An introduction to dynamical modeling with applications to performance

News item

Tuberculosis is a global disease with much significant impact. Well over two million deaths are recorded each year due to Mycobacterium tuberculosis. The work by Colijn et al. (J Theor Biol 247, 765–774, 2007) examined two mathematical models (delay differential equation and network models) that explored the role of the contact structure of the population and found that in declining epidemics, localized outbreaks may occur as a result of contact heterogeneities even in the absence of host or strain variability. In this work, we carry out several qualitative studies (including the use of Sturm sequences for obtaining conditions on bifurcations) on the differential equation model with a time delay in Ref. Colijn et al., J Theor Biol 247, 765–774 (2007), and explicitly obtain threshold conditions for disease eradication as well as existence of bifurcations depending on a critical delay value which characterize the duration of fast latency. Further analysis showed the existence of a transcritical bifurcation when the reproduction number was less than one with the disease-free equilibrium co-existing with just one endemic state that was shown to be stable; the period of fast latency and the reinfection progression rate were shown to be very significant in the dynamics of the disease viz a viz the individuals who progressed to active TB within 5 years of their infection.

Advertisement involves significant amount of money and remain as one of the major issues bothering companies and other investment sectors. Advertising is a marketing strategy directed to the consumers through any media in order to present and promote product, services and any other cause. Sale of products occurs with or without advertisement and may increase or decrease depending on the market situation. The question is, is there a mathematical model that can surely confirm the effects of advertising on sales of products? The aim of this paper is to develop a first order linear differential equation model that seeks to justify the impacts of advertising on the sale of products in Ghanaian market. Two basic variables, the advertisement and sales of product were considered for the model. A system of two differential equations from the two basic variables were developed and converted to first order non-homogeneous linear differential equation. The geometrical method was employed to obtain qualitative information about the solution of the first order non-homogeneous linear differential equation. The MATLAB software package was used to plots the graphs to illustrate the behaviour of the sale of products with respect to the time of advertisement. The findings revealed that differential equation model can be used to justify the advertising impacts on the sale of products.

Complex intraindividual variability observed in psychology may be well described using differential equations. It is difficult, however, to apply differential equation models in psychological contexts, as time series are frequently short, poorly sampled, and have large proportions of measurement and dynamic error. Furthermore, current methods for differential equation modeling usually consider data that are atypical of many psychological applications. Using embedded and observed data matrices, a statistical approach to differential equation modeling is presented. This approach appears robust to many characteristics common to psychological time series.

Stock price analysis is one of the most important issues concerning investments and financial decision-making. Thus, stock price analysis and estimation models can be very useful in the estimation of a firm’s financial development. The aim of this paper is to propose a model of differential equations that will be able to be applied in the case of stock price analysis and estimation. The differential equations model will be based on Lanchester’s combat model, a mathematical theory of war. In the field of business, such operations research models have been used in cases such as competition analysis and resources allocation optimization. The case to be examined in this paper refers to the healthcare services index stocks of the Athens Stock Exchange. A 7 × 7 differential equations model was developed to analyze the examined firms’ stocks.

Dynamics of the mechanical system is described by the matrix-second-order differential equation, and the coefficient matrices in the equation have some special properties such as symmetry and positivity. These properties should be reflected to the design method of controllers for mechanical systems. In this paper we discuss control problems of mechanical systems in the matrix-second-order differential equation model by paying attention to their potential and kinetic energy. First we introduce a concept called generalized energy function and then by using this new concept we discuss observability and controllability and propose a design method of feedback controllers of the mechanical systems.

Accurate reporting and prediction of PM2.5 concentration is very important for improving public health. In this article, we use spectral clustering algorithm to cluster 15 cities in the Pearl River Delta. On this basis, we propose a special difference equation model, especially the use of nonlinear diffusion equations to characterize the temporal and spatial dynamic characteristics of PM2.5 propagation between and within clusters for real‐time prediction. For example, through the analysis of PM2.5 concentration data for 91 consecutive days in the Pearl River Delta, and according to different accuracy definitions, the average prediction accuracy of the difference equation model in all city clusters is 97% or 88%. The mean absolute error (MAE) of the forecast data for each urban agglomeration is within 7 units ( ). Experimental results show that the difference equation model can effectively reduce the prediction time and improve the prediction accuracy. Therefore, based on the spectral clustering algorithm and the difference equation model, the fastest prediction speed and the best prediction result can be obtained, and the problem of PM2.5 concentration prediction can be effectively solved. The research can provide decision support for local air pollution early warning and urban integrated management.

In this article, the due date setting problem consisting of due date quoting, order insertion and rescheduling with relevant sequence-dependent setups and a frozen zone is considered. A differential equation model is introduced. The differential equation can be solved analytically and requires low computing effort. Explicit expressions are gained concerning the earliest possible due date (due date quoting problem) and the latest possible insertion position (rescheduling problem) in order to meet the customer requested due date. The differential equation model is compared with an approach based on a recently introduced hybrid binomial inverse hypergeometric probability distribution for the due date quoting problem. The analytical results of the proposed model are very close to the expectation values (these cannot be explicitly calculated, and high computation effort is required) of the recently introduced approach.

Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability, we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.

Mathematical models to study quorum sensing (QS) have become an important tool to explore all aspects of this type of bacterial communication. A wide spectrum of mathematical tools and methods such as dynamical systems, stochastics, and spatial models can be employed. In this chapter, we focus on giving an overview of models consisting of differential equations (DE), which can be used to describe changing quantities, for example, the dynamics of one or more signaling molecule in time and space, often in conjunction with bacterial growth dynamics. The chapter is divided into two sections: ordinary differential equations (ODE) and partial differential equations (PDE) models of QS. Rates of change are represented mathematically by derivatives, i.e., in terms of DE. ODE models allow describing changes in one independent variable, for example, time. PDE models can be used to follow changes in more than one independent variable, for example, time and space. Both types of models often consist of systems (i.e., more than one equation) of equations, such as equations for bacterial growth and autoinducer concentration dynamics. Almost from the onset, mathematical modeling of QS using differential equations has been an interdisciplinary endeavor and many of the works we revised here will be placed into their biological context.

Agent-based models provide a flexible framework that is frequently used for modelling many biological systems, including cell migration, molecular dynamics, ecology and epidemiology. Analysis of the model dynamics can be challenging due to their inherent stochasticity and heavy computational requirements. Common approaches to the analysis of agent-based models include extensive Monte Carlo simulation of the model or the derivation of coarse-grained differential equation models to predict the expected or averaged output from the agent-based model. Both of these approaches have limitations, however, as extensive computation of complex agent-based models may be infeasible, and coarse-grained differential equation models can fail to accurately describe model dynamics in certain parameter regimes. We propose that methods from the equation learning field provide a promising, novel and unifying approach for agent-based model analysis. Equation learning is a recent field of research from data science that aims to infer differential equation models directly from data. We use this tutorial to review how methods from equation learning can be used to learn differential equation models from agent-based model simulations. We demonstrate that this framework is easy to use, requires few model simulations, and accurately predicts model dynamics in parameter regions where coarse-grained differential equation models fail to do so. We highlight these advantages through several case studies involving two agent-based models that are broadly applicable to biological phenomena: a birth-death-migration model commonly used to explore cell biology experiments and a susceptible-infected-recovered model of infectious disease spread.

A recent study presented a new differential equation model, MultiVelo, which simultaneously performs on chromatin accessibility and gene expression from single-cell data to reveals regulatory network of epigenomic and transcriptomic changes. However, it is unclear how regulatory network changes with perturbations. Here, we extend the ordinal differential equation models to time-delay differential equation models with controller to investigate the stability of regulatory network with perturbations.

Differential equation models can be used to describe the relationships between the current state of a system of constructs (e.g., stress) and how those constructs are changing (e.g., based on variable-like experiences). The following article describes a differential equation model based on the concept of a reservoir. With a physical reservoir, such as one for water, the level of the liquid in the reservoir at any time depends on the contributions to the reservoir (inputs) and the amount of liquid removed from the reservoir (outputs). This reservoir model might be useful for constructs such as stress, where events might "add up" over time (e.g., life stressors, inputs), but individuals simultaneously take action to "blow off steam" (e.g., engage coping resources, outputs). The reservoir model can provide descriptive statistics of the inputs that contribute to the "height" (level) of a construct and a parameter that describes a person's ability to dissipate the construct. After discussing the model, we describe a method of fitting the model as a structural equation model using latent differential equation modeling and latent distribution modeling. A simulation study is presented to examine recovery of the input distribution and output parameter. The model is then applied to the daily self-reports of negative affect and stress from a sample of older adults from the Notre Dame Longitudinal Study on Aging.

In this paper, we design learning law with high-order internal models for fractional order differential equations to track the varying reference accurately by adopting a few iterations in a finite time interval. We establish sufficient conditions of convergence for the P-type and PD-type updating law for different fractional order differential equations. Finally, we give some numerical examples to demonstrate the validity of the designed method.

The paper deals with the criteria for the closed-loop stability of a noise control system in a duct. To study the stability of the system, the model of delay differential equation is derived from the propagation of acoustic wave governed by a partial differential equation of hyperbolic type. Then, a simple feedback controller is designed, and its closed-loop stability is analyzed on the basis of the derived model of delay differential equation. The obtained criteria reveal the influence of the controller gain and the positions of a sensor and an actuator on the closed-loop stability. Finally, numerical simulations are presented to support the theoretical results.