Temporal Model for Dengue Disease with Treatment

Laurencia Ndelamo Massawe^1*, Estomih S. Massawe¹, Oluwole D. Makinde²

¹Mathematics Department, University of Dar es Salaam, Dar es Salaam, Tanzania.
²Faculty of Military Science, Stellenbosch University, Saldanha, South Africa.

DOI: 10.4236/aid.2015.51003                

Abstract

This paper examines the effect of treatment of Dengue fever disease. A non linear mathematical model for the problem is proposed and analysed quantitatively using the stability theory of the differential equations. The results show that the disease-free equilibrium point is locally andglobally asymptotically stable if the reproduction number (R₀) is less than unity. The additive compound matrices approach is used to show that the dengue fever model’s endemic equilibrium point is locally asymptotically stable when trace, determinant and determinant of second additive compound matrix of the Jacobian matrix are all negative. However, treatment will have a control of dengue fever disease. Numerical simulation of the model is implemented to investigate the sensitivity of certain key parameters on the dengue fever disease with treatment.

Keywords

Dengue Fever Disease, Treatment of Dengue Fever Disease, Equilibrium Stability, Reproduction Number, Sensitivity Index

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Semenza, J.C. and Menne, B.B. (2009) Climate Change and Infectious Diseases in Europe. The Lancet Infectious Diseases, 9, 365-375. http://dx.doi.org/10.1016/S1473-3099(09)70104-5
[2]	Rodrigues, H.S., Monteiro, M.T.T., Torres, D.F.M and Zinober, A. (2011) Dengue Disease, Basic Reproduction Number and Control. International Journal of Computer Mathematics, 1-13.
[3]	Lenhart, S. and Workman, J.T. (2007) Optimal Control Applied to Biological Models. Chapman& Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton.
[4]	Chikaki, E. and Ishikawa, H. (2009) A Dengue Transmission Model in Thailand Considering Sequential Infections with All Four Serotypes. J Infect Devctries, 3, 711-722.
[5]	Thome, R.C., Yang, H.M. and Esteva, L. (2010) Optimal Control of Aedes aegypti Mosquitoes by the Sterile Insect Technique and Insecticide. Mathematical Biosciences, 223, 12-23. http://dx.doi.org/10.1016/j.mbs.2009.08.009
[6]	Rodrigues, H.S., Monteiro, M.T.T. and Torres, D.F.M. (2010) Insecticide Control in a Dengue Epidemics Model. In: Simos, T., Ed., AIP Conference of Proceedings of the Numerical Analysis and Applied Mathematics, 1281, 979-982.
[7]	Centers for Disease Control and Prevention (2011) Division of Vector Borne and Infectious Diseases, Prevention, How to Reduce Your Risk of Dengue Infection. http://www.cdc.gov/Dengue/prevention/index.html.
[8]	Rodrigues, H.S., Monteiro, M.T.T. and Torres, D.F.M. (2012) Modeling and Optimal Control Applied to a Vector Borne Disease. International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE, 2012, 1063-1070.
[9]	Rodrigues, H.S., Monteiro, M.T.T. and Torres, D.F.M. (2013) Bioeconomic Perspectives to an Optimal Control Dengue Model. International Journal of Computer Mathematics, 2013.
[10]	Rodrigues, H.S., Monteiro, M.T.T. and Torres, D.F.M. (2013) Sensitivity Analysis in a Dengue Epidemiological Model. Conference Papers in Mathematics, 2013.
[11]	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. http://dx.doi.org/10.1016/S0025-5564(02)00108-6
[12]	Ratera, S., Massawe, E.S. and Makinde, O.D. (2012) Modelling the Effect of Screening and Treatment on Transmission of HIV/AIDS Infection in Population. American Journal of Mathematics and Statistics, 2, 75-88. http://dx.doi.org/10.5923/j.ajms.20120204.03
[13]	Ozair, M., Lashari, A.A., Jung, I.H., Seo, Y.I. and Kim, B.N. (2013) Stability Analysis of a Vector-Borne Disease with Variable Human Population. Research Article Stability, 2013, 1-12
[14]	LaSalle, J.P. (1976) The Stability of Dynamical Systems. Regional Conference Series in Applied Mathematics, SIAM, Philadelphia.
[15]	Tumwiine, J., Mugisha, J.Y.T. and Luboobi, L.S. (2007) 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. http://dx.doi.org/10.1016/j.amc.2006.12.084
[16]	Lee, K.S. and Lashari, A.A. (2014) Global Stability of a Host-Vector Model for Pine Wilt Disease with Nonlinear Incidence Rate. Abstract and Applied Analysis, 2014, 1-11
[17]	McCluskey, C.C. and van den Driessche, P. (2004) Global Analysis of Two Tuberculosis Models. Journal of Differential Equations, 16, 139-166. http://dx.doi.org/10.1023/B:JODY.0000041283.66784.3e