On partial Caputo fractional models

Abstract A partial Caputo fractional model mixes, in the same formulation, classical and Riemann–Liouville terms, in the realm of epidemiology and sociology. The Riemann–Liouville part introduces non-Markovian behavior and makes the hazard risk function of transition between the compartments lower as time advances. As a generalization of standard Caputo fractional models, where all of the terms are fractional, here we investigate the mathematical theory on incomplete fractionalization. Essentially, the goal is to prove that partially fractional initial-value problems are well posed, with existence, uniqueness, and continuity of solution with respect to input data. We present fixed-point results, a Cauchy–Kovalevskaya theorem on fractional power series, and Gronwall- and Nagumo-like arguments for uniqueness. Continuity of solutions uses bounds with the Mittag-Leffler function. Linear equations are also addressed, exhibiting global existence, global power-series representation, and certain closed-form solutions with Mikusiński operational calculus, refined bounds, and stability properties. The stochastic counterpart of partial Caputo models is introduced, with several results.