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 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!
Best Answer
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 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.