This work studies a mathematical model for the dynamics of Chagas disease, a parasitic disease that affects humans and domestic mammals throughout rural areas in Central and South America. It presents a modified version of the model found in Spagnuolo et al. [A model for Chagas disease with controlled spraying, J. Biol. Dyn. 5 (2011), pp. 299-317] with a delayed logistic growth term, which captures an overshoot, beyond the vector carrying capacity, in the total vector population when the blood meal supply is large. It studies the steady states of the system in the case of constant coefficients without spraying, and the analysis shows that for given-averaged parameters, the endemic equilibrium is stable and attracting. The numerical simulations of the model dynamics with time-dependent coefficients are shown when interruptions in the annual insecticide spraying cycles are taken into account. Simulations show that when there are spraying schedule interruptions, spraying may become ineffective when the blood meal supply is large.

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This work presents a new mathematical model for the domestic transmission of Chagas disease, a parasitic disease affecting humans and other mammals throughout Central and South America. The model takes into account congenital transmission in both humans and domestic mammals as well as oral transmission in domestic mammals. The model has time-dependent coefficients to account for seasonality and consists of four nonlinear differential equations, one of which has a delay, for the populations of vectors, infected vectors, infected humans, and infected mammals in the domestic setting. Computer simulations show that congenital transmission has a modest effect on infection while oral transmission in domestic mammals substantially contributes to the spread of the disease. In particular, oral transmission provides an alternative to vector biting as an infection route for the domestic mammals, who are key to the infection cycle. This may lead to high infection rates in domestic mammals even when the vectors have a low preference for biting them, and ultimately results in high infection levels in humans.

We consider characteristics-based finite element methods for solving nonlinear, degenerate, advection-diffusion equations. These equations have applications in the simulation of petroleum reservoirs and groundwater aquifers and in the modeling of free boundary problems. Standard finite element Galerkin methods have been studied for these equations. In this paper, we analyze the characteristics-based finite element methods for them. The main difficulty in the analysis is that the equations are degenerate and the solution lacks regularity. Here we develop a technique that respects the degeneracy and the known minimal regularity. This technique is based on the Green operator for standard elliptic equations and is developed directly for the degenerate advection-diffusion equations. We concentrate our analysis on the modified method of characteristics (MMOC) and one of its variants, the modified method of characteristics with adjusted advection (MMOCAA), which conserves mass. We derive error estimates in various norms. The extension to other variants is discussed. The present technique is also applied to nondegenerate problems; error estimates previously obtained for the MMOC are derived under much weaker regularity assumptions on the solution, and the error estimates for the MMOCAA appear new even in the nondegenerate case. Finally, numerical results are presented to show the sharpness of the error estimates derived.