The Living Thing / Notebooks : Fractional order differential equations and super diffusive systems

Super diffusive” systems, non-Markov processes… Classically, (stochastic or deterministic) ODEs are “memoryless” in the sense that the current state (and not the history) of the system determines the future states/distribution of states.

One way you can destroy this is by using fractional derivatives in the formulation of the equation. (Why this choice, as opposed to putting in explicit integrals over the history of the process, I have no idea. Perhaps it leads to more elegant paraterisation or solutions?)

I’ll make this precise later, but want to note some evocative similarities to other branching processes which I usually study in discrete index and/or state space.

Popoular in modelling Dengue and phamacokinetics, whatever that is. Connections to to Lévy flights.

To learn: connection to long memory models. Why not presume a state filter model and learn that?

Refs

AhEl07
Ahmed, E., & Elgazzar, A. S.(2007) On fractional order differential equations model for nonlocal epidemics. Physica A: Statistical Mechanics and Its Applications, 379(2), 607–614. DOI.
BeRT15
Bendahmane, M., Ruiz-Baier, R., & Tian, C. (2015) Turing pattern dynamics and adaptive discretization for a super-diffusive Lotka-Volterra model. Journal of Mathematical Biology, 72(6), 1441–1465. DOI.
HaSD11
Hanert, E., Schumacher, E., & Deleersnijder, E. (2011) Front dynamics in fractional-order epidemic models. Journal of Theoretical Biology, 279(1), 9–16. DOI.
PoRT11
Pooseh, S., Rodrigues, H. S., & Torres, D. F. M.(2011) Fractional derivatives in Dengue epidemics. arXiv:1108.1683 [Math, Q-Bio], 739–742. DOI.
SaRC15
Sardar, T., Rana, S., & Chattopadhyay, J. (2015) A mathematical model of dengue transmission with memory. Communications in Nonlinear Science and Numerical Simulation, 22(1–3), 511–525. DOI.