With COVID19 becoming a pandemic I saw some researchers trying to model it with fractional differential equations (FDE) instead of ordinary differential equations (ODE). From a technical standpoint I wondered what's the reason in doing so. I don't see how FDEs appear "naturally" when modelling diseases. For example lets look at a simple SIR-model


$$\frac{dS}{dt} = -\beta IS $$
$$\frac{dI}{dt}=\beta IS - \gamma I$$
$$\frac{dR}{dt} = \gamma I$$
With $S$ being the amount of susceptible persons, $I$ being the amount of infectious persons and $R$ being the amount of recovered persons. $\beta > 0$ the average number of contacts per person and $\gamma>0$ being the transition rate.


What would be the benefit of looking at it as a fractional differential equation with parameter $\alpha \in (0,1)$? (Let's say e.g. with a Caputo-Derivative)


I know, that a major difference between ODE and FDE is, that the solution of the FDE is not "local" in a sense that the solution in a point $t\_1$ depends on the values of the solution on the whole intervall $[0,t\_1]$. Whereas in ODE this is not really the case, we don't necessarily need the information about the values of the solution on the whole interval $[0,t\_1]$. Often this is interpreted as the solution having a "memory" but why is it good for modelling epidemics, that the solution, i.e. the amount of susceptible, infectious and recovered people has a "memory"?


Apart from that, are there any technical advantages from using FDE instead of ODE? I know, that we have slightly different results regarding the stability of equilibrium points. (if we look at $u'=f(u)$, with equilibrium point $u\_\*$, then by the principle of linearized stability, we know that $u\_\*$ is asymptotically stable if the Eigenvalues $\lambda$ of $f'(u\_\*)$ have negative real part. Whereas for FDE of order $\alpha$ we have asymptotic stability if $|arg(\lambda)|>\frac{\alpha \pi}{2}$). (see [this](https://arxiv.org/abs/1512.04989) paper for a proof of this statement). However I don't see how this would "help" us in the model given above.


**Question**
Why should we use FDEs instead of ODEs, when it comes to modelling infectious diseases? What are FDEs able to do, that ODEs are not? Is there a way to show that FDEs appear "naturally" in modelling diseases? (Literature hints are always appreciated!) Thanks in advance!


P.S. I already asked this question on Math stackexchange two weeks ago, however I thought mathoverflow may be better suited for this question. If its not I apologize in advance!



Why should we model infectious diseases with fractional differential equations?
There are two issues here: Firstly, fractional derivatives are non-local operators, so they can be used to model processes with a "memory", where the prior history governs the future evolution. Secondly, the exponent of the fractional derivative can be used as a fit parameter to improve the agreement with data.


Studies in the literature based on the [compartmental models](https://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology) for the spread of an infectious diseases typically compare a bit better with data when the ODE is substituted by a FDE. The improvement is not large and might well be due mainly to the additional fit parameter. The memory effect might as well be included by introducing additional compartments, I don't think there is a truly compelling reason to prefer FDE over ODE.


