Fractional Epidemic Modeling: Theoretical Constructions and Estimation Strategies

This paper presents a generalized epidemic modeling framework based on g-tempered Caputo fractional derivatives with discrete time delays. The proposed approach incorporates nonlocal memory effects, nonlinear temporal scaling, and delayed epidemiological responses within a unified mathematical structure. The introduction of the nonlinear time transformation g(t) and the tempering parameter λ eliminates the unrealistic infinite-memory behavior associated with classical power-law kernels while simultaneously introducing new challenges related to parameter identifiability and inverse problems. We investigate the structural properties of the resulting dynamical systems and show that the associated inverse problem is inherently ill-posed. To illustrate the practical implications of these results, the framework is applied to a delayed SIQR epidemiological model. Numerical simulations are performed using a generalized L1-type scheme adapted to delayed fractional histories, and a multi-phase parameter estimation procedure is proposed to address the ill-posedness of the reconstruction problem. The results demonstrate the ability of the model to capture both short- and long-term memory effects in epidemic evolution while highlighting the challenges of statistical identifiability in generalized fractional systems.