<p>The study introduces a mathematical model for the transmission dynamics of canine Chagas disease caused by <i>Trypanosoma cruzi</i> within a host–vector framework. The interaction between triatomine vectors and peridomestic dogs is described by a deterministic delay differential equation model incorporating biologically relevant time delays associated with vector development and infection processes. The vector population is structured into developmental stages, while the adult population is further stratified based on feeding behavior to reflect epidemiological relevance. The qualitative properties of the model are rigorously analyzed, including positivity, boundedness, and existence of solutions, ensuring biological feasibility. Equilibrium points are derived, and the basic reproduction number is computed using the next-generation matrix method to characterize the threshold dynamics of disease transmission. A sensitivity analysis of the reproduction number is also performed to identify key epidemiological parameters influencing disease spread. First-order and second-order nonstandard finite difference (NSFD) schemes are developed to preserve essential qualitative properties such as stability, positivity, and boundedness of the continuous model. Numerical simulations are conducted to validate the analytical findings and to investigate the influence of delay parameters, transmission rates, and discretization effects on the system dynamics. The results indicate that delay mechanisms play a significant role in reducing effective transmission and altering transient dynamics, while transmission parameters strongly influence disease persistence and spread. Overall, the proposed framework provides useful insights into the complex dynamics of canine Chagas disease and offers a reliable tool for exploring control strategies and long-term epidemiological behavior.</p>