Reaction‐Superdiffusion Systems in Epidemiology, an Application of Fractional Calculus

Spatially extended stochastic processes in epidemiology lead to classical reaction‐diffusion process, when infection spreads only locally. This notion can be generalized using fractional derivatives, especially fractional Laplacian operators, leading to Lévy flights and sub‐ or super‐diffusion. Especially super‐diffusion is a more realistic mechanism of spreading epidemics than ordinary diffusion.


INTRODUCTION
Classical derivatives of integer order have been generalized historically in various ways to derivatives of fractional order [1,2] and especially [3,4] with more reference and results. Especially, the Weyl-derivative generahzing the derivative in Fourier space is of interest here. It is linked with the Riemann-Liouville via the Marchaud regularization [2]. In this way the Laplacian operator can be generalized to a fractional Laplacian describing L6vy flights rather than ordinary random walks. This leads to the notion of sub-and super-diffusion, well applicable in reaction-diffusion systems [6]. In epidemiological systems especially the super-diffusion case is of interest as description of more realistic spreading than normal diffusion on regular lattices. To understand even basic epidemiological processes it is often necessary to investigate well the spatial spreading since all epidemic processes happen on spatially restricted networks [7]. We have previously studied epidemic processes with reinfection on regular lattices [13] as they also appear in the physics literature [10]. A crucial question in such systems is in how far basic notions like finite spreading and phase diagrams hold not only for ordinary diffusion but also in the super-diffusion case [7,8]. Wider processes with multi strain interaction [12,5] could be treated similarly. As our prime example here we will investigate the susceptibleinfected-susceptible SIS epidemic, which leads in the framework of reaction diffusion processes to the well know Kolmogorov-Fisher equation [11,9].

FROM STOCHASTIC EPIDEMIC MODELS TO REACTION-DIFFUSION PROCESSES
The SIS epidemics is an autocatalytic process given by the reaction scheme

S + I -h I + I I -^ S
and can be described via a master equation to capture the population noise of the epidemiological model (see [14] for a more detailed description of the SIS process). The stochastic spatially extended SIS epidemic process on general lattice or network topologies is given by the following dynamics for the probability p of the state of a network d_ dt for variables /,• G {0,1} and adjacency matrix (Jij). Local quantities like the expectation value of infected at a single lattice point, which in reaction diffusion systems corresponds to the local density u{x,t) are given by (2) For such quantities dynamics can be derived using the original dynamics of the stochastic process description for p{h,h, •••JNJ)-In such dynamics for local quantities there appears the discretized diffusion operator in the case of lattice models where we identify the growth rate r = /3g -a, the carrying capacity k= (1 -^) and diffusion constant X = P-Often the carrying capacity is simply set to unity, as well as the diffusion constant.

FRACTIONAL CALCULUS: FROM DIFFUSION TO SUPER-DIFFUSION
Here we concentrate on the diffusion part of Eq. (7), hence we look at a Fokker-Planck type equation for simple diffusion, which also describes a random walker in space (see Fig. 1 a), there for graphical reasons in two spatial dimensions)  The Fourier transform of the probability p{x,t) := p{x,t\xo = 0,to = 0) of the Wiener process, Eq. (9), is simply and the Fokker-Planck equation is in Fourier space given by which now can be easily generalized to other powers of k than the power of 2 for normal diffusion. Fractional calculus, i.e. the generahzation of first, second etc. derivatives to derivatives of non-integer order, hence fractional, gives us the tools to generahze also the Laplace operator in diffusion systems. Historically, many ways of generahzing integer derivatives have been suggested. Here we use the generalization in terms of the Fourier expansion of a function, hence the Weyl-derivative, which is linked via the Marchaud-regularization to the Riemann-Liouvillederivative. Hence, to describe super-diffusion we generalize the Fourier representation of the diffusion process to j U G (0

^x-y[pix,t) -p{y,t)] = p{y,t) -p{x,t) -{p{y -(y -x) ,t) -p{x -(y -x) ,t))
= p{y,t)-2p{x,t)+p{2x-y,t) , c c with transition rate w-^^y = ,_ j*;^^ for j U G (0,1) and W;^\y = ,_ j*;^^ ^x-y for j U G [1,2). The solution in real space representation for f > fo is given by An example for a process with long range jumps we plot in Fig. 1 b) a L6vy flight with exponent jj. = 1.5, again as in a) for graphical reasons in two spatial dimensions.