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We formulate a compounded random walk that is physically well defined on both finite and infinite domains, and samples space-dependent forces throughout jumps. The governing evolution equation for the walk limits to a space-fractional Fokker-Planck equation valid on bounded domains, and recovers the well known superdiffusive space-fractional diffusion equation on infinite domains. This compounded random walk, and its associated fractional Fokker-Planck equation, provides a major advance for modelling space-fractional diffusion through potential fields and on finite domains., Comment: 6 pages, 3 figures

We have derived the governing equations for an SIR model with delay terms in both the infectivity and recovery of the disease. The equations are derived by modelling the dynamics as a continuous time random walk, where individuals move between the classic SIR compartments. With an appropriate choice of distributions for the infectivity and recovery processes delay terms are introduced into the governing equations in a manner that ensures the physicality of the model. This provides novel insight into the underlying dynamics of an SIR model with time delays. The SIR model with delay infectivity and recovery allows for a more diverse range of dynamical behaviours. The model accounts for an incubation effect without the need to introduce new compartments., Comment: 17 pages, 6 figures

Compartment models with delay terms are widely used across a range of disciplines. The motivation to include delay terms varies across different contexts. In epidemiological and pharmacokinetic models, the delays are often used to represent an incubation period. In this work, we derive a compartment model with delay terms from an underlying non-Markov stochastic process. Delay terms arise when waiting times are drawn from a delay exponential distribution. This stochastic process approach allows us to preserve the physicality of the model, gaining understanding into the conditions under which delay terms can arise. By providing the conditions under which the delay exponential function is a probability distribution, we establish a critical value for the delay terms. An exact stochastic simulation method is introduced for the generalized model, enabling us to utilize the simulation in scenarios where intrinsic stochasticity is significant, such as when the population size is small. We illustrate the applications of the model and validate our simulation algorithm on examples drawn from epidemiology and pharmacokinetics., Comment: 20 pages, 4 figures

In this work, exact solutions are derived for an integer- and fractional-order time-delayed diffusion equation with arbitrary initial conditions. The solutions are obtained using Fourier transform methods in conjunction with the known properties of delay functions. It is observed that the solutions do not exhibit infinite speed of propagation for smooth initial conditions that are bounded and positive. Sufficient conditions on the initial condition are also established such that the finite time blowup of the solutions can be explicitly calculated. Examples are provided that highlight the contrasting behaviours of these exact solutions with the known dynamics of solutions to the standard diffusion equation., Comment: Errors were discovered in the analysis, significant revisions are being made to the manuscript

Transport phenomena play a vital role in various fields of science and engineering. In this work, exact solutions are derived for advection equations with integer- and fractional-order time derivatives and a constant time-delay in the spatial derivative. Solutions are obtained, for arbitrary separable initial conditions, by incorporating recently introduced delay functions in a separation of variables approach. Examples are provided showing oscillatory and translatory behaviours that are fundamentally different to standard propagating wave solutions., Comment: Letter

Our study focuses on fractional order compartment models derived from underlying physical stochastic processes, providing a more physically grounded approach compared to models that use the dynamical system approach by simply replacing integer-order derivatives with fractional order derivatives. In these models, inherent stochasticity becomes important, particularly when dealing with the dynamics of small populations far from the continuum limit of large particle numbers. The necessity for stochastic simulations arises from deviations of the mean states from those obtained from the governing equations in these scenarios. To address this, we introduce an exact stochastic simulation algorithm designed for fractional order compartment models, based on a semi-Markov process. We have considered a fractional order resusceptibility SIS model and a fractional order recovery SIR model as illustrative examples, highlighting significant disparities between deterministic and stochastic dynamics when the total population is small. Beyond its modeling applications, the algorithm presented serves as a versatile tool for solving fractional order differential equations via Monte Carlo simulations., Comment: 26 pages, 7 figures

Explicit numerical finite difference schemes for partial differential equations are well known to be easy to implement but they are particularly problematic for solving equations whose solutions admit shocks, blowups and discontinuities. Here we present an explicit numerical scheme for solving non-linear advection-diffusion equations admitting shock solutions that is both easy to implement and stable. The numerical scheme is obtained by considering the continuum limit of a discrete time and space stochastic process for non-linear advection-diffusion. The stochastic process is well posed and this guarantees the stability of the scheme. Several examples are provided to highlight the importance of the formulation of the stochastic process in obtaining a stable and accurate numerical scheme., Comment: 15 pages, 2 figures

There has recently been considerable interest in using a nonstandard piecewise approximation to formulate fractional order differential equations as difference equations that describe the same dynamical behaviour and are more amenable to a dynamical systems analysis. Unfortunately, due to mistakes in the fundamental papers, the difference equations formulated through this process do not capture the dynamics of the fractional order equations. We show that the correct application of this nonstandard piecewise approximation leads to a one parameter family of fractional order differential equations that converges to the original equation as the parameter tends to zero. A closed formed solution exists for each member of this family and leads to the formulation of a difference equation that is of increasing order as time steps are taken. Whilst this does not lead to a simplified dynamical analysis it does lead to a numerical method for solving the fractional order differential equation. The method is shown to be equivalent to a quadrature based method, despite the fact that it has not been derived from a quadrature. The method can be implemented with non-uniform time steps. An example is provided showing that the difference equation can correctly capture the dynamics of the underlying fractional differential equation.

Fractional-order SIR models have become increasingly popular in the literature in recent years, however unlike the standard SIR model, they often lack a derivation from an underlying stochastic process. Here we derive a fractional-order infectivity SIR model from a stochastic process that incorporates a time-since-infection dependence on the infectivity of individuals. The fractional derivative appears in the generalised master equations of a continuous time random walk through SIR compartments, with a power-law function in the infectivity. We show that this model can also be formulated as an infection-age structured Kermack-McKendrick integro-differential SIR model. Under the appropriate limit the fractional infectivity model reduces to the standard ordinary differential equation SIR model., Comment: 16 pages, no figures

We demonstrate an efficient high-precision optical spectroscopy technique for single trapped ions with non-closed transitions. In a double-shelving technique, the absorption of a single photon is first amplified to several phonons of a normal motional mode shared with a co-trapped cooling ion of a different species, before being further amplified to thousands of fluorescence photons emitted by the cooling ion using the standard electron shelving technique. We employ this extension of the photon recoil spectroscopy technique to perform the first high precision absolute frequency measurement of the $^{2}$D$_{3/2}$ $\rightarrow$ $^{2}$P$_{1/2}$ transition in $^{40}$Ca$^{+}$, resulting in a transition frequency of $f=346\, 000\, 234\, 867(96)$ kHz. Furthermore, we determine the isotope shift of this transition and the $^{2}$S$_{1/2}$ $\rightarrow$ $^{2}$P$_{1/2}$ transition for $^{42}$Ca$^{+}$, $^{44}$Ca$^{+}$ and $^{48}$Ca$^{+}$ ions relative to $^{40}$Ca$^{+}$ with an accuracy below 100 kHz. Improved field and mass shift constants of these transitions as well as changes in mean square nuclear charge radii are extracted from this high resolution data.

We derive the generalized master equation for reaction-diffusion on networks from an underlying stochastic process, the continuous time random walk (CTRW). The non-trivial incorporation of the reaction process into the CTRW is achieved by splitting the derivation into two stages. The reactions are treated as birth-death processes and the first stage of the derivation is at the single particle level, taking into account the death process, whilst the second stage considers an ensemble of these particles including the birth process. Using this model we have investigated different types of pattern formation across the vertices on a range of networks. Importantly, the CTRW defines the Laplacian operator on the network in a non \emph{ad-hoc} manner and the pattern formation depends on the structure of this Laplacian. Here we focus attention on CTRWs with exponential waiting times for two cases; one in which the rate parameter is constant for all vertices and the other where the rate parameter is proportional to the vertex degree. This results in nonsymmetric and symmetric CTRW Laplacians respectively. In the case of symmetric Laplacians, pattern formation follows from the Turing instability. However in nonsymmetric Laplacians, pattern formation may be possible with or without a Turing instability., Comment: 27 pages, 10 figures