Mathematical analysis and dynamical systems : modeling Highland malaria in western Kenya

Wairimu J. Mathematical analysis and dynamical systems : modeling Highland malaria in western Kenya.; 2012.


The objective of this thesis is to model highland malaria in western Kenya using dynamical systems. Two mathematical models are formulated ; one, on differentiated susceptibility and differentiated infectivity in a metapopulation setting with age structure, the other, a saturated vector feeding rate model with disease induced deaths and varying host and vector populations. In the first model, we consider the different ecosystems identified as malaria hotspots in the western Kenya highlands and consider the ecosystems as different patches. The population in each patch is classified as, either child or, adult. The model will aid in examining the role of ecosystem heterogeneity and age structure to the persistent malaria epidemics in the highlands. We formulate the differentiated susceptibility and infectivity model that extend to multiple patches the well known epidemiological models in one patch. Classifying the hot spots as n patches, we give its mathematical analysis using the theory of triangular system, monotone non-linear dynamical systems, and Lyapunov-Lasalle invariance principle techniques. Key to our analysis is the definition of a reproductive number, Ro, the number of new infections caused by one individual in an otherwise fully susceptible population throughout the duration of the infectious period. The existence and stability of disease-free and endemic equilibrium is established. We prove that the disease free state of the systems is globally asymptotically stable when the basic reproduction number Ro<1, and when Ro>1 an endemic equilibrium is established which is locally and globally asymptotically stable. The model shows that the age structuring reduces the magnitude of infection. Using relevant data we did some simulation, to demonstrate the role played by metapopulation and age structuring on the incidence and Ro. In the second part we formulate a model for malaria with saturation on the vector feeding rates that lead to a nonlinear function in the infection term. The vector feeding rate is assumed, as in the predator prey models, to rise linearly as a function of the host-vector ratio until it reaches a threshold Qv, after which the vector feeds freely at its desired rate. The two populations are variable and drive malaria transmission, such that when the vectors are fewer than hosts, the rate of feeding is determined by the vectors feeding desire, whereas, when the hosts are more than the vectors, the feeding rate is limited by host availability and other feeding sources may have to be sought by the vector. Malaria induced deaths are introduced in the host population, while the vector is assumed to survive with the parasite till its death. We prove that the Disease Free Equilibrium is locally and globally asymptotically stable if Ro<1 and when Ro>1, an endemic equilibrium emerges, which is unique, locally and globally asymptotically stable. The role of the saturated mosquito feeding rate is explored with simulation showing the crucial role it plays especially on the basic reproduction number


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