Malaria is an infectious disease caused by the Plasmodium parasite and transmitted between humans through bites of female Anopheles mosquitoes. A mathematical model describes the dynamics of malaria and human population compartments in terms of mathematical equations and these equations represent the relations between relevant properties of the compartments. The aim of the study is to understand the important parameters in the transmission and spread of endemic malaria disease, and try to find appropriate solutions and strategies for its prevention and control by applying mathematical modelling. The malaria model is developed based on basic mathematical modelling techniques leading to a system of ordinary differential equations (ODEs). Qualitative analysis of the model applies dimensional analysis, scaling, and perturbation techniques in addition to stability theory for ODE systems. We also derive the equilibrium points of the model and investigate their stability. Our results show that if the reproduction number, R0, is less than 1, the disease-free equilibrium point is stable, so that the disease dies out. If R0 is larger than 1, then the disease-free equilibrium is unstable. In that case, the endemic state has a unique equilibrium, re-invasion is always possible, and the disease persists within the human population. Numerical simulations have been carried out applying the numerical software Matlab. These simulations show the behavior of the populations in time and the stability of disease-free and endemic equilibrium points.
Published in | American Journal of Applied Mathematics (Volume 3, Issue 2) |
DOI | 10.11648/j.ajam.20150302.12 |
Page(s) | 36-46 |
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), 2015. Published by Science Publishing Group |
Malaria, Endemic Model, Reproduction Number, Equilibrium Points, Numerical Simulation
[1] | Addo, D. E., “Mathematical model for the control of Malaria”, Master Thesis, University of Cape Coast, 2009. |
[2] | Bush, A. O., Fernandez, J. C., Esch, G.W., Seed, J.R., Parasitism, “The Diversity and Ecology of Animal Parasites”, First ed., Cambridge University Press, Cambridge, 2001. |
[3] | Ngwa, G. A. “Modelling the dynamics of endemic malaria in growing populations”, Discrete Contin. Dyn. Syst. Ser. B, vol. 4, pp. 1173-1202, 2004. |
[4] | Macdonald G., “The Epidemiology and Control of Malaria”, Oxford university press, 1957. |
[5] | Yang, H. M., “Malaria transmission model for different levels of acquired immunity and temperature-dependent parameters (vector)”, Revista de Saude Publica, vol. 34, pp. 223-231, 2000. |
[6] | Ferreira, M. U., H. M. Yang, “Assessing the effects of global warming and local social and economic conditions on the malaria transmission”, Revista de Saude Publica, vol. 34, pp. 214-222, 2000. |
[7] | WHO, “Investing in health research for development”, Technical Report, World Health Organization, Geneva, 1996. |
[8] | Koella, J. C. and R. Antia, “Epidemiological models for the spread of anti-malarial resistance”, Malaria Journal, vol. 2, 2003. |
[9] | Aron J. L., “Acquired immunity dependent upon exposure in an SIRS epidemic model”, Journal of Mathematical Biosciences, vol. 88, pp. 37-47, 1988. |
[10] | Aron J. L., “Mathematical modelling of immunity to Malaria”, Journal of Mathematical Bio- sciences, vol. 90, pp. 385-396, 1988. |
[11] | Welch, J. Li, R. M., U. S. Nair, T. L. Sever, D. E. Irwin, C. Cordon-Rosales, N. Padilla, “Dynamic malaria models with environmental changes”, in Proceedings of the Thirty-fourth southeastern symposium on system theory, Huntsville, pp. 396-400. |
[12] | Tumwiine J., L.S. Luboobi, J.Y.T. Mugisha, “Modelling the effect of treatment and mosquitoes control on malaria transmission”, International Journal of Management and Systems, vol. 21, pp. 107-124, 2005. |
[13] | Grimwade K., N. French, D. D. Mbatha, D. D. Zungu, M. Dedicoat, C. F. Gilks (2004). “HIV infection as a cofactor for severe falciparum malaria in adults living in a region of unstable malaria transmission in South Africa”, Journal, vol.18, pp. 547-554. |
[14] | Bacaer N. and C. Sokhna. “A reaction-diffusion system modeling the spread of resistance to an antimalarial drug”, Math. Biosci. Engrg, vol. 2, pp. 227-238, 2005 |
[15] | Ngwa, G.A. and W.S. Shu, “A Mathematical model for endemic malaria with variable human and mosquito populations”, Mathematical and Computer Modeling Journal, vol. 32, pp. 747-763, 2000. |
[16] | Bailey N.T.J., “The biomathematics of malaria”, Charles Gri, London, 1982. |
[17] | Plemmons W. R., “Mathematical study of malaria models of Ross and Ngwa”, Master Thesis, University of Florida, pp. 1-69, 2006. |
[18] | 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. |
[19] | Ross R., “The Prevention of Malaria”, John Murray, 1911. |
[20] | 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. |
[21] | Lin, C.C., Segal L.A., “Mathematics Applied to Deterministic Problems in the Natural Sciences”, SIAM Classics in Applied Mathematics vol. 1, 1988. |
[22] | Krogstad. H.E., Dawed, M.Y., Tegegne, T.T., “Alternative analysis of the Michaelis-Menten Equations”, Teaching Mathematics and its Applications, vol. 30(3) pp. 138-146, 2011. |
[23] | Diekmann O., J.A.P. Heesterbeek, and J.A.J. Metz, On the dentition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., vol. 28, pp. 365–382, 1990. |
APA Style
Abadi Abay Gebremeskel, Harald Elias Krogstad. (2015). Mathematical Modelling of Endemic Malaria Transmission. American Journal of Applied Mathematics, 3(2), 36-46. https://doi.org/10.11648/j.ajam.20150302.12
ACS Style
Abadi Abay Gebremeskel; Harald Elias Krogstad. Mathematical Modelling of Endemic Malaria Transmission. Am. J. Appl. Math. 2015, 3(2), 36-46. doi: 10.11648/j.ajam.20150302.12
AMA Style
Abadi Abay Gebremeskel, Harald Elias Krogstad. Mathematical Modelling of Endemic Malaria Transmission. Am J Appl Math. 2015;3(2):36-46. doi: 10.11648/j.ajam.20150302.12
@article{10.11648/j.ajam.20150302.12, author = {Abadi Abay Gebremeskel and Harald Elias Krogstad}, title = {Mathematical Modelling of Endemic Malaria Transmission}, journal = {American Journal of Applied Mathematics}, volume = {3}, number = {2}, pages = {36-46}, doi = {10.11648/j.ajam.20150302.12}, url = {https://doi.org/10.11648/j.ajam.20150302.12}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20150302.12}, abstract = {Malaria is an infectious disease caused by the Plasmodium parasite and transmitted between humans through bites of female Anopheles mosquitoes. A mathematical model describes the dynamics of malaria and human population compartments in terms of mathematical equations and these equations represent the relations between relevant properties of the compartments. The aim of the study is to understand the important parameters in the transmission and spread of endemic malaria disease, and try to find appropriate solutions and strategies for its prevention and control by applying mathematical modelling. The malaria model is developed based on basic mathematical modelling techniques leading to a system of ordinary differential equations (ODEs). Qualitative analysis of the model applies dimensional analysis, scaling, and perturbation techniques in addition to stability theory for ODE systems. We also derive the equilibrium points of the model and investigate their stability. Our results show that if the reproduction number, R0, is less than 1, the disease-free equilibrium point is stable, so that the disease dies out. If R0 is larger than 1, then the disease-free equilibrium is unstable. In that case, the endemic state has a unique equilibrium, re-invasion is always possible, and the disease persists within the human population. Numerical simulations have been carried out applying the numerical software Matlab. These simulations show the behavior of the populations in time and the stability of disease-free and endemic equilibrium points.}, year = {2015} }
TY - JOUR T1 - Mathematical Modelling of Endemic Malaria Transmission AU - Abadi Abay Gebremeskel AU - Harald Elias Krogstad Y1 - 2015/02/13 PY - 2015 N1 - https://doi.org/10.11648/j.ajam.20150302.12 DO - 10.11648/j.ajam.20150302.12 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 36 EP - 46 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20150302.12 AB - Malaria is an infectious disease caused by the Plasmodium parasite and transmitted between humans through bites of female Anopheles mosquitoes. A mathematical model describes the dynamics of malaria and human population compartments in terms of mathematical equations and these equations represent the relations between relevant properties of the compartments. The aim of the study is to understand the important parameters in the transmission and spread of endemic malaria disease, and try to find appropriate solutions and strategies for its prevention and control by applying mathematical modelling. The malaria model is developed based on basic mathematical modelling techniques leading to a system of ordinary differential equations (ODEs). Qualitative analysis of the model applies dimensional analysis, scaling, and perturbation techniques in addition to stability theory for ODE systems. We also derive the equilibrium points of the model and investigate their stability. Our results show that if the reproduction number, R0, is less than 1, the disease-free equilibrium point is stable, so that the disease dies out. If R0 is larger than 1, then the disease-free equilibrium is unstable. In that case, the endemic state has a unique equilibrium, re-invasion is always possible, and the disease persists within the human population. Numerical simulations have been carried out applying the numerical software Matlab. These simulations show the behavior of the populations in time and the stability of disease-free and endemic equilibrium points. VL - 3 IS - 2 ER -