Modeling Noisy Data with Differential Equations Using Observed and Expected Matrices

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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.

Time-varying coefficient estimation in differential equation models with noisy time-varying covariates

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.

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.

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.

This study examines different approaches to modeling change over time, comparing growth curve models with difference equation models. Growth curve models provide a descriptive framework for capturing longitudinal trends but primarily serve as phenomenological tools that summarize observed trajectories. In contrast, difference equation models offer a process-based perspective, modeling system dynamics iteratively and allowing researchers to explore change processes. This study’s empirical analyses highlight both the advantages and limitations of such models. Simulations demonstrate the effectiveness of difference equations in capturing system dynamics, but high levels of variability in early time periods can make the models fit somewhat poorly in some cases. Findings suggest that while growth curve models remain valuable for summarizing change, difference equation models provide deeper insights into the mechanisms driving temporal processes. Future research should explore hybrid modeling approaches that integrate these frameworks, using dynamical systems theory to enhance psychological and organizational research. An R package to assist with these analyses is available at https://github.com/smspain/DifferenceEquationsInR

The tenth special issue of the Japanese Journal of Statistics and Data Science (JJSD) focuses on statistics for stochastic processes. With the advancement of information and communication technology and measurement technology, a vast amount of diverse time series data have been readily obtained. Consequently, the significance of stochastic processes as mathematical models for random phenomena evolving in time or space has increased dramatically, and statistics for various types of stochastic processes have been rapidly developing. In fact, a multitude of studies on statistics for stochastic processes such as ARMA, CARMA, COGARCH, diffusion processes, fractional Brownian motions, GARCH, jump-diffusion processes, Lèvy processes, Markov processes, point processes, random fields, stochastic differential equations and stochastic partial differential equations have been conducted. Nonetheless, because of the mathematical complexity of statistics for stochastic processes, numerous theoretical and computational challenges remain to be addressed. This special feature contains ten original papers contributed by leading mathematical statisticians and data scientists on current topics in statistics for stochastic processes. The first paper examines the efficient particle Markov chain Monte Carlo method. The second and third papers respectively address the estimation problems of Hawkes processes and Cox proportional excess hazard models. The fourth and fifth papers investigate parametric estimation for probability partial differential equation models. The latter five papers focus on statistical inference for stochastic differential equation models driven by various stochastic processes, including the Wiener process, fractional Brownian motion, β-stable process and Lèvy process. The ten papers in the special feature are summarized as follows: Awaya and Omori propose an efficient particle Markov chain Monte Carlo methodology for the rolling-window estimation of state space models. They provide theoretical justifications for their proposed method and demonstrate the efficiency of B Masayuki Uchida

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.

New delay differential equation models for heating systems with pipes

We find concrete evidence for a recently discovered form of intermittency, referred to as in–out intermittency, in both partial differential equation (PDE) and ordinary differential equation (ODE) models of mean field dynamos. This type of intermittency [introduced in P. Ashwin, E. Covas, and R. Tavakol, Nonlinearity 9, 563 (1999)] occurs in systems with invariant submanifolds and, as opposed to on–off intermittency which can also occur in skew product systems, it requires an absence of skew product structure. By this we mean that the dynamics on the attractor intermittent to the invariant manifold cannot be expressed simply as the dynamics on the invariant subspace forcing the transverse dynamics; the transverse dynamics will alter that tangential to the invariant subspace when one is far enough away from the invariant manifold. Since general systems with invariant submanifolds are not likely to have skew product structure, this type of behavior may be of physical relevance in a variety of dynamical settings. The models employed here to demonstrate in–out intermittency are axisymmetric mean-field dynamo models which are often used to study the observed large-scale magnetic variability in the Sun and solar-type stars. The occurrence of this type of intermittency in such models may be of interest in understanding some aspects of such variabilities.

In many areas, researchers might think that a differential equation model is required, but one might be forced to use an approximate difference equation model if data is only available at discrete points in time. In this paper, a detailed comparison is given of the behavior of continuous and discrete models for two representative time-delay models, namely a model for HIV and an extended logistic growth model. For each model, there are seven different time-delay versions because there are seven different positions to include time delays. For the seven different time-delay versions of each model, proofs are given of necessary and sufficient conditions for the existence and stability of equilibrium points and for the existence of Andronov–Hopf bifurcations in the differential equations and Neimark–Sacker bifurcations in the difference equations. We show that only five of the seven time-delay versions have bifurcations and that all bifurcation versions have supercritical limit cycles with one having a repelling cycle and four having attracting cycles. Numerical simulations are used to illustrate the analytical results and to show that critical times for Neimark–Sacker bifurcations are less than critical times for Andronov–Hopf bifurcations but converge to them as the time step of the discretization tends to zero.

Maximum profile likelihood estimation of differential equation parameters through model based smoothing state estimates