This paper presents a unified analytical and computational framework for the study of typhoid fever transmission dynamics governed by a Caputo fractional-order compartmental model of order κ∈ (0, 1]. The population is stratified into five epidemiological classes, namely susceptible (S), asymptomatic (A), symptomatic (I), hospitalised (H), and recovered (R), and the governing system explicitly incorporates asymptomatic transmission, treatment dynamics, and temporary immunity with waning. The use of the Caputo fractional derivative is motivated by the well-documented existence of chronic asymptomatic Salmonella Typhi carriers, whose heavy-tailed sojourn times in the carrier state are naturally encoded by the Mittag–Leffler waiting-time distribution arising from the fractional operator. A complete qualitative analysis of the fractional system is carried out: the basic reproduction number R0 is derived via the next-generation matrix method; local and global asymptotic stability of both the disease-free equilibrium E0 (when R0≤1) and the endemic equilibrium E* (when R0>1) are established using fractional Lyapunov theory and the LaSalle invariance principle; and the normalised sensitivity indices of R0 are computed to identify transmission-amplifying and transmission-suppressing parameters. Existence, uniqueness, and Ulam–Hyers stability of solutions are established via Banach and Leray–Schauder fixed-point arguments. To complement the analytical results, a fractional physics-informed neural network (PINN) framework is developed to simultaneously reconstruct compartmental trajectories and identify unknown biological parameters from sparse synthetic observations. PINN embeds the L1-Caputo discretisation directly into the training residuals and employs a four-stage Adam–L-BFGS optimisation strategy to recover five trainable parameters Θ = ϕ, μ, σ, ψ, β across three fractional orders κ∈1. 0, 0. 95, 0. 9. The estimated parameters show strong agreement with the true values at the classical limit κ=1. 0 (MAPE=2. 27%), with the natural mortality rate μ recovered with APE≤0. 51% and the transmission rate β with APE≤3. 63% across all fractional orders, confirming the structural identifiability of the model. Pairwise correlation analysis of the learned parameters establishes the absence of equifinality, validating that β can be reliably included in the trainable set. Noise robustness experiments under Gaussian perturbations of 1%, 3%, and 5% demonstrate graceful degradation (MAPE: 0. 82%→3. 10%→7. 31%), confirming the reliability of the proposed framework under realistic observational conditions.
Mani et al. (Tue,) studied this question.