Fractional derivatives in biomathematical models with memory: A critical discussion

In recent years, the application of fractional derivatives in mathematical models has gained significant popularity. In the case of time-fractional derivatives, one of the main reasons for their use lies in their nonlocal property, which can overcome the limitations of ordinary differential equation models that are purely local and might fail to describe memory-dependent processes. The most common approach, often called fractionalisation, is based on the direct replacement of the classical derivatives in the ODE models, with fractional ones. When they are compared to real data, fractionalised models are often shown to provide better fitting results. The most common interpretation of this is that fractionalised models keep track of the history, while local models do not. However, while the physical meaning of a classical derivative is clear, the same cannot be said for fractional derivatives. Therefore, the relationship between modelling assumptions and mathematical equations remains unclear. Here, we introduce and critically discuss the fractionalisation approach by considering two representative examples of fractionalised biomathematical models. In our discussion, we address several properties of fractional operators that may impose significant limitations in their applications. However, the key question on which we would like to reflect is: is a fractionalised model still a model?