Mathematical Theory of Structured Population Dynamics and its Applications

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Our main concern in this research is mathematical analysis and model developments for structured population models in demography, epidemiology and theoretical biology. Main results are as follows:[1] Chagas disease (American trypanosomiasis) is transmitted by the vector population and also by blood transfusion and characterized by the existence of the acute short duration stage and the long chronic stage. We have formulated the duration-dependent population model for the Chagas disease and examined the existence and stability of the steady state. Depending on parameter values, the backward bifurcation of the endemic steady stat e can occur, which is an important fact to be considered in the preventive policy.[2] We consider an age-duration-structured population model for HIV infection in a homosexual community. W e have shown sufficient conditions for backward or forward bifurcation to occur when the basic reproduction ratio crosses the unity. We have given sufficient conditions for th … More e local stability of the endemic steady states.[3] We have considered a SIR type epidemic model for the spread of horizontally and vertically transmitted infectious diseases in an age-structured population. We proved that an endemic steady state exists if and only if the basic reproduction ratio is grater than unity, while the disease-free steady state is globally asymptotically stable it is less than unity. We also show that the locally stable endemic steady states are forwardly bifurcated from the disease-free steady state when the basic reproduction number crosses the unity.[4] We have studied mathematical structure of the SIR epidemic model for the spread of directly transmitted infectious diseases in age-structured host populations, in which the basic system is formulated as infinite-dimensional homogeneous dynamical systems. We consider the case that the host population is described by the stable population model and proved the linearized stability principle such that as far as we concern the asymptotic behavior of the basic system, it is sufficient to examine the normalized system induced by assuming that the host population has already attained the stable population. Less