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Chagas disease is caused by the parasite Trypanosoma cruzi, which is spread primarily by domestic vectors in the reduviid family. This work presents a model of the dynamics of Chagas disease in a rural village. The model consists of a nonlinear delay logistic-type differential equation for the population of total vectors and three nonlinear differential equations for the populations of infected vectors, infected humans, and infected domestic mammals. Steady state solutions for the model are derived and analyzed. Stability numbers are provided with necessary and sufficient conditions for local asymptotic stability and partial results for global asymptotic stability. Numerical simulation results are presented, verifying the theoretical results.