Fractional modelling of typhoid with a piecewise modified Mittag-Leffler kernel: Memory and seasonality

Abstract

Typhoid fever remains a persistent public health challenge in endemic regions where seasonal climatic variations and inadequate sanitation infrastructure sustain year-round transmission. We develop a fractional-order mathematical model using the modified Atangana-Baleanu-Caputo ($\m$) derivative to capture epidemiological memory effects: immunological recall, sub-exponential environmental pathogen decay, and sustained behavioral adaptations, which classical integer-order models systematically neglect. The model integrates seasonally varying transmission rates $\beta(t) = \beta_0[1 + \beta_1\cos(2\pi(t - t_{\text{peak}})/365)]$ reflecting monsoon-driven contamination cycles, natural immunity waning (rate $\omega_r$) and treatment failure (rate $\epsilon_r$), relapse dynamics via rate $\tau_r$ capturing recrudescent infections, a hygienic compartment $\H$ representing sustained protective behaviors, and environmental bacteria $\B$ as the primary waterborne reservoir. Mathematical analysis establishes existence, uniqueness, positivity, and boundedness of solutions, with the basic reproduction number $\mathcal{R}_0 = \mathcal{R}_0^{\text{env}} + \mathcal{R}_0^{\text{dir}}$ decomposing environmental versus direct transmission routes and confirming waterborne contamination as the dominant mechanism. Three climatically distinct scenarios are investigated---dry/low contamination, monsoon peak, and post-monsoon persistence---each exhibiting different seasonal $\mathcal{R}_0$ ranges and intervention opportunities. The fractional extension reveals profound memory-induced suppression of epidemic potential: as memory strength increases (fractional order $\alpha$ decreases from unity), the reproduction number drops dramatically, crossing the epidemic threshold at a critical value $\alpha_c$ that delineates endemic persistence from natural elimination regimes. Numerical simulations demonstrate that strong memory substantially accelerates the decay of infected compartments while promoting faster convergence to protective states compared to classical predictions. These findings reveal that classical integer-order models systematically overestimate typhoid burden and intervention requirements, while populations with strong immunological or behavioral memory can suppress transmission below critical thresholds even under intense seasonal forcing. The results advocate incorporating fractional-order dynamics into typhoid forecasting models and prioritizing memory-enhancing interventions---long-duration vaccines, sustained hygiene education, and pre-monsoon water treatment campaigns---to achieve disease elimination in endemic settings without requiring indefinite high-intensity control efforts.