A Liouville–Caputo Fractional Co-Infection Model: Theoretical Analysis, Ulam-Type Stability, and Numerical Simulation

This paper investigates a fractional-order mathematical model for the co-infection dynamics of pneumonia and typhoid fever using the Liouville–Caputo derivative. We establish the existence, uniqueness, non-negativity, and boundedness of solutions using Banach’s fixed point theorem and fractional comparison principles. The Hyers–Ulam and generalized Ulam–Hyers–Rassias stability of the system are rigorously proved; this stability analysis is epidemiologically significant because it guarantees that small perturbations in initial conditions or model parameters—inevitable in real-world data collection—do not lead to unbounded deviations in disease trajectory predictions. To approximate solutions numerically, we develop a Laplace-Based Optimized Decomposition Method (LODM) and validate its convergence against a modified predictor–corrector scheme. The LODM provides a semi-analytical series solution, while the predictor–corrector method serves as a numerical benchmark; this dual approach ensures reliability of simulations. Numerical simulations illustrate the influence of the fractional order ξ on system dynamics. Quantitative comparison between ξ=1 (integer order) and ξ