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Mathematical Modelling of Malaria with Treatment

Published online by Cambridge University Press:  03 June 2015

Mini Ghosh
Mini Ghosh*
Affiliation:
School of Advanced Sciences, VIT University, Chennai Campus, Chennai, India
*
*Corresponding author. Email: [email protected]
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Abstract

This paper proposes a Susceptible-Infective-Susceptible (SIS) model to study the malaria transmission with treatment by considering logistic growth of mosquito population. In this work, it is assumed that the treatment rate is proportional to the number of infectives below the capacity and is constant when the number of infectives is greater than the capacity. We find that the system exhibits backward bifurcation if the capacity is small and it gives bi-stable equilibria which makes the system more sensitive to the initial conditions. The existence and stability of the equilibria of the model are discussed in-detail and numerical simulations are presented to illustrate the numerical results.

Keywords

92D3037N25Malariatreatmentsimulation
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 6 , December 2013 , pp. 857 - 871
Copyright
Copyright © Global-Science Press 2013

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References

[1]World Health Organization (WHO) and WHO Global Malaria Programme. See , .Google Scholar
[2]Martcheva, M. and Hoppensteadt, F., India’s approach to eliminate Plasmodium falciparum malaria: a modelling perspective, J. Biological Systems, 18(4) (2010), pp. 867–891.CrossRefGoogle Scholar
[3]Das, S. C., Bhuyan, M., Baruah, L. and Talukdar, P. K., Mosquito survey in Tripura, Indian J. Malariology, 28 (1991), pp. 129–134.Google ScholarPubMed
[4]Sharma, V. P., Re-emergence of malaria in India, Indian J. Med. Research, 103 (1996), pp. 26–45.Google Scholar
[5]Sharma, V. P., Fighting malaria in India, Current Science, 75 (1998), pp. 1127–1140.Google Scholar
[6]Sharma, V. P., Battling the malaria iceberg with chloroquine in India, Malaria J., 6 (2007), pp. 105.CrossRefGoogle ScholarPubMed
[7]Site, M., Malaria in India, .Google Scholar
[8]Hamer, D. H., Singh, M. P., Wylie, B. J., Yeboah-Antwi, K., Tuchman, J., Desai, M., Udhayakumar, V., Gupta, P., Brooks, M. I. M. I., Shukla, M. M., Awasthy, K., Sabin, L., Macleod, W. B., Dash, A. P. and Singh, N., Burden of malaria in pregnancy in Jharkhand State, India, Malaria J., 8 (2009), pp. 209.Google Scholar
[9]Singh, V., Mishra, N., Awasthi, G., Dash, A. P. and Das, A., Why is it important to study malaria epidemiology in India? Trends Paras, 25(10) (2009), pp. 452–457.Google Scholar
[10]Sharma, V. P., Artemisinin drugs in the treatment of P. Falciparum Malaria in India, Curr. Sci., 90(10) (2006), pp. 1323–1324.Google Scholar
[11]Ross, R., The prevention of Malaria, Eng. Edn. John Murray London, 4-10 (1911).Google Scholar
[12]Radcliffe, J., The initial geographical spread of host-vector and carrier borne epidemics, J. Appl. Prob., 10 (1973), pp. 703–717.CrossRefGoogle Scholar
[13]Radcliffe, J., The periodicity of endemic malaria, J. Appl. Prob., 11(3) (1997), pp. 562–567.Google Scholar
[14]Dietz, K., Molineaux, L. and Thomas, A., A Malaria model tested in the African Savannah, Bull. Wld. Hlth. Org., 50 (1974), pp. 347–357.Google Scholar
[15]Molineaux, L., Dietz, K. and Thomas, A., Further epidemiological evaluation of a malaria model, Bull. Wld. Hlth. Org., 56 (1978), pp. 565–571.Google Scholar
[16]Bailey, N. T. J., The Biomathematics of Malaria, Griffin, London, 1982.Google Scholar
[17]Ngwa, G. A. and Shu, W. S., A mathematical model for endemic malaria with variable human and mosquito populations, Math. Comput. Model., 32 (2000), pp. 747–763.Google Scholar
[18]Chitnis, N., Cushing, J. M. and Hyman, J. M., Bifurcation Analysis of mathematical model for malaria transmission, SIAM J. Appl. Math., 67(1) (2006), pp. 24–45.Google Scholar
[19]Chitnis, N., Hyman, J. M. and Cushing, J. M., Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull Math Biol., 70 (2008), pp. 1272–1296.CrossRefGoogle ScholarPubMed
[20]Mandal, Sandip, Sarkar, Ramrup and Sinha, Somdatta, Mathematical models of malaria-a review, Malaria J., 10 (2011), pp. 202.CrossRefGoogle ScholarPubMed
[21]Kim, Byul Nim, Nah, Kyeongah, Chu, Chaeshin, Ryu, Sang Uk, Kang, Yong Han and Kim, Yongkuk, Optimal control strategy of plasmodium vivax Malaria transmission in Korea, Osong Public Health and Research Perspectives, 3(3) (2012), pp. 128–136.Google Scholar
[22]Diaz, H., Ramirez, A. A., Olarte, A. and Clavijo, C., A model for the control of malaria using genetically modified vectors, J. Theoretical Biology, 276(1) (2011), pp. 57–66.Google Scholar
[23]Lashari, Abid Ali and Zaman, Gul, Optimal control of a vector borne disease with horizontal transmission, Nonlinear Anal. Real World Appl., 13(1) (2012), pp. 203–212.Google Scholar
[24]Ghosh, Mini, Lashari, Abid Ali and Li, Xue-Zhi, Biological control of malaria: a mathematical model, Appl. Math. Comput., 219 (15), pp. 7923–7939.Google Scholar
[25]Wang, W. D., Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), pp. 58–71.Google Scholar
[26]Gao, L. Q. and Hethcote, H. W., Disease transmission models with density dependent demographics, J. Math. Bio., 32 (1992), pp. 717–731.Google Scholar
[27]Ermentrout, B., Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, 1st edition, SIAM, Philadelphia, PA, 2002.CrossRefGoogle Scholar