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Compartment models are a widely used class of models that are useful when considering the flow of objects, people, or energy between different labeled states, referred to as compartments. Classic examples include SIR models in epidemiology and many pharmacokinetic models used in pharmacology. These models are formulated as sets of coupled ordinary differential equations, but in recent years there has been increasing interest in generalizations involving fractional differential equations. The majority of such generalizations have been performed in an ad hoc manner by replacing integer order derivatives with fractional derivatives. Such an approach does allow for the incorporation of history effects into the models, but may be problematic in a number of ways, such as breaking conservation of matter. To overcome these problems we have developed a systematic approach for the inclusion of fractional derivatives into compartment models by deriving the deterministic governing equations from an underlying physical stochastic process. This derivation also reveals the connection between these fractional order models and age-structured models. Unlike the ad hoc addition of fractional derivatives, our approach ensures that the model remains physically reasonable at all times and provides for an easy interpretation of all the parameters in the model. Illustrative examples, drawn from epidemiology, pharmacokinetics, and in-host virus dynamics, are provided.

There has been considerable recent interest in certain integral transform operators with non-singular kernels and their ability to be considered as fractional derivatives. Two such operators are the Caputo–Fabrizio operator and the Atangana–Baleanu operator. Here we present solutions to simple initial value problems involving these two operators and show that, apart from some special cases, the solutions have an intrinsic discontinuity at the origin. The intrinsic nature of the discontinuity in the solution raises concerns about using such operators in modelling. Solutions to initial value problems involving the traditional Caputo operator, which has a singularity inits kernel, do not have these intrinsic discontinuities.

A standard reaction-diffusion equation consists of two additive terms, a diffusion term and a reaction rate term. The latter term is obtained directly from a reaction rate equation which is itself derived from known reaction kinetics, together with modelling assumptions such as the law of mass action for well-mixed systems. In formulating a reaction-subdiffusion equation, it is not sufficient to know the reaction rate equation. It is also necessary to know details of the reaction kinetics, even in well-mixed systems where reactions are not diffusion limited. This is because, at a fundamental level, birth and death processes need to be dealt with differently in subdiffusive environments. While there has been some discussion of this in the published literature, few examples have been provided, and there are still very many papers being published with Caputo fractional time derivatives simply replacing first order time derivatives in reaction-diffusion equations. In this paper, we formulate clear examples of reaction-subdiffusion systems, based on; equal birth and death rate dynamics, Fisher-Kolmogorov, Petrovsky and Piskunov (Fisher-KPP) equation dynamics, and Fitzhugh-Nagumo equation dynamics. These examples illustrate how to incorporate considerations of reaction kinetics into fractional reaction-diffusion equations. We also show how the dynamics of a system with birth rates and death rates cancelling, in an otherwise subdiffusive environment, are governed by a mass-conserving tempered time fractional diffusion equation that is subdiffusive for short times but standard diffusion for long times.

Reaction-diffusion equations are widely used as the governing evolution equations for modeling many physical, chemical, and biological processes. Here we derive reaction-diffusion equations to model transport with reactions on a one-dimensional domain that is evolving. The model equations, which have been derived from generalized continuous time random walks, can incorporate complexities such as subdiffusive transport and inhomogeneous domain stretching and shrinking. Inhomogeneously growing domains are frequently encountered in biological phenomena involving stochastic transport, such as tumor growth and morphogen gradient formation. A method for constructing analytic expressions for short-time moments of the position of the particles is developed and moments calculated from this approach are shown to compare favorably with results from random walk simulations and numerical integration of the reaction transport equation. The results show the important role played by the initial condition. In particular, it strongly affects the time dependence of the moments in the short-time regime by introducing additional drift and diffusion terms. We also discuss how our reaction transport equation could be applied to study the spreading of a population on an evolving interface. From a more general perspective, our findings help to mitigate the scarcity of analytic results for reaction-diffusion problems in geometries displaying nonuniform growth. They are also expected to pave the way for further results, including the treatment of first-passage problems associated with encounter-controlled reactions in such domains.

The extension of fractional power series solutions for linear fractional differential equations with variable coefficients is considered. Generalized series expansions involving integer powers and fractional powers in the independent variable have recently been shown to provide solutions to certain linear fractional order differential equations. In these studies the generalized series could be formulated as the Cauchy product of a standard power series with integer powers and a fractional power series with powers as multiples of a fractional exponent. Here we show that generalized fractional powers series can provide solutions to a wider class of problems if there is no requirement that the series can be formulated as a Cauchy product.

We construct a time-fractional geometric Fokker–Planck equation from the diffusion limit of a continuous time random walk with a power law waiting time density, and a biased multiplicative jump length density dependent on the particles’ current position. The bias is related to a force, and an associated potential, through a steady state Boltzmann distribution. The limit of the random walk, with a force derived from a logarithmic potential, defines a stochastic process that is a fractional generalization of geometric Brownian motion. We have investigated the moments for this process. In geometric Brownian motion the expectation of the logarithm of the position of the particle scales linearly with time. In the fractional generalization, this scales as a sub-linear power law in time, similar to anomalous scaling of the mean square displacement in subdiffusion. In financial applications this could be observed as a sub-linear scaling in the logarithmic return.

We derive the generalized master equations for a stochastic process representing populations entering, leaving, or waiting in compartments at discrete times. This discrete time compartment model limits to a fractional order continuous time compartment model for a particular choice of waiting time distribution and the appropriate limiting process. We demonstrate that the discrete time master equations can be used to provide an explicit numerical method that can be employed for solving the fractional order compartment equations. The advantage of this approach is that the numerical scheme has a physical interpretation, it is stable, and it is easy to implement. The method can be applied to a wide class of fractional order compartment model equations that arise in a broad range of applications.

A class of discrete time random walks has recently been introduced to provide a stochastic process based numerical scheme for solving fractional order partial differential equations, including the fractional subdiffusion equation. Here we develop a Monte Carlo method for simulating discrete time random walks with Sibuya power law waiting times, providing another approximate solution of the fractional subdiffusion equation. The computation time scales as a power law in the number of time steps with a fractional exponent simply related to the order of the fractional derivative. We also provide an explicit form of a subordinator for discrete time random walks with Sibuya power law waiting times. This subordinator transforms from an operational time, in the expected number of random walk steps, to the physical time, in the number of time steps.

A generalised advection equation with a time fractional derivative is derived from a continuous time random walk on a one-dimensional lattice, with power law distributed waiting times. We consider walks governed by a two-sided jump density and walks governed by a one-sided jump density. With the two-sided density, the particle can jump in both directions on the lattice, whereas with the one-sided density the particle cannot jump in one of these directions. The master equations describing the evolution of the probability density for the position of the particle are different for each of the jump densities. However in an advective limit both master equations limit to a common generalized advection equation with time fractional derivatives. We have also considered the stochastic processes in a discrete time setting, again arriving at different discrete time master equations for each of the jump densities. The discrete time master equations can be used to provide different numerical approximations to the solutions of the fractional generalized advection equation. The approximations allow us to compare the efficacy of the one-sided and two-sided densities.

Here we present a numerical method for the solution of time fractional partial differential equations (fPDEs). The method is based on constructing a sequence of integrable approximations to the fPDE, such that in the appropriate limit the original fractional PDE is obtained. Convergence results are obtained in general and examples are given for simple cases such as the fractional heat equation and the fractional diffusion-wave equation.

Compartment models have been used to describe the time evolution of a system undergoing reactions between populations in different compartments. The governing equations are a set of coupled ordinary differential equations. In recent years fractional order derivatives have been introduced in compartment models in an ad hoc way, replacing ordinary derivatives with fractional derivatives. This has been motivated by the utility of fractional derivatives in incorporating history effects, but the ad hoc inclusion can be problematic for flux balance. To overcome these problems we have derived fractional order compartment models from an underlying physical stochastic process. In general, our fractional compartment models differ from ad hoc fractional models and our derivation ensures that the fractional derivatives have a physical basis in our models. Some illustrative examples, drawn from epidemiology, pharmacokinetics, and in-host virus dynamics, are provided.

In the standard continuous time random walk the initial state is taken as a non-equilibrium state, in which the random walking particle has just arrived at a given site. Here we consider generalizations of the continuous time random walk to accommodate arbitrary initial states. One such generalization provides information about the initial state through the introduction of a first waiting time density that is taken to be different from subsequent waiting time densities. Another generalization provides information about the initial state through the prior history of the arrival flux density. The master equations have been derived for each of these generalizations. They are different in general but they are shown to limit to the same master equation in the case of an equilibrium initial state. Under appropriate conditions they also reduce to the master equation for the standard continuous time random walk with the non-equilibrium initial state. The diffusion limit of the generalized master equations is also considered, with Mittag-Leffler waiting time densities, resulting in the same fractional Fokker–Planck equation for different initial conditions.

The Min protein system creates a dynamic spatial pattern in Escherichia coli cells where the proteins MinD and MinE oscillate from pole to pole. MinD positions MinC, an inhibitor of FtsZ ring formation, contributing to the mid-cell localization of cell division. In this paper, Fourier analysis is used to decompose experimental and model MinD spatial distributions into time-dependent harmonic components. In both experiment and model, the second harmonic component is responsible for producing a mid-cell minimum in MinD concentration. The features of this harmonic are robust in both experiment and model. Fourier analysis reveals a close correspondence between the time-dependent behaviour of the harmonic components in the experimental data and model. Given this, each molecular species in the model was analysed individually. This analysis revealed that membrane-bound MinD dimer shows the mid-cell minimum with the highest contrast when averaged over time, carrying the strongest signal for positioning the cell division ring. This concurs with previous data showing that the MinD dimer binds to MinC inhibiting FtsZ ring formation. These results show that non-linear interactions of Min proteins are essential for producing the mid-cell positioning signal via the generation of second-order harmonic components in the time-dependent spatial protein distribution.

Over the past several decades, there has been a proliferation of epidemiological models with ordinary derivatives replaced by fractional derivatives in an ad hoc manner. These models may be mathematically interesting, but their relevance is uncertain. Here we develop an SIR model for an epidemic, including vital dynamics, from an underlying stochastic process. We show how fractional differential operators arise naturally in these models whenever the recovery time from the disease is power-law distributed. This can provide a model for a chronic disease process where individuals who are infected for a long time are unlikely to recover. The fractional order recovery model is shown to be consistent with the Kermack-McKendrick age-structured SIR model, and it reduces to the Hethcote-Tudor integral equation SIR model. The derivation from a stochastic process is extended to discrete time, providing a stable numerical method for solving the model equations.We have carried out simulations of the fractional order recovery model showing convergence to equilibrium states. The number of infecteds in the endemic equilibrium state increases as the fractional order of the derivative tends to zero.

There is growing evidence that many neurodegenerative disease processes involve the proliferation, accumulation and spread of pathogenic proteins. The transport of proteins in the brain is typically hindered on small scales by micro-domain traps and binding sites but it may be enhanced on larger scales by neuronal pathways identified as white matter transport networks. We have introduced a mathematical network model to simulate a pathogenic protein neurodegenerative disease in the brain taking into account the anomalous transport. The proliferation and accumulation of pathogenic proteins is modelled using a set of reaction kinetics equations on the nodes of a network. Transport of the proteins on the network is modelled as a continuous time random walk with power law distributed waiting times on the nodes. This power law waiting time distribution is shown to be consistent with anomalously slowed diffusion on local scales but transport is enhanced on larger scales by the jumps between nodes. The model reveals that the disease spreads as a propagating front throughout the brain. The anomalous behaviour leads to a lessor variation in the concentration of pathogenic proteins. The enhanced transport on the network ensures that the approach to equilibrium is dominated by the short time behaviour of the waiting time density, hence the effects of subdiffusion are not as pronounced as in a spatial continuum.

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.

We have introduced a new explicit numerical method, based on a discrete stochastic process, for solving a class of fractional partial differential equations that model reaction subdiffusion. The scheme is derived from the master equations for the evolution of the probability density of a sum of discrete time random walks. We show that the diffusion limit of the master equations recovers the fractional partial differential equation of interest. This limiting procedure guarantees the consistency of the numerical scheme. The positivity of the solution and stability results are simply obtained, provided that the underlying process is well posed. We also show that the method can be applied to standard reaction-diffusion equations. This work highlights the broader applicability of using discrete stochastic processes to provide numerical schemes for partial differential equations, including fractional partial differential equations.

There has recently been considerable interest in using a nonstandardpiecewise approximation to formulate fractional order differential equations asdifference equations that describe the same dynamical behaviour and are moreamenable to a dynamical systems analysis. Unfortunately, due to mistakes in thefundamental papers, the difference equations formulated through this process donot capture the dynamics of the fractional order equations. We show that thecorrect application of this nonstandard piecewise approximation leads to a oneparameter family of fractional order differential equations that converges tothe original equation as the parameter tends to zero. A closed formed solutionexists for each member of this family and leads to the formulation of adifference equation that is of increasing order as time steps are taken. Whilstthis does not lead to a simplified dynamical analysis it does lead to anumerical method for solving the fractional order differential equation. Themethod is shown to be equivalent to a quadrature based method, despite the factthat it has not been derived from a quadrature. The method can be implementedwith non-uniform time steps. An example is provided showing that the differenceequation can correctly capture the dynamics of the underlying fractionaldifferential equation.

Continuous time random walks, which generalize random walks by adding a stochastic time between jumps, provide a useful description of stochastic transport at mesoscopic scales. The continuous time random walk model can accommodate certain features, such as trapping, which are not manifest in the standard macroscopic diffusion equation. The trapping is incorporated through a waiting time density, and a fractional diffusion equation results from a power law waiting time. A generalized continuous time random walk model with biased jumps has been used to consider transport that is also subject to an external force. Here we have derived the master equations for continuous time random walks with space- and time-dependent forcing for two cases: when the force is evaluated at the start of the waiting time and at the end of the waiting time. The differences persist in low order spatial continuum approximations; however, the two processes are shown to be governed by the same Fokker-Planck equations in the diffusion limit. Thus the fractional Fokker-Planck equation with space- and time-dependent forcing is robust to these changes in the underlying stochastic process.

The continuous time random walk, introduced in the physics literature by Montroll and Weiss, has been widely used to model anomalous diffusion in external force fields. One of the features of this model is that the governing equations for the evolution of the probability density function, in the diffusion limit, can generally be simplified using fractional calculus. This has in turn led to intensive research efforts over the past decade to develop robust numerical methods for the governing equations, represented as fractional partial differential equations. Here we introduce a discrete time random walk that can also be used to model anomalous diffusion in an external force field. The governing evolution equations for the probability density function share the continuous time random walk diffusion limit. Thus the discrete time random walk provides a novel numerical method for solving anomalous diffusion equations in the diffusion limit, including the fractional Fokker-Planck equation. This method has the clear advantage that the discretisation of the diffusion limit equation, which is necessary for numerical analysis, is itself a well defined physical process. Some examples using the discrete time random walk to provide numerical solutions of the probability density function for anomalous subdiffusion, including forcing, are provided.

The TGF-b superfamily cytokine MIC-1/GDF15 circulates in the blood of healthy humans. Its levels rise substantially in cancer and other diseases and this may sometimes lead to development of an anorexia/cachexia syndrome. This is mediated by a direct action of MIC- 1/GDF15 on feeding centres in the hypothalamus and brainstem. More recent studies in germline gene deleted mice also suggest that this cytokine may play a role in physiological regulation of energy homeostasis. To further characterize the role of MIC-1/GDF15 in physiological regulation of energy homeostasis in man, we have examined diurnal and food associated variation in serum levels and whether variation in circulating levels relate to BMI in human monozygotic twin pairs. We found that the within twin pair differences in serum MIC-1/GDF15 levels were significantly correlated with within twin pair differences in BMI, suggesting a role for MIC-1/GDF15 in the regulation of energy balance in man. MIC-1/ GDF15 serum levels altered slightly in response to a meal, but comparison with variation its serum levels over a 24hour period suggested that these changes are likely to be due to bimodal diurnal variation which can alter serum MIC-1/GDF15 levels by about plus or minus 10% from the mesor. The lack of a rapid and substantial postprandial increase in MIC-1/ GDF15 serum levels suggests that MIC1/GDF15 is unlikely to act as a satiety factor. Taken together, our findings suggest that MIC-1/GDF15 may be a physiological regulator of energy homeostasis in man, most probably due to actions on long-term regulation of energy homeostasis.

We demonstrate an efficient high-precision optical spectroscopy technique for single trapped ions with nonclosed 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 cotrapped 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 D3/22→P1/22 transition in calcium, resulting in a transition frequency of f=346 000 234 867(96) kHz. Furthermore, we determine the isotope shift of this transition and the S1/22→P1/22 transition for Ca+42, Ca+44, and Ca+48 ions relative to Ca+40 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.

The stepping direction of linear molecular motors is usually defined by a spatial asymmetry of the motor, its track, or both. Here we present a model for a molecular walker that undergoes biased directional motion along a symmetric track in the presence of a temporally symmetric chemical cycle. Instead of using asymmetry, directionality is achieved by persistence. At small load force the walker can take on average thousands of steps in a given direction until it stochastically reverses direction. We discuss a specific experimental implementation of a synthetic motor based on this design and find, using Langevin and Monte Carlo simulations, that a realistic walker can work against load forces on the order of picoNewtons with an efficiency of ∼18%, comparable to that of kinesin. In principle, the walker can be turned into a permanent motor by externally monitoring the walker's momentary direction of motion, and using feedback to adjust the direction of a load force. We calculate the thermodynamic cost of using feedback to enhance motor performance in terms of the Shannon entropy, and find that it reduces the efficiency of a realistic motor only marginally. We discuss the implications for natural protein motor performance in the context of the strong performance of this design based only on a thermal ratchet.

Oscillations of the Min protein system are involved in the correct midcell placement of the divisome during Escherichia coli cell division. Based on molecular interactions of the Min system, we formulated a mathematical model that reproduces Min patterning during cell growth and division. Specifically, the increase in the residence time of MinD attached to the membrane as its own concentration increases, is accounted for by dimerisation of membrane- bound MinD and its interaction with MinE. Simulation of this system generates unparalleled correlation between the waveshape of experimental and theoretical MinD distributions, suggesting that the dominant interactions of the physical system have been successfully incorporated into the model. For cells where MinD is fully-labelled with GFP, the model reproduces the stationary localization of MinD-GFP for short cells, followed by oscillations from pole to pole in larger cells, and the transition to the symmetric distribution during cell filamentation. Cells containing a secondary, GFP-labelled MinD display a contrasting pattern. The model is able to account for these differences, including temporary midcell localization just prior to division, by increasing the rate constant controlling MinD ATPase and heterotetramer dissociation. For both experimental conditions, the model can explain how cell division results in an equal distribution of MinD and MinE in the two daughter cells, and accounts for the temperature dependence of the period of Min oscillations. Thus, we show that while other interactions may be present, they are not needed to reproduce the main characteristics of the Min system in vivo.

We have developed a mathematical model for in-host virus dynamics that includes spatial chemotaxis and diffusion across a two-dimensional surface representing the vaginal or rectal epithelium at primary HIV infection. A linear stability analysis of the steady state solutions identified conditions for Turing instability pattern formation. We have solved the model equations numerically using parameter values obtained from previous experimental results for HIV infections. Simulations of the model for this surface show hot spots of infection. Understanding this localization is an important step in the ability to correctly model early HIV infection. These spatial variations also have implications for the development and effectiveness of microbicides against HIV.