Mathematical Analysis of Basic Reproduction Number for the Spread and Control of Malaria Model with Non-Drug Compliant Humans

T. S.  Faniran; E. O. Ayoola

T. S. Faniran Department of Computer and Physical Sciences, Lead City University, Ibadan, Nigeria.
E. O. Ayoola Department of Mathematics, University of Ibadan, Ibadan, Nigeria.

Keywords: non-drug compliance, basic reproduction number, stability.

Abstract

Malaria arises when there is an infection of a host by Plasmodium falciparum that causes malaria in humans. Non-drug compliance results from not taking medication as prescribed by doctors. Previous research had concentrated on mathematical modeling of transmission dynamics of malaria without considering some infectious humans who do not comply to drug. This study is therefore designed to model transmission dynamics of malaria taking into consideration some infectious humans who do not comply to drug. The model is formulated using nonlinear ordinary differential equations. The human population is partitioned into Susceptible human $(S_H)$, Exposed Human $E_H$, Infectious human $(I_H)$, Non-drug compliant human $I_{NH}$ and Recovered human $(R_H)$. Using next generation matrix, the reproduction number $R_0$ is obtained. This is used to analyse the global stability of the disease-free equilibria and local stability of the endemic equilibria of the model. The global stability of the disease-free equilibria and the local stability of the endemic equilibrium of the model are established through the construction of suitable Lyapunov function and analysis of characteristic equation. It is shown that the disease-free equilibrium is globally asymptotically stable whenever $R_o< 1$. It is also shown that the endemic equilibrium becomes stable through the Routh-Hurwitz stability criteria.

References

[1] Alles, H.K., Mendis, K.N., & Carter, R. Malaria mortality rates in South Asia and in Africa:implications for malaria control Parasitology Today 14, 369-375 (1998).
[2] Aneke, S.J. Mathematical modeling of drug resistant malaria parasites and vector populations.Mathematical Methods in the Applied Sciences 90, 385-396 (2012).
[3] Antino, B., Corbett, Y., Catelli, F. & Taramelli, D. Pathogenesis of malaria in tissues and blood.Mediterranean Journal of Hematology and Infectious Diseases 2035-3006 (2012).
[4] Aron, J.L. Mathematical modeling of immunity to malaria. Mathematical Biosciences 90, 385-396 (1982).
[5] Bakary, T., Sangare, B. & Traore, S. Mathematical modeling of malaria transmission with structured vector population and seasonality. Journal of Applied Mathematics 15, (2017).
[6] Blayneh, K., Cao, Y. & Kwon, H. Optimal control of vector-borne diseases. Treatment and prevention. Discrete Continuous Dyn Syst Ser B 11, 587-611 (2009).
[7] Chitnis, N., Cushing, J.M & Hyman, J.M. Bifurcation analysis of a mathematical model for malaria transmission. SIAM Journal of Applied Mathematics 67, 24-45 (2018).
[8] Coutinho, F.A.B., Burattini, M.N., Lopez, L.F. & Massad, E. An approximate threshold condition for non-autonomous system:an application to a vector-borne infection. Mathematics and Computer in Simulation 70, 149-158 (2005).
[9] Gemperli, A., Vounatsou P., Sogoba, N. & Smith, T. Malaria mapping using transmission models: application to survey data. American Journal of Epidemiology 163, 289-297, (2006).
[10] Ishikawa, H., Ishii, A., Nagai, N., Ohmae, H., Mazakazu, H., Shuguri, S. & Leafasia, J. A mathematical model for the transmission of plasmodium vivax malaria. Parasitology International 52, 81-93 (2013).
[11] Jongwutiwes, S., Putaporntip, C., Iwasaki, T., Sata T. & Kanbara, H. Naturally acquired plasmodium knowlesi in malaria in human, Thailand. Emerging Infectious Diseases 10, 2211-2232 (2010).
[12] Khalil, H. Nonlinear systems. Prentice Hall (2002).
[13] Laxminarayan, R. Act now or later? Economics of malaria resistance. The American Journal of Tropical Medicine and Hyiegene 71, 187-195 (2004).
[14] Lasalle, J.P. The stability of dynamical systems. Philadelphia, PA: SIAM (1976).
[15] Lancaster, P. Theory of matrices. Academic Press, New York (1969).
[16] Lieshonta, M.V., Kovats, R.S., Livermore, M.T.J & Martens, P. Climate change and malaria: analysis of the SRES climate and socio-economic scenarios. Global Environmental Change 14, 87-99 (2004).
[17] Murray, C.J.L, Rosenfeld, L.C., Lim, S.S., Andrews, K.G., Foreman, K.J., Haring, D., Fullman, N., Naghari, M., Lozano, R. & Lopez, A.D. Global malaria mortality between 1980 and 2010: a systematic analysis. The Lancet 379, 413-431 (2012).
[18] Murray, J.D. Mathematical Biology I, An introduction. Springer-Verlag, Berlin (2002).
[19] Macdonald, G. The epidemiology and control of malaria. Oxford: Oxford University Press (1957).
[20] Ousmane, K., Traore, B. & Sangare, B. Mathematical modeling of malaria transmission global dynamics: taking into account the immatured stages of the vectors. Advances in Difference Equations 22, (2018).
[21] Tumwiine, J., Mugisha, J.Y.T & Luboobi, L.S. On oscillatory pattern of malaria dynamics in a population with temporary immunity. Computational and Mathematical Mathematical Methods in Medicine 8, 191-203 (2007).
[22] Tumwiine, J., Luboobi, L.S. & Mugisha, J.Y.T. Modeling the effect of treatment and mosquito control on malaria transmission. International Journal of Management and Systems 21, 107-124 (2005).
[23] Tumwiine, J., Mugisha, J.Y.T & Luboobi, L.S. A mathematical model for the dynamics of malaria in a human host and mosquito vector with temporary immunity. Applied Mathematics and Computation 189, 1953-1965 (2007).
[24] Wedajo, A.J., Bole, B.K, Koya, P.R. Analysis of SIR mathematical model for malaria disease with the inclusion of infected immigrants. IOSR Journal of Mathematics 14, 10-21 (2018).
[25] World Health Organization. World malaria report. WHO Library Cataloguing-in-publication data (2013).
[26] World Health Organization. Malaria control in complex emergencies, an inter-agency field handbook. WHO Library Cataloguing-in-publication data (2005).