Applied and Computational Mathematics
Volume 4, Issue 3, June 2015, Pages: 192-206
Received: May 13, 2015;
Accepted: May 26, 2015;
Published: Jun. 8, 2015
Views 4201 Downloads 125
Laurencia Ndelamo Massawe, Faculty of Science, Technology and Environmental Studies, the Open University of Tanzania, Dar es Salaam, Tanzania
Estomih S. Massawe, Mathematics Department, University of Dar es salaam, Dar es Salaam, Tanzania
Oluwole Daniel Makinde, Faculty of Military Science, Stellenbosch University, Saldanha, South Africa
In this paper a mathematical model for the transmission dynamics of dengue fever disease is presented. We present a SITR (susceptible, infected, treated, recovery) and ASI (aquatic, susceptible, infected) epidemic model to describe the interaction between human and dengue fever mosquito populations. In order to assess the transmission of Dengue fever disease, the susceptible population is divided into two, namely, careful and careless human susceptible population. The model presents four possible equilibria: two disease-free and two endemic equilibrium.The results show that the disease-free equilibrium point is locally and globally asymptotically stable if the reproduction number is less than unity. Endemic equilibrium point is locally and globally asymptotically stable under certain conditions using additive compound matrix and Lyapunov method respectively. Sensitivity analysis of the model is implemented in order to investigate the sensitivity of certain key parameters of dengue fever disease with treatment, Careful and Careless Susceptibles on the transmission of Dengue fever Disease.
Laurencia Ndelamo Massawe,
Estomih S. Massawe,
Oluwole Daniel Makinde,
Modelling Infectiology of Dengue Epidemic, Applied and Computational Mathematics.
Vol. 4, No. 3,
2015, pp. 192-206.
World Health Organization. (Online) (Cited 2015). Available From URL: http://www.who.Int/mediacentre/factsheets/fs 117/en/.
Rodrigues ,H.S., Monteiro M.T.T., Torres, D.F.M. and Zinober, A.( 2010). Control of dengue disease Computational and Mathematical Methods in Science and Engineering , 816–822.
Seidu, B. and Makinde, O.D. ( 2014). Optimal Control of HIV/AIDS in the Workplace in the Presence of Careless Individuals. Computational and Mathematical Methods in Medicine , 1- 19
El hia, M., Balatif O., Rachik, M. and Bouyaghroumni, J. (2013). Application of optimal control theory to an SEIR model with immigration of infectives. International Journal of Computer Science Issues 10 (2), 1694-0784
Laarabi, H., Labriji ,E.H., Rachik, M. and Kaddar, A.(2012). Optimal control of an epidemic model with a saturated incidence rate. Nonlinear Analysis: Modelling and Control 17(4) , 448–459
Lenhart, S. and Workman, J.T.(2007). Optimal Control Applied to Biological Models, Mathematical and Computational Biology Series, Chapman and Hall/CRC, London, UK.
Mwamtobe, P.M., Abelman, S., Tchuenche, J.M., and Kasambara, A. (2014) Optimal Control of Intervention Strategies for Malaria Epidemic in Karonga District, Malawi. Abstract and Applied Analysis,1-20
Ozair, M., Lashari, A.A., Jung, Il. H., and Okosun, K.O. (2012). Stability Analysis and Optimal Control of a Vector-Borne Disease with Nonlinear Incidence. Discrete Dynamics in Nature and Society, 1-21.
Rodrigues, H.S., Monteiro, M.T.T., and Torres, D.F.M. (2011) Dengue disease, basic reproduction number and control. International Journal of Computer Mathematics, 1–13
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,1063-1070.
Thome, R.C.A., Yang, H.M., and Esteva, L. (2010) Optimal control of Aedes aegypti mosquitoes by the sterile insect technique and insecticide. Math. Biosci.223, 12-23
Massawe, L.N., Massawe, E.S., and Makinde, O.D.(2015). Temporal model for dengue disease with treatment. Advances in Infectious Diseases, 5, 21-36
Rodrigues, H.S., Monteiro, M.T.T., and Torres, D.F.M (2013). Sensitivity Analysis in a Dengue Epidemiological Model. Conference Papers in Mathematics, vol. 2013
Driessche P van den and Watmough, J. (2002). “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, 180, 29–48
Lashari, A.A., Hattaf, K., Zaman, G., and Li, Xue-Zhi. (2013) Backward bifurcation and optimal control of a vector borne disease, Appl. Math. Inf. Sci. 7 (1) 301-309
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(4), 75-88.
Dumont, Y. Chiroleu, F., and Domerg, C.(2008). “On a temporal model for the Chikungunya disease: modelling, theory and numerics,” Mathematical Biosciences, 213(1), 80–91.
LaSalle, J.P. (1976). the Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1976.
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, 1-11
McCluskey, C.C., and Driessche, P van den. (2004). Global analysis of tuberclosis models, Journal of Differential Equations, 16, 139–166.