Modeling the Dynamics of Malaria Transmission with Bed Net Protection Perspective


We propose and analyze an epidemiological model to evaluate the effectiveness of bed nets as a prophylactic measure in malaria-endemic areas. The main purpose in this work is the modeling of the aggressiveness of anopheles mosquitoes relative to the way humans use to protect themselves against bites of mosquitoes. This model is a system of several differential equations: the number of equations depends on the particular assumptions of the model. We compute the basic reproduction number, and show that if, the disease free equilibrium (DFE) is globally asymptotically stable on the non-negative orthant. If, the system admits a unique endemic equilibrium (EE) that is globally and asymptotically stable. Numerical simulations are presented corresponding to scenarios typical of malaria-endemic areas, based on data collected in the literature. Finally, we discuss the relative effectiveness of different kinds of bed nets.

Share and Cite:

Kamgang, J. , Kamla, V. and Tchoumi, S. (2014) Modeling the Dynamics of Malaria Transmission with Bed Net Protection Perspective. Applied Mathematics, 5, 3156-3205. doi: 10.4236/am.2014.519298.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] WHO (2013) World Malaria Report 2013. Technical Report, WHO.
[2] Gollin, D. and Zimmermann, C. (2007) Malaria: Disease Impacts and Long-Run Income Differences. IZA Discussion Papers 2997, Institution for the Study of Labor (IZA).
[3] Ross, R. (1911) The Prevention of Malaria. John Murray, London.
[4] Barbour, A.D. (1978) MacDonald’s Model and the Transmission of Bilharzia. Transactions of the Royal Society of Tropical Medicine and Hygiene, 72, 6-15.
[5] Ngwa, A.G. and Shu, W.S. (2000) A Mathematical Model for Endemic Malaria with Variable Human and Mosqioto Population. Mathematical and Computer Modelling, 32, 747-763.
[6] Chitnis, N. (2005) Using Mathematical Models in Controlling the Spread of Malaria. Ph.D. Thesis, University of Arizona, Tucson.
[7] Zongo, P. (2009) Modélisation mathématique de la dynamique de transmission du paludisme. Ph.D. Thesis, Universite de Ouagadougou, Ouagadougou.
[8] Fontenille, D., Lochouarn, L., Diagne, N., Sokhna, C., Lemasson, J.J., Diatta, M., Konate, L., Faye, F., Rogier, C. and Trape, J.F. (1997) High Annual and Seasonal Variations in Malaria Transmission by Anophelines and Vector Species Composition in Dielmo, a Holoendemic Area in Senegal. American Journal of Tropical Medicine and Hygiene, 56, 247-253.
[9] Rogier, C., Tall, A., Diagne, N., Fontenille, D., Spiegel, A. and Trape, J.F. (2000) Plasmodium falciparum Clinical Malaria: Lessons from Longitudinal Studies in Senegal. Parassitologia, 41, 255-259.
[10] van den Driessche, P. and Watmough, J. (2002) Reproduction Numbers and Sub-Threshold Endemic Equilibria for Compartmental Models of Disease Transmission. Mathematical Biosciences, 180, 29-48.
[11] Carnevale, P. and Vincent, R. (2009) Les anophèles, Biologie, transmission du Paludisme et lutte antivectorielle. IRD.
[12] Kamgang, J.C. and Sallet, G. (2008) Computation of Threshold Conditions for Epidemiological Models and Global Stability of the Disease Free Equilibrium. Mathematical Biosciences, 213, 1-12.
[13] Bame, N., Bowong, S., Mbang, J., Sallet, G. and Tewa, J.J. (2008) Global Stability for SEIS Models with n Latent Classes. Mathematical Biosciences and Engineering, 5, 20-33.
[14] Bowong, S. and Tewa, J.J. (2009) Mathematical Analysis of a Tuberculosis Model with Differential Infectivity. Communications in Nonlinear Science and Numerical Simulation, 14, 4010-4021.
[15] Perelson, A.S., Kirschner, D.E. and De Boer, R. (1993) Dynamics of HIV Infection of CD4+ T Cells. Mathematical Biosciences, 114, 81-125.
[16] Guo, H., Li, M.Y. and Shuai, Z. (2006) Global Stability of the Endemic Equilibrium of Multigroup Models. Canadian Applied Mathematics Quarterly, 14, 259-284.
[17] Guo, H., Li, M.Y. and Shuai, Z. (2008) A Graph-Theoretic Approach to the Method of Global Lyapunov Functions. Proceedings of the American Mathematical Society, 136, 2793-2802.
[18] Korobeinikov, A. (2001) A Lyapunov Function for Leslie-Gower Predator-Prey Models. Applied Mathematics Letters, 14, 697-699.
[19] Korobeinikov, A. (2004) Lyapunov Functions and Global Properties for SEIR and SEIS Models. Mathematical Medicine and Biology, 21, 75-83.
[20] Korobeinikov, A. and Maini, P.K. (2004) A Lyapunov Function and Global Properties for SIR and SEIR Epidemiological Models with Nonlinear Incidence. Mathematical Biosciences and Engineering, 1, 57-60.
[21] Korobeinikov, A. and Wake, G.C. (2002) Lyapunov Functions and Global Stability for SIR, SIRS, and SIS Epidemiological Models. Applied Mathematics Letters, 15, 955-960.
[22] Ma, Z., Liu, J. and Li, J. (2003) Stability Analysis for Differential Infectivity Epidemic Models. Nonlinear Analysis: Real World Applications, 4, 841-856.
[23] McCluskey, C.C. (2006) Lyapunov Functions for Tuberculosis Models with Fast and Slow Progression. Mathematical Biosciences and Engineering, 3, 603-614.
[24] McCluskey, C.C. (2003) A Model of HIV/AIDS with Staged Progression and Amelioration. Mathematical Biosciences, 181, 1-16.
[25] McCluskey, C.C. (2005) A Strategy for Constructing Lyapunov Functions for Non-Autonomous Linear Differential Equations. Linear Algebra and Its Applications, 409, 100-110.
[26] McCluskey, C.C. and van den Driessche, P. (2004) Global Analysis of Two Tuberculosis Models. Journal of Dynamics and Differential Equations, 16, 139-166.
[27] Tewa, J.J., Dimi, J.L. and Bowong, S. (2009) Lyapunov Functions for a Dengue Disease Transmission Model. Chaos, Solitons & Fractals, 39, 936-941.
[28] Tewa, J.J., Fokouop, R., Mewoli, B. and Bowong, S. (2012) Mathematical Analysis of a General Class of Ordinary Differential Equations Coming from Within-Hosts Models of Malaria with Immune Effectors. Applied Mathematics and Computation, 218, 7347-7361.
[29] Bhatia, N.P. and Szegö, G.P. (1970) Stability Theory of Dynamical Systems. Springer-Verlag, Berlin.
[30] LaSalle, J.P. (1968) Stability Theory for Ordinary Differential Equations. Stability Theory for Ordinary Differential Equations. Journal of Differential Equations, 41, 57-65.
[31] LaSalle, J.P. (1976) The Stability of Dynamical Systems. Society for Industrial and Applied Mathematics, Philadelphia.
[32] LaSalle, J.P. (1976) Stability Theory and Invariance Principles. Dynamical Systems, Vol. I, Academic Press, New York, 211-222.
[33] Anguelov, R., Dumont, Y., Lubuma, J. and Shillor, M. (2013) Dynamically Consistent Nonstandard Finite Difference Schemes for Epidemiological Models. Journal of Computational and Applied Mathematics, 255, 161-182.
[34] Kamgang, J.C. and Sallet, G. (2005) Global Asymptotic Stability for the Disease Free Equilibrium for Epidemiological Models. Comptes Rendus Mathematique, 341, 433-438.
[35] Berman, A. and Plemmons, R.J. (1994) Nonnegative Matrices in the Mathematical Sciences, Volume 9. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia.
[36] Jacquez, J.A. and Simon, C.P. (1993) Qualitative Theory of Compartmental Systems. SIAM Review, 35, 43-79.
[37] Luenberger, D.G. (1979) Introduction to Dynamic Systems. Theory, Models, and Applications. John Wiley & Sons Ltd., Hoboken.
[38] McCluskey, C.C. (2007) Global Stability for a Class of Mass Action Systems Allowing for Latency in Tuberculosis. Journal of Mathematical Analysis and Applications, 338, 518-535.
[39] Li, J., Blakeley, D. and Smith, R.J. (2011) The Failure of . Computational and Mathematical Methods in Medicine, 2011, Article ID: 527610.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.