, J. Latorre, R. Escosa, M. Masia, M. Fuentes, S. Mas-Coma, and M. Bargues, Malaria Resurgence Risk in Southern Europe: Climate Assessment in an HistoricallyEendemic Area of Rice Fields at the Mediterranean Shore of Spain, Malaria Journal 9, 221-236 (2010).

J. Sachs and P. Malaney, The Economic and Social Burden of Malaria, Nature 415, 680-685 (2002).

M. I. Teboh-Ewungkem and T. Yuster, A Within-vector Mathematical Model of Plasmodium Falciparum and Implications of Incomplete Fertilization on Optimal Gametocyte Sex Ratio, Journal of Theoretical Biology 264, 273-86 (2010).

M. I. Teboh-Ewungkem, Malaria Control: The Role of Local Communities as Seen through a Mathematical Model in a Changing Population-Cameroon, Chapter 4, 103-140, in Advances in Disease Epidemiology (Nova Science Publishers, 2009). 31

J. N. Wilford, Malaria is a Likely Killer in King Tuts Post-mortem, Technical Report 16, The New York Times, accessed March 2011.

World Health Organisation, The World Malaria Report, WHO Press, acessed March 2011 (2010).

World Health Organisation, The World Health Report, WHO Press (2009).

World Health Organization, 10 Facts on Malaria, WHO Press (2009).

S. Wyborny, Parasites: The Malaria Parasite, (KidHaven Press, 2005), 1st


XiaoLing Li, GuangPing Hu, Stability and Hopf bifurcation on a neuron network with discrete and distributed delays, Appl. Math. Sci., 2077-2084 (42), 2011.

G. Zhou, N. Minakawa, A. K. Githeko, and G. Yan, Association Between Climate Variability and Malaria Epidemics in the East African Highlands, Proceedings of the National Academy of Sciences 101, 2375-2380 (2004).

J. A. Patz and S. H. Olson, Malaria Risk and Temperature: Influences from Global Climate Change and Local Land use Practices, Proceedings of the National Academy of Sciences 103, 5635-5636 (2006).

DOI: http://dx.doi.org/10.23755/rm.v31i0.319


  • There are currently no refbacks.

Copyright (c) 2017 Muhammad A. Yau, Bootan Rahman

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

Ratio Mathematica - Journal of Mathematics, Statistics, and Applications. ISSN 1592-7415; e-ISSN 2282-8214.