Accelerated maximum likelihood parameter estimation for stochastic biochemical systems

ABSTRACT: BACKGROUND: A prerequisite for the mechanistic simulation of a biochemical system is detailed knowledge of its kinetic parameters. Despite recent experimental advances, the estimation of unknown parameter values from observed data is still a bottleneck for obtaining accurate simulation results. Many methods exist for parameter estimation in deterministic biochemical systems; methods for discrete stochastic systems are less well developed. Given the probabilistic nature of stochastic biochemical models, a natural approach is to choose parameter values that maximize the probability of the observed data with respect to the unknown parameters, a.k.a. the maximum likelihood parameter estimates (MLEs). MLE computation for all but the simplest models requires the simulation of many system trajectories that are consistent with experimental data. For models with unknown parameters, this presents a computational challenge, as the generation of consistent trajectories can be an extremely rare occurrence. RESULTS: We have developed Monte Carlo Expectation-Maximization with Modified Cross-Entropy Method (MCEM2): an accelerated method for calculating MLEs that combines advances in rare event simulation with a computationally efficient version of the Monte Carlo expectation-maximization (MCEM) algorithm. Our method requires no prior knowledge regarding parameter values, and it automatically provides a multivariate parameter uncertainty estimate. We applied the method to five stochastic systems of increasing complexity, progressing from an analytically tractable pure-birth model to a computationally demanding model of yeast-polarization. Our results demonstrate that MCEM2 substantially accelerates MLE computation on all tested models when compared to a stand-alone version of MCEM. Additionally, we show how our method identifies parameter values for certain classes of models more accurately than two recently proposed computationally efficient methods. CONCLUSIONS: This work provides a novel, accelerated version of a likelihood-based parameter estimation method that can be readily applied to stochastic biochemical systems. In addition, our results suggest opportunities for added efficiency improvements that will further enhance our ability to mechanistically simulate biological processes.     

A two-stage model for the SIR outbreak: accounting for the discrete and stochastic nature of the epidemic at the initial contamination stage

The evolution of an infectious disease outbreak in an isolated population is split into two stages: a stochastic Markov process describing the initial contamination and a linked deterministic dynamical system with random initial conditions for the continued development of the outbreak. The initial contamination stage is well approximated by the randomized SI (susceptible/infected) model. We obtain the probability density function for the early behavior of the epidemic. This provides an appropriate distribution for the initial conditions with which to describe the subsequent deterministic evolution of the system. We apply the method of matching asymptotic expansions to link the two stages. This allows us to estimate the standard deviation of the number of infectives in the developed outbreak, and the statistical characteristics of the outbreak time. The potential trajectories caused by the stochastic nature of the contamination stage show greatest divergence at the initial and fade-out stages and coincide most tightly just after the peak of the epidemic. The time to the peak of the outbreak is not strongly dependent on the initial trajectory.     

A Stochastic Tick-Borne Disease Model: Exploring the Probability of Pathogen Persistence

We formulate and analyse a stochastic epidemic model for the transmission dynamics of a tick-borne disease in a single population using a continuous-time Markov chain approach. The stochastic model is based on an existing deterministic metapopulation tick-borne disease model. We compare the disease dynamics of the deterministic and stochastic models in order to determine the effect of randomness in tick-borne disease dynamics. The probability of disease extinction and that of a major outbreak are computed and approximated using the multitype Galton–Watson branching process and numerical simulations, respectively. Analytical and numerical results show some significant differences in model predictions between the stochastic and deterministic models. In particular, we find that a disease outbreak is more likely if the disease is introduced by infected deer as opposed to infected ticks. These insights demonstrate the importance of host movement in the expansion of tick-borne diseases into new geographic areas.     

A stochastic SIR model with contact-tracing: large population limits and statistical inference

This paper is devoted to the presentation and study of a specific stochastic epidemic model accounting for the effect of contact-tracing on the spread of an infectious disease. Precisely, one considers here the situation in which individuals identified as infected by the public health detection system may contribute to detecting other infectious individuals by providing information related to persons with whom they have had possibly infectious contacts. The control strategy, which consists of examining each individual who has been able to be identified on the basis of the information collected within a certain time period, is expected to efficiently reinforce the standard random-screening-based detection and considerably ease the epidemic. In the novel modelling of the spread of a communicable infectious disease considered here, the population of interest evolves through demographic, infection and detection processes, in a way that its temporal evolution is described by a stochastic Markov process, of which the component accounting for the contact-tracing feature is assumed to be valued in a space of point measures. For adequate scalings of the demographic, infection and detection rates, it is shown to converge to the weak deterministic solution of a PDE system, as a parameter n, interpreted as the population size, roughly speaking, becomes larger. From the perspective of the analysis of infectious disease data, this approximation result may serve as a key tool for exploring the asymptotic properties of standard inference methods such as maximum likelihood estimation. We state preliminary statistical results in this context. Eventually, relations of the model with the available data of the HIV epidemic in Cuba, in which country a contact-tracing detection system has been set up since 1986, is investigated and numerical applications are carried out.     

A mechanistic and data-driven reconstruction of the time-varying reproduction number: Application to the COVID-19 epidemic

The effective reproduction number R_eff is a critical epidemiological parameter that characterizes the transmissibility of a pathogen. However, this parameter is difficult to estimate in the presence of silent transmission and/or significant temporal variation in case reporting. This variation can occur due to the lack of timely or appropriate testing, public health interventions and/or changes in human behavior during an epidemic. This is exactly the situation we are confronted with during this COVID-19 pandemic. In this work, we propose to estimate R_eff for the SARS-CoV-2 (the etiological agent of the COVID-19), based on a model of its propagation considering a time-varying transmission rate. This rate is modeled by a Brownian diffusion process embedded in a stochastic model. The model is then fitted by Bayesian inference (particle Markov Chain Monte Carlo method) using multiple well-documented hospital datasets from several regions in France and in Ireland. This mechanistic modeling framework enables us to reconstruct the temporal evolution of the transmission rate of the COVID-19 based only on the available data. Except for the specific model structure, it is non-specifically assumed that the transmission rate follows a basic stochastic process constrained by the observations. This approach allows us to follow both the course of the COVID-19 epidemic and the temporal evolution of its R_eff(t). Besides, it allows to assess and to interpret the evolution of transmission with respect to the mitigation strategies implemented to control the epidemic waves in France and in Ireland. We can thus estimate a reduction of more than 80% for the first wave in all the studied regions but a smaller reduction for the second wave when the epidemic was less active, around 45% in France but just 20% in Ireland. For the third wave in Ireland the reduction was again significant (>70%).     

Modelling Holling type II functional response in deterministic and stochastic food chain models with mass conservation

The Rosenzweig-MacArthur predator-prey model is the building block in modeling food chain, food webs and ecosystems. There are a number of hidden assumptions involved in the derivation. For instance the prey population growth is logistic without predation but also with predation. In order to reveal these we will start with modelling a resource-predator-prey system in a closed spatially homogeneous environment. This allows us to keep track of the nutrient flow. With an instantaneous remineralisation of the products excreted in the environment by the populations and dead body mass there is conservation of mass. This allows for a model dimension reduction and yields the mass balance predator-prey model. When furthermore the searching and handling processes are much faster that the population changing rates, the trophic interaction is described by a Holling type II functional response, also assumed in the Rosenzweig-MacArthur model. The derivation uses an extended deterministic model with number of searching and handling predators as model variables where the ratio of the predator/prey body masses is used as a mechanistic time-scale parameter. This extended model is also used as a starting point for the derivation of a stochastic model. We will investigate the stochastic effects of random switching between searching and handling of the predators and predator dying. Prey growth by consumption of ambient resources is still deterministic and therefore the stochastic model is hybrid. The transient dynamics is studied by numerical Monte Carlo simulations and also the quasi-equilibrium distribution for the population quantities is calculated. The body mass of the prey individual is the scaling parameter in the stochastic model formulation. This allows for a quantification of the mean-field approximation criterion for the justification of replacement of the stochastic by a deterministic model.     

Screening stochastic Life Cycle assessment inventory models

A screening methodology is presented that utilizes the linear structure within the deterministic life cycle inventory (LCI) model. The methodology ranks each input data element based upon the amount it contributes toward the final output. The identified data elements along with their position in the deterministic model are then sorted into descending order based upon their individual contributions. This enables practitioners and model users to identify those input data elements that contribute the most in the inventory stage. Percentages of the top ranked data elements are then selected, and their corresponding data quality index (DQI) value is upgraded in the stochastic LCI model. Monte Carlo computer simulations are obtained and used to compare the output variance of the original stochastic model with modified stochastic model. The methodology is applied to four real-world beverage delivery system LCA inventory models for verification. This research assists LCA practitioners by streamlining the conversion process when converting a deterministic LCI model to a stochastic model form. Model users and decision-makers can benefit from the reduction in output variance and the increase in ability to discriminate between product system alternatives.     

Moment fitting for parameter inference in repeatedly and partially observed stochastic biological models

The inference of reaction rate parameters in biochemical network models from time series concentration data is a central task in computational systems biology. Under the assumption of well mixed conditions the network dynamics are typically described by the chemical master equation, the Fokker Planck equation, the linear noise approximation or the macroscopic rate equation. The inverse problem of estimating the parameters of the underlying network model can be approached in deterministic and stochastic ways, and available methods often compare individual or mean concentration traces obtained from experiments with theoretical model predictions when maximizing likelihoods, minimizing regularized least squares functionals, approximating posterior distributions or sequentially processing the data. In this article we assume that the biological reaction network can be observed at least partially and repeatedly over time such that sample moments of species molecule numbers for various time points can be calculated from the data. Based on the chemical master equation we furthermore derive closed systems of parameter dependent nonlinear ordinary differential equations that predict the time evolution of the statistical moments. For inferring the reaction rate parameters we suggest to not only compare the sample mean with the theoretical mean prediction but also to take the residual of higher order moments explicitly into account. Cost functions that involve residuals of higher order moments may form landscapes in the parameter space that have more pronounced curvatures at the minimizer and hence may weaken or even overcome parameter sloppiness and uncertainty. As a consequence both deterministic and stochastic parameter inference algorithms may be improved with respect to accuracy and efficiency. We demonstrate the potential of moment fitting for parameter inference by means of illustrative stochastic biological models from the literature and address topics for future research.     

Bayesian analysis of severe acute respiratory syndrome: the 2003 Hong Kong epidemic

This paper analyzes data arising from a Severe Acute Respiratory Syndrome (SARS) epidemic in Hong Kong in 2003 involving 1755 cases. A discrete time stochastic model that uses a back-projection approach is proposed. Markov Chain Monte Carlo (MCMC) methods are developed for estimation of model parameters. The algorithm is further extended to integrate numerically over unobserved variables of the model. Applying the method to SARS data from Hong Kong, a value of 3.88 with a posterior standard deviation of 0.09 was estimated for the basic reproduction number. An estimate of the transmission parameter at the beginning of the epidemic was also obtained as 0.149 with a posterior standard deviation of 0.003. A reduction in the transmission parameter during the course of the epidemic forced the effective reproduction number to cross the threshold value of one, seven days after control interventions were introduced. At the end of the epidemic, the effective reproduction number was as low as 0.001 suggesting that the epidemic was brought under control by the intervention measures introduced.     

Goodness-of-fit methods for generalized linear mixed models

We develop graphical and numerical methods for checking the adequacy of generalized linear mixed models (GLMMs). These methods are based on the cumulative sums of residuals over covariates or predicted values of the response variable. Under the assumed model, the asymptotic distributions of these stochastic processes can be approximated by certain zero-mean Gaussian processes, whose realizations can be generated through Monte Carlo simulation. Each observed process can then be compared, both visually and analytically, to a number of realizations simulated from the null distribution. These comparisons enable one to assess objectively whether the observed residual patterns reflect model misspecification or random variation. The proposed methods are particularly useful for checking the functional form of a covariate or the link function. Extensive simulation studies show that the proposed goodness-of-fit tests have proper sizes and are sensitive to model misspecification. Applications to two medical studies lead to improved models.     

Noise-induced neural impulses

The firing pattern of neural pulses often show the following features: the shapes of individual pulses are nearly identical and frequency independent; the firing frequency can vary over a broad range; the time period between pulses shows a stochastic scatter. This behaviour cannot be understood on the basis of a deterministic non-linear dynamic process, e.g. the Bonhoeffer-van der Pol model. We demonstrate in this paper that a noise term added to the Bonhoeffer-van der Pol model can reproduce the firing patterns of neurons very well. For this purpose we have considered the Fokker-Planck equation corresponding to the stochastic Bonhoeffer-van der Pol model. This equation has been solved by a new Monte Carlo algorithm. We demonstrate that the ensuing distribution functions represent only the global characteristics of the underlying force field: lines of zero slope which attract nearby trajectories prove to be the regions of phase space where the distributions concentrate their amplitude. Since there are two such lines the distributions are bimodal representing repeated fluctuations between two lines of zero slope. Even in cases where the deterministic Bonhoeffer-van der Pol model does not show limit cycle behaviour the stochastic system produces a limit cycle. This cycle can be identified with the firing of neural pulses.     

Stochastic modeling of nonlinear epidemiology

The objectives of this paper to analyse, model and simulate the spread of an infectious disease by resorting to modern stochastic algorithms. The approach renders it possible to circumvent the simplifying assumption of linearity imposed in the majority of the past works on stochastic analysis of epidemic processes. Infectious diseases are often transmitted through contacts of those infected with those susceptible; hence the processes are inherently nonlinear. According to the classical model of Kermack and McKendrick, or the SIR model, three classes of populations are involved in two types of processes: conversion of susceptibles (S) to infectives (I) and conversion of infectives to removed (R). The master equations of the SIR process have been formulated through the probabilistic population balance around a particular state by considering the mutually exclusive events. The efficacy of the present methodology is mainly attributable to its ability to derive the governing equations for the means, variances and covariance of the random variables by the method of system-size expansion of the nonlinear master equations. Solving these equations simultaneously along with rates associated influenza epidemic data yields information concerning not only the means of the three populations but also the minimal uncertainties of these populations inherent in the epidemic. The stochastic pathways of the three different classes of populations during an epidemic, i.e. their means and the fluctuations around these means, have also been numerically simulated independently by the algorithm derived from the master equations, as well as by an event-driven Monte Carlo algorithm. The master equation and Monte Carlo algorithms have given rise to the identical results.     

A two-component model for counts of infectious diseases

We propose a stochastic model for the analysis of time series of disease counts as collected in typical surveillance systems on notifiable infectious diseases. The model is based on a Poisson or negative binomial observation model with two components: a parameter-driven component relates the disease incidence to latent parameters describing endemic seasonal patterns, which are typical for infectious disease surveillance data. An observation-driven or epidemic component is modeled with an autoregression on the number of cases at the previous time points. The autoregressive parameter is allowed to change over time according to a Bayesian changepoint model with unknown number of changepoints. Parameter estimates are obtained through the Bayesian model averaging using Markov chain Monte Carlo techniques. We illustrate our approach through analysis of simulated data and real notification data obtained from the German infectious disease surveillance system, administered by the Robert Koch Institute in Berlin. Software to fit the proposed model can be obtained from http://www.statistik.lmu.de/ approximately mhofmann/twins.     

Coexistence of multiple pathogen strains in stochastic epidemic models with density-dependent mortality

Stochastic differential equations that model an SIS epidemic with multiple pathogen strains are derived from a system of ordinary differential equations. The stochastic model assumes there is demographic variability. The dynamics of the deterministic model are summarized. Then the dynamics of the stochastic model are compared to the deterministic model. In the deterministic model, there can be either disease extinction, competitive exclusion, where only one strain persists, or coexistence, where more than one strain persists. In the stochastic model, all strains are eventually eliminated because the disease-free state is an absorbing state. However, if the population size and the initial number of infected individuals are sufficiently large, it may take a long time until all strains are eliminated. Numerical simulations of the stochastic model show that coexistence cases predicted by the deterministic model are an unlikely occurrence in the stochastic model even for short time periods. In the stochastic model, either disease extinction or competitive exclusion occur. The initial number of infected individuals, the basic reproduction numbers, and other epidemiological parameters are important determinants of the dominant strain in the stochastic epidemic model.     

Waiting times to appearance and dominance of advantageous mutants: estimation based on the likelihood

The germinal center reaction (GCR) of vertebrate immunity provides a remarkable example of evolutionary succession, in which an advantageous phenotype arises as a spontaneous mutation from the parental type and eventually displaces the parental type altogether. In the case of the immune response to the hapten (4-hydroxy-3-nitrophenyl)acetyl (NP), as with several other designed immunogens, the process is dominated by a single key mutation, which greatly simplifies the modeling of and analysis of data. We developed a two-stage model of this process in which the primary stage represents the appearance and establishment of the mutant population as a stochastic process while the second stage represents the growth and dominance of the clone as a deterministic process, conditional on its time of establishment from stage one. We applied this model to the analysis of population samples from several germinal center (GC) reactions and used maximum-likelihood methods to estimate the waiting times to arrival and to dominance of the mutant clone. We determined the sampling properties of the maximum-likelihood estimates using Monte Carlo methods and compared them to their asymptotic distributions. The methods we present here are well-suited for use in the analysis of other systems, such as tumor growth and the experimental evolution of bacteria.     

A stochastic SIR model with contact-tracing: large population limits and statistical inference

This paper is devoted to the presentation and study of a specific stochastic epidemic model accounting for the effect of contact-tracing on the spread of an infectious disease. Precisely, one considers here the situation in which individuals identified as infected by the public health detection system may contribute to detecting other infectious individuals by providing information related to persons with whom they have had possibly infectious contacts. The control strategy, which consists of examining each individual who has been able to be identified on the basis of the information collected within a certain time period, is expected to efficiently reinforce the standard random-screening-based detection and considerably ease the epidemic. In the novel modelling of the spread of a communicable infectious disease considered here, the population of interest evolves through demographic, infection and detection processes, in a way that its temporal evolution is described by a stochastic Markov process, of which the component accounting for the contact-tracing feature is assumed to be valued in a space of point measures. For adequate scalings of the demographic, infection and detection rates, it is shown to converge to the weak deterministic solution of a PDE system, as a parameter n, interpreted as the population size, roughly speaking, becomes larger. From the perspective of the analysis of infectious disease data, this approximation result may serve as a key tool for exploring the asymptotic properties of standard inference methods such as maximum likelihood estimation. We state preliminary statistical results in this context. Eventually, relations of the model with the available data of the HIV epidemic in Cuba, in which country a contact-tracing detection system has been set up since 1986, is investigated and numerical applications are carried out.     

Assessing the variability of stochastic epidemics.

In predicting the course of individual realizations of an epidemic it is important to know the magnitude of the variability of such realizations about their mean. In this paper and in the context of the general stochastic epidemic, some methods of obtaining approximate estimates of this variability are investigated; one is a multivariate normal approximation based on an asymptotic Gaussian diffusion process, and another uses an approximating linear stochastic process. The extension of these methods to the more detailed models used to describe the transmission dynamics of HIV infection and AIDS is discussed.     

心肌细胞团搏动整数倍节律的非线性动力学机制

利用心肌细胞耦合模型研究心肌整数倍节律的动力学机理。确定性模型仿真揭示了心肌细胞团同步搏动加周期分岔的节律变化规律;随机模型仿真发现在加周期分岔序列中分岔点附近会出现整数倍节律,其中,0-1整数倍节律产生于从静息到周期1的Hopf分岔点附近,1-2整数倍节律产生于周期1和周期2极限环间的加周期分岔点附近;对系统相空间轨道的分析进一步揭示出整数倍节律是由系统运动在相邻的两个轨道之间随机跃迁形成的。上述分析结果不仅阐明了心肌整数倍节律的机理,并且揭示了各种整数倍节律与加周期分岔序列中相邻节律的内在联系,为重新认识心律变化的规律开辟了新的途径。     

Model for radial dependence of frequency distributions for energy imparted in nanometer volumes from HZE particles

This paper develops a deterministic model of frequency distributions for energy imparted (total energy deposition) in small volumes similar to DNA molecules from high-energy ions of interest for space radiation protection and cancer therapy. Frequency distributions for energy imparted are useful for considering radiation quality and for modeling biological damage produced by ionizing radiation. For high-energy ions, secondary electron (delta-ray) tracks originating from a primary ion track make dominant contributions to energy deposition events in small volumes. Our method uses the distribution of electrons produced about an ion's path and incorporates results from Monte Carlo simulation of electron tracks to predict frequency distributions for ions, including their dependence on radial distance. The contribution from primary ion events is treated using an impact parameter formalism of spatially restricted linear energy transfer (LET) and energy-transfer straggling. We validate our model by comparing it directly to results from Monte Carlo simulations for proton and alpha-particle tracks. We show for the first time frequency distributions of energy imparted in DNA structures by several high-energy ions such as cosmic-ray iron ions. Our comparison with results from Monte Carlo simulations at low energies indicates the accuracy of the method.     

Parameter Estimation Methods for Chaotic Intercellular Networks

We have investigated simulation-based techniques for parameter estimation in chaotic intercellular networks. The proposed methodology combines a synchronization–based framework for parameter estimation in coupled chaotic systems with some state–of–the–art computational inference methods borrowed from the field of computational statistics. The first method is a stochastic optimization algorithm, known as accelerated random search method, and the other two techniques are based on approximate Bayesian computation. The latter is a general methodology for non–parametric inference that can be applied to practically any system of interest. The first method based on approximate Bayesian computation is a Markov Chain Monte Carlo scheme that generates a series of random parameter realizations for which a low synchronization error is guaranteed. We show that accurate parameter estimates can be obtained by averaging over these realizations. The second ABC–based technique is a Sequential Monte Carlo scheme. The algorithm generates a sequence of “populations”, i.e., sets of randomly generated parameter values, where the members of a certain population attain a synchronization error that is lesser than the error attained by members of the previous population. Again, we show that accurate estimates can be obtained by averaging over the parameter values in the last population of the sequence. We have analysed how effective these methods are from a computational perspective. For the numerical simulations we have considered a network that consists of two modified repressilators with identical parameters, coupled by the fast diffusion of the autoinducer across the cell membranes.