Let C be a nonempty closed convex subset of a real Hilbert space, and let T , S : C → C , where T is nonexpansive and S is qualitatively strongly quasi‐nonexpansive with I d − S demiclosed at zero. We propose a projected double‐inertial Halpern method in which a convex combination of T and S is evaluated at a two‐step extrapolated point constructed from the two most recent displacement vectors. The inertial parameters are governed by a computable bound on the effective extrapolation error. The projection ensures that the method is well defined on a general closed convex domain, while the bound controls the effective inertial perturbation. Strong convergence is established to the metric projection of the anchor onto the appropriate fixed‐point set in three parameter regimes: the common fixed point regime, the T ‐dominant regime, and the S ‐dominant regime. The convergence analysis is based on a mixed defect functional that simultaneously captures the norm losses associated with both mappings and their discrepancy. This approach preserves the qualitative strong quasi‐nonexpansivity assumption and does not require a uniform quadratic modulus. The original noninertial Halpern iteration and single‐inertial variants are recovered as special cases. An application to the best‐approximation formulation of convex feasibility is presented, together with numerical experiments illustrating the computational behavior of the proposed method. MSC2020 Classification: Primary 47H10, 47H09, 47J25, 47J26, 46B20
Typhoid fever remains a major public health problem in settings with inadequate vaccination, unsafe water, poor sanitation, and environmental contamination. This study developed a data‐driven Caputo fractional‐order model incorporating imperfect vaccination, direct and environmental transmission, waning immunity, and pathogen contamination. Positivity, boundedness, existence, uniqueness, Hyers–Ulam stability, and local stability of the disease‐free equilibrium were established. The effective reproduction number, , was derived, with the disease‐free equilibrium stable when . The model was fitted to 90 monthly laboratory‐confirmed Typhi and Paratyphi cases recorded from October 2016 to March 2024, yielding an estimated fractional order of α = 0.75. Numerical results showed that fractional memory significantly altered disease progression, with smaller α reducing exposed, infectious, and environmental pathogen burdens. Treatment produced the fastest reduction in human infection, whereas sanitation and pathogen removal were most effective in controlling environmental contamination. The findings supported integrated typhoid control combining vaccination, treatment, safe water, sanitation, and environmental decontamination.
This paper investigates the direct segregated boundary‐domain integrodifferential equations (BDIDEs) associated with the Dirichlet initial boundary value problem (IBVP) for nonstationary diffusion partial differential equations (PDEs) with variable coefficient, in appropriate Sobolev‐type Banach‐valued spaces. By employing a stationary parametrix (Levi function), we transformed the IBVP into an equivalent system of BDIDEs. Subsequently, the properties of the parametrix‐based potential operators are analyzed in Sobolev‐type Banach‐valued spaces, and the equivalence between the original problem and the transformed system is rigorously established.
Typhoid fever remains a major public health concern in many low- and middle-income countries, where the co-circulation of Salmonella Typhi and Salmonella Paratyphi, coupled with environmental contamination, sustains disease transmission. This study develops a data-driven Caputo fractional-order mathematical model to investigate the environmental co-transmission dynamics of these two pathogens while accounting for memory effects inherent in disease progression and environmental persistence. Model parameters are estimated by fitting the fractional-order system to surveillance data using nonlinear least-squares estimation. Analytical results establish the disease-free equilibrium, derive the effective reproduction number, and prove the local asymptotic and Hyers–Ulam stability of the proposed fractional-order system. Numerical simulations demonstrate excellent agreement between the fitted model and observed Typhi and Paratyphi incidence data, confirming the capability of the proposed framework to reproduce the observed epidemiological trends. Comparative simulations further show that co-transmission substantially increases the infectious burden and environmental contamination compared with single-pathogen transmission. Sensitivity analyses of intervention strategies reveal that strengthening preventive measures, improving treatment, and enhancing environmental decontamination substantially reduce disease burden and environmental pathogen concentrations, with combined interventions providing the greatest epidemiological benefit. The estimated fractional order indicates that memory effects play an important role in describing typhoid transmission dynamics. The proposed framework provides a realistic and data-driven tool for understanding environmental typhoid transmission and supports evidence-based planning of integrated public health interventions in endemic settings.
