On the interpretation of Caputo fractional compartmental models

In the last decade, hundreds of papers have been published on fractional modeling. The approach is often similar, with the motivation of incorporating a "memory effect": take the integer-order differential-equation model, and replace the ordinary derivative to the left-hand side by a fractional operator/derivative. Does this make sense? Which is the memory there? What is the physical and biological interpretation of fractional models? This paper aims at investigating these important issues. With fluxes, transitions, and concepts from survival analysis (hazard function), we give a meaning for Caputo compartmental models and their non-Markovian property. Essentially, a fractional index is connected with the instantaneous risk of leaving a compartment, conditioned on the past stay in there. We distinguish between Caputo models that are purely fractional (i.e., the usual ones) and partially fractional (i.e., a mix of ordinary and Riemann–Liouville components), and we explore initialization as well. We build new Euler-type numerical schemes, rooted in probability, and assess them for different models: a simple death process, logistic growth, SIR equations, the dynamics of an HIV infection, and the evolution of the smoking habit. A detailed discussion on the pros and cons of the fractional methodology is made along the article.