An optimal investment consumption model for retirees with no health insurance

Retirees meet a number of problems as they are growing older which needs persistent attention. Hence, without a doubt, the outcomes of the financial markets influence the choices that people make when nearing retirement. In our model, the stock price dynamics follow Geometric Brownian motion (GBM) and our goal was to optimize the expected discounted utility of consumption and terminal wealth whilst considering health expenses. The investment return process comprises risk free asset and risky assets, and the health expenses. We choose power utility functions where comprehensive solutions for Hyperbolic Absolute Risk Aversion (HARA) utility functions are obtained and optimal investment, consumption and health expenditure strategies are derived by applying dynamic programming and variable change technique on the Hamilton-Jacobi-Bellman (HJB) equations. In our numerical results it showed various effects of some economic and market parameters on the optimal investment, consumption and health expense strategies. The inflation price market risk governs the amount invested in stock, bond and also how much to be put in health to sustain a given period of the retiree's lifetime. As the health welfare rate R increases, the proportion of wealth invested in the stock increases. We also investigated the effects of the high correlation coefficients and low correlation coefficients on consumption and income rate respectively. As the constant variance discounting coefficient increases, seasoned enterprise annuity retirees decrease their allocation to the risky assets. Finally, a numerical example is presented to depict the effects of financial parameters on the optimal investment strategy with health expenditure.

Multi-stage trajectory tracking of robot manipulators under stochastic environments

For robot manipulators composed of Lagrange subsystems driven by direct current (DC) motors under stochastic environments, multi-stage trajectory tracking is investigated in this paper. The main challenge is how to achieve the end-effector drive of manipulators from a given initial state to a final state. First, the inverse kinematics method and the partition of the task space are adopted to tackle multi-stage trajectory planning. Second, the adaptive backstepping technique is used to design tracking controller for stochastic Lagrangian subsystems. Then, based on the state-dependent switching signal, a multi-stage switched controller is designed for trajectory tracking of robot manipulators. All signals in the close-loop error switched system are bounded in probability, and the tracking error in mean square can be made arbitrarily small enough by parameters-tuning The effectiveness of the proposed control method is illustrated by simulation results.

Dynamic behaviors of a stochastic virus infection model with Beddington-DeAngelis incidence function, eclipse-stage and Ornstein-Uhlenbeck process

In this paper, we present a virus infection model that incorporates eclipse-stage and Beddington-DeAngelis function, along with perturbation in infection rate using logarithmic Ornstein-Uhlenbeck process. Rigorous analysis demonstrates that the stochastic model has a unique global solution. Through construction of appropriate Lyapunov functions and a compact set, combined with the strong law of numbers and Fatou's lemma, we obtain the existence of the stationary distribution under a critical condition, which indicates the long-term persistence of T-cells and virions. Moreover, a precise probability density function is derived around the quasi-equilibrium of the model, and spectral radius analysis is employed to identify critical condition for elimination of the virus. Finally, numerical simulations are presented to validate theoretical results, and the impact of some key parameters such as the speed of reversion, volatility intensity and mean infection rate are investigated.

A stochastic differential equation model for predator-avoidance fish schooling

This paper presents a mathematical model based on stochastic differential equations (SDEs) to depict the dynamics of a predator-prey system in an aquatic environment characterized by schooling behavior among the prey. The model employs a particle-like approach, incorporating attractive and repulsive forces, akin to phenomena observed in molecular physics, to capture the interactions among the constituent units. Two hunting tactics of the predator, center-attacking and nearest-attacking strategies, are integrated into the model. Numerical simulations of this model unveil four distinct predator-avoidance patterns exhibited by schooling prey: Split and Reunion, Split and Separate into Two Groups, Scattered, and Maintain Formation and Distance. Our results also confirm the effectiveness of large groups of schooling prey in mitigating predation risk, consistent with real-life observations in natural aquatic ecosystems. These findings validate the accuracy and applicability of our model.

Bayesian inference on the Allee effect in cancer cell line populations using time-lapse microscopy images

The Allee effect describes the phenomenon that the per capita reproduction rate increases along with the population density at low densities. Allee effects have been observed at all scales, including in microscopic environments where individual cells are taken into account. This is great interest to cancer research, as understanding critical tumour density thresholds can inform treatment plans for patients. In this paper, we introduce a simple model for cell division in the case where the cancer cell population is modelled as an interacting particle system. The rate of the cell division is dependent on the local cell density, introducing an Allee effect. We perform parameter inference of the key model parameters through Markov Chain Monte Carlo, and apply our procedure to two image sequences from a cervical cancer cell line. The inference method is verified on in silico data to accurately identify the key parameters, and results on the in vitro data strongly suggest an Allee effect.

Statistical Analysis of Two-Compartment Pharmacokinetic Models with Drug Non-adherence

Poor drug adherence is considered one of major barriers to achieving the clinical and public health benefits of many pharmacotherapies. In the current paper, we aim to investigate the impact of dose omission on the plasma concentrations of two-compartment pharmacokinetic models with two typical routes of drug administration, namely the intravenous bolus and extravascular first-order absorption. First, we reformulate the classical two-compartment pharmacokinetic models with a new stochastic feature, where a binomial random model of dose intake is integrated. Then, we formalize the explicit expressions of expectation and variance for trough concentrations and limit concentrations, with the latter proved of the existence and uniqueness for steady-state distribution. Moreover, we mathematically demonstrate the strict stationarity and ergodicity of trough concentrations as a Markov chain. In addition, we numerically simulate the impact of drug non-adherence to different extents on the variability and regularity of drug concentration and compare the drug pharmacokinetic preference between one and two compartment pharmacokinetic models. The results of sensitivity analysis also suggest the drug non-adherence as one of the most sensitive model parameters to the expectation of limit concentration. Our modelling and analytical approach can be integrated into the chronic disease models to estimate or quantitatively predict the therapy efficacy with drug pharmacokinetics presumably affected by random dose omissions.

Weak selection and stochastic evolutionary stability in a stochastic replicator dynamics

It is a very challenging problem whether natural selection is able to effectively resist the continuous disturbance of environmental noise such that the direction or outcome of evolution determined by the deterministic selection pressure will not be changed. By analyzing the impact of weak selection on the evolutionary stability of a stochastic replicator dynamics with n possible pure strategies, we found that the weak selection is able to enhance the evolutionary stability, that is, under weak selection, the stochastic evolutionary stability of the system is determined by the mean payoff matrix. This finding strongly implies that the weak selection should be regarded as an important mechanism to ensure evolutionary stability in stochastic environments.

Stochastic analysis of a COVID-19 model with effects of vaccination and different transition rates: Real data approach

This paper presents a stochastic model for COVID-19 that takes into account factors such as incubation times, vaccine effectiveness, and quarantine periods in the spread of the virus in symptomatically contagious populations. The paper outlines the conditions necessary for the existence and uniqueness of a global solution for the stochastic model. Additionally, the paper employs nonlinear analysis to demonstrate some results on the ergodic aspect of the stochastic model. The model is also simulated and compared to deterministic dynamics. To validate and demonstrate the usefulness of the proposed system, the paper compares the results of the infected class with actual cases from Iraq, Bangladesh, and Croatia. Furthermore, the paper visualizes the impact of vaccination rates and transition rates on the dynamics of infected people in the infected class.

Fast and precise inference on diffusivity in interacting particle systems

Particle systems made up of interacting agents is a popular model used in a vast array of applications, not the least in biology where the agents can represent everything from single cells to animals in a herd. Usually, the particles are assumed to undergo some type of random movements, and a popular way to model this is by using Brownian motion. The magnitude of random motion is often quantified using mean squared displacement, which provides a simple estimate of the diffusion coefficient. However, this method often fails when data is sparse or interactions between agents frequent. In order to address this, we derive a conjugate relationship in the diffusion term for large interacting particle systems undergoing isotropic diffusion, giving us an efficient inference method. The method accurately accounts for emerging effects such as anomalous diffusion stemming from mechanical interactions. We apply our method to an agent-based model with a large number of interacting particles, and the results are contrasted with a naive mean square displacement-based approach. We find a significant improvement in performance when using the higher-order method over the naive approach. This method can be applied to any system where agents undergo Brownian motion and will lead to improved estimates of diffusion coefficients compared to existing methods.

Path integral control of a stochastic multi-risk SIR pandemic model

In this paper a Feynman-type path integral control approach is used for a recursive formulation of a health objective function subject to a fatigue dynamics, a forward-looking stochastic multi-risk susceptible-infective-recovered (SIR) model with risk-group's Bayesian opinion dynamics toward vaccination against COVID-19. My main interest lies in solving a minimization of a policy-maker's social cost which depends on some deterministic weight. I obtain an optimal lock-down intensity from a Wick-rotated Schrödinger-type equation which is analogous to a Hamiltonian-Jacobi-Bellman (HJB) equation. My formulation is based on path integral control and dynamic programming tools facilitates the analysis and permits the application of algorithm to obtain numerical solution for pandemic control model.

Understanding noise in cell signalling in the prospect of drug-targets

The introduction of noise to signals can alter central regulatory switches of cellular processes leading to diseases. Noise is inherently present in the cellular signalling system and plays a decisive role in the input-output (I/O) relation. The current study aims to understand the noise tolerance of motif structures in the cell signalling processes. The vulnerability of a node to noise could be a significant factor in causing signalling error and need to be controlled. We developed stochastic differential equation (SDE) based mathematical models for different network motifs with two nodes and studied the association between motif structure and signal-noise relation. A two-dimensional parameter space analysis on motif sensitivity with noise and input signal variation was performed to classify and rank the motifs. Identifying sensitive motifs and their high druggability infers their significance in screening potential drug-target candidates. Finally, we proposed a theoretical framework to identify nodes from a network as potential drug targets. We applied this mathematical formalism to three cancer networks to identify drug-targets and validated them with existing databases.

Dynamical analysis of a stochastic non-autonomous SVIR model with multiple stages of vaccination

In this paper, we analyze the dynamics of a new proposed stochastic non-autonomous SVIR model, with an emphasis on multiple stages of vaccination, due to the vaccine ineffectiveness. The parameters of the model are allowed to depend on time, to incorporate the seasonal variation. Furthermore, the vaccinated population is divided into three subpopulations, each one representing a different stage. For the proposed model, we prove the mathematical and biological well-posedness. That is, the existence of a unique global almost surely positive solution. Moreover, we establish conditions under which the disease vanishes or persists. Furthermore, based on stochastic stability theory and by constructing a suitable new Lyapunov function, we provide a condition under which the model admits a non-trivial periodic solution. The established theoretical results along with the performed numerical simulations exhibit the effect of the different stages of vaccination along with the stochastic Gaussian noise on the dynamics of the studied population.

Modeling ATP-mediated endothelial cell elongation on line patterns

Endothelial cell (EC) migration is crucial for a wide range of processes including vascular wound healing, tumor angiogenesis, and the development of viable endovascular implants. We have previously demonstrated that ECs cultured on 15-μm wide adhesive line patterns exhibit three distinct migration phenotypes: (a) “running” cells that are polarized and migrate continuously and persistently on the adhesive lines with possible spontaneous directional changes, (b) “undecided” cells that are highly elongated and exhibit periodic changes in the direction of their polarization while maintaining minimal net migration, and (c) “tumbling-like” cells that migrate persistently for a certain amount of time but then stop and round up for a few hours before spreading again and resuming migration. Importantly, the three migration patterns are associated with distinct profiles of cell length. Because of the impact of adenosine triphosphate (ATP) on cytoskeletal organization and cell polarization, we hypothesize that the observed differences in EC length among the three different migration phenotypes are driven by differences in intracellular ATP levels. In the present work, we develop a mathematical model that incorporates the interactions between cell length, cytoskeletal (F-actin) organization, and intracellular ATP concentration. An optimization procedure is used to obtain the model parameter values that best fit the experimental data on EC lengths. The results indicate that a minimalist model based on differences in intracellular ATP levels is capable of capturing the different cell length profiles observed experimentally.

Physical-stochastic continuous-time identification of a forced Duffing oscillator

Despite the simplicity of the Duffing oscillator, its dynamical behaviour is extremely rich. Hence, the Duffing equations are used to describe the dynamic behaviour of many real-world nonlinear systems for a wide range of frequency bands and amplitude of the excitation signal in basic sciences and engineering. For example, the Duffing oscillator has been successfully used to model a variety of physical processes such as stiffening springs, beam buckling, nonlinear electronic circuits, superconducting Josephson parametric amplifiers, and ionisation waves in plasmas etc. Therefore, the identification of the Duffing oscillator model/parameters directly from the measured input-output data is a topic of active research in many scientific fields In this paper, we use the concept of stochastic differential equations (SDEs) to identify a model of the Duffing oscillator. SDE-based grey-box models allow us to capture the underlying mathematical structure describing the physics of the system (e.g. the original Duffing equations) using the drift term and explicitly handling of model uncertainty (or the process noise) using the diffusion term whereas the measurement uncertainty is modelled using the measurement noise term respectively. In this paper, we propose a slight variation of the maximum likelihood estimation framework used for the identification of SDEs based grey-box models yielding improved performance for long-term predictions. The proposed framework is combined with an iterative residual analysis to develop a grey-box model of the forced Duffing oscillator. The benchmark data from the so-called Brussels "Silverbox system", which is an electrical circuit mimicking the forced Duffing oscillator dynamics is used for the identification purpose. Finally, the identified model performance (the simulation errors) is compared with the existing results available in the literature.

Approximation of SDEs: a stochastic sewing approach

We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of Lê (Electron J Probab 25:55, 2020. 10.1214/20-EJP442). This approach allows one to exploit regularization by noise effects in obtaining convergence rates. In our first application we show convergence (to our knowledge for the first time) of the Euler–Maruyama scheme for SDEs driven by fractional Brownian motions with non-regular drift. When the Hurst parameter is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H\in (0,1)$$\end{document}H∈(0,1) and the drift is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}^\alpha $$\end{document}Cα, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in [0,1]$$\end{document}α∈[0,1] and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >1-1/(2H)$$\end{document}α>1-1/(2H), we show the strong \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_p$$\end{document}Lp and almost sure rates of convergence to be \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$((1/2+\alpha H)\wedge 1) -\varepsilon $$\end{document}((1/2+αH)∧1)-ε, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document}ε>0. Our conditions on the regularity of the drift are optimal in the sense that they coincide with the conditions needed for the strong uniqueness of solutions from Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016. 10.1016/j.spa.2016.02.002). In a second application we consider the approximation of SDEs driven by multiplicative standard Brownian noise where we derive the almost optimal rate of convergence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/2-\varepsilon $$\end{document}1/2-ε of the Euler–Maruyama scheme for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}^\alpha $$\end{document}Cα drift, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ,\alpha >0$$\end{document}ε,α>0.

Approximation of SDEs: a stochastic sewing approach

We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of Lê (Electron J Probab 25:55, 2020. 10.1214/20-EJP442). This approach allows one to exploit regularization by noise effects in obtaining convergence rates. In our first application we show convergence (to our knowledge for the first time) of the Euler-Maruyama scheme for SDEs driven by fractional Brownian motions with non-regular drift. When the Hurst parameter is H ∈ ( 0 , 1 ) and the drift is C α , α ∈ [ 0 , 1 ] and α > 1 - 1 / ( 2 H ) , we show the strong L p and almost sure rates of convergence to be ( ( 1 / 2 + α H ) ∧ 1 ) - ε , for any ε > 0 . Our conditions on the regularity of the drift are optimal in the sense that they coincide with the conditions needed for the strong uniqueness of solutions from Catellier and Gubinelli (Stoch Process Appl 126(8):2323-2366, 2016. 10.1016/j.spa.2016.02.002). In a second application we consider the approximation of SDEs driven by multiplicative standard Brownian noise where we derive the almost optimal rate of convergence 1 / 2 - ε of the Euler-Maruyama scheme for C α drift, for any ε , α > 0 .

A general theory of coexistence and extinction for stochastic ecological communities

We analyze a general theory for coexistence and extinction of ecological communities that are influenced by stochastic temporal environmental fluctuations. The results apply to discrete time (stochastic difference equations), continuous time (stochastic differential equations), compact and non-compact state spaces and degenerate or non-degenerate noise. In addition, we can also include in the dynamics auxiliary variables that model environmental fluctuations, population structure, eco-environmental feedbacks or other internal or external factors. We are able to significantly generalize the recent discrete time results by Benaim and Schreiber (J Math Biol 79:393-431, 2019) to non-compact state spaces, and we provide stronger persistence and extinction results. The continuous time results by Hening and Nguyen (Ann Appl Probab 28(3):1893-1942, 2018a) are strengthened to include degenerate noise and auxiliary variables. Using the general theory, we work out several examples. In discrete time, we classify the dynamics when there are one or two species, and look at the Ricker model, Log-normally distributed offspring models, lottery models, discrete Lotka-Volterra models as well as models of perennial and annual organisms. For the continuous time setting we explore models with a resource variable, stochastic replicator models, and three dimensional Lotka-Volterra models.

Memory-driven movement model for periodic migrations

We propose a model for memory-based movement of an individual. The dynamics are modeled by a stochastic differential equation, coupled with an eikonal equation, whose potential depends on the individual's memory and perception. Under a simple periodic environment, we discover that both long and short-term memory with appropriate time scales are essential for producing expected periodic migrations.

A practical guide to pseudo-marginal methods for computational inference in systems biology

For many stochastic models of interest in systems biology, such as those describing biochemical reaction networks, exact quantification of parameter uncertainty through statistical inference is intractable. Likelihood-free computational inference techniques enable parameter inference when the likelihood function for the model is intractable but the generation of many sample paths is feasible through stochastic simulation of the forward problem. The most common likelihood-free method in systems biology is approximate Bayesian computation that accepts parameters that result in low discrepancy between stochastic simulations and measured data. However, it can be difficult to assess how the accuracy of the resulting inferences are affected by the choice of acceptance threshold and discrepancy function. The pseudo-marginal approach is an alternative likelihood-free inference method that utilises a Monte Carlo estimate of the likelihood function. This approach has several advantages, particularly in the context of noisy, partially observed, time-course data typical in biochemical reaction network studies. Specifically, the pseudo-marginal approach facilitates exact inference and uncertainty quantification, and may be efficiently combined with particle filters for low variance, high-accuracy likelihood estimation. In this review, we provide a practical introduction to the pseudo-marginal approach using inference for biochemical reaction networks as a series of case studies. Implementations of key algorithms and examples are provided using the Julia programming language; a high performance, open source programming language for scientific computing (https://github.com/davidwarne/Warne2019_GuideToPseudoMarginal).

Spectral density-based and measure-preserving ABC for partially observed diffusion processes. An illustration on Hamiltonian SDEs

Approximate Bayesian computation (ABC) has become one of the major tools of likelihood-free statistical inference in complex mathematical models. Simultaneously, stochastic differential equations (SDEs) have developed to an established tool for modelling time-dependent, real-world phenomena with underlying random effects. When applying ABC to stochastic models, two major difficulties arise: First, the derivation of effective summary statistics and proper distances is particularly challenging, since simulations from the stochastic process under the same parameter configuration result in different trajectories. Second, exact simulation schemes to generate trajectories from the stochastic model are rarely available, requiring the derivation of suitable numerical methods for the synthetic data generation. To obtain summaries that are less sensitive to the intrinsic stochasticity of the model, we propose to build up the statistical method (e.g. the choice of the summary statistics) on the underlying structural properties of the model. Here, we focus on the existence of an invariant measure and we map the data to their estimated invariant density and invariant spectral density. Then, to ensure that these model properties are kept in the synthetic data generation, we adopt measure-preserving numerical splitting schemes. The derived property-based and measure-preserving ABC method is illustrated on the broad class of partially observed Hamiltonian type SDEs, both with simulated data and with real electroencephalography data. The derived summaries are particularly robust to the model simulation, and this fact, combined with the proposed reliable numerical scheme, yields accurate ABC inference. In contrast, the inference returned using standard numerical methods (Euler-Maruyama discretisation) fails. The proposed ingredients can be incorporated into any type of ABC algorithm and directly applied to all SDEs that are characterised by an invariant distribution and for which a measure-preserving numerical method can be derived.

Data-driven inference for stationary jump-diffusion processes with application to membrane voltage fluctuations in pyramidal neurons

The emergent activity of biological systems can often be represented as low-dimensional, Langevin-type stochastic differential equations. In certain systems, however, large and abrupt events occur and violate the assumptions of this approach. We address this situation here by providing a novel method that reconstructs a jump-diffusion stochastic process based solely on the statistics of the original data. Our method assumes that these data are stationary, that diffusive noise is additive, and that jumps are Poisson. We use threshold-crossing of the increments to detect jumps in the time series. This is followed by an iterative scheme that compensates for the presence of diffusive fluctuations that are falsely detected as jumps. Our approach is based on probabilistic calculations associated with these fluctuations and on the use of the Fokker-Planck and the differential Chapman-Kolmogorov equations. After some validation cases, we apply this method to recordings of membrane noise in pyramidal neurons of the electrosensory lateral line lobe of weakly electric fish. These recordings display large, jump-like depolarization events that occur at random times, the biophysics of which is unknown. We find that some pyramidal cells increase their jump rate and noise intensity as the membrane potential approaches spike threshold, while their drift function and jump amplitude distribution remain unchanged. As our method is fully data-driven, it provides a valuable means to further investigate the functional role of these jump-like events without relying on unconstrained biophysical models.

Synchronization of stochastic mean field networks of Hodgkin-Huxley neurons with noisy channels

In this work we are interested in a mathematical model of the collective behavior of a fully connected network of finitely many neurons, when their number and when time go to infinity. We assume that every neuron follows a stochastic version of the Hodgkin-Huxley model, and that pairs of neurons interact through both electrical and chemical synapses, the global connectivity being of mean field type. When the leak conductance is strictly positive, we prove that if the initial voltages are uniformly bounded and the electrical interaction between neurons is strong enough, then, uniformly in the number of neurons, the whole system synchronizes exponentially fast as time goes to infinity, up to some error controlled by (and vanishing with) the channels noise level. Moreover, we prove that if the random initial condition is exchangeable, on every bounded time interval the propagation of chaos property for this system holds (regardless of the interaction intensities). Combining these results, we deduce that the nonlinear McKean-Vlasov equation describing an infinite network of such neurons concentrates, as time goes to infinity, around the dynamics of a single Hodgkin-Huxley neuron with chemical neurotransmitter channels. Our results are illustrated and complemented with numerical simulations.

Tree-Based Learning of Regulatory Network Topologies and Dynamics with Jump3

Inference of gene regulatory networks (GRNs) from time series data is a well-established field in computational systems biology. Most approaches can be broadly divided in two families: model-based and model-free methods. These two families are highly complementary: model-based methods seek to identify a formal mathematical model of the system. They thus have transparent and interpretable semantics but rely on strong assumptions and are rather computationally intensive. On the other hand, model-free methods have typically good scalability. Since they are not based on any parametric model, they are more flexible than model-based methods, but also less interpretable.In this chapter, we describe Jump3, a hybrid approach that bridges the gap between model-free and model-based methods. Jump3 uses a formal stochastic differential equation to model each gene expression but reconstructs the GRN topology with a nonparametric method based on decision trees. We briefly review the theoretical and algorithmic foundations of Jump3, and then proceed to provide a step-by-step tutorial of the associated software usage.

Stochastic discrete Hamiltonian variational integrators

Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and its corresponding variational principle. Our approach permits to recast in a unified framework a number of integrators previously studied in the literature, and presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators are symplectic; they preserve integrals of motion related to Lie group symmetries; and they include stochastic symplectic Runge-Kutta methods as a special case. Several new low-stage stochastic symplectic methods of mean-square order 1.0 derived using this approach are presented and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to nonsymplectic methods.

Stochastic Mirror Descent Dynamics and Their Convergence in Monotone Variational Inequalities

We examine a class of stochastic mirror descent dynamics in the context of monotone variational inequalities (including Nash equilibrium and saddle-point problems). The dynamics under study are formulated as a stochastic differential equation, driven by a (single-valued) monotone operator and perturbed by a Brownian motion. The system's controllable parameters are two variable weight sequences, that, respectively, pre- and post-multiply the driver of the process. By carefully tuning these parameters, we obtain global convergence in the ergodic sense, and we estimate the average rate of convergence of the process. We also establish a large deviations principle, showing that individual trajectories exhibit exponential concentration around this average.

Stationary moments, diffusion limits, and extinction times for logistic growth with random catastrophes

A central problem in population ecology is understanding the consequences of stochastic fluctuations. Analytically tractable models with Gaussian driving noise have led to important, general insights, but they fail to capture rare, catastrophic events, which are increasingly observed at scales ranging from global fisheries to intestinal microbiota. Due to mathematical challenges, growth processes with random catastrophes are less well characterized and it remains unclear how their consequences differ from those of Gaussian processes. In the face of a changing climate and predicted increases in ecological catastrophes, as well as increased interest in harnessing microbes for therapeutics, these processes have never been more relevant. To better understand them, I revisit here a differential equation model of logistic growth coupled to density-independent catastrophes that arrive as a Poisson process, and derive new analytic results that reveal its statistical structure. First, I derive exact expressions for the model's stationary moments, revealing a single effective catastrophe parameter that largely controls low order statistics. Then, I use weak convergence theorems to construct its Gaussian analog in a limit of frequent, small catastrophes, keeping the stationary population mean constant for normalization. Numerically computing statistics along this limit shows how they transform as the dynamics shifts from catastrophes to diffusions, enabling quantitative comparisons. For example, the mean time to extinction increases monotonically by orders of magnitude, demonstrating significantly higher extinction risk under catastrophes than under diffusions. Together, these results provide insight into a wide range of stochastic dynamical systems important for ecology and conservation.

Fluctuating periodic solutions and moment boundedness of a stochastic model for the bone remodeling process

In this work, we model osteoclast-osteoblast population dynamics with random environmental fluctuations in order to understand the random variations of the bone remodeling process in real life. For this purpose, we construct a stochastic differential model for the interactions between the osteoclast and osteoblast cell populations using the parameter perturbation technique. We prove the existence of a globally attractive positive unique solution for the stochastically perturbed system. Also, the stochastic boundedness of the solution is demonstrated using its p-th order moments for p ≥ 1. Finally, we show that the introduction of noise in the deterministic model provides a fluctuating periodic solution. Numerical evidence supports our theoretical results and a discussion of the results is carried out.

Data-driven modelling of social forces and collective behaviour in zebrafish

Zebrafish are rapidly emerging as a powerful model organism in hypothesis-driven studies targeting a number of functional and dysfunctional processes. Mathematical models of zebrafish behaviour can inform the design of experiments, through the unprecedented ability to perform pilot trials on a computer. At the same time, in-silico experiments could help refining the analysis of real data, by enabling the systematic investigation of key neurobehavioural factors. Here, we establish a data-driven model of zebrafish social interaction. Specifically, we derive a set of interaction rules to capture the primary response mechanisms which have been observed experimentally. Contrary to previous studies, we include dynamic speed regulation in addition to turning responses, which together provide attractive, repulsive and alignment interactions between individuals. The resulting multi-agent model provides a novel, bottom-up framework to describe both the spontaneous motion and individual-level interaction dynamics of zebrafish, inferred directly from experimental observations.

The impact of inter-annual rainfall variability on African savannas changes with mean rainfall

Savannas are mixed tree-grass ecosystems whose dynamics are predominantly regulated by resource competition and the temporal variability in climatic and environmental factors such as rainfall and fire. Hence, increasing inter-annual rainfall variability due to climate change could have a significant impact on savannas. To investigate this, we used an ecohydrological model of stochastic differential equations and simulated African savanna dynamics along a gradient of mean annual rainfall (520-780 mm/year) for a range of inter-annual rainfall variabilities. Our simulations produced alternative states of grassland and savanna across the mean rainfall gradient. Increasing inter-annual variability had a negative effect on the savanna state under dry conditions (520 mm/year), and a positive effect under moister conditions (580-780 mm/year). The former resulted from the net negative effect of dry and wet extremes on trees. In semi-arid conditions (520 mm/year), dry extremes caused a loss of tree cover, which could not be recovered during wet extremes because of strong resource competition and the increased frequency of fires. At high mean rainfall (780 mm/year), increased variability enhanced savanna resilience. Here, resources were no longer limiting and the slow tree dynamics buffered against variability by maintaining a stable population during 'dry' extremes, providing the basis for growth during wet extremes. Simultaneously, high rainfall years had a weak marginal benefit on grass cover due to density-regulation and grazing. Our results suggest that the effects of the slow tree and fast grass dynamics on tree-grass interactions will become a major determinant of the savanna vegetation composition with increasing rainfall variability.

Modeling a SI epidemic with stochastic transmission: hyperbolic incidence rate

In this paper a stochastic susceptible-infectious (SI) epidemic model is analysed, which is based on the model proposed by Roberts and Saha (Appl Math Lett 12: 37-41, 1999), considering a hyperbolic type nonlinear incidence rate. Assuming the proportion of infected population varies with time, our new model is described by an ordinary differential equation, which is analogous to the equation that describes the double Allee effect. The limit of the solution of this equation (deterministic model) is found when time tends to infinity. Then, the asymptotic behaviour of a stochastic fluctuation due to the environmental variation in the coefficient of disease transmission is studied. Thus a stochastic differential equation (SDE) is obtained and the existence of a unique solution is proved. Moreover, the SDE is analysed through the associated Fokker-Planck equation to obtain the invariant measure when the proportion of the infected population reaches steady state. An explicit expression for invariant measure is found and we study some of its properties. The long time behaviour of deterministic and stochastic models are compared by simulations. According to our knowledge this incidence rate has not been previously used for this type of epidemic models.

Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient

We prove strong convergence of order [Formula: see text] for arbitrarily small [Formula: see text] of the Euler-Maruyama method for multidimensional stochastic differential equations (SDEs) with discontinuous drift and degenerate diffusion coefficient. The proof is based on estimating the difference between the Euler-Maruyama scheme and another numerical method, which is constructed by applying the Euler-Maruyama scheme to a transformation of the SDE we aim to solve.

Seasonality can induce coexistence of multiple bet-hedging strategies in Dictyostelium discoideum via storage effect

The social amoeba Dictyostelium discoideum has been recently suggested as an example of bet-hedging in microbes. In the presence of resources, amoebae reproduce as unicellular organisms. Resource depletion, however, leads to a starvation phase in which the population splits between aggregators, which form a fruiting body made of a stalk and resistant spores, and non-aggregators, which remain as vegetative cells. Spores are favored when starvation periods are long, but vegetative cells can exploit resources in environments where food replenishes quickly. The investment in aggregators versus non-aggregators can therefore be understood as a bet-hedging strategy that evolves in response to stochastic starvation times. A genotype (or strategy) is defined by the balance between each type of cells. In this framework, if the ecological conditions on a patch are defined in terms of the mean starvation time (i.e. time between the onset of starvation and the arrival of a new food pulse), a single genotype dominates each environment, which is inconsistent with the huge genetic diversity observed in nature. Here we investigate whether seasonality, represented by a periodic, wet-dry alternation in the mean starvation times, allows the coexistence of several strategies in a single patch. We study this question in a non-spatial (well-mixed) setting in which different strains compete for a common pool of resources over a sequence of growth-starvation cycles. We find that seasonality induces a temporal storage effect that can promote the stable coexistence of multiple genotypes. Two conditions need to be met in our model. First, there has to be a temporal niche partitioning (two well-differentiated habitats within the year), which requires not only different mean starvation times between seasons but also low variance within each season. Second, each season's well-adapted strain has to grow and create a large enough population that permits its survival during the subsequent unfavorable season, which requires the number of growth-starvation cycles within each season to be sufficiently large. These conditions allow the coexistence of two bet-hedging strategies. Additional tradeoffs among life-history traits can expand the range of coexistence and increase the number of coexisting strategies, contributing toward explaining the genetic diversity observed in D. discoideum. Although focused on this cellular slime mold, our results are general and may be easily extended to other microbes.

MIMICKING COUNTERFACTUAL OUTCOMES TO ESTIMATE CAUSAL EFFECTS

In observational studies, treatment may be adapted to covariates at several times without a fixed protocol, in continuous time. Treatment influences covariates, which influence treatment, which influences covariates, and so on. Then even time-dependent Cox-models cannot be used to estimate the net treatment effect. Structural nested models have been applied in this setting. Structural nested models are based on counterfactuals: the outcome a person would have had had treatment been withheld after a certain time. Previous work on continuous-time structural nested models assumes that counterfactuals depend deterministically on observed data, while conjecturing that this assumption can be relaxed. This article proves that one can mimic counterfactuals by constructing random variables, solutions to a differential equation, that have the same distribution as the counterfactuals, even given past observed data. These "mimicking" variables can be used to estimate the parameters of structural nested models without assuming the treatment effect to be deterministic.

ADAPTIVE METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS VIA NATURAL EMBEDDINGS AND REJECTION SAMPLING WITH MEMORY

Adaptive time-stepping with high-order embedded Runge-Kutta pairs and rejection sampling provides efficient approaches for solving differential equations. While many such methods exist for solving deterministic systems, little progress has been made for stochastic variants. One challenge in developing adaptive methods for stochastic differential equations (SDEs) is the construction of embedded schemes with direct error estimates. We present a new class of embedded stochastic Runge-Kutta (SRK) methods with strong order 1.5 which have a natural embedding of strong order 1.0 methods. This allows for the derivation of an error estimate which requires no additional function evaluations. Next we derive a general method to reject the time steps without losing information about the future Brownian path termed Rejection Sampling with Memory (RSwM). This method utilizes a stack data structure to do rejection sampling, costing only a few floating point calculations. We show numerically that the methods generate statistically-correct and tolerance-controlled solutions. Lastly, we show that this form of adaptivity can be applied to systems of equations, and demonstrate that it solves a stiff biological model 12.28x faster than common fixed timestep algorithms. Our approach only requires the solution to a bridging problem and thus lends itself to natural generalizations beyond SDEs.

Approaches for modeling within subject variability in pharmacometric count data analysis: dynamic inter-occasion variability and stochastic differential equations

Parameter variation in pharmacometric analysis studies can be characterized as within subject parameter variability (WSV) in pharmacometric models. WSV has previously been successfully modeled using inter-occasion variability (IOV), but also stochastic differential equations (SDEs). In this study, two approaches, dynamic inter-occasion variability (dIOV) and adapted stochastic differential equations, were proposed to investigate WSV in pharmacometric count data analysis. These approaches were applied to published count models for seizure counts and Likert pain scores. Both approaches improved the model fits significantly. In addition, stochastic simulation and estimation were used to explore further the capability of the two approaches to diagnose and improve models where existing WSV is not recognized. The results of simulations confirmed the gain in introducing WSV as dIOV and SDEs when parameters vary randomly over time. Further, the approaches were also informative as diagnostics of model misspecification, when parameters changed systematically over time but this was not recognized in the structural model. The proposed approaches in this study offer strategies to characterize WSV and are not restricted to count data.

Incorporating prior knowledge induced from stochastic differential equations in the classification of stochastic observations

In classification, prior knowledge is incorporated in a Bayesian framework by assuming that the feature-label distribution belongs to an uncertainty class of feature-label distributions governed by a prior distribution. A posterior distribution is then derived from the prior and the sample data. An optimal Bayesian classifier (OBC) minimizes the expected misclassification error relative to the posterior distribution. From an application perspective, prior construction is critical. The prior distribution is formed by mapping a set of mathematical relations among the features and labels, the prior knowledge, into a distribution governing the probability mass across the uncertainty class. In this paper, we consider prior knowledge in the form of stochastic differential equations (SDEs). We consider a vector SDE in integral form involving a drift vector and dispersion matrix. Having constructed the prior, we develop the optimal Bayesian classifier between two models and examine, via synthetic experiments, the effects of uncertainty in the drift vector and dispersion matrix. We apply the theory to a set of SDEs for the purpose of differentiating the evolutionary history between two species.

Stochastic nonlinear mixed effects: a metformin case study

In nonlinear mixed effect (NLME) modeling, the intra-individual variability is a collection of errors due to assay sensitivity, dosing, sampling, as well as model misspecification. Utilizing stochastic differential equations (SDE) within the NLME framework allows the decoupling of the measurement errors from the model misspecification. This leads the SDE approach to be a novel tool for model refinement. Using Metformin clinical pharmacokinetic (PK) data, the process of model development through the use of SDEs in population PK modeling was done to study the dynamics of absorption rate. A base model was constructed and then refined by using the system noise terms of the SDEs to track model parameters and model misspecification. This provides the unique advantage of making no underlying assumptions about the structural model for the absorption process while quantifying insufficiencies in the current model. This article focuses on implementing the extended Kalman filter and unscented Kalman filter in an NLME framework for parameter estimation and model development, comparing the methodologies, and illustrating their challenges and utility. The Kalman filter algorithms were successfully implemented in NLME models using MATLAB with run time differences between the ODE and SDE methods comparable to the differences found by Kakhi for their stochastic deconvolution.

Mechanistic Hierarchical Gaussian Processes

The statistics literature on functional data analysis focuses primarily on flexible black-box approaches, which are designed to allow individual curves to have essentially any shape while characterizing variability. Such methods typically cannot incorporate mechanistic information, which is commonly expressed in terms of differential equations. Motivated by studies of muscle activation, we propose a nonparametric Bayesian approach that takes into account mechanistic understanding of muscle physiology. A novel class of hierarchical Gaussian processes is defined that favors curves consistent with differential equations defined on motor, damper, spring systems. A Gibbs sampler is proposed to sample from the posterior distribution and applied to a study of rats exposed to non-injurious muscle activation protocols. Although motivated by muscle force data, a parallel approach can be used to include mechanistic information in broad functional data analysis applications.

Stochastic differential equations as a tool to regularize the parameter estimation problem for continuous time dynamical systems given discrete time measurements

In this paper we consider the problem of estimating parameters in ordinary differential equations given discrete time experimental data. The impact of going from an ordinary to a stochastic differential equation setting is investigated as a tool to overcome the problem of local minima in the objective function. Using two different models, it is demonstrated that by allowing noise in the underlying model itself, the objective functions to be minimized in the parameter estimation procedures are regularized in the sense that the number of local minima is reduced and better convergence is achieved. The advantage of using stochastic differential equations is that the actual states in the model are predicted from data and this will allow the prediction to stay close to data even when the parameters in the model is incorrect. The extended Kalman filter is used as a state estimator and sensitivity equations are provided to give an accurate calculation of the gradient of the objective function. The method is illustrated using in silico data from the FitzHugh-Nagumo model for excitable media and the Lotka-Volterra predator-prey system. The proposed method performs well on the models considered, and is able to regularize the objective function in both models. This leads to parameter estimation problems with fewer local minima which can be solved by efficient gradient-based methods.

Locally Adaptive Bayes Nonparametric Regression via Nested Gaussian Processes

We propose a nested Gaussian process (nGP) as a locally adaptive prior for Bayesian nonparametric regression. Specified through a set of stochastic differential equations (SDEs), the nGP imposes a Gaussian process prior for the function's mth-order derivative. The nesting comes in through including a local instantaneous mean function, which is drawn from another Gaussian process inducing adaptivity to locally-varying smoothness. We discuss the support of the nGP prior in terms of the closure of a reproducing kernel Hilbert space, and consider theoretical properties of the posterior. The posterior mean under the nGP prior is shown to be equivalent to the minimizer of a nested penalized sum-of-squares involving penalties for both the global and local roughness of the function. Using highly-efficient Markov chain Monte Carlo for posterior inference, the proposed method performs well in simulation studies compared to several alternatives, and is scalable to massive data, illustrated through a proteomics application.