Modeling the Dynamics of Endemic Malaria Transmission with the Effects of Control Measure
American Journal of Applied Mathematics
Volume 8, Issue 3, June 2020, Pages: 158-170
Received: Dec. 27, 2019;
Accepted: Jun. 3, 2020;
Published: Jun. 17, 2020
Views 43 Downloads 51
Dereje Gutema Edossa, Department of Mathematics, Wollega University, Nekemte, Ethiopia
Alemu Geleta Wedajo, Department of Mathematics, Wollega University, Nekemte, Ethiopia
Purnachandra Rao Koya, Department of Mathematics, Wollega University, Nekemte, Ethiopia
Malaria is an infectious disease caused by Plasmodium parasite and is transmitted among humans through bites of female Anopheles mosquitoes. It is estimated 216 million people suffered from malaria in 2016, with over 400,000 deaths mainly in sub-Saharan Africa. A number of control measures have been put in place: most importantly the insecticide treated net (ITN) and indoor residual sprayings (IRS) of insecticide. Currently, the emergence and spread of resistance in mosquito populations against insecticides is the most common and widely spread .It is also poses a key obstacle to malaria control as well as jeopardizing the effects of the most efficient malaria control interventions. A mathematical model that incorporates the evolution of insecticide resistance and its impact on endemic malaria transmission i.e., effects of indoor residual sprayings (IRS) on the insecticide resistant and sensitive malaria vector strains as a control strategy is incorporated and analyzed. The object of the study is to understand qualitatively the factor that have more influence for the emergence and spread of resistance of malaria vectors against IRS and their impacts on the efficacy of IRS. Based on a Ross-Macdonald derivation of malaria model the effective reproduction number〖 R〗_e isused to assess the effects of IRS in the qualitative analysis of the model. The existence and stability of the disease-free and endemic equilibria of the model are studied. It is established that the malaria can be brought under control as long as R_(e )is kept below the threshold value. Numerical simulations studies are conducted so as to determine the role played by key parameters of the model. The public health implications of the results include: (i) every effort should be taken to minimize the evolution of insecticide resistance due to malaria control interventions failure and (ii) at least a combination of two types of different control measures and followed by rotation of intervention strategies could be more realistic to minimize the number of resistant malaria vector strains and essential in reducing the malaria burden in the community.
Dereje Gutema Edossa,
Alemu Geleta Wedajo,
Purnachandra Rao Koya,
Modeling the Dynamics of Endemic Malaria Transmission with the Effects of Control Measure, American Journal of Applied Mathematics.
Vol. 8, No. 3,
2020, pp. 158-170.
Centers for Disease Control and Prevention. CDC - Malaria (accessed August 8, 2011) http://www.cdc.gov/MALARIA.
Worrall E, Basu S, Hanson K. Is malaria a disease of poverty? A review of the literature. Tropical Medicine & International Health.2005;10:1047-1059
Hawass Z, Gad YZ, Ismail S, Khairat R, Fathalla D, Hasan N, et al. Ancestry and pathology in King Tutankhamun’s family. JAMA.2010;303:638-647. DOI: 10.1001/jama. 2010.121
WHO. World Malaria Report 2017. Geneva, Switzerland: World Health Organization; 2017
Bhatt S, Weiss D, Cameron E, Bisanzio D, Mappin B, Dalrymple U, et al. The effect of malaria control on Plasmodium falciparum in Africa between 2000 and 2015. Nature. 2015; 526:207-211
WHO. Global Plan for Insecticide Resistance Management in Malaria Vectors. Geneva, Switzerland: World Health Organization; 2012. 132 pp. Available from: www.who.int/malaria/vector_control/ivm/gpirm/en/
Davidson G. Insecticide resistance in Anopheles sundaicus. Nature.1957;4598:1333-1335
Gilbert LI, Gill SS. Insect Control: Biological and Synthetic Agents. London, United Kingdom: Academic Press; 2010. 470 p
Ross R. “The Prevention of Malaria”, John Murray, 1911.
Macdonald G., “The Epidemiology and Control of Malaria”, Oxford university press, 1957.
J. L. Aron and R. M. May. The population dynamics of malaria. In The Population Dynamics of Infectious Diseases: Theory and Applications, pages 139–179. Springer, 1982.
S. Gupta, J. Swinton, and R. M. Anderson. Proceedings of the Royal Society of London. Series B: Biological Sciences, 256(1347):231–238, 1994.
J. C. Koella. Acta Tropica, 49(1):1–25, 1991.
G. Macdonald. Proceedings of the Royal Society of Medicine, 48(4):295–302, 1955.
G. Macdonald. The Epidemiology and Control of Malaria. Oxford University Press, 1957
S. Ruan, D. Xiao, and J. C. Beier. Bulletin of Mathematical Biology, 70(4):1098–1114, 2008.
D. L. Smith, K. E. Battle, S. I. Hay, C. M. Barker, T. W. Scott, and F. E. McKenzie. PLoS Pathogens, 8(4):e1002588, 2012
Aron J. L., “Acquired immunity dependent upon exposure in a SIRS epidemic model”, Journal ofMathematical Biosciences, vol. 88, pp. 37-47, 1988.
Aron J. L., “Mathematical modeling of immunity to Malaria”, Journal of Mathematical Bio- sciences, vol. 90, pp. 385-396, 1988.
Tumwiine J., Mugisha J., Luboobi L., “A mathematical model for the dynamics of malaria in a human host and mosquito vector with temporary immunity”, Journal of Applied Mathematics and Computation, vol. 189, pp. 1953-1965, 2005.
Tumwiine J., L. S. Luboobi, J. Y. T. Mugisha, “Modeling the effect of treatment and mosquitoes’ control on malaria transmission”, International Journal of Management and Systems, vol. 21, pp. 107-124, 2005.
Yang H., Wei H., Li X., “Global stability of an epidemic model for vector borne disease”, J Syst Sci Complex Journal, vol. 23, pp. 279-292, 2010.
Fekadu Tadege Kobe & Purnachandra Rao Koya, ’Spread of Malaria Disease Using Intervention Strategies ‘’Journal of multi disciplinary Engineering Science and Technology (JMEST), vol.2, Issse5, May-2015.
E. C. Ibezim and U. odo, Current trends in malarial chemotherapy, African Journal of Biotechnology, 7 No.4 (2008), 349-356, 2014.
O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, on the definition and computation of the basic reproduction ratio in models for infectious diseases in heterogeneous populations, J. Math. Biol, 28 (1990), 365-382, doi: 10.1007/BF00178324.156 S..
P. van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, pp. 29–48, 2002.
J. C. Koella and R. Antia, “Epidemiological models for the spread of anti-malarial resistance,” Malaria Journal, vol. 2, pp. 1–11, 2003
H. Khalil. Nonlinear Systems, Prentice Hall, 2002.
Merkin DR. Introduction to the theory of stability. Springer-Verlag New York, Inc; 1 997.
Chiyaka C., Tchuenche J. M., Garira W., and Dube S. A mathematical analysis of the effects of control strategies on the transmission dynamics of malaria. Appl. Math. Comput., 195(2):641 – 662, 2008.
Chiyaka C., Garira W., and Dube S. Transmission model of endemic human malaria in a partially immune population. Math. Comput. Modelling, 46(5-6):806 – 822, 2007.
Rowland M. Activity and mating competitiveness of gamma HCH/dieldrin resistant and susceptible male and virgin female Anopheles gambiae and An. stephensi mosquitoes, with assessment of an insecticide-rotation strategy. Medical and Veterinary Entomology. 1991; 5:207-222.