Malaria Transmission Dynamics Incorporating Asymptomatic Human Compartment

Joseph Caleb Shaorga¹

Section:Research Paper, Product Type: Journal-Paper
Vol.9 , Issue.3 , pp.1-10, Jun-2022

Online published on Jun 30, 2022

Copyright © Joseph Caleb Shaorga . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

View this paper at   Google Scholar | DPI Digital Library

XML View     PDF Download

How to Cite this Paper

Abstract :
This research work presents a six-dimensional system of ordinary differential equations that captures dynamics of malaria incorporating asymptomatic human compartment. The model was analyzed for the Disease Free Equilibrium (DFE) and Endemic Equilibrium (EE). The next generation matrix technique is employed to obtain an explicit formula for the reproduction number a threshold parameter for the model. The model result show that the DFE is asymptotically stable when reproduction number is less than one and unstable otherwise. The existence of a unique endemic equilibrium point when reproduction number is greater than one is also determined. The numerical simulations were performed to confirm analytic results that shows increased proportion of recovered humans when both asymptomatic and infected humans are given equal consideration for treatment.

Key-Words / Index Term :
Malaria, Susceptible, Infected, Asymptomatic, Local stability, Antibody, Basic Reproduction Number, Incidence rate, Force of Infection, and Saturated Incidence Rate

References :
[1] T.A. Abeku, “Response to malaria epidemics in Africa,” Emerging infectious diseases, Vol. 13, Issue.5, pp. 681-6, 2007.
[2] L. Acedo, G.C. González-Parra, & A.J. Arenas, “An exact global solution for the classical SIRS epidemic model,” Nonlinear Analysis-real World Applications, Vol.11, pp.1819-1825, 2010
[3] M.E. Alexander, & S.M. Moghadas, “Bifurcation Analysis of an SIRS Epidemic Model with Generalized Incidence,” SIAM J. Appl. Math., Vol.65, pp.1794-1816, 2005.
[4] R.M. Anderson, & R.M. May, “Contagious diseases of humans: dynamics and control,” Oxford university Press, London , pp. 88-97, 1991.
[5] J.T. Bousema, L.C. Gouagna, C.J. Drakeley, A.M. Meutstege, B.A. Okech, I.N.J. Akim, J.C. Beier, J.I. Githure, & R.W. Sauerwein, “Plasmodium falciparum gametocyte carriage in asymptomatic children in western Kenya,” Malaria Journal, Vol.3, Issue.18, pp.1-6, 2004.
[6] Center for Disease Control (CDC), “Malaria Global Health,” Center for Disease Control Annual Report, U.S. pp. 3, 2016.
[7] C. Chiyaka, W. Garira, & S. Dube, “Modelling immune response and drug theraphy in humanmalaria infection,” Computational and Mathematical Methods in Medicine, Vol.9, Issue.2, pp.143-163, 2008.
[8] A.F.Cowman, D. Berry, & J. Baum, “The circular and molecular basis for malaria parasite invasion of the human red blood cell,” Journal of Cell Biology, Vol.198, Issue.6, pp. 96-171, 2012.
[9] B.T. Crookston, S.C. Alder, I. Boakye, R.M. Merrill, J.H. Amuasi, C.A. Porucznik, J.B. Stanford, T.T. Dickerson, K.A. Dearden, D.C. Hale, J. Sylverken, B.S. Snow, A. Osei-Akoto, D. Ansong, “Exploring the relationship between chronic undernutrition and asymptomatic malaria in Ghanaian children,” Malaria Journal, Vol.9, Issue.39, pp. 1-7, 2010.
[10] Cucunub, Z.M., Guerra, A.P., Rahirant, S.J., Rivera, J.A., Corts, L.J., & Nicholls, R.S., “Asymptomatic Plasmodium spp. Infection in Tierralta, Colombia,” Mem Inst Oswaldo Cruz, Vol.103, pp.668-673, 2008.
[11] L. Cui, G. Yan, J. Sattabongkot, Y. Coa, B. Chen, X. Chen, Q. Fan, S. Jongwutiwes, J. Sirichaisinthop, M.P. Kyaw, X.Z. Su, H. Yang, Z. Yang, B. Wang, D. Zhong, & G. Zhou, “Malaria in the Greater Mekong Subregion: Heterogeneity and complexity. ActaTropica, Vol.121 Issue.3, pp.227-239, 2012.
[12] O. Diekmann, J.A.P. Heesterbeek, & J.A.J. Mez, “On the defination and computation of the basic production radio inmodels for contagious diseases in hetrogenous populations,” J.Math. Biol, Vol.28, pp.365-382, 1990.
[13] K. Dietz, L. Molineaux, & A. Thomas, “A malaria model tested in the african savannah,” BullWorld Health Organ, Vol.50 pp.374-357, 1974.
[14] L. Esteve, & M.Matias, “A Model for Vector Transmitted Disease with Saturation Incidence,” Journal of Biological System, Vol.9 Issue.4 pp.235-245, 2001
[15] R.G. Feachem, A.A. Phillips, J. Hwang, C. Cotter, B. Wiegosz, B.M. Greenwood, O. Sabot, M.H. Rodriguez, R.R. Adeyasnghe, T.A. Ghebreyesus, & R.W. Snow, “Shrinking the malaria map: progress and prospects,” Lancet, Vol.376, Issue.9752, pp.166-178, 2010.
[16] F.F. Ferri, “Protozoal infections: Ferri’s color Atlas and Text of Clinical Medicine,” Elsevier Health Sciences, Vol.26, Issue.2, pp.243–59, 2009.
[17] F.E. Cox, “History of the discovery of the malaria parasites and their vectors,” Parasites Vectors, Vol.3, Issue.5, pp.1-9, 2010.
[18] P.W. Gething, A.P. Patil, D.L. Smith, C.A. Guerra, I.R. Elyazar, G.L. Johnson, A.J. Tatem, & S.I. Hay, “A new world malaria map: Plasmodium falciparum endemicity in 2010,” Malaria Journal, Vol.10, Issue.1, pp.1-16, 2011.
[19] B. Greenwood, & T. Mutabingwa, “Malaria in 2002”. Nature, Vol.415 Issue.6872, pp. 670-672, 2002.
[20] C.A. Guerra, S.I. Hay, L.S. Lucioparedes, P.W. Gikandi, A.J. Tatem A.M. Noor, & R.W. Snow, “Assembling a global database of malaria parasite prevalence for the Malaria Atlas Project,” Malaria Journal, Vol.6, Issue.6, pp.1-13, 2007.
[21] K.Harper, & G. Armelagos, “The changing disease-scape in the third epidiological transition,” International Journal of Environmental Research and Public Health, Vol.7, Issue.2, pp. 675-697. 2012
[22] I. Harris, W.W. Sharrock, L.M. Bain, K.A. Gray, A. Bobogare, L. Boaz, K. Lilley, D. Krause, A. Valley, M.L. Johnson, M.L., Gatton, G.D. Shanks, & Q. Cheng, “A large proportion of asymptomatic Plasmodium infections with low and sub-microscope parasite densities in the low transmission setting of Temotu Province, Solomon Islands: challenges for malaria diagnostics in an elimination setting,” Malar Journal, Vol.9, Issue.254, pp. 1-8, 2010.
[23] C.C. John, M.A. Riedesel, N.G. Magak, K.A. Lindblade, D.M. Menge, J.S. Hodges, J.M. Vulule, & W. Akhwale, “Possible interruption of malaria transission, highland Kenya, 2007/2008. Infect Dis, Vol.15. pp.1917-1924, 2009.
[24] P. Kajfasz, “Malaria prevention,” International Maritime Health, Vol.60, Issue.1-2. Pp. 67-70, 2009.
[25] G. Killeen, U. Fillinger, L. Gouagna, & B. Knols, “Eradication of Anopheles gambiae from Brazil: Lessons for Malaria control in Africa?” Lancet Contagious Diseases, Vol.2, Issue.10, 618-62, 2012.
[26] S.K. Kiyeny, “Using Mathematical Model to Illustrate the Spread of Malaria,” University of Nairobi Publishers, Nairobi, pp. 38-41, 2014.
[27] A.A. Lashari, S. Aly, K. Hattaf, G. Zaman, I.H. Jung, & X. Li, “Presensation of Malaria of Epidemics Using Multiple Optimal Controls,” Journal and Applied Mathematics, Vol.2012, Issue.5-7, 1-17, 2012.
[28] S.P. Layne, & I.J. Bigio, “Raman Spectroscopy of Bacillus Megaterium Using an Optical Multi-Channel Analyzer,” Physica Scripta, Vol.33, pp.91 – 96, 1986.
[29] G. Magombedze, C. Chiyaka, & Z. Mukandavire, “Optimal control of malaria chemotherapy”, Nonlinear Analysis: Modelling and Control, Vol.16, Issue.4, pp. 415-434, 2011.
[30] S.Males, O. Gaye, & A. Gracia, “Long-term asymptomatic carriage of plasmodium Falciparum protects from malaria attacks: a prospective study among SeneGhalese children,” ClinInfect Dis, Vol.46, pp.516-522, 2008.
[31] S. Mandal, R.R. Sarkar, & S. Sinha, “Mathematical models of of malaria– a review,” Malaria Journal, Vol.10, Issue. 202, pp. 1-19, 2011.
[32] C.J. Murray, L.C. Rosenfeld, S.S. Lim, K.G. Andrew, K.J. Foreman, D. Haring, N. Fullman, M. Naghavi, R. Lozano, & A.D. Lopez, “Global malaria mortality between 1980 and 2010: a systematic analysis,” The Lancet, Vol.379, pp.413-431, 2012
[33] G. Macdonald, “The epidemology and control of malaria,” Oxford University Press, London, pp.52-66, 1957
[34] B. Nadjm, & R.H. Behrens, “Malaria: an update for physicians,” Infect Dis Clin North Am. Vol.26, Issue.2, pp. 243-59, 2012
[35] K.O. Okosun, & O.D. Makinde, “Modeling the impact of drugs resistance in malariatransmition and its optimal control analysis,” International Journal of the Physical Sciences, Vol.6, Issue.28, pp.6479-6487, 2011
[36] S. Olaniyi, & O.S. Obabiyi, “Mathematical Model for Malaria Transmission Dynamics in Human and Mosquito Population with Nonlinear Forces of Infection,” International Journal of Pure and Applied Mathematics, Vol.88, Issue.1, pp.125-156, 2013.
[37] S. Owusu-Agyei, K.A. Koram, J.K. Baird, G.C. Utz, F.N. Binka, F.K. Nkrumah, D.J. Fryauff, & S.L. Hoffman, “Incidence of symtomatic and asymptomatic Plasmodium falciparum infection following curative therapy in adult residents of northern Ghana,” American Journal of Tropical Medicine and Hygiene, Vol.65, pp.197-203, 2001.
[38] C. Provost, “World Malaria Day: Which countries are the hardest hit? Get the full data,” The Guardian. Retrieved 2012-05-03.
[39] M. Quincampoix, “Differential inclusions and Target problems,” SIAM journal on control and optimization, Vol.30, Issue.2, pp.324-335, 1992.
[40] J. Tumwiine, J.V. Mugisha, & L.S. Luboobi, “A mathematical model for the dynamics of malaria in a human host and mosquito vector with temporary immunity,” Applied Mathematical and Computation, Vol.189, pp.1953-1965, 2007.
[41] US Centers for Disease Control and Prevention (2010). Eradication of malaria in the United States (19471951). Retrieved 2012-05-02.
[42] P. Van den Driessche, & J. Watmough, “Reproduction numbers and subthreshold Endemic equilibria for compartmental models of disease transmition,” Math. Bois, Vol.180, pp.29-48, 2002.
[43] J.P. Vanderberg, “Reflections on early malaria vaccine studies, the first successful human malaria vaccination, and beyond,” Vaccine, Vol.27, Issue.1, pp.2–9, 2009.
[44] M. Vidyasagar, “Decomposition teckniques for largescale system with non-additive interactions: stability and stabilizability,” IEE Transactions on Automatic control, Vol.25, Issue.4, pp.773-779, 1980.
[45] U.S. Welch, T.L. Sever, D.E. Irwin, C.C. Gordon-Rosales, & N. Padilla, “Dynamic Malaria models with Eviromentals Charger,” in Proceedings of the Thirty-fourth southeastern symposium on system theory, Huntsville, pp. 396-400, pp. 396-400. 2002.
[46] World Malaria Report, “Investing in health research for development”, Technical Report, World Health Organization, Geneva, pp. 1-4, 2012.
[47] M. Zhien, Z. Yicang, & M. Jianhong, “Modeling and dynamics of infectous disease,” World scientific publishing Co Pte Ltd. Singapore, pp.26-29, 2009.