Typhoid fever remains a major public-health problem, particularly in regions with poor sanitation, unsafe water, and limited healthcare. Motivated by persistent transmission, treatment failure, and relapse, this study proposes a deterministic typhoid fever model with direct and environmental transmission, treatment, and relapse mechanisms. The model is formulated in the modified Atangana–Baleanu–Caputo (mABC) fractional framework to incorporate memory effects. Its main qualitative properties, including positivity, boundedness, existence, and uniqueness of solutions, are established. Numerically, the Laplace–Adomian decomposition method (LADM) is applied to obtain approximate solutions for different fractional orders. The results show that the fractional order strongly affects transient disease dynamics while preserving biologically meaningful behavior. A deep neural network (DNN) surrogate is also trained on the generated trajectories and assessed through standard diagnostics. In addition, parameter estimation is performed using clean and noisy synthetic data, showing good agreement between fitted and reference dynamics.