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In this paper, we introduce a new method based on a library of chemical reactions for constructing a system of ordinary differential equations from stochastic simulations arising from an agent-based model. The advantage of this approach is that this library respects any coupling between systems components, whereas the SINDy algorithm (introduced by Brunton et al.) treats the individual components as decoupled from one another. Another advantage of our approach is that we can use a non-negative least squares algorithm to find the non-negative rate constants in a very robust, stable and simple manner. We illustrate our ideas on an agent-based model of tumour growth on a 2D lattice. [ABSTRACT FROM AUTHOR]

Mathematics provides us with tools to capture and explain phenomena in everyday biology, even at the nanoscale. The most regularly applied technique to biology is differential equations. In this article, we seek to present how differential equation models of biological phenomena, particularly the flow through ion channels, can be used to motivate and teach differential equations. Ion channels on the cell membrane allow the passage of ions from one side of the membrane to the other. The movement of these ions drives crucial processes such as the beating of our hearts. Using a system of two ordinary differential equations it is possible to capture the movement across ion channels that are opening and closing. Then using standard undergraduate techniques, we can predict how these channels behave in the long-term. In this work, we discuss how this example can be used to create tangible links to mathematical equations and motivate the teaching of techniques such as differentiation, integration, algebraic manipulation and equilibrium analysis. Furthermore, we show how a simple reformulation of this model into a stochastic setting using Gillespie's Stochastic Simulation Algorithm can allow us to capture the noise in ion channel flow. [ABSTRACT FROM AUTHOR]

In order to construct an algorithm for homogeneous diffusive motion that lives on a sphere, we consider the equivalent process of a randomly rotating spin vector of constant length. By introducing appropriate sets of random variables based on cross products, we construct families of methods with increasing efficacy that exactly preserve the spin modulus for every realisation. This is done by exponentiating an antisymmetric matrix whose entries are these random variables that are Gaussian in the simplest case. [ABSTRACT FROM AUTHOR]

We study pseudospectra of matrices arising in models of ion channel dynamics, bimolecular reactions, and Michaelis--Menten enzyme kinetics. We show that matrices for ion channel models and for bimolecular reactions are examples of a special class known as twisted Toeplitz matrices. Using Michaelis--Menten enzyme kinetics as an example, we suggest more complicated reactions cannot be modeled simply by the twisted Toeplitz ideas currently in the literature, and thus we propose a generalization to what we call block twisted Toeplitz matrices. Furthermore, we show that certain simple cases of ion channel models always exhibit a phenomenon known as localization. This result comes by studying the symbol of the operator and an application of a theorem of Trefethen and Chapman. We further observe that a larger magnitude of the so-called twist-ratio, a number that comes from the symbol, is associated with stronger localization. For certain bimolecular reactions, specialized to a dimerization, we show by the same theorem that there are large regions of parameter space where localization is guaranteed and that this is always guaranteed in the limit as the matrix size becomes very large. However, we also find examples for these reactions where the hypotheses of the theorem of Trefethen and Chapman do not hold, and yet in numerical experiments, we still observe localization. Finally we return to the first of our matrix models for a more detailed study to give further insight into important applications to cardiac ion channels and in particular sodium channels. We demonstrate behavior similar to the so-called cutoff phenomenon of Markov processes, and we show that the stationary distribution, which is so crucial to applications of these models to ion channels and especially sodium channels, can be extremely localized. [ABSTRACT FROM AUTHOR]

Stochastic differential equations (SDE) are a powerful tool to model biological regulatory processes with intrinsic and extrinsic noise. However, numerical simulations of SDE models may be problematic if the values of noise terms are negative and large, which is not realistic for biological systems since the molecular copy numbers or protein concentrations should be non-negative. To address this issue, we propose the composite Patankar-Euler methods to obtain positive simulations of SDE models. A SDE model is separated into three parts, namely, the positive-valued drift terms, negative-valued drift terms, and diffusion terms. We first propose the deterministic Patankar-Euler method to avoid negative solutions generated from the negative-valued drift terms. The stochastic Patankar-Euler method is designed to avoid negative solutions generated from both the negative-valued drift terms and diffusion terms. These Patankar-Euler methods have the strong convergence order of a half. The composite Patankar-Euler methods are the combinations of the explicit Euler method, deterministic Patankar-Euler method, and stochastic Patankar-Euler method. Three SDE system models are used to examine the effectiveness, accuracy, and convergence properties of the composite Patankar-Euler methods. Numerical results suggest that the composite Patankar-Euler methods are effective methods to ensure positive simulations when any appropriate stepsize is used. [ABSTRACT FROM AUTHOR]

The jaguar (Panthera onca) is the dominant predator in Central and South America, but is now considered near‐threatened. Estimating jaguar population size is difficult, due to uncertainty in the underlying dynamical processes as well as highly variable and sparse data. We develop a stochastic temporal model of jaguar abundance in the Peruvian Amazon, taking into account prey availability, under various climate change scenarios. The model is calibrated against existing data sets and an elicitation study in Pacaya Samiria. In order to account for uncertainty and variability, we construct a population of models over four key parameters, namely three scaling parameters for aquatic, small land, and large land animals and a hunting index. We then use this population of models to construct probabilistic evaluations of jaguar populations under various climate change scenarios characterized by increasingly severe flood and drought events and discuss the implications on jaguar numbers. Results imply that jaguar populations exhibit some robustness to extreme drought and flood, but that repeated exposure to these events over short periods can result in rapid decline. However, jaguar numbers could return to stability—albeit at lower numbers—if there are periods of benign climate patterns and other relevant factors are conducive. [ABSTRACT FROM AUTHOR]

Quantitative modelling is increasingly important in cancer research, helping to integrate myriad diverse experimental data into coherent pictures of the disease and able to discriminate between competing hypotheses or suggest specific experimental lines of enquiry and new approaches to therapy. Here, we review a diverse set of mathematical models of cancer cell plasticity (a process in which, through genetic and epigenetic changes, cancer cells survive in hostile environments and migrate to more favourable environments, respectively), tumour growth, and invasion. Quantitative models can help to elucidate the complex biological mechanisms of cancer cell plasticity. In this review, we discuss models of plasticity, tumour progression, and metastasis under three broadly conceived mathematical modelling techniques: discrete, continuum, and hybrid, each with advantages and disadvantages. An emerging theme from the predictions of many of these models is that cell escape from the tumour microenvironment (TME) is encouraged by a combination of physiological stress locally (e.g., hypoxia), external stresses (e.g., the presence of immune cells), and interactions with the extracellular matrix. We also discuss the value of mathematical modelling for understanding cancer more generally. [ABSTRACT FROM AUTHOR]

Highlights • An optimal control-ordinary differential equation model is constructed for basic fibroblast growth factor and binding agent interactions. • Theoretical predictions for optimal bFGF delivery are proposed for chronic wounds. • Insight is provided into the priority of minimising bFGF delivery and optimising bound bFGF distribution. Abstract In this paper, we present an ordinary differential equation model depicting the interactions of basic fibroblast growth factor (bFGF) and its binding agents in a chronic wound. The delivery of bFGF was treated as a control variable and is coupled to an objective functional. By optimising the objective functional with respect to the control, predictions for optimal delivery rates of bFGF are proposed. The optimal control is then validated by comparing the cost of the objective functional for the optimal delivery rate and several alternative delivery rates. This paper addresses two objectives of effective drug delivery to chronic wounds. The first is to provide insight for the priority of delivering bFGF: to minimise the quantity of bFGF, or to optimise the distribution of bound bFGF. For effective concentrations of bound bFGF, the optimisation of bound bFGF must be prioritised over the minimisation of bFGF delivered. The second objective is to comment on the effect of the proteolytic environment within the wound, with the concentration of bound bFGF starting to decrease late in the treatment period for highly proteolytic environments. This will lead to long term complications with wound closure after the treatment has been completed. Also, it was found that for highly proteolytic environments, the cost of delivering bFGF increased. The need for optimal drug delivery is made apparent by the burden of chronic wounds on the medical industry across the developed world. [ABSTRACT FROM AUTHOR]

Stochastic differential equations (SDEs) are a powerful tool to model fluctuations and uncertainty in complex systems. Although numerical methods have been designed to simulate SDEs effectively, it is still problematic when numerical solutions may be negative, but application problems require positive simulations. To address this issue, we propose balanced implicit Patankar–Euler methods to ensure positive simulations of SDEs. Instead of considering the addition of balanced terms to explicit methods in existing balanced methods, we attempt the deletion of possible negative terms from the explicit methods to maintain positivity of numerical simulations. The designed balanced terms include negative-valued drift terms and potential negative diffusion terms. The proposed method successfully addresses the issue of divisions with very small denominators in our recently designed stochastic Patankar method. Stability analysis shows that the balanced implicit Patankar–Euler method has much better stability properties than our recently designed composite Patankar–Euler method. Four SDE systems are used to examine the effectiveness, accuracy, and convergence properties of balanced implicit Patankar–Euler methods. Numerical results suggest that the proposed balanced implicit Patankar–Euler method is an effective and efficient approach to ensure positive simulations when any appropriate stepsize is used in simulating SDEs of biological regulatory systems. [ABSTRACT FROM AUTHOR]

There has been considerable recent work on the development of energy conserving one-step methods that are not symplectic. Here we extend these ideas to stochastic Hamiltonian problems with additive noise and show that there are classes of Runge-Kutta methods that are very effective in preserving the expectation of the Hamiltonian, but care has to be taken in how the Wiener increments are sampled at each timestep. Some numerical simulations illustrate the performance of these methods. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we extend the ideas of Brugnano, Iavernaro and Trigiante in their development of HBVM () methods to construct symplectic Runge–Kutta methods for all values of and with . However, these methods do not see the dramatic performance improvement that HBVMs can attain. Nevertheless, in the case of additive stochastic Hamiltonian problems an extension of these ideas, which requires the simulation of an independent Wiener process at each stage of a Runge–Kutta method, leads to methods that have very favourable properties. These ideas are illustrated by some simple numerical tests for the modified midpoint rule. [Copyright &y& Elsevier]

In the last decade, Australia has experienced an overall decline in red cell demand, but there has been an increased need for phenotyped matched red cells. Lifeblood and mathematicians from Queensland universities have developed a probabilistic model to determine the percentage of the donor panel that would need extended antigen typing to meet this increasing demand, and an estimated timeline to achieve the optimum required phenotyped (genotyped) panel. Mathematical modelling, based on Multinomial distributions, was used to provide guidance on the percentage of typed donor panel needed, based on recent historical blood request data and the current donor panel size. Only antigen combinations determined to be uncommon, but not rare, were considered. Simulations were run to attain at least 95% success percentage. Modelling predicted a target of 38% of the donor panel, or 205,000 donors, would need to be genotyped to meet the current demand. If 5% of weekly returning donors were genotyped, this target would be reached within 12 years. For phenotyping, 35% or 188,000 donors would need to be phenotyped to meet Lifeblood's demand. With the current level of testing, this would take eight years but could be performed within three years if testing was increased to 9% of weekly returning donors. An additional 26,140 returning donors need to be phenotyped annually to maintain this panel. This mathematical model will inform business decisions and assist Lifeblood in determining the level of investment required to meet the desired timeline to achieve the optimum donor panel size. [ABSTRACT FROM AUTHOR]

We study reflectionless properties at the boundary for the wave equation in one space dimension and time, in terms of a well-known matrix that arises from a simple discretisation of space. It is known that all matrix functions of the familiar second difference matrix representing the Laplacian in this setting are the sum of a Toeplitz matrix and a Hankel matrix. The solution to the wave equation is one such matrix function. Here, we study the behaviour of the corresponding waves that we call Toeplitz waves and Hankel waves. We show that these waves can be written as certain linear combinations of even Bessel functions of the first kind. We find exact and explicit formulae for these waves. We also show that the Toeplitz and Hankel waves are reflectionless on even, respectively odd, traversals of the domain. Our analysis naturally suggests a new method of computer simulation that allows control, so that it is possible to choose — in advance — the number of reflections. An attractive result that comes out of our analysis is the appearance of the well-known shift matrix, and also other matrices that might be thought of as Hankel versions of the shift matrix. By revealing the algebraic structure of the solution in terms of shift matrices, we make it clear how the Toeplitz and Hankel waves are indeed reflectionless at the boundary on even or odd traversals. Although the subject of the reflectionless boundary condition has a long history, we believe the point of view that we adopt here in terms of matrix functions is new. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the Stochastic Adams-Bashforth (SAB) and Stochastic Adams-Moulton (SAM) methods as an extension of the τ-leaping framework to past information. Using the Θ-trapezoidal τ-leap method of weak order two as a starting procedure, we show that the k-step SAB method with k ≥ 3 is order three in the mean and correlation, while a predictor-corrector implementation of the SAM method is weak order three in the mean but only order one in the correlation. These convergence results have been derived analytically for linear problems and successfully tested numerically for both linear and non-linear systems. A series of additional examples have been implemented in order to demonstrate the efficacy of this approach. [ABSTRACT FROM AUTHOR]

In this paper, we study the class of mixed-index time fractional differential equations in which different components of the problem have different time fractional derivatives on the left-hand side. We prove a theorem on the solution of the linear system of equations, which collapses to the well-known Mittag–Leffler solution in the case that the indices are the same and also generalises the solution of the so-called linear sequential class of time fractional problems. We also investigate the asymptotic stability properties of this class of problems using Laplace transforms and show how Laplace transforms can be used to write solutions as linear combinations of generalised Mittag–Leffler functions in some cases. Finally, we illustrate our results with some numerical simulations. [ABSTRACT FROM AUTHOR]

Dynamical properties of numerically approximated discrete systems may become inconsistent with those of the corresponding continuous-time system. We present a qualitative analysis of the dynamical properties of two-species Lotka-Volterra and Ricker-type predator-prey systems under discrete and continuous settings. By creating an arbitrary time discretisation, we obtain stability conditions that preserve the characteristics of continuous-time models and their numerically approximated systems. Here, we show that even small changes to some of the model parameters may alter the system dynamics unless an appropriate time discretisation is chosen to return similar dynamical behaviour to what is observed in the corresponding continuous-time system. We also found similar dynamical properties of the Ricker-type predator-prey systems under certain conditions. Our results demonstrate the need for preliminary analysis to identify which dynamical properties of approximated discretised systems agree or disagree with the corresponding continuous-time systems. [ABSTRACT FROM AUTHOR]

In this paper, a class of new Magnus-type methods is proposed for non-commutative Itô stochastic differential equations (SDEs) with semi-linear drift term and semi-linear diffusion terms, based on Magnus expansion for non-commutative linear SDEs. We construct a Magnus-type Euler method, a Magnus-type Milstein method and a Magnus-type Derivative-free method, and give the mean-square convergence analysis of these methods. Numerical tests are carried out to present the efficiency of the proposed methods compared with the corresponding underlying methods and the specific performance of the simulation Itô integral algorithms is investigated. [ABSTRACT FROM AUTHOR]

Abstract: Time delay is an important aspect in the modelling of genetic regulation due to slow biochemical reactions such as gene transcription and translation, and protein diffusion between the cytosol and nucleus. In this paper we introduce a general mathematical formalism via stochastic delay differential equations for describing time delays in genetic regulatory networks. Based on recent developments with the delay stochastic simulation algorithm, the delay chemical master equation and the delay reaction rate equation are developed for describing biological reactions with time delay, which leads to stochastic delay differential equations derived from the Langevin approach. Two simple genetic regulatory networks are used to study the impact of intrinsic noise on the system dynamics where there are delays. [Copyright &y& Elsevier]

In this paper we give an overview of some very recent work, as well as presenting a new approach, on the stochastic simulation of multi-scaled systems involving chemical reactions. In many biological systems (such as genetic regulation and cellular dynamics) there is a mix between small numbers of key regulatory proteins, and medium and large numbers of molecules. In addition, it is important to be able to follow the trajectories of individual molecules by taking proper account of the randomness inherent in such a system. We describe different types of simulation techniques (including the stochastic simulation algorithm, Poisson Runge–Kutta methods and the balanced Euler method) for treating simulations in the three different reaction regimes: slow, medium and fast. We then review some recent techniques on the treatment of coupled slow and fast reactions for stochastic chemical kinetics and present a new approach which couples the three regimes mentioned above. We then apply this approach to a biologically inspired problem involving the expression and activity of LacZ and LacY proteins in E. coli, and conclude with a discussion on the significance of this work. [Copyright &y& Elsevier]

The article discusses a study regarding the use of calibrate populations of models (POMs) to data density to unlock data sets from cardiac electrophysiology. Topics discussed include the impact of the approach on the different hypotheses about variability underlying complex systems, the application of Courtemanche-Ramirez-Nattel (CRN) model to construct silico POMs, and the improvement of calibration of populations of models to data through Sequential Monte Carlo (SMC).

In this work, we discuss the use of a local model developed previously [1] that describes the multiphase flow of gaseous species and liquid water within a single coal seam to investigate the gas production from a spatially heterogeneous production field. The field is located within the Surat Basin in Queensland, and is composed of a total of 80 production wells spread over a region covering approximately 36 km 2 . However, not every well is producing gas at any one time and so in this work we take a subset of 42 wells that are the top-producing wells in terms of total gas volume. We utilise a population of models approach to understand the variability in the underlying physical processes, and as a mechanism for dealing with the spatial heterogeneity that arises due to geological variation across the field. We are able to simultaneously obtain a family of parameter sets for each of these wells, in which each set in the family yields a predicted cumulative total gas production curve that matches the measured cumulative production curve for a given well to within an allowable limit of error. By analysing the results of this population of models approach we can identify the similarities between wells based on the parameter distributions, and understand the sensitivity of key model parameters. We show by example that high correlation between wells based on their parameter values may be an indicator of their similarity. A combinatorial sum of the predicted gas production is compared against the individual gas volumes (given in terms of percentage of the total volume) measured at the compression facility as a way of further calibrating a subpopulation of models. [ABSTRACT FROM AUTHOR]

In this work we discuss the development of a mathematical model to predict the shift in gas composition observed over time from a producing CSG (coal seam gas) well, and investigate the effect that physical properties of the coal seam have on gas production. A detailed (local) one-dimensional, two-scale mathematical model of a coal seam has been developed. The model describes the competitive adsorption and desorption of three gas species (CH 4 , CO 2 and N 2 ) within a microscopic, porous coal matrix structure. The (diffusive) flux of these gases between the coal matrices (microscale) and a cleat network (macroscale) is accounted for in the model. The cleat network is modelled as a one-dimensional, volume averaged, porous domain that extends radially from a central well. Diffusive and advective transport of the gases occurs within the cleat network, which also contains liquid water that can be advectively transported. The water and gas phases are assumed to be immiscible. The driving force for the advection in the gas and liquid phases is taken to be a pressure gradient with capillarity also accounted for. In addition, the relative permeabilities of the water and gas phases are considered as functions of the degree of water saturation. [ABSTRACT FROM AUTHOR]

Representing a renewable source for cell replacement, neural stem cells have received substantial attention in recent years. The neurosphere assay represents a method to detect the presence of neural stem cells, however owing to a deficiency of specific and definitive markers to identify them, their quantification and the rate they expand is still indefinite. Here we propose a mathematical interpretation of the neurosphere assay allowing actual measurement of neural stem cell symmetric division frequency. The algorithm of the modeling demonstrates a direct correlation between the overall cell fold expansion over time measured in the sphere assay and the rate stem cells expand via symmetric division. The model offers a methodology to evaluate specifically the effect of diseases and treatments on neural stem cell activity and function. Not only providing new insights in the evaluation of the kinetic features of neural stem cells, our modeling further contemplates cancer biology as cancer stem-like cells have been suggested to maintain tumor growth as somatic stem cells maintain tissue homeostasis. Indeed, tumor stem cell's resistance to therapy makes these cells a necessary target for effective treatment. The neurosphere assay mathematical model presented here allows the assessment of the rate malignant stem-like cells expand via symmetric division and the evaluation of the effects of therapeutics on the self-renewal and proliferative activity of this clinically relevant population that drive tumor growth and recurrence. [ABSTRACT FROM AUTHOR]