Formulation of a Vector SIS Malaria Model in a Patchy Environment with Two Age Classes
- 1 School of mathematics, University of Nairobi, Nairobi, Kenya
- 2 INRIA, Metz and University of Lorraine, Metz, France
- 3 School of mathematics, University of Nairobi, Nairobi, Kenya
Abstract
We formulate an SIS model describing transmission of highland malaria in Western Kenya. The host population is classified as children, age 1- 5 years and adults, above 5 years. The susceptibility and infectivity of an individual depend on age class and residence. The large scale system with 6 n equations is reduced into a compact form of 3 n equations by a change of variables. Then 3 n equations are vectorialized using the matrix theory to get a one dimension, compact form of the system, equation in . Using Vidyasagar theorem [1] , the graph of the reduced system is shown to be strongly connected and the system is a monotone dynamical system. This means that circulation of malaria parasites among the species and among the patches is strongly connected, hence transmission is sustained. We show that for then-dimensional age structured system the positive orthant is positively invariant for all positive values of the variables.
- Vidyasagar, M. (1980) Decomposition Techniques for Large Scale Sytems with Nonadditive Interactions: Stability and Stabilization. IEEE Transactions on Automatic Control, 25, 773-779. http://dx.doi.org/10.1109/TAC.1980.1102422
- Githeko, A.K., Ayisi, J.M., Odada, P.K., Atieli, F.K., Ndenga, B.A., Githure, J.I. and Yan, G. (2006) Topography and Malaria Transmission Heterogeneity in Western Kenya Highlands: Prospects for Focal Vector Control. Malaria Journal, 5, 107. http://dx.doi.org/10.1186/1475-2875-5-107
- Wolfgang, M.R., Willem, T., Richard, C. and Bart, K. (2002) Host-Specific Cues Cause Differential Attractiveness of Kenyan Men to the African Malaria Vector Anopheles Gambiae. Malaria Journal, 1, 17. http://dx.doi.org/10.1186/1475-2875-1-17
- Smith, D.L., Guerra, C.A., Snow, R.W. and Simon, H.I. (2007) Standardizing Estimates of the Plasmodium Falciparum Parasite Rate. Malaria Journal, 6, 131. http://dx.doi.org/10.1186/1475-2875-6-131
- Tumwiine, J., Mugisha, J.Y.T. and Luboobi, L.S. (2007) On Oscillatory Pattern of Malaria Dynamics in a Population with Temporary Immunity. Computational and Mathematical Methods in Medicine, 8, 191-203.
- Wanjala, C.L., Waitumbi, J., Zhou, G. and Githeko, A.K. (2011) Identification of Malaria Transmission and Epidemic Hotspots in the Western Kenya Highlands: Its Application to Malaria Epidemic Prediction. Parasites and Vectors, 4, 81. http://dx.doi.org/10.1186/1756-3305-4-81
- Hyman, J.M. and Li, J. (2006) Differential Susceptibility and Infectivity Epidemic Models. Mathematical Biosciences and Engineering, 3, 89-100. http://dx.doi.org/10.3934/mbe.2006.3.89
- Pongsumpun, P. and Tang, I.M. (2003) Transmission of Dengue Haemorrhagic Fever in an Age Structured Population. Mathematical and Computer Modeling, 37, 949-961. http://dx.doi.org/10.1016/S0895-7177(03)00111-0
- Gao, D. and Ruan, S. (2012) A Multipatch Malaria Model with Logistic Growth Populations. SIAM: SIAM Journal on Applied Mathematics, 72, 819-841. http://dx.doi.org/10.1137/110850761
- Auger, P., Kouokam, E., Sallet, G., Tchuente, M. and Tsanou, B. (2008) The Ross-Macdonald Mode in a Patchy Environment. Mathematical Biosciences, 216, 123-131. http://dx.doi.org/10.1016/j.mbs.2008.08.010
- Ross, R. (1911) The Prevention of Malaria. Springer-Verlag, Berlin.
- Lutambi, A.M., Penny, M.A., Smith, N. and Chitnis N. (2013) Mathematical Modeling of Mosquito Dispersal in a Heterogenous Environment. Mathematical Biosciences, 241, 198-216.