Formulation of a Vector SIS Malaria Model in a Patchy Environment with Two Age Classes

Affiliation(s)

School of mathematics, University of Nairobi, Nairobi, Kenya.

INRIA, Metz and University of Lorraine, Metz, France.

School of mathematics, University of Nairobi, Nairobi, Kenya.

INRIA, Metz and University of Lorraine, Metz, France.

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.

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

KEYWORDS

Highland Malaria, Differentiated Susceptibility and Infectivity, Monotone Dynamical Systems, Age Structure

Highland Malaria, Differentiated Susceptibility and Infectivity, Monotone Dynamical Systems, Age Structure

Cite this paper

Wairimu, J. , Gauthier, S. and Ogana, W. (2014) Formulation of a Vector SIS Malaria Model in a Patchy Environment with Two Age Classes.*Applied Mathematics*, **5**, 1535-1545. doi: 10.4236/am.2014.510147.

Wairimu, J. , Gauthier, S. and Ogana, W. (2014) Formulation of a Vector SIS Malaria Model in a Patchy Environment with Two Age Classes.

References

[1] 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

[2] 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

[3] 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

[4] 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

[5] 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.

[6] 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

[7] 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

[8] 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

[9] 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

[10] 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

[11] Ross, R. (1911) The Prevention of Malaria. Springer-Verlag, Berlin.

[12] 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.

[13] Kelly, D.W. and Thompson, C.E. (2000) Epidemiology and Optimal Foraging: Modelling the Ideal Free Distribution of Insect Vectors. Parasitology, 120, 319-327.

[14] Arino, J, Davis, J.R., Hartley, D., Jordan R., Miller, J. and van den Driessche, P. (2005) A Multispecies Epidemic Model with Spatial Dynamics. Mathematical Medical Biology, 22, 129-142.

[15] Berman, A.P. and Robert, J. (1979) Nonnegative Matrices in the Mathematical Sciences, Volume 9 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia. Revised Reprint of the 1979 Original.

[16] Smith, H.L. (1995) Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Volume 41. Mathematical Surveys and Monographs.

[17] Hirsch, M.W. (1982) Systems of Differential Equations That Are Competitive or Cooperative I: Limit Sets. SIAM: SIAM Journal on Applied Mathematics, 13, 167-179. http://dx.doi.org/10.1137/0513013

[18] Hirsch, M.W. (1988) Systems of Differential Equations That Are Competitive or Cooperative III: Competing Species. SIAM: SIAM Journal on Applied Mathematics, 1, 51-71.

[19] Hirsch, M.W. and Smith, H.L. (2005) Monotone Dynamical Systems. Handbook of Differential Equations: Ordinary Differential Equations. Vol. II, Elsevier B. V., Amsterdam, 239-357.

[20] Arino, J. (2009) Diseases in Metapopulations. In: Ma, Z., Zhou, Y. and Wu, J., Eds., Modeling and Dynamics of Infectious Diseases in Series in Contemporary Applied Mathematics, World Scientific, Singapore City, 65-123. Also CDM Preprint Series Report 2008-04.

[1] 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

[2] 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

[3] 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

[4] 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

[5] 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.

[6] 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

[7] 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

[8] 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

[9] 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

[10] 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

[11] Ross, R. (1911) The Prevention of Malaria. Springer-Verlag, Berlin.

[12] 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.

[13] Kelly, D.W. and Thompson, C.E. (2000) Epidemiology and Optimal Foraging: Modelling the Ideal Free Distribution of Insect Vectors. Parasitology, 120, 319-327.

[14] Arino, J, Davis, J.R., Hartley, D., Jordan R., Miller, J. and van den Driessche, P. (2005) A Multispecies Epidemic Model with Spatial Dynamics. Mathematical Medical Biology, 22, 129-142.

[15] Berman, A.P. and Robert, J. (1979) Nonnegative Matrices in the Mathematical Sciences, Volume 9 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia. Revised Reprint of the 1979 Original.

[16] Smith, H.L. (1995) Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Volume 41. Mathematical Surveys and Monographs.

[17] Hirsch, M.W. (1982) Systems of Differential Equations That Are Competitive or Cooperative I: Limit Sets. SIAM: SIAM Journal on Applied Mathematics, 13, 167-179. http://dx.doi.org/10.1137/0513013

[18] Hirsch, M.W. (1988) Systems of Differential Equations That Are Competitive or Cooperative III: Competing Species. SIAM: SIAM Journal on Applied Mathematics, 1, 51-71.

[19] Hirsch, M.W. and Smith, H.L. (2005) Monotone Dynamical Systems. Handbook of Differential Equations: Ordinary Differential Equations. Vol. II, Elsevier B. V., Amsterdam, 239-357.

[20] Arino, J. (2009) Diseases in Metapopulations. In: Ma, Z., Zhou, Y. and Wu, J., Eds., Modeling and Dynamics of Infectious Diseases in Series in Contemporary Applied Mathematics, World Scientific, Singapore City, 65-123. Also CDM Preprint Series Report 2008-04.