A novel simulation-based analysis of a stochastic HIV model with the time delay using high order spectral collocation technique

The economic impact of Human Immunodeficiency Virus (HIV) goes beyond individual levels and it has a significant influence on communities and nations worldwide. Studying the transmission patterns in HIV dynamics is crucial for understanding the tracking behavior and informing policymakers about the possible control of this viral infection. Various approaches have been adopted to explore how the virus interacts with the immune system. Models involving differential equations with delays have become prevalent across various scientific and technical domains over the past few decades. In this study, we present a novel mathematical model comprising a system of delay differential equations to describe the dynamics of intramural HIV infection. The model characterizes three distinct cell sub-populations and the HIV virus. By incorporating time delay between the viral entry into target cells and the subsequent production of new virions, our model provides a comprehensive understanding of the infection process. Our study focuses on investigating the stability of two crucial equilibrium states the infection-free and endemic equilibriums. To analyze the infection-free equilibrium, we utilize the LaSalle invariance principle. Further, we prove that if reproduction is less than unity, the disease free equilibrium is locally and globally asymptotically stable. To ensure numerical accuracy and preservation of essential properties from the continuous mathematical model, we use a spectral scheme having a higher-order accuracy. This scheme effectively captures the underlying dynamics and enables efficient numerical simulations.

Numerical simulation of a fractional stochastic delay differential equations using spectral scheme: a comprehensive stability analysis

The fractional stochastic delay differential equation (FSDDE) is a powerful mathematical tool for modeling complex systems that exhibit both fractional order dynamics and stochasticity with time delays. The purpose of this study is to explore the stability analysis of a system of FSDDEs. Our study emphasizes the interaction between fractional calculus, stochasticity, and time delays in understanding the stability of such systems. Analyzing the moments of the system's solutions, we investigate stochasticity's influence on FSDDS. The article provides practical insight into solving FSDDS efficiently using various numerical techniques. Additionally, this research focuses both on asymptotic as well as Lyapunov stability of FSDDS. The local stability conditions are clearly presented and also the effects of a fractional orders with delay on the stability properties are examine. Through a comprehensive test of a stability criteria, practical examples and numerical simulations we demonstrate the complexity and challenges concern with the analyzing FSDDEs.

Transient analysis and reliability modeling of fault-tolerant system operating under admission control policy with double retrial features and working vacation

Maintaining a high-level reliability and efficiency without interruption are the key concerns for many real-time machining systems. Using the redundancy and repair facility features, we develop a double retrial orbit queueing model for the fault-tolerant machining system (FTMS) operating under the restriction of admission of repair jobs based on threshold policy and working vacation. The provision of primary and secondary orbits is made so that the failed units can wait there based on the facility available in case the repairman is occupied. From the orbits, the failed units retry to get the repairman free so that the repair job can be accomplished. Chapman-Kolmogorov equations for the system states of FTMS have been constructed to evaluate the transient reliability and queueing indices using the spectral technique. The sensitivity along with the relative sensitivity analysis of crucial system parameters, have facilitated. The impacts of parameter variability on the system metrics and total expected cost are examined for illustrative examples.

Intermolecular interaction of diamine-diol binary system: A mini-review

The highly selective chemical reaction of carbon dioxide with organic amines is considered to a mature technology and a feasible initial path for carbon capture. In order to solve the disadvantages of high volatility, equipment corrosion and high energy consumption of traditional organic amines, amine alcohol "mixture based" solution has been developed and showed excellent carbon dioxide absorption capacity, which is due to the positive effect of intermolecular interaction in amine alcohol "mixture based" solution system on thermodynamic properties. However, the influencing factors of the intermolecular force in multicomponent solution system are complex, including the chemical, physical, structural effects. Therefore, it is necessary to comprehensively use a variety of characterization methods to systematically understand the form of intermolecular interaction in multicomponent solution system. This review systematically discusses the determination of intermolecular interactions in diamine-diol multicomponent solutions by three mainstream research methods, theoretical calculation method, spectral method, and thermodynamic method, aiming to provide a theoretical reference for the industrial production, the supplement to experimental data, and construction and understanding of theoretical models of multicomponent solution system.

SBTD: A Novel Method for Detecting Topological Associated Domains from Hi-C Data

The development of Hi-C technology has generated terabytes of chromatin interaction data, which bring possibilities for insight study of chromatin structure. Several studies revealed that mammalian chromosomes are folded into topological associated domains (TADs), which are conserved across cell types. Accurate detection of topological associated domains is now a vital process for revealing the relationship between the structure and function of genome organization. Unfortunately, the current TAD detection methods require massive computing resources, careful parameter adjustment and/or encounter inconsistent results. In this paper, we propose a novel method, Spectral-Based TAD Detector (SBTD), and evaluate its performance with a set of widely accepted statistical methods. We treat the chromatin interaction matrix as a graph and first introduce cosine similarity as a measure of the interaction patterns between bins. The results show that SBTD identifies higher quality TADs than the popular methods (DomainCaller, TopDom and SpectralTAD) and the internal bins of TADs identified by SBTD have higher correlation. Besides, The TADs identified by SBTD show a highly similar histone modification signal enrichment pattern at the boundary as reported in the previous literature. Finally, the motif enrichment analysis shows that compared with the background region, the DNA motifs of known insulator proteins are significantly enriched in the TAD boundary region identified by our method, which proves the high performance of our proposed method. Overall, SBTD is much more effective than existing methods with only one easy-to-adjust parameter, cluster number, for which we provide optimization guidelines.

A fuzzy fractional model of coronavirus (COVID-19) and its study with Legendre spectral method

The virus which belongs to the family of the coronavirus was seen first in Wuhan city of China. As it spreads so quickly and fastly, now all over countries in the world are suffering from this. The world health organization has considered and declared it a pandemic. In this presented research, we have picked up the existing mathematical model of corona virus which has six ordinary differential equations involving fractional derivative with non-singular kernel and Mittag-Leffler law. Another new thing is that we study this model in a fuzzy environment. We will discuss why we need a fuzzy environment for this model. First of all, we find out the approximate value of ABC fractional derivative of simple polynomial function ( t - a ) n . By using this approximation we will derive and developed the Legendre operational matrix of fractional differentiation for the Mittag-Leffler kernel fractional derivative on a larger interval [ 0 , b ] , b ⩾ 1 , b ∈ N . For the numerical investigation of the fuzzy mathematical model, we use the collocation method with the addition of this newly developed operational matrix. For the feasibility and validity of our method we pick up a particular case of our model and plot the graph between the exact and numerical solutions. We see that our results have good accuracy and our method is valid for the fuzzy system of fractional ODEs. We depict the dynamics of infected, susceptible, exposed, and asymptotically infected people for the different integer and fractional orders in a fuzzy environment. We show the effect of fractional order on the suspected, exposed, infected, and asymptotic carrier by plotting graphs.

A novel mathematical approach of COVID-19 with non-singular fractional derivative

We analyze a proposition which considers new mathematical model of COVID-19 based on fractional ordinary differential equation. A non-singular fractional derivative with Mittag-Leffler kernel has been used and the numerical approximation formula of fractional derivative of function ( t - a ) n is obtained. A new operational matrix of fractional differentiation on domain [0, a], a ≥ 1, a ∈ N by using the extended Legendre polynomial on larger domain has been developed. It is shown that the new mathematical model of COVID-19 can be solved using Legendre collocation method. Also, the accuracy and validity of our developed operational matrix have been tested. Finally, we provide numerical evidence and theoretical arguments that our new model can estimate the output of the exposed, infected and asymptotic carrier with higher fidelity than the previous models, thereby motivating the use of the presented model as a standard tool for examining the effect of contact rate and transmissibility multiple on number of infected cases are depicted with graphs.

A posteriori error estimates of spectral method for nonlinear parabolic optimal control problem

In this paper, we investigate the spectral approximation of optimal control problem governed by nonlinear parabolic equations. A spectral approximation scheme for the nonlinear parabolic optimal control problem is presented. We construct a fully discrete spectral approximation scheme by using the backward Euler scheme in time. Moreover, by using an orthogonal projection operator, we obtain L2(H1)-L2(L2) a posteriori error estimates of the approximation solutions for both the state and the control. Finally, by introducing two auxiliary equations, we also obtain L2(L2)-L2(L2) a posteriori error estimates of the approximation solutions for both the state and the control.

Validation of a Spectral Method for Quantitative Measurement of Color in Protein Drug Solutions

A quantitative spectral method has been developed to precisely measure the color of protein solutions. In this method, a spectrophotometer is utilized for capturing the visible absorption spectrum of a protein solution, which can then be converted to color values (L*a*b*) that represent human perception of color in a quantitative three-dimensional space. These quantitative values (L*a*b*) allow for calculating the best match of a sample's color to a European Pharmacopoeia reference color solution. In order to qualify this instrument and assay for use in clinical quality control, a technical assessment was conducted to evaluate the assay suitability and precision. Setting acceptance criteria for this study required development and implementation of a unique statistical method for assessing precision in 3-dimensional space. Different instruments, cuvettes, protein solutions, and analysts were compared in this study. The instrument accuracy, repeatability, and assay precision were determined. The instrument and assay are found suitable for use in assessing color of drug substances and drug products and is comparable to the current European Pharmacopoeia visual assessment method.

LAY ABSTRACT

In the biotechnology industry, a visual assessment is the most commonly used method for color characterization, batch release, and stability testing of liquid protein drug solutions. Using this method, an analyst visually determines the color of the sample by choosing the closest match to a standard color series. This visual method can be subjective because it requires an analyst to make a judgment of the best match of color of the sample to the standard color series, and it does not capture data on hue and chroma that would allow for improved product characterization and the ability to detect subtle differences between samples. To overcome these challenges, we developed a quantitative spectral method for color determination that greatly reduces the variability in measuring color and allows for a more precise understanding of color differences. In this study, we established a statistical method for assessing precision in 3-dimensional space and demonstrated that the quantitative spectral method is comparable with respect to precision and accuracy to the current European Pharmacopoeia visual assessment method.

Theoretical evaluation of the acoustic field in an ultrasonic bioreactor

Ultrasound-assisted bioreactors that provide mechanical conditioning to cells have broad applicability in tissue engineering, but biological experiments with ultrasound are very sensitive to environmental conditions. A mathematical model was developed to complement experimental measurements, as well as to describe ultrasonic fields existing in regions where measurements are impossible, specifically, within microporous tissue engineering scaffolds. The model uniquely combines Biot theory to predict the ultrasonic field in the scaffold with an electromechanical transducer model to couple the mechanical stimulation experienced by cells to the external electrical input. In the specific example examined here, cells immobilized on scaffolds are subjected to different forms of ultrasonic stimulation due to the formation of standing wave fields and vertical high-pressure bands. The model confirms the sensitivity of the supplied acoustic power to the liquid level in sonobioreactors and identifies the input electrical impedance as a method of detecting resonance effects.