Itai Dattner

Optimal rate of direct estimators in systems of ordinary differential equations linear in functions of the parameters

Many processes in biology, chemistry, physics, medicine, and engineering are modeled by a system of differential equations. Such a system is usually characterized via unknown parameters and estimating their ‘true’ value is thus required. In this paper we focus on the quite common systems for which the derivatives of the states may be written as sums of products of a function of the states and a function of the parameters. For such a system linear in functions of the unknown parameters we present a necessary and sufficient condition for identifiability of the parameters. We develop an estimation approach that bypasses the heavy computational burden of numerical integration and avoids the estimation of system states derivatives, drawbacks from which many classic estimation methods suffer. We also suggest an experimental design for which smoothing can be circumvented. The optimal rate of the proposed estimators, i.e., their √n-consistency, is proved and simulation results illustrate their excellent finite sample performance and compare it to other estimation approaches.

Time-course window estimator for ordinary differential equations linear in the parameters

In many applications obtaining ordinary differential equation descriptions of dynamic processes is scientifically important. In both, Bayesian and likelihood approaches for estimating parameters of ordinary differential equations, the speed and the convergence of the estimation procedure may crucially depend on the choice of initial values of the parameters. Extending previous work, we show in this paper how using window smoothing yields a fast estimator for systems that are linear in the parameters. Using weak assumptions on the measurement error, we prove that the proposed estimator is

Modelling and parameter inference of predator-prey dynamics in heterogeneous environments using the direct integral approach.

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Modern statistical tools for inference and prediction of infectious diseases using mathematical models

n-consistent. The estimator does not require an initial guess for the parameters and is computationally fast and, therefore, it can serve as a good initial estimate for more efficient estimators. In simulation studies and on real data we illustrate the performance of the proposed estimator.

A two-stage approach for estimating the parameters of an age-group epidemic model from incidence data

The inverse problem of parameter estimation from noisy observations is a major challenge in statistical inference for dynamical systems. Parameter estimation is usually carried out by optimizing some criterion function over the parameter space. Unless the optimization process starts with a good initial guess, the estimation may take an unreasonable amount of time, and may converge to local solutions, if at all. In this article, we introduce a novel technique for generating good initial guesses that can be used by any estimation method. We focus on the fairly general and often applied class of systems linear in the parameters. The new methodology bypasses numerical integration and can handle partially observed systems. We illustrate the performance of the method using simulations and apply it to real data.

Statistical properties of the Hough transform estimator in the presence of measurement errors

Most bacterial habitats are topographically complex in the micro scale. Important examples include the gastrointestinal and tracheal tracts, and the soil. Although there are myriad theoretical studies that explore the role of spatial structures on antagonistic interactions (predation, competition) among animals, there are many fewer experimental studies that have explored, validated and quantified their predictions. In this study, we experimentally monitored the temporal dynamic of the predatory bacterium Bdellovibrio bacteriovorus, and its prey, the bacterium Burkholderia stabilis in a structured habitat consisting of sand under various regimes of wetness. We constructed a dynamic model, and estimated its parameters by further developing the direct integral method, a novel estimation procedure that exploits the separability of the states and parameters in the model. We also verified that one of our parameter estimates was consistent with its known, directly measured value from the literature. The ability of the model to fit the data combined with realistic parameter estimates indicate that bacterial predation in the sand can be described by a relatively simple model, and stress the importance of prey refuge on predation dynamics in heterogeneous environments.

Accelerated least squares estimation for systems of ordinary differential equations

In this paper, we study application of Le Cams one‐step method to parameter estimation in ordinary differential equation models. This computationally simple technique can serve as an alternative to numerical evaluation of the popular non‐linear least squares estimator, which typically requires the use of a multistep iterative algorithm and repetitive numerical integration of the ordinary differential equation system. The one‐step method starts from a preliminary n‐consistent estimator of the parameter of interest and next turns it into an asymptotic (as the sample size n→∞) equivalent of the least squares estimator through a numerically straightforward procedure. We demonstrate performance of the one‐step estimator via extensive simulations and real data examples. The method enables the researcher to obtain both point and interval estimates. The preliminary n‐consistent estimator that we use depends on non‐parametric smoothing, and we provide a data‐driven methodology for choosing its tuning parameter and support it by theory. An easy implementation scheme of the one‐step method for practical use is pointed out.

Consistency of direct integral estimator for partially observed systems of ordinary differential equations linear in the parameters

Infectious diseases are a major cause of morbidity and mortality worldwide; in modern times, diseases such as HIV and malaria remain a significant public health burden leading to death of millions each year. Interestingly, both human evolution and the collapse of nations have been attributed to disease outbreaks. Famous examples include the so-called Antonine plagues in the Roman Empire, the collapse of the Chinese Han Empire, the defeat of the Aztecs by Cortez and the fall of the Inca Empire. The idea that mathematical modeling can be a powerful tool to study infectious disease dates back to the 18th century when Daniel Bernoulli studied how variolation can be used to reduce the burden of smallpox. At the beginning of the 20th century, the seminal model that Ross developed combined with the work of Kermack and McKendrick laid the theoretical pillars of mathematical epidemiology. More so, both landmark studies have shown the practical usefulness of using mathematical models to combat infectious diseases. Ross’s work revealed that mosquitoes transmit malaria; nonetheless, it was essential to construct a mathematical model in order to establish that it is sufficient to reduce the mosquito population (not eliminate) in order to control malaria. Ross’s work corroborated that epidemics are governed by threshold dynamics. A few years later Kermark and Mckendrick developed the well-known, susceptible, infected, recovered framework (SIR), which is the basis of most modern infectious disease models. The simplest version of the SIR formulation can be described using ordinary differential equations. The analysis of the model has led to the fundamental concept of the basic reproduction number, defined as the average number of people infected by an infected individual over the disease infectivity period, in an entirely susceptible population. The basic reproduction number (R0) serves as a threshold parameter and can be represented using the model parameters. In addition, the mathematical analysis of the SIR model reveals that the size of the outbreak depends on the initial fraction of susceptibles, and on R0 and will not depend on the initial number of infectives. In our days, infectious disease modeling is considered a fundamental tool for studying, understanding and revealing the fundamental mechanisms that influence the spread of epidemics. Moreover, modeling provides an essential tool for testing different possible public health policies to minimize the impact of outbreaks in the form of testing different control strategies (e.g. vaccination) and prediction of future disease trends. In fact, models are often the only means of identifying and testing policy alternatives, which cannot be tested experimentally due to various reasons such as limited budget and ethical constraints. The idea that mathematical models of infectious diseases should be confronted with real data, combined with the fact that all models are, at best, simplified representations of the real world raises the need for uncertainty quantification. In particular, statistical methodologies should be used in order to allow testing and validating which assumptions agree better with reality. Indeed, statistical learning (inference, prediction) of dynamic systems involves ‘‘standard’’ statistical problems such as studying the identifiability of a model, estimating model parameters, predicting future states of the system, testing hypotheses and choosing the ‘‘best’’ model. However, modern dynamic systems are typically very complex: nonlinear, high dimensional and only partly measured. Moreover, data may be sparse and noisy. Thus, statistical learning of dynamical systems is not a trivial task in practice which leads to difficulties in understanding the process and in generating valid inference and reliable predictions.