Mathematical Modelling of In-Vivo Dynamics of HIV Subject to the Influence of the CD8+ T-Cells

There have been many mathematical models aimed at analysing the in-vivo dynamics of HIV. However, in most cases the attention has been on the interaction between the HIV virions and the CD4+ T-cells. This paper brings in the intervention of the CD8+ T-cells in seeking, destroying, and killing the infected CD4+ T-cells during early stages of infection. The paper presents and analyses a five-component in-vivo model and applies the results in investigating the in-vivo dynamics of HIV in presence of the CD8+ T-cells. We prove the positivity and the boundedness of the model solutions. In addition, we show that the solutions are biologically meaningful. Both the endemic and virions- free equilibria are determined and their stability investigated. In addition, the basic reproductive number is derived by the next generation matrix method. We prove that the virions-free equilibrium state is locally asymptotically stable if and only if R0 < 1 and unstable otherwise. The results show that at acute infection the CD8+ T-cells play a paramount role in reducing HIV viral replication. We also observe that the model exhibits backward and trans-critical bifurcation for some set of parameters for R0 . This is a clear indication that having R0 is not sufficient condition for virions depletion.

Extension of Smoothed Particle Hydrodynamics (SPH), Mathematical Background of Vortex Blob Method (VBM) and Moving Particle Semi-Implicit (MPS)

SPH has a reasonable mathematical background. Although VBM and MPS are similar to SPH, their ma-thematical backgrounds seem fragile. VBM has some problems in treating the viscous diffusion of vortices but is known as a practical method for calculating viscous flows. The mathematical background of MPS is also not sufficient. Not with standing, the numerical results seem reasonable in many cases. The problem common in both VBM and MPS is that the space derivatives necessary for calculating viscous diffusion are not estimated reasonably, although the treatment of advection is mathematically correct. This paper discusses a method to estimate the above mentioned problem of how to treat the space derivatives. The numerical results show the comparison among FDM (Finite Difference Method), SPH and MPS in detail. In some cases, there are big differences among them. An extension of SPH is also given.

Parameter Identification in Traveling Wave Solutions of a Modified Fisher’s Equation

In this work, we focus on the inverse problem of determining the parameters in a partial differential equation from given numerical solutions. For this purpose, we consider a modified Fisher’s equation that includes a relaxation time in relating the flux to the gradient of the density and an added cubic non-linearity. We show that such equations still possess traveling wave solutions by using standard methods for nonlinear dynamical systems in which fixed points in the phase plane are found and their stability characteristics are classified. A heteroclinic orbit in the phase plane connecting a saddle point to a node represents the traveling wave solution. We then design parameter estimation/discovery algorithms for this system including a few based on machine learning methods and compare their performance.

Dynamics and Equilibria of N Point Charges on a 2D Ellipse or a 3D Ellipsoid

We consider the so-called Thomson problem which refers to finding the equilibrium distribution of a finite number of mutually repelling point charges on the surface of a sphere, but for the case where the sphere is replaced by a spheroid or ellipsoid. To get started, we first consider the problem in two dimensions, with point charges on circles (for which the equilibrium distribution is intuitively obvious) and ellipses. We then generalize the approach to the three-dimensional case of an ellipsoid. The method we use is to begin with a random distribution of charges on the surface and allow each point charge to move tangentially to the surface due to the sum of all Coulomb forces it feels from the other charges. Deriving the proper equations of motion requires using a projection operator to project the total force on each point charge onto the tangent plane of the surface. The position vectors then evolve and find their final equilibrium distribution naturally. For the case of ellipses and ellipsoids or spheroids, we find that multiple distinct equilibria are possible for certain numbers of charges, depending on the starting conditions. We characterize these based on their total potential energies. Some of the equilibria found turn out to represent local minima in the potential energy landscape, while others represent the global minimum. We devise a method based on comparing the moment-of-inertia tensors of the final configurations to distinguish them from one another.

Schur Forms and Normal-Nilpotent Decompositions

Real and complex Schur forms have been receiving increasing attention from the fluid mechanics community recently,especially related to vortices and turbulence.Several decompositions of the velocity gradient tensor,such as the triple decomposition of motion(TDM)and normal-nilpotent decomposition(NND),have been proposed to analyze the local motions of fluid elements.However,due to the existence of different types and non-uniqueness of Schur forms,as well as various possible definitions of NNDs,confusion has spread widely and is harming the research.This work aims to clean up this confusion.To this end,the complex and real Schur forms are derived constructively from the very basics,with special consideration for their non-uniqueness.Conditions of uniqueness are proposed.After a general discussion of normality and nilpotency,a complex NND and several real NNDs as well as normal-nonnormal decompositions are constructed,with a brief comparison of complex and real decompositions.Based on that,several confusing points are clarified,such as the distinction between NND and TDM,and the intrinsic gap between complex and real NNDs.Besides,the author proposes to extend the real block Schur form and its corresponding NNDs for the complex eigenvalue case to the real eigenvalue case.But their justification is left to further investigations.

Radiative Blood-Based Hybrid Copper-Graphene Nanoliquid Flows along a Source-Heated Leaning Cylinder

Variant graphene,graphene oxides(GO),and graphene nanoplatelets(GNP)dispersed in blood-based copper(Cu)nanoliquids over a leaning permeable cylinder are the focus of this study.These forms of graphene are highly beneficial in the biological and medical fields for cancer therapy,anti-infection measures,and drug delivery.The non-Newtonian Sutterby(blood-based)hybrid nanoliquid flows are generalized within the context of the Tiwari-Das model to simulate the effects of radiation and heating sources.The governing partial differential equations are reformulated into a nonlinear set of ordinary differential equations using similar transformational expressions.These equations are then transformed into boundary value problems through a shooting technique,followed by the implementation of the bvp4c tool in MATLAB.The influences of various parameters on the model’s nondimensional velocity and temperature profiles,reduced skin friction,and reduced Nusselt number are presented for detailed discussions.The results indicated that Cu-GNP/blood and Cu-GO/blood hybrid nanofluids exhibit the lowest and highest velocity distributions,respectively,for increased nanoparticles volume fraction,curvature parameter,Sutterby fluid parameter,Hartmann number,and wall permeability parameter.Conversely,opposite trends are observed for the temperature distribution for all considered parameters,except the mixed convection parameter.Increases in the reduced skin friction magnitude and the reduced Nusselt number with higher values of graphene/GO/GNP nanoparticle volume fraction are also reported.Finally,GNP is identified as the superior heat conductor,with an average increase of approximately 5%and a peak of 7.8%in the reduced Nusselt number compared to graphene and GO nanoparticles in the Cu/blood nanofluids.

Random Green's Function Method for Large-Scale Electronic Structure Calculation

We report a linear-scaling random Green's function(rGF) method for large-scale electronic structure calculation. In this method, the rGF is defined on a set of random states and is efficiently calculated by projecting onto Krylov subspace. With the rGF method, the Fermi–Dirac operator can be obtained directly, avoiding the polynomial expansion to Fermi–Dirac function. To demonstrate the applicability, we implement the rGF method with the density-functional tight-binding method. It is shown that the Krylov subspace can maintain at small size for materials with different gaps at zero temperature, including H_(2)O and Si clusters. We find with a simple deflation technique that the rGF self-consistent calculation of H_(2)O clusters at T = 0 K can reach an error of~ 1 me V per H_(2)O molecule in total energy, compared to deterministic calculations. The rGF method provides an effective stochastic method for large-scale electronic structure simulation.

Report for Type 2 Bayes-Fuzzy Estimation in No-Data Problem

It is well known that the system (1 + 1) can be unequal to 2, because this system has both observation error and system error. Furthermore, we must provide our mustered service within our cool head and warm heart, where two states of nature are existing upon us. Any system is regarded as the two-dimensional variable error model. On the other hand, we consider that the fuzziness is existing in this system. Though we can usually obtain the fuzzy number from the possibility theory, it is not fuzzy but possibility, because the possibility function is as same as the likelihood function, and we can obtain the possibility measure by the maximal likelihood method (i.e. max product method proposed by Dr. Hideo Tanaka). Therefore, Fuzzy is regarded as the only one case according to Vague, which has both some state of nature in this world and another state of nature in the other world. Here, we can consider that Type 1 Vague Event in other world can be obtained by mapping and translating from Type 1 fuzzy Event in this world. We named this estimation as Type 1 Bayes-Fuzzy Estimation. When the Vague Events were abnormal (ex. under War), we need to consider that another world could exist around other world. In this case, we call it Type 2 Bayes-Fuzzy Estimation. Where Hori et al. constructed the stochastic different equation upon Type 1 Vague Events, along with the general following probabilistic introduction method from the single regression model, multi-regression model, AR model, Markov (decision) process, to the stochastic different equation. Furthermore, we showed that the system theory approach is Possibility Markov Process, and that the making decision approach is Sequential Bayes Estimation, too. After all, Type 1 Bays-Fuzzy estimation is the special case in Bayes estimation, because the pareto solutions can exist in two stochastic different equations upon Type 2 Vague Events, after we ignore one equation each other (note that this is Type 1 case), we can obtain both its system solution and its decision solution. Here, it is noted that Type 2 Vague estimation can be applied to the shallow abnormal decision problem with possibility reserved judgement. However, it is very important problem that we can have no idea for possibility reserved judgement under the deepest abnormal envelopment (ex. under War). Expect for this deepest abnormal decision problem, Bayes estimation can completely cover fuzzy estimation. In this paper, we explain our flowing study and further research object forward to this deepest abnormal decision problem.

Using Extreme Value Theory Approaches to Estimate High Quantiles for Stroke Data

This paper aims to explore the application of Extreme Value Theory (EVT) in estimating the conditional extreme quantile for time-to-event outcomes by examining the functional relationship between ambulatory blood pressure trajectories and clinical outcomes in stroke patients. The study utilizes EVT to analyze the functional connection between ambulatory blood pressure trajectories and clinical outcomes in a sample of 297 stroke patients. The 24-hour ambulatory blood pressure measurement curves for every 15 minutes are considered, acknowledging a censored rate of 40%. The findings reveal that the sample mean excess function exhibits a positive gradient above a specific threshold, confirming the heavy-tailed distribution of data in stroke patients with a positive extreme value index. Consequently, the estimated conditional extreme quantile indicates that stroke patients with higher blood pressure measurements face an elevated risk of recurrent stroke occurrence at an early stage. This research contributes to the understanding of the relationship between ambulatory blood pressure and recurrent stroke, providing valuable insights for clinical considerations and potential interventions in stroke management.

Acoustic radiation force imaging sonoelastography for noninvasive staging of liver fibrosis

AIM:To investigate the diagnostic accuracy of acoustic radiation force impulse (ARFI) imaging as a noninvasive method for the assessment of liver fibrosis in chronic hepatitis C (CHC) patients.METHODS:We performed a prospective blind com-parison of ARFI elastography,APRI index and FibroMax in a consecutive series of patients who underwent liver biopsy for CHC in University Hospital Bucharest. His-topathological staging of liver fibrosis according to the METAVIR scoring system served as the reference. A to-tal of 74 patients underwent ARFI elastography,APRI index,FibroMax and successful liver biopsy. RESULTS:The noninvasive tests had a good correlation with the liver biopsy results. The most powerful test in predicting fibrosis was ARFI elastography. The diagnostic accuracy of ARFI elastography,expressedas area under receiver operating characteristic curve (AUROC) had a validity of 90.2% (95% CI AUROC = 0.831-0.972,P < 0.001) for the diagnosis of significant f ibrosis (F ≥ 2). ARFI sonoelastography predicted even better F3 or F4 fibrosis (AUROC = 0.993,95% CI = 0.979-1).CONCLUSION:ARFI elastography had very good accuracy for the assessment of liver fibrosis and was superior to other noninvasive methods (APRI Index,FibroMax) for staging liver fibrosis.

A new type of conserved quantity of Mei symmetry for the motion of mechanico electrical coupling dynamical systems

We obtain a new type of conserved quantity of Mei symmetry for the motion of mechanico--electrical coupling dynamical systems under the infinitesimal transformations. A criterion of Mei symmetry for the mechanico-electrical coupling dynamical systems is given. Simultaneously, the condition of existence of the new conserved quantity of Mei symmetry for mechanico-electrical coupling dynamical systems is obtained. Finally, an example is given to illustrate the application of the results.

Mei symmetry and Mei conserved quantity of nonholonomic systems with unilateral Chetaev type in Nielsen style

This paper studies the Mei symmetry and Mei conserved quantity for nonholonomic systems of unilateral Chetaev type in Nielsen style. The differential equations of motion of the system above are established. The definition and the criteria of Mei symmetry, loosely Mei symmetry, strictly Mei symmetry for the system are given in this paper. The existence condition and the expression of Mei conserved quantity are deduced directly by using Mei symmetry. An example is given to illustrate the application of the results.

Lagrange equations of nonholonomic systems with fractional derivatives

This paper obtains Lagrange equations of nonholonomic systems with fractional derivatives. First, the exchanging relationships between the isochronous variation and the fractional derivatives are derived. Secondly, based on these exchanging relationships, the Hamilton＇s principle is presented for non-conservative systems with fractional derivatives. Thirdly, Lagrange equations of the systems are obtained. Furthermore, the d＇Alembert-Lagrange principle with fractional derivatives is presented, and the Lagrange equations of nonholonomic systems with fractional derivatives are studied. An example is designed to illustrate these results.

Towards an Artificial Intelligence Framework for Data-Driven Prediction of Coronavirus Clinical Severity

The virus SARS-CoV2,which causes the Coronavirus disease COVID-19 has become a pandemic and has spread to every inhabited continent.Given the increasing caseload,there is an urgent need to augment clinical skills in order to identify from among the many mild cases the few that will progress to critical illness.We present a first step towards building an artificial intelligence(AI)framework,with predictive analytics(PA)capabilities applied to real patient data,to provide rapid clinical decision-making support.COVID-19 has presented a pressing need as a)clinicians are still developing clinical acumen given the disease’s novelty,and b)resource limitations in a rapidly expanding pandemic require difficult decisions relating to resource allocation.The objectives of this research are:(1)to algorithmically identify the combinations of clinical characteristics of COVID-19 that predict outcomes,and(2)to develop a tool with AI capabilities that will predict patients at risk for more severe illness on initial presentation.The predictive models learn from historical data to help predict specifically who will develop acute respiratory distress syndrome(ARDS),a severe outcome in COVID-19.Our experimental results based on two hospitals in Wenzhou,Zhejang,China identify features most predictive of ARDS in COVID-19 initial presentation which would not have stood out to clinicians.A mild increase in elevated alanine aminotransferase(ALT)(a liver enzyme)),a presence of myalgias(body aches),and an increase in hemoglobin,in this order,are the clinical features,on presentation,that are the most predictive.Those two centers’COVID-19 case series symptoms on initial presentation can help predict severe outcomes.Predictive models that learned from historical data of patients from two Chinese hospitals achieved 70%to 80%accuracy in predicting severe cases.

Conformal invariance and conserved quantity of Hamilton systems

This paper studies conformal invariance and conserved quantities of Hamilton system. The definition and the determining equation of conformal invariance for Hamilton system are provided. The relationship between the conformal invariance and the Lie symmetry are discussed, and the necessary and sufficient condition that the conformal invariance would be the Lie symmetry of the system under the infinitesimal one-parameter transformation group is deduced. It gives the conserved quantities of the system and an example for illustration.

Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type for Lagrange systems

Based on the invariance of differential equations under infinitesimal transformations of group, Lie symmetries, exact invariants, perturbation to the symmetries and adiabatic invariants in form of non-Noether for a Lagrange system are presented. Firstly, the exact invariants of generalized Hojman type led directly by Lie symmetries for a Lagrange system without perturbations are given. Then, on the basis of the concepts of Lie symmetries and higher order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for the system with the action of small disturbance is investigated, the adiabatic invariants of generalized Hojman type for the system are directly obtained, the conditions for existence of the adiabatic invariants and their forms are proved. Finally an example is presented to illustrate these results.

Lie Symmetrical Perturbation and Adiabatic Invariants of Generalized Hojman Type for Disturbed Nonholonomic Systems

For a nonholonomic mechanics system with the action of small disturbance, the Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type are studied under general infinitesimal transformations of groups in which the generalized coordinates and time are variable. On the basis of the invariance of disturbed nonholonomic dynamical equations under general infinitesimal transformations, the determining equations, the constrained restriction equations and the additional restriction equations of Lie symmetries of the system are constructed, which only depend on the variables t, qs and q^.s. Based on the definition of higher-order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for a nonholonomic system with the action of small disturbance is investigated, and the Lie symmetrical adiabatic invariants, the weakly Lie symmetrical adiabatic invariants and the strongly Lie symmetrical adiabatic invariants of generalized Hojman type of disturbed nonholonomic systems are obtained. An example is given to illustrate applications of the results.

Lie Symmetrical Perturbation and Adiabatic Invariants of Generalized Hojman Type for Relativistic Birkhoffian Systems

For a relativistic Birkhoffian system, the Lie symmetrical perturbation and adiabatic invariants of generalized Bojman type are studied under general infinitesimal transformations. On the basis of the invariance of relativistic Birkhotfian equations under general infinitesimal transformations,Lie symmetrical transformations of the system are constructed, which only depend on the Birkhoffian variables. The exact invariants in the form of generalized Hojman conserved quantities led by the Lie symmetries of relativistic Birkhoffian system without perturbations are given. Based on the definition of higher-order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for relativistic Birkhoffian system with the action of small disturbance is investigated, and a new type of adiabatic invariants of the system is obtained. In the end of the paper, an example is given to illustrate the application of the results.

Special Lie symmetry and Hojman conserved quantity of Appell equations in a dynamical system of relative motion

Special Lie symmetry and the Hojman conserved quantity for Appell equations in a dynamical system of relative motion are investigated. The definition and the criterion of the special Lie symmetry of Appell equations in a dynamical system of relative motion under infinitesimal group transformation are presented. The expression of the equation for the special Lie symmetry of Appell equations and the Hojman conserved quantity, deduced directly from the special Lie symmetry in a dynamical system of relative motion, are obtained. An example is given to illustrate the application of the results.

Routh Order Reduction Method of Relativistic Birkhoffian Systems

Routh order reduction method of the relativistic Birkhoffian equations is studied. For a relativistic Birkhoffian system, the cyclic integrals can be found by using the perfect differential method. Through these cyclic integrals, the order of the system can be reduced. If the relativistic Birkhoffian system has a cyclic integral, then the Birkhoffian equations can be reduced at least by two degrees and the Birkhoffian form can be kept. The relations among the relativistic Birkhoffian mechanics, the relativistic Hamiltonian mechanics, and the relativistic Lagrangian mechanics are discussed, and the Routh order reduction method of the relativistic Lagrangian system is obtained. And an example is given to illustrate the application of the result.