Scientific Research

An Academic Publisher

**Mathematical Analysis of a Large Scale Vector SIS Malaria Model in a Patchy Environment** ()

We answer the stability question of the large scale
SIS model describing transmission of highland malaria in Western Kenya in a
patchy environment, formulated in [1]. There are two equilibrium states and their
stability depends on the basic reproduction number, R_{o} [2]. If R_{o }≤1, the disease-free steady solution is
globally asymptotically stable and the disease always dies out. If R_{o }>1, there exists a unique endemic equilibrium
which is globally stable and the disease persists. Application is done on data
from Western Kenya. The age structure reduces the level of infection and the
populations settle to the equilibrium faster than in the model without age
structure.

Keywords

Share and Cite:

*Applied Mathematics*,

**5**, 1913-1926. doi: 10.4236/am.2014.513185.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | Josephine, W.K., Gauthier, S. and Ogana, W. (2013) Formulation of a Vector SIS Malaria Model in a Patchy Environment with Two Age Classes. Applied Mathematics, 222, 4444. |

[2] | Diekmann, O., Heesterbeek, J.A.P. and Metz, J.A.J. (1990) On the Definition and the Computation of the Basic Reproduction Ratio R0 in Models for Infectious Diseases in Heterogeneous Populations. Journal of Mathematical Biology, 28, 365-382. http://dx.doi.org/10.1007/BF00178324 |

[3] | Diekmann, O. and Heesterbeek, J.A.P. (2000) Mathematical Epidemiology of Infectious Diseases in Mathematical and Computational Biology. Wiley Series, Hoboken. |

[4] |
Van Den Driessche, P. and Watmough, J. (2002) Reproduction Numbers and Subthreshold Endemic Equilibria for Compartmental Models of Disease Transmission. Mathematical Biosciences, 180, 29-48. http://dx.doi.org/10.1016/S0025-5564(02)00108-6 |

[5] | Varga, R.S. (1962) Matrix Iterative Analysis. Prentice-Hall, Upper Saddle River. |

[6] |
Thieme, H.R. (2009) Spectral Bound and Reproduction Number for Infinite-Dimensional Population Structure and Time-Heterogeneity. SIAM Journal on Applied Mathematics, 70, 188-211. http://dx.doi.org/10.1137/080732870 |

[7] | Varga, R.S. (1960) Factorisation and Normalised Iterative Methods Boundary Problems in Differential Equation. University of Wisconsin Press, Madison. |

[8] | La Salle, J.P. (1976) The Stability of Dynamical Systems. Society for Industrial and Applied Mathematics. Regional Conference Series in Applied Mathematics. |

[9] |
Hirsch, M.W. (1982) Systems of Differential Equations that Are Competitive or Cooperative: I. Limit Sets. SIAM Journal on Mathematical Analysis, 13, 167-179. http://dx.doi.org/10.1137/0513013 |

[10] | Berman, A. and Plemmons, R.J. (1994) Nonnegative Matrices in the Mathematical Sciences, Volume 9 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia. |

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

[12] |
Chitnis, N., Hyman, J.M. and Cushing, J.M. (2008) Determining Important Parameters in the Spread of Malaria through the Sensitivity Analysis of a Mathematical Model. Bulletin of Mathematical Biology, 70, 1272-1296. http://dx.doi.org/10.1007/s11538-008-9299-0 |

[13] | Githeko, A.K., Branding-Bennet, D., Beier, M., Atieli, F., Owaga, M. and Collins, F.H. (1992) The Reservoir of Plasmodium Falciparum Malaria in a Holoendemic Area of Western Kenya. Transactions of the Royal Society of Tropical Medicine and Hygiene, 86, 335-358. |

[14] |
Ndenga, B., Githeko, A., Omukunda, E., Munyekenye, G., Atieli, H., Wamai, P., Mbogo, C., Minakawa, N., Zhou, G. and Yan, G. (2006) Population Dynamics of Malaria Vectors in Western Kenya Highlands. Journal of Medical Entomology, 43, 200-206. http://dx.doi.org/10.1603/0022-2585(2006)043[0200:PDOMVI]2.0.CO;2 |

[15] | UNICEF (2010) Kenya Statistics. Technical Report, United Nation, New York City. |

[16] |
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 |

[17] | Balls, M.J., Bodker, R., Thomas, C.J., Kisinza, W., Msangeni, H.A. and Lindsay, S.W. (2004) Effect of Topography on the Risk of Malaria Infection in the Usambara Mountains, Tanzania. Transactions of the Royal Society of Tropical Medicine and Hygiene, 98, 400-408. |

[18] |
Mukabana, W.R., Takken, W., Richard, C. and Knols, B.G.J. (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 |

[19] |
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 |

[20] | Bowong, S., Dumont, Y. and Tewa, J.J. (2013) A Patchy Model for Chikungunya-Like Diseases. Biomath, 2, 1-19. |

Copyright © 2020 by authors and Scientific Research Publishing Inc.

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