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Some numerical methods approximate the Jacobian using finite differences as part of their process. The finite difference method of approximation resembles the limit definition of the derivative. Since a computer cannot handle the mathematical concept of limits, a sufficiently small number has to be chosen as an increment. However, when the increments are chosen outside of certain acceptable ranges, we need to account for error stemming from issues regarding representability. In this thesis we will look at the mathematical limitations of the increments. We will also go over how they are chosen in 3 different solvers, and suggest ways to improve upon them., Inom matematik så används något som kallas derivator för att beräkna förändring. Ett exempel är hastighet som är positionens derivata. När man uppskattar hastighet så dividerar man förflyttningssträckan med tidsåtgången. För att ge en mer precis uppskattning för hastigheten i stunden så kan man då minska tiden man beräknar över. När detta görs på datorer så stöter man på problem när stegen blir för små. Detta grundar sig i hur datorer lagrar information. Det blir helt enkelt problem när man försöker summera ett stort och litet tal. Lite förenklat så kan datorn inte se skillnaden på hastigheten vid 15 minuter efter fyra och 15 minuter och 113 miljontedels sekunder efter fyra. Detta arbete ser över de matematiska begränsningarna för hur denna tidsskillnad kan väljas, och hur det praktiskt görs i 3 olika simuleringsalgoritmer. Vi föreslår sedan hur förbättringar skulle kunna göras.

Macrophages are immune cells which play a key role in the reaction to biomaterials. They exhibit a functional phenotype (or state) induced by the stimulus received and conditions of the microenvironment. This polarization process is governed by specific cytokines that are released by the macrophage itself, as well as produced by other cellular activation mechanisms. Cytokines act as phenotype markers within a heterogeneous range whose extremes are historically identified as pro-inflammatory or M1 and anti-inflammatory or M2. In such a context, this work aims to propose a predictive modeling approach for the simulation of the response to a pro-inflammatory stimulus in macrophages. This will allow us to subsequently simulate the immune reaction induced by the presence of biomaterials at the cellular level, with the final goal to build a digital twin of the inflammatory response in a foreign body reaction. To do that, existing Ordinary Differential Equation (ODE) and Agent Based (AB) models have been considered and validated with in-vitro experimental data. Preliminary results highlight a better agreement of the AB approach over the ODE models taken into account in this work. This specific scheme is making simplified assumptions on spatial resolution and diffusion of inflammation (both cytokine and macrophages). However, the good agreement that we have observed in this simplified model encourages the use of a more advanced and comprehensive hybrid simulation platform based on AB modeling which implements a more thorough description of the intracellular pathways and the microenvironment.

We extend recent results on the existence of positive periodic solutions for singular ordinary differential equations. The nonlinearity and the impulses satisfy weaker conditions and we present new examples., Ministerio de Economía, Industria y Competitividad, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

In this paper, we deal with a classical object, namely, a nonhyperbolic limit cycle in a system of smooth autonomous ordinary differential equations. While the existence of a center manifold near such a cycle was assumed in several studies on cycle bifurcations based on periodic normal forms, no proofs were available in the literature until recently. The main goal of this paper is to give an elementary proof of the existence of a periodic smooth locally invariant center manifold near a nonhyperbolic cycle in finite-dimensional ordinary differential equations by using the Lyapunov-Perron method. In addition, we provide several explicit examples of analytic vector fields admitting (non)-unique, (non)-C∞-smooth and (non)-analytic periodic center manifolds.

The research presented in this study investigates the performance of two different approaches, namely the staggered approach and the coupled approach, in simulating growth glioblastoma cells in microfluidic devices. The main objective is to compare the accuracy and computational efficiency of these approaches in various cases. The objective of this research is to enhance the efficiency and versatility of the coupled approach by introducing modifications that decouple highly coupled nonlinear equations in order to develop a staggered approach. The study focuses on identifying the appropriate scenarios and conditions where each approach can be effectively implemented. To achieve these goals, we begin by analyzing the coupled approach and identifying its limitations in terms of computational efficiency and flexibility. By decoupling the highly coupled equations, we aim to alleviate these limitations and improve the overall performance of analysis, specifically in terms of computational time. Through extensive experimentation and analysis, we establish the suitable situations and conditions where each approach excels.

We focus on Taylor Series Methods (TSM) and Automatic Differentiation (AD) for the numerical solution of Ordinary Differential Equations (ODE) characterized by a vector field given by a finite composition of elementary and standard functions. We show that computational advantages are achieved if a kind of pre-processing said Exact Quadratization (EQ) is applied to the ODE before applying the TSM and the AD. In particular, when the ODE function is given by a formal polynomial (i.e. with real powers) of n variables and m monomials, the computational complexity required by our EQ based method for the calculation of the k-th order Taylor coefficient is O(k) whereas by using the existing AD methods it amounts to O(k2).

Introduction: Dynamic crop growth models are an important tool to predict complex traits, like crop yield, for modern and future genotypes in their current and evolving environments, as those occurring under climate change. Phenotypic traits are the result of interactions between genetic, environmental, and management factors, and dynamic models are designed to generate the interactions producing phenotypic changes over the growing season. Crop phenotype data are becoming increasingly available at various levels of granularity, both spatially (landscape) and temporally (longitudinal, time-series) from proximal and remote sensing technologies. Methods: Here we propose four phenomenological process models of limited complexity based on differential equations for a coarse description of focal crop traits and environmental conditions during the growing season. Each of these models defines interactions between environmental drivers and crop growth (logistic growth, with implicit growth restriction, or explicit restriction by irradiance, temperature, or water availability) as a minimal set of constraints without resorting to strongly mechanistic interpretations of the parameters. Differences between individual genotypes are conceptualized as differences in crop growth parameter values. Results: We demonstrate the utility of such low-complexity models with few parameters by fitting them to longitudinal datasets from the simulation platform APSIM-Wheat involving in silico biomass development of 199 genotypes and data of environmental variables over the course of the growing season at four Australian locations over 31 years. While each of the four models fits well to particular combinations of genotype and trial, none of them provides the best fit across the full set of genotypes by trials because different environmental drivers will limit crop growth in different trials and genotypes in any specific trial will not necessarily experience the same environmental limitation. Di

The paper deals with the asymptotic expressions for the solutions along with their first derivatives, the distribution of the eigenvalues and the normalized eigenvector for large eigenvalues corresponding to a system of second order differential equations having turning points at both ends of the interval, under certain suitable boundary conditions.

This report is the user’s manual for a package of MATLAB™ scripts and functions, developed for recursive prediction error identification of nonlinear state space systems. The identified state space model incorporates delay, which allows a treatment of general nonlinear networked identification, as well as of general nonlinear systems with delay. The core of the package is an implementation of two output error identification algorithms. The algorithms are based on a continuous time, structured black box state space model of a nonlinear system. The present revision adds a new algorithm, where also the output is determined via a parameterized measurement equation in the states and inputs. The software can only be run off-line, i.e. no true real time operation is possible. The algorithms are however implemented so that true on-line operation can be obtained by extraction of the main algorithmic loop. The user must then provide the real time environment. The software package contains scripts and functions that allow the user to either input live measurements or to generate test data by simulation. The scripts and functions for the setup and execution of the identification algorithms are somewhat more general than what is described in the references. The functionality for display of results include scripts for plotting of e.g. data, parameters, prediction errors, eigenvalues and the condition number of the Hessian. The estimated model obtained at the end of a run can be simulated and the model output plotted, alone or together with the data used for identification. Model validation is supported by two methods apart from the display functionality. First, a calculation of the RPEM loss function can be performed, using parameters obtained at the end of an identification run. Secondly, the accuracy as a function of the output signal amplitude can be assessed.

This report is the user’s manual for a package of MATLAB™ scripts and functions, developed for recursive prediction error identification of nonlinear state space systems. The identified state space model incorporates delay, which allows a treatment of general nonlinear networked identification, as well as of general nonlinear systems with delay. The core of the package is an implementation of two output error identification algorithms. The algorithms are based on a continuous time, structured black box state space model of a nonlinear system. The present revision adds a new algorithm, where also the output is determined via a parameterized measurement equation in the states and inputs. The software can only be run off-line, i.e. no true real time operation is possible. The algorithms are however implemented so that true on-line operation can be obtained by extraction of the main algorithmic loop. The user must then provide the real time environment. The software package contains scripts and functions that allow the user to either input live measurements or to generate test data by simulation. The scripts and functions for the setup and execution of the identification algorithms are somewhat more general than what is described in the references. The functionality for display of results include scripts for plotting of e.g. data, parameters, prediction errors, eigenvalues and the condition number of the Hessian. The estimated model obtained at the end of a run can be simulated and the model output plotted, alone or together with the data used for identification. Model validation is supported by two methods apart from the display functionality. First, a calculation of the RPEM loss function can be performed, using parameters obtained at the end of an identification run. Secondly, the accuracy as a function of the output signal amplitude can be assessed.

This report is the user’s manual for a package of MATLAB™ scripts and functions, developed for recursive prediction error identification of nonlinear state space systems. The identified state space model incorporates delay, which allows a treatment of general nonlinear networked identification, as well as of general nonlinear systems with delay. The core of the package is an implementation of two output error identification algorithms. The algorithms are based on a continuous time, structured black box state space model of a nonlinear system. The present revision adds a new algorithm, where also the output is determined via a parameterized measurement equation in the states and inputs. The software can only be run off-line, i.e. no true real time operation is possible. The algorithms are however implemented so that true on-line operation can be obtained by extraction of the main algorithmic loop. The user must then provide the real time environment. The software package contains scripts and functions that allow the user to either input live measurements or to generate test data by simulation. The scripts and functions for the setup and execution of the identification algorithms are somewhat more general than what is described in the references. The functionality for display of results include scripts for plotting of e.g. data, parameters, prediction errors, eigenvalues and the condition number of the Hessian. The estimated model obtained at the end of a run can be simulated and the model output plotted, alone or together with the data used for identification. Model validation is supported by two methods apart from the display functionality. First, a calculation of the RPEM loss function can be performed, using parameters obtained at the end of an identification run. Secondly, the accuracy as a function of the output signal amplitude can be assessed.

This report is the user’s manual for a package of MATLAB™ scripts and functions, developed for recursive prediction error identification of nonlinear state space systems. The identified state space model incorporates delay, which allows a treatment of general nonlinear networked identification, as well as of general nonlinear systems with delay. The core of the package is an implementation of two output error identification algorithms. The algorithms are based on a continuous time, structured black box state space model of a nonlinear system. The present revision adds a new algorithm, where also the output is determined via a parameterized measurement equation in the states and inputs. The software can only be run off-line, i.e. no true real time operation is possible. The algorithms are however implemented so that true on-line operation can be obtained by extraction of the main algorithmic loop. The user must then provide the real time environment. The software package contains scripts and functions that allow the user to either input live measurements or to generate test data by simulation. The scripts and functions for the setup and execution of the identification algorithms are somewhat more general than what is described in the references. The functionality for display of results include scripts for plotting of e.g. data, parameters, prediction errors, eigenvalues and the condition number of the Hessian. The estimated model obtained at the end of a run can be simulated and the model output plotted, alone or together with the data used for identification. Model validation is supported by two methods apart from the display functionality. First, a calculation of the RPEM loss function can be performed, using parameters obtained at the end of an identification run. Secondly, the accuracy as a function of the output signal amplitude can be assessed.

This paper proposes the Unscented Kalman Filter (UKF) as a tool to achieve reliable real-time estimates of plasma insulinemia from noisy sampled glucose measurements. The approach suitably exploits Hovorka's glucose-insulin model and the filter allows to simultaneously estimate also some of the model parameters. Hovorka's model in its original form is a nonlinear ordinary differential equation system; here it is endowed with an additive state noise accounting for the unavoidable uncertainties affecting the glucose-insulin dynamics. The problem is tackled as a state filtering problem for a continuous-discrete state-space model. In our simulation, we evaluated the two representative filters: the Extended Kalman Filter (EKF) and the UKF. Our simulation results indicate that when the initial states for the filters are significantly deviated from the true values, the estimation accuracy of the UKF becomes better than the EKF whilst, when the relatively precise initial filter value is available, the use of the EKF is sufficient.

The book is devoted to the qualitative study of differential equations defined by piecewise linear (PWL) vector fields, mainly continuous, and presenting two or three regions of linearity. The study focuses on the more common bifurcations that PWL differential systems can undergo, with emphasis on those leading to limit cycles. Similarities and differences with respect to their smooth counterparts are considered and highlighted. Regarding the dimensionality of the addressed problems, some general results in arbitrary dimensions are included. The manuscript mainly addresses specific aspects in PWL differential systems of dimensions 2 and 3, which are sufficinet for the analysis of basic electronic oscillators. The work is divided into three parts. The first part motivates the study of PWL differential systems as the natural next step towards dynamic complexity when starting from linear differential systems. The nomenclature and some general results for PWL systems in arbitrary dimensions are introduced. In particular, a minimal representation of PWL systems, called canonical form, is presented, as well as the closing equations, which are fundamental tools for the subsequent study of periodic orbits. The second part contains some results on PWL systems in dimension 2, both continuous and discontinuous, and both with two or three regions of linearity. In particular, the focus-center-limit cycle bifurcation and the Hopf-like bifurcation are completely described. The results obtained are then applied to the study of different electronic devices. In the third part, several results on PWL differential systems in dimension 3 are presented. In particular, the focus-center-limit cycle bifurcation is studied in systems with two and three linear regions, in the latter case with symmetry. Finally, the piecewise linear version of the Hopf-pitchfork bifurcation is introduced. The analysis also includes the study of degenerate situations. Again, the above results are applied to the, El libro está dedicado al estudio cualitativo de las ecuaciones diferenciales definidas por medio de campos de vectores lineales a trozos (abreviado, PWL), principalmente continuos, y que presentan dos o tres regiones de linealidad. El análisis se centra en el tipo de bifurcaciones más usuales que pueden aparecer para estos sistemas diferenciales, haciendo especial énfasis en aquellos que derivan en círculos límite. Se consideran y se destacan también las similitudes y diferencias respecto a sus contrapartidas diferenciables. En relación a la dimensión de los problemas que se abordan, los autores incluyen algunos resultados generales en dimensión arbitraria, aunque el texto está destinado principalmente a tratar aspectos específicos de los sistemas diferenciales PWL en dimensiones 2 y 3. Así, el trabajo queda dividido en tres partes fundamentales: la primera, dedicada al estudio general de estos sistemas diferenciales, como continuación natural hacia la complejidad dinámica cuando se parte de los sistemas diferenciales lineales; y la segunda y la tercera, al análisis de los sistemas diferenciales en dimensión 2 y en dimensión 3, respectivamente, aplicándose los resultados recogidos al estudio de diferentes osciladores electrónicos, en ambos casos. Esta monografía va dirigida, por un lado, a los investigadores interesados en la teoría cualitativa de sistemas diferenciales lineales a trozos o a aquellos que modelan el mundo real por medio de este tipo de sistemas diferenciales. Por otro lado, puede servir como libro de texto para un curso de postgrado sobre sistemas dinámicos definidos mediante sistemas diferenciales PWL.

We present some generalized Lipschitz conditions which imply uniqueness of solutions for scalar ODEs. We illustrate the applicability of our results with examples not covered by earlier Lipschitz–type uniqueness tests

This report is the user’s manual for a package of MATLAB™ scripts and functions, developed for recursive prediction error identification of nonlinear state space systems. The identified state space model incorporates delay, which allows a treatment of general nonlinear networked identification, as well as of general nonlinear systems with delay. The core of the package is an implementation of two output error identification algorithms. The algorithms are based on a continuous time, structured black box state space model of a nonlinear system. The present revision adds a new algorithm, where also the output is determined via a parameterized measurement equation in the states and inputs. The software can only be run off-line, i.e. no true real time operation is possible. The algorithms are however implemented so that true on-line operation can be obtained by extraction of the main algorithmic loop. The user must then provide the real time environment. The software package contains scripts and functions that allow the user to either input live measurements or to generate test data by simulation. The scripts and functions for the setup and execution of the identification algorithms are somewhat more general than what is described in the references. The functionality for display of results include scripts for plotting of e.g. data, parameters, prediction errors, eigenvalues and the condition number of the Hessian. The estimated model obtained at the end of a run can be simulated and the model output plotted, alone or together with the data used for identification. Model validation is supported by two methods apart from the display functionality. First, a calculation of the RPEM loss function can be performed, using parameters obtained at the end of an identification run. Secondly, the accuracy as a function of the output signal amplitude can be assessed.

This paper proposes the Unscented Kalman Filter (UKF) as a tool to achieve reliable real-time estimates of plasma insulinemia from noisy sampled glucose measurements. The approach suitably exploits Hovorka's glucose-insulin model and the filter allows to simultaneously estimate also some of the model parameters. Hovorka's model in its original form is a nonlinear ordinary differential equation system; here it is endowed with an additive state noise accounting for the unavoidable uncertainties affecting the glucose-insulin dynamics. The problem is tackled as a state filtering problem for a continuous-discrete state-space model. In our simulation, we evaluated the two representative filters: the Extended Kalman Filter (EKF) and the UKF. Our simulation results indicate that when the initial states for the filters are significantly deviated from the true values, the estimation accuracy of the UKF becomes better than the EKF whilst, when the relatively precise initial filter value is available, the use of the EKF is sufficient.

In this work we consider a given root of a family of n-degree polynomials as a one-variable function that depends only on the independent term. Then we prove that this function satisfies several ordinary differential equations (ODE). More concretely, it satisfies several simple separated variables ODE, a first order generalized Abel ODE of degree n−1 and an (n−1)-th order linear ODE. Although some of our results are not new, our approach is simple and self-contained. For n=2,3 and 4 we recover, from these ODE, the classical formulas for solving these polynomials.

This reports is intended as a users manual for a package of MATLAB™ scripts and functions, developed for recursive prediction error identification of nonlinear state space systems and nonlinear static systems. The core of the package is the implementation of three output error identification and scaling algorithms. The first algorithm is based on a continuous time, structured black box state space model of a nonlinear system. An RPEM algorithm for recursive identification of nonlinear static systems, that re-uses the parameterization of the nonlinear ODE model, is also included in the software package. The present revision adds a third algorithm, where also the output is determined via a parameterized measurement equation in the states and inputs. The software can only be run off-line, i.e. no true real time operation is possible. The algorithm is however implemented so that true on-line operation can be obtained by extraction of the main algorithmic loop. The user must then provide the real time environment. The software package contains scripts and functions that allow the user to either input live measurements or to generate test data by simulation. The scripts and functions for the setup and execution of the identification algorithms are somewhat more general than what is described in the references. There is e.g. support for automatic re-initiation of the algorithms using the parameters obtained at the end of a previous identification run. This allows for multiple runs through a set of data, something that is useful for data sets that are too short to allow complete convergence. The re-initiation step also allows the user to modify the degrees of the polynomial model structure and tospecify terms that are to be excluded from the model. This makes it possible to iteratively re-fine the estimated model using multiple runs. The functionality for display of results include scripts for plotting of data, parameters, prediction errors, eigenvalues and the condition nu

This reports is intended as a users manual for a package of MATLAB™ scripts and functions, developed for recursive prediction error identification of nonlinear state space systems and nonlinear static systems. The core of the package is the implementation of three output error identification and scaling algorithms. The first algorithm is based on a continuous time, structured black box state space model of a nonlinear system. An RPEM algorithm for recursive identification of nonlinear static systems, that re-uses the parameterization of the nonlinear ODE model, is also included in the software package. The present revision adds a third algorithm, where also the output is determined via a parameterized measurement equation in the states and inputs. The software can only be run off-line, i.e. no true real time operation is possible. The algorithm is however implemented so that true on-line operation can be obtained by extraction of the main algorithmic loop. The user must then provide the real time environment. The software package contains scripts and functions that allow the user to either input live measurements or to generate test data by simulation. The scripts and functions for the setup and execution of the identification algorithms are somewhat more general than what is described in the references. There is e.g. support for automatic re-initiation of the algorithms using the parameters obtained at the end of a previous identification run. This allows for multiple runs through a set of data, something that is useful for data sets that are too short to allow complete convergence. The re-initiation step also allows the user to modify the degrees of the polynomial model structure and tospecify terms that are to be excluded from the model. This makes it possible to iteratively re-fine the estimated model using multiple runs. The functionality for display of results include scripts for plotting of data, parameters, prediction errors, eigenvalues and the condition nu

In this paper, we prove that, in the deep limit, the stochastic gradient descent on a ResNet type deep neural network, where each layer shares the same weight matrix, converges to the stochastic gradient descent for a Neural ODE and that the corresponding value/loss functions converge. Our result gives, in the context of minimization by stochastic gradient descent, a theoretical foundation for considering Neural ODEs as the deep limit of ResNets. Our proof is based on certain decay estimates for associated Fokker-Planck equations.

This reports is intended as a users manual for a package of MATLAB™ scripts and functions, developed for recursive prediction error identification of nonlinear state space systems and nonlinear static systems. The core of the package is the implementation of three output error identification and scaling algorithms. The first algorithm is based on a continuous time, structured black box state space model of a nonlinear system. An RPEM algorithm for recursive identification of nonlinear static systems, that re-uses the parameterization of the nonlinear ODE model, is also included in the software package. The present revision adds a third algorithm, where also the output is determined via a parameterized measurement equation in the states and inputs. The software can only be run off-line, i.e. no true real time operation is possible. The algorithm is however implemented so that true on-line operation can be obtained by extraction of the main algorithmic loop. The user must then provide the real time environment. The software package contains scripts and functions that allow the user to either input live measurements or to generate test data by simulation. The scripts and functions for the setup and execution of the identification algorithms are somewhat more general than what is described in the references. There is e.g. support for automatic re-initiation of the algorithms using the parameters obtained at the end of a previous identification run. This allows for multiple runs through a set of data, something that is useful for data sets that are too short to allow complete convergence. The re-initiation step also allows the user to modify the degrees of the polynomial model structure and tospecify terms that are to be excluded from the model. This makes it possible to iteratively re-fine the estimated model using multiple runs. The functionality for display of results include scripts for plotting of data, parameters, prediction errors, eigenvalues and the condition nu

This reports is intended as a users manual for a package of MATLAB™ scripts and functions, developed for recursive prediction error identification of nonlinear state space systems and nonlinear static systems. The core of the package is the implementation of three output error identification and scaling algorithms. The first algorithm is based on a continuous time, structured black box state space model of a nonlinear system. An RPEM algorithm for recursive identification of nonlinear static systems, that re-uses the parameterization of the nonlinear ODE model, is also included in the software package. The present revision adds a third algorithm, where also the output is determined via a parameterized measurement equation in the states and inputs. The software can only be run off-line, i.e. no true real time operation is possible. The algorithm is however implemented so that true on-line operation can be obtained by extraction of the main algorithmic loop. The user must then provide the real time environment. The software package contains scripts and functions that allow the user to either input live measurements or to generate test data by simulation. The scripts and functions for the setup and execution of the identification algorithms are somewhat more general than what is described in the references. There is e.g. support for automatic re-initiation of the algorithms using the parameters obtained at the end of a previous identification run. This allows for multiple runs through a set of data, something that is useful for data sets that are too short to allow complete convergence. The re-initiation step also allows the user to modify the degrees of the polynomial model structure and tospecify terms that are to be excluded from the model. This makes it possible to iteratively re-fine the estimated model using multiple runs. The functionality for display of results include scripts for plotting of data, parameters, prediction errors, eigenvalues and the condition nu

This reports is intended as a users manual for a package of MATLAB™ scripts and functions, developed for recursive prediction error identification of nonlinear state space systems and nonlinear static systems. The core of the package is the implementation of three output error identification and scaling algorithms. The first algorithm is based on a continuous time, structured black box state space model of a nonlinear system. An RPEM algorithm for recursive identification of nonlinear static systems, that re-uses the parameterization of the nonlinear ODE model, is also included in the software package. The present revision adds a third algorithm, where also the output is determined via a parameterized measurement equation in the states and inputs. The software can only be run off-line, i.e. no true real time operation is possible. The algorithm is however implemented so that true on-line operation can be obtained by extraction of the main algorithmic loop. The user must then provide the real time environment. The software package contains scripts and functions that allow the user to either input live measurements or to generate test data by simulation. The scripts and functions for the setup and execution of the identification algorithms are somewhat more general than what is described in the references. There is e.g. support for automatic re-initiation of the algorithms using the parameters obtained at the end of a previous identification run. This allows for multiple runs through a set of data, something that is useful for data sets that are too short to allow complete convergence. The re-initiation step also allows the user to modify the degrees of the polynomial model structure and tospecify terms that are to be excluded from the model. This makes it possible to iteratively re-fine the estimated model using multiple runs. The functionality for display of results include scripts for plotting of data, parameters, prediction errors, eigenvalues and the condition nu

Coronavirus Disease 2019 (COVID-19) was identified in late 2019 and world health Organization (WHO) declared as a pandemic on March 11, 2019. World top researchers, physician and pharmacists are trying to find out remedy but it is still in research phase. COVID-19 spread through the air by coughing or sneezing also depends on environment. In this paper, our main goal is to COVID-19 threat analysis in South Asian people based on their habits, culture, consciousness etc. compare to Europe and North American culture. The research work is formulated in three steps. First, we formulate a dynamic infection transmission model by considering the fertility rate, mortality rate, transmission rate, and cure rate of the COVD-19 caused death rate as variables. Second, we define the variables of the model based on the census of south Asia. Finally, we analyze the threat that COVID-19 can cause to the population of crowded country likes Bangladesh, India etc. in south Asia.

[EN] There are few studies developed about the general factor of personality (GFP) dynamics. This paper uses a dynamical mathematical model, the response model, to predict the short-term effects of a dose of alcohol on GFP and reports the results of an alcohol intake experiment. The GFP dynamical mechanism of change is based on the unique trait personality theory (UTPT). This theory proposes the existence of GFP, which occupies the apex of the hierarchy of personality. An experiment with 37 volunteers was performed. All the participants completed The five-adjective scale of the general factor of personality (GFP-FAS) in trait-format (GFP-T) and state-format (GFP-S) before alcohol consumption. The participants in the experimental group (28) received 26.51 g of alcohol and a slight food, while the participants in the control group (9) just received the food. Every participant filled the GFP-S each 7 minutes. The results show that GFP is modified by a single dose of alcohol: both the high scores of GFP-T and the high scores of GFP-S explain the most part of the alcohol impact. Moreover, they prove that the response model calibration to the GFP-S scores reproduces the biphasic GFP dynamics as a consequence of an alcohol dose intake described by the literature. In fact, the results also demonstrate that the response model provides the UTPT prediction: the high scores of GFP-T predict a stronger stimulant-like effect and a stronger inhibitor effect. Thus, the response model is a useful mathematical tool to predict those individuals inclined to the alcohol misuse.

[EN] There are few studies developed about the general factor of personality (GFP) dynamics. This paper uses a dynamical mathematical model, the response model, to predict the short-term effects of a dose of alcohol on GFP and reports the results of an alcohol intake experiment. The GFP dynamical mechanism of change is based on the unique trait personality theory (UTPT). This theory proposes the existence of GFP, which occupies the apex of the hierarchy of personality. An experiment with 37 volunteers was performed. All the participants completed The five-adjective scale of the general factor of personality (GFP-FAS) in trait-format (GFP-T) and state-format (GFP-S) before alcohol consumption. The participants in the experimental group (28) received 26.51 g of alcohol and a slight food, while the participants in the control group (9) just received the food. Every participant filled the GFP-S each 7 minutes. The results show that GFP is modified by a single dose of alcohol: both the high scores of GFP-T and the high scores of GFP-S explain the most part of the alcohol impact. Moreover, they prove that the response model calibration to the GFP-S scores reproduces the biphasic GFP dynamics as a consequence of an alcohol dose intake described by the literature. In fact, the results also demonstrate that the response model provides the UTPT prediction: the high scores of GFP-T predict a stronger stimulant-like effect and a stronger inhibitor effect. Thus, the response model is a useful mathematical tool to predict those individuals inclined to the alcohol misuse.

From the viewpoint of the approach of the theory of dynamic inversion, an input reconstruction problem in a differential system of special type is under investigation. The first equation of the system is a linear stochastic Ito equation, whereas the second is a linear ordinary equation containing an unknown disturbance. The statement when the reconstruction is performed from the discrete information on several realizations of the stochastic process being a solution of the first equation is considered. The problem is reduced to an inverse problem for the system of ordinary differential equations, which includes, instead of the stochastic equation, the equation describing the dynamics of the mathematical expectation of the desired process. A finite-step software-oriented solving algorithm based on the method of auxiliary feedback controlled models is designed; an estimate for its convergence rate with respect to the number of measurable realizations is obtained. An illustrating example is given, for which the calibration of algorithm’s parameters is discussed. © 2020 the Author(s), licensee AIMS Press.

This research investigates the suitability of employing the observability Gramian as a computational instrument in analyzing microgrid system operations. An observable system permits the use of a limited sensor network to provide fault detection, abnormal activity detection, and system calibration with partial information. Microgrids can be modeled using systems of ordinary differential equations (ODEs). The observability Gramian is calculated using the solution to the ODEs with perturbed initial conditions. The observability is determined by the minimum eigenvalue of the Gramian matrix. A large minimum eigenvalue implies that the system is strongly observable, while a small value means the system is weakly observable or practically unobservable. Employing contactless sensors allows a non-invasive method of monitoring the system without interruption. As a case study, the empirical observability Gramian was numerically computed using a 9-bus microgrid model solving a system of nonlinear ODEs with 6 state variables and 12 parameters. The case study first determines the observability of various sensor network configurations, and then an observability analysis for fault detection introduces variations in system parameters. This case study showed that, depending on the sensor network configuration, microgrids can be observable in the presence of unknown parameter variation., http://archive.org/details/observabilityana1094564181, Captain, United States Army, Approved for public release; distribution is unlimited.

In this work, we establish a theory of Calculus based on the new concept of displacement. We develop all the concepts and results necessary to go from the definition to differential equations, starting with topology and measure and moving on to differentiation and integration. We find interesting notions on the way, such as the integral with respect to a path of measures or the displacement derivative. We relate both of these two concepts by a Fundamental Theorem of Calculus. Finally, we develop the necessary framework in order to study displacement equations by relating them to Stieltjes differential equations

En este trabajo, se investiga la estabilidad de dos modelos epidemiologicos de la obesidad. El primer modelo solo admite la influencia social, es decir el ambiente en que se desplazan los individuos, el segundo modelo le agrega control para reducir la obesidad y el exceso de peso. El punto de equilibrio libre de obesidad es local y asintóticamente estable si R0 < 1 donde R0 es una tasa básica de reemplazo. Para el modelo con control, se muestra la estabilidad del mismo y se aprecia numericamente que acelera la disminución de individuos con obesidad., In this work, we analyze the stability of two epidemiological models of obesity. The first model only admits social influence, that is, the environment in which individuals live, the second model adds control to reduce obesity and excess weight. The obesity-free equilibrium point is local and asymptotically stable if R0 < 1 where R0 is a basic replacement rate. For the model with control, its stability is shown and it is appreciated numerically that it accelerates the decrease of obese individuals.

The stability of two one-link flexible arms for grasping and orientation control of an object is studied. Flexible arms are modelled by Euler–Bernoulli beam model and the overall system is represented by a hybrid partial differential equation (PDE)–ordinary differential equation (ODE) model. The authors' primary concern is the stability analysis of this hybrid PDE–ODE model. In particular, they use the frequency domain method and prove the exponential stability of this system under their previously proposed boundary controller. In addition, they discuss the robustness of the closed-loop system with respect to the several disturbances including disturbances distributed over the arms and disturbances acting at boundaries. Finally, simulation results are presented to investigate the theoretical results.

In this report an equation is derived to calculate the dynamic effect of primary and secondary aircraft cabin air contamination. The equation is applied in order to understand implications and hazards. Primary contamination is from an outside source in form of normal low level contamination or high level contamination in a failure case. Secondary contamination originates from deposited material released into the cabin by a trigger event. The dynamic effect is described as an initial value problem (IVP) of a system governed by a nonhomogeneous linear first order ordinary differential equation (ODE). More complicated excitations are treated as a sequence of IVPs. The ODE is solved from first principles. Spreadsheets are provided with sample calculations that can be adapted to user needs. The method is not limited to a particular principle of the environmental control system (ECS) or contamination substance. The report considers cabin air recirculation and several locations of contamination sources, filters, and deposit points (where contaminants can accumulate and from where they can be released). This is a level of detail so far not considered in the cabin air literature. Various primary and secondary cabin contamination scenarios are calculated with plausible input parameters taken from popular passenger aircraft. A large cabin volume, high air exchange rate, large filtered air recirculation rate, and high absorption rates at deposit points lead to low contamination concentration at given source strength. Especially high contamination concentrations would result if large deposits of contaminants are released in a short time. The accuracy of the results depends on the accuracy of the input parameters. Five different approaches to reduce the contaminant concentration in the aircraft cabin are discussed and evaluated. More effective solutions involve higher implementation efforts. The method and the spreadsheets allow predicting cabin air contamination concentrations i

In this paper, we present an ordinary differential equation model depicting the interactions of basic fibroblast growth factor (bFGF) and its binding agents in a chronic wound. The delivery of bFGF was treated as a control variable and is coupled to an objective functional. By optimising the objective functional with respect to the control, predictions for optimal delivery rates of bFGF are proposed. The optimal control is then validated by comparing the cost of the objective functional for the optimal delivery rate and several alternative delivery rates. This paper addresses two objectives of effective drug delivery to chronic wounds. The first is to provide insight for the priority of delivering bFGF: to minimise the quantity of bFGF, or to optimise the distribution of bound bFGF. For effective concentrations of bound bFGF, the optimisation of bound bFGF must be prioritised over the minimisation of bFGF delivered. The second objective is to comment on the effect of the proteolytic environment within the wound, with the concentration of bound bFGF starting to decrease late in the treatment period for highly proteolytic environments. This will lead to long term complications with wound closure after the treatment has been completed. Also, it was found that for highly proteolytic environments, the cost of delivering bFGF increased. The need for optimal drug delivery is made apparent by the burden of chronic wounds on the medical industry across the developed world.

Proponemos y analizamos un modelo matemático que describe la dinámica de los mosquitos infestados por bacterias.La introducción de ciertas bacterias en la población de mosquitos apunta a disminuir gradualmente la transmisión de enfermedades de huesped-vector. Esta es una buena estrategia de control biológico., We propose and analyze a mathematical model in ordinary differential equations to describe the dynamics of mosquitoes infested by bacteria. The introduction of some bacteria in mosquitoes population aims to diminish gradually the transmission of vector host-diseases. This is a good strategy of biological control.

Background: Helicobacter pylori causes gastric cancer in 1-2% of cases but is also beneficial for protection against allergies and gastroesophageal diseases. An estimated 85% of H. pylori-colonized individuals experience no detrimental effects. To study the mechanisms promoting host tolerance to the bacterium in the gastrointestinal mucosa and systemic regulatory effects, we investigated the dynamics of immunoregulatory mechanisms triggered by H. pylori using a high-performance computing-driven ENteric Immunity SImulator multiscale model. Immune responses were simulated by integrating an agent-based model, ordinary, and partial differential equations. Results: The outputs were analyzed using 2 sequential stages: The first used a partial rank correlation coefficient regression-based and the second a metamodel-based global sensitivity analysis. The influential parameters screened from the first stage were selected to be varied for the second stage. The outputs from both stages were combined as a training dataset to build a spatiotemporal metamodel. The Sobol indices measured time-varying impact of input parameters during initiation, peak, and chronic phases of infection. The study identified epithelial cell proliferation and epithelial cell death as key parameters that control infection outcomes. In silico validation showed that colonization with H. pylori decreased with a decrease in epithelial cell proliferation, which was linked to regulatory macrophages and tolerogenic dendritic cells. Conclusions: The hybrid model of H. pylori infection identified epithelial cell proliferation as a key factor for successful colonization of the gastric niche and highlighted the role of tolerogenic dendritic cells and regulatory macrophages in modulating the host responses and shaping infection outcomes.

In this report an equation is derived to calculate the dynamic effect of primary and secondary aircraft cabin air contamination. The equation is applied in order to understand implications and hazards. Primary contamination is from an outside source in form of normal low level contamination or high level contamination in a failure case. Secondary contamination originates from deposited material released into the cabin by a trigger event. The dynamic effect is described as an initial value problem (IVP) of a system governed by a nonhomogeneous linear first order ordinary differential equation (ODE). More complicated excitations are treated as a sequence of IVPs. The ODE is solved from first principles. Spreadsheets are provided with sample calculations that can be adapted to user needs. The method is not limited to a particular principle of the environmental control system (ECS) or contamination substance. The report considers cabin air recirculation and several locations of contamination sources, filters, and deposit points (where contaminants can accumulate and from where they can be released). This is a level of detail so far not considered in the cabin air literature. Various primary and secondary cabin contamination scenarios are calculated with plausible input parameters taken from popular passenger aircraft. A large cabin volume, high air exchange rate, large filtered air recirculation rate, and high absorption rates at deposit points lead to low contamination concentration at given source strength. Especially high contamination concentrations would result if large deposits of contaminants are released in a short time. The accuracy of the results depends on the accuracy of the input parameters. Five different approaches to reduce the contaminant concentration in the aircraft cabin are discussed and evaluated. More effective solutions involve higher implementation efforts. The method and the spreadsheets allow predicting cabin air contamination concentrations i

Background: Helicobacter pylori causes gastric cancer in 1-2% of cases but is also beneficial for protection against allergies and gastroesophageal diseases. An estimated 85% of H. pylori-colonized individuals experience no detrimental effects. To study the mechanisms promoting host tolerance to the bacterium in the gastrointestinal mucosa and systemic regulatory effects, we investigated the dynamics of immunoregulatory mechanisms triggered by H. pylori using a high-performance computing-driven ENteric Immunity SImulator multiscale model. Immune responses were simulated by integrating an agent-based model, ordinary, and partial differential equations. Results: The outputs were analyzed using 2 sequential stages: The first used a partial rank correlation coefficient regression-based and the second a metamodel-based global sensitivity analysis. The influential parameters screened from the first stage were selected to be varied for the second stage. The outputs from both stages were combined as a training dataset to build a spatiotemporal metamodel. The Sobol indices measured time-varying impact of input parameters during initiation, peak, and chronic phases of infection. The study identified epithelial cell proliferation and epithelial cell death as key parameters that control infection outcomes. In silico validation showed that colonization with H. pylori decreased with a decrease in epithelial cell proliferation, which was linked to regulatory macrophages and tolerogenic dendritic cells. Conclusions: The hybrid model of H. pylori infection identified epithelial cell proliferation as a key factor for successful colonization of the gastric niche and highlighted the role of tolerogenic dendritic cells and regulatory macrophages in modulating the host responses and shaping infection outcomes.

Proponemos y analizamos un modelo matemático que describe la dinámica de los mosquitos infestados por bacterias.La introducción de ciertas bacterias en la población de mosquitos apunta a disminuir gradualmente la transmisión de enfermedades de huesped-vector. Esta es una buena estrategia de control biológico., We propose and analyze a mathematical model in ordinary differential equations to describe the dynamics of mosquitoes infested by bacteria. The introduction of some bacteria in mosquitoes population aims to diminish gradually the transmission of vector host-diseases. This is a good strategy of biological control.

We consider propagation of surface TE waves in a circular metal-dielectric waveguide filled with nonlinear (Kerr nonlinearity) metamaterial medium. Analysis is reduced to solving a nonlinear transmission eigenvalue problem for an ordinary differential equation; eigenvalues of the problem correspond to propagation constants of the waveguide. For the numerical solution, a method is proposed based on solving an auxiliary Cauchy problem (a version of the shooting method). As a result of comprehensive numerical modeling, new propagation regimes are discovered.

Ordinary differential equations (ODEs) are a widespread formalism for the mathematical modeling of natural and engineering systems, whose analysis is generally performed by means of numerical integration methods. However, real-world models are often characterized by stiffness, a circumstance that can lead to prohibitive execution times. In such cases, the practical viability of many computational tools—e.g., sensitivity analysis—is hampered by the necessity to carry out a large number of simulations. In this work, we present ginSODA, a general-purpose black-box numerical integrator that distributes the calculations on graphics processing units, and allows to run massive numbers of numerical integrations of ODE systems characterized by stiffness. By leveraging symbolic differentiation, meta-programming techniques, and source code hashing, ginSODA automatically builds highly optimized binaries for the CUDA architecture, preventing code re-compilation and allowing to speed up the computation with respect to the sequential execution. ginSODA also provides a simplified Python interface, which allows to define a system of ODEs and the test to be performed in a few lines of code. According to our results, ginSODA provides up to a 25× speedup with respect to the sequential execution

Background: Helicobacter pylori causes gastric cancer in 1-2% of cases but is also beneficial for protection against allergies and gastroesophageal diseases. An estimated 85% of H. pylori-colonized individuals experience no detrimental effects. To study the mechanisms promoting host tolerance to the bacterium in the gastrointestinal mucosa and systemic regulatory effects, we investigated the dynamics of immunoregulatory mechanisms triggered by H. pylori using a high-performance computing-driven ENteric Immunity SImulator multiscale model. Immune responses were simulated by integrating an agent-based model, ordinary, and partial differential equations. Results: The outputs were analyzed using 2 sequential stages: The first used a partial rank correlation coefficient regression-based and the second a metamodel-based global sensitivity analysis. The influential parameters screened from the first stage were selected to be varied for the second stage. The outputs from both stages were combined as a training dataset to build a spatiotemporal metamodel. The Sobol indices measured time-varying impact of input parameters during initiation, peak, and chronic phases of infection. The study identified epithelial cell proliferation and epithelial cell death as key parameters that control infection outcomes. In silico validation showed that colonization with H. pylori decreased with a decrease in epithelial cell proliferation, which was linked to regulatory macrophages and tolerogenic dendritic cells. Conclusions: The hybrid model of H. pylori infection identified epithelial cell proliferation as a key factor for successful colonization of the gastric niche and highlighted the role of tolerogenic dendritic cells and regulatory macrophages in modulating the host responses and shaping infection outcomes.

In this report an equation is derived to calculate the dynamic effect of primary and secondary aircraft cabin air contamination. The equation is applied in order to understand implications and hazards. Primary contamination is from an outside source in form of normal low level contamination or high level contamination in a failure case. Secondary contamination originates from deposited material released into the cabin by a trigger event. The dynamic effect is described as an initial value problem (IVP) of a system governed by a nonhomogeneous linear first order ordinary differential equation (ODE). More complicated excitations are treated as a sequence of IVPs. The ODE is solved from first principles. Spreadsheets are provided with sample calculations that can be adapted to user needs. The method is not limited to a particular principle of the environmental control system (ECS) or contamination substance. The report considers cabin air recirculation and several locations of contamination sources, filters, and deposit points (where contaminants can accumulate and from where they can be released). This is a level of detail so far not considered in the cabin air literature. Various primary and secondary cabin contamination scenarios are calculated with plausible input parameters taken from popular passenger aircraft. A large cabin volume, high air exchange rate, large filtered air recirculation rate, and high absorption rates at deposit points lead to low contamination concentration at given source strength. Especially high contamination concentrations would result if large deposits of contaminants are released in a short time. The accuracy of the results depends on the accuracy of the input parameters. Five different approaches to reduce the contaminant concentration in the aircraft cabin are discussed and evaluated. More effective solutions involve higher implementation efforts. The method and the spreadsheets allow predicting cabin air contamination concentrations i

We consider modeling non-autonomous dynamical systems for a group of subjects. The proposed model involves a common baseline gradient function and a multiplicative time-dependent subject-specific effect that accounts for phase and amplitude variations in the rate of change across subjects. The baseline gradient function is represented in a spline basis and the subject-specific effect is modeled as a polynomial in time with random coefficients. We establish appropriate identifiability conditions and propose an estimator based on the hierarchical likelihood. We prove consistency and asymptotic normality of the proposed estimator under a regime of moderate-to-dense observations per subject. Simulation studies and an application to the Berkeley Growth Data demonstrate the effectiveness of the proposed methodology.

This report is the user's manual for a package of MATLAB™ scripts and functions, developed for recursive prediction error identification of nonlinear state space systems. The identified state space model incorporates delay, which allows a treatment of general nonlinear networked identification, as well as of general nonlinear systems with delay. The core of the package is an implementation of an output error identification algorithm. The algorithm is based on a continuous time, structured black box state space model of a nonlinear system. The software can only be run off-line, i.e. no true real time operation is possible. The algorithms are however implemented so that true on-line operation can be obtained by extraction of the main algorithmic loop. The user must then provide the real time environment. The software package contains scripts and functions that allow the user to either input live measurements or to generate test data by simulation. The scripts and functions for the setup and execution of the identification algorithms are somewhat more general than what is described in the references. The functionality for display of results include scripts for plotting of e.g. data, parameters, prediction errors, eigenvalues and the condition number of the Hessian. The estimated model obtained at the end of a run can be simulated and the model output plotted, alone or together with the data used for identification. Model validation is supported by two methods apart from the display functionality. First, a calculation of the RPEM loss function can be performed, using parameters obtained at the end of an identification run. Secondly, the accuracy as a function of the output signal amplitude can be assessed.

Vortex induced vibrations (VIV) for energy extraction have been revisited in last years by both marine power and wind power communities. Even though vortex induced vibrations have been focus of research for many years, energy extraction from vortex induced vibrations is relevantly new field which needs more detailed investigation and modeling. To this end, there has been recent experimental and modeling parametric studies where VIV was modeled by solution of one-degree-of-freedom ordinary differential equation spring system where engineering modeling of vortex induced vibration for energy extraction has been investigated based on a spring system with the forces defined from forced oscillation experiments where full coupling of free oscillations were not taken into account. Herein a Computational Fluid Dynamics (CFD) modeling of a circular cylinder will be studied to compare forced and free vibrations in the context of vortex-induced energy extraction. The model is essentially solved by partial differential isothermal incompressible Navier-Stokes equations to model fully mathematical model of the fluid-structure interaction of vortex induced vibration. The comparison between forced and free oscillation response studies of this paper will serve to improve the scientific knowledge where vortex induced vibration modeling are comparatively more limited. The preliminary results are presented herein for forced and free oscillations for the Reynolds number regimes Re = 100 and Re = 3800 in two dimensions for combinations of amplitudes and frequency of oscillations in the context of energy extraction modeling.

This report is the user's manual for a package of MATLAB™ scripts and functions, developed for recursive prediction error identification of nonlinear state space systems. The identified state space model incorporates delay, which allows a treatment of general nonlinear networked identification, as well as of general nonlinear systems with delay. The core of the package is an implementation of an output error identification algorithm. The algorithm is based on a continuous time, structured black box state space model of a nonlinear system. The software can only be run off-line, i.e. no true real time operation is possible. The algorithms are however implemented so that true on-line operation can be obtained by extraction of the main algorithmic loop. The user must then provide the real time environment. The software package contains scripts and functions that allow the user to either input live measurements or to generate test data by simulation. The scripts and functions for the setup and execution of the identification algorithms are somewhat more general than what is described in the references. The functionality for display of results include scripts for plotting of e.g. data, parameters, prediction errors, eigenvalues and the condition number of the Hessian. The estimated model obtained at the end of a run can be simulated and the model output plotted, alone or together with the data used for identification. Model validation is supported by two methods apart from the display functionality. First, a calculation of the RPEM loss function can be performed, using parameters obtained at the end of an identification run. Secondly, the accuracy as a function of the output signal amplitude can be assessed.

This report is the user's manual for a package of MATLAB™ scripts and functions, developed for recursive prediction error identification of nonlinear state space systems. The identified state space model incorporates delay, which allows a treatment of general nonlinear networked identification, as well as of general nonlinear systems with delay. The core of the package is an implementation of an output error identification algorithm. The algorithm is based on a continuous time, structured black box state space model of a nonlinear system. The software can only be run off-line, i.e. no true real time operation is possible. The algorithms are however implemented so that true on-line operation can be obtained by extraction of the main algorithmic loop. The user must then provide the real time environment. The software package contains scripts and functions that allow the user to either input live measurements or to generate test data by simulation. The scripts and functions for the setup and execution of the identification algorithms are somewhat more general than what is described in the references. The functionality for display of results include scripts for plotting of e.g. data, parameters, prediction errors, eigenvalues and the condition number of the Hessian. The estimated model obtained at the end of a run can be simulated and the model output plotted, alone or together with the data used for identification. Model validation is supported by two methods apart from the display functionality. First, a calculation of the RPEM loss function can be performed, using parameters obtained at the end of an identification run. Secondly, the accuracy as a function of the output signal amplitude can be assessed.