Models of Chagas' disease: Stability, thresholds and asymptotic behavior

Mathematical models of Chagas' disease are analyzed to characterize stability properties of equilibria as well as the threshold parameters that determine the existence of endemic or disease-free states. The models are of two mathematical types: systems of ordinary differential equations and models of hyperbolic systems of partial differential equations. Both types incorporate the structure of the population imposed by the characteristics of Chagas' disease: two infective stages (corresponding to the acute and chronic phases of the disease) with no cure rate, as well as the dynamics of the vector population and the effect of blood transfusion transmission.; Models using partial differential equations incorporate age structure and time since infection as well as the effect of horizontal transmission. The vector transmission rate is assumed constant. Here, existence and stability properties of the steady-states are proved by the application of semigroup methods and parameters are estimated using finite dimensional approximations to the model reformulated as an abstract semilinear equation in a particular Banach space. Data for the parameter estimation were obtained from bibliographical sources from Latin America. Parameters are estimated for a population from Brazil. Chagas' disease is endemic in areas where human populations grow rapidly and in which several reservoir species coexist. These two factors have been included in some of the models. Ordinary differential equations are used to model the transmission dynamics of the disease in heterogeneous populations subdivided either by subgroups with different susceptibility to infection and subgroups determined by geographic and socioeconomic factors, or subdivided in groups of hosts of different biological species (comprising the human, domestic and wild cycles of the transmission of Chagas' disease).; Ordinary and partial differential equation models show the existence of an endemic equilibrium point which is asymptotically stable when the basic reproductive number is greater than one. This number depends upon the contact rate of vectors and hosts and the horizontal transmission rate, and it is exceedingly high. Conclusions are drawn and discussed in this respect and guidelines are suggested relevant to the implementation of eradication policies.