THE MATHEMATICAL MODELS AND GENERALIZED VARIATIONAL PRINCIPLES OF NONLINEAR ANALYSIS FOR PERFORATED THIN PLATES

On foe basis of the Kirchoff-Karman hypothses for the nonlinear bending of thin plates, the three kinds of boundary value problems of nonlinear analysis for perforated fhin plates are presented under the differenr in-plane boundary conditions and the corresponding generalized varialional principles are established. One can see that all mathematical models presented in this paper are completely new ones and differ from the ordinary von Karman theory. These mathematical models can be applied to the nonlinear analysis and the Stability analysis of perforaled thin plates in arbitraryplane boundary conditions.

MATHEMATICAL MODELING AND BIFURCATION ANALYSIS FOR A BIOLOGICAL MECHANISM OF CANCER DRUG RESISTANCE

Drug resistance is one of the most intractable issues in targeted therapy for cancer diseases.It has also been demonstrated to be related to cancer heterogeneity,which promotes the emergence of treatment-refractory cancer cell populations.Focusing on how cancer cells develop resistance during the encounter with targeted drugs and the immune system,we propose a mathematical model for studying the dynamics of drug resistance in a conjoint heterogeneous tumor-immune setting.We analyze the local geometric properties of the equilibria of the model.Numerical simulations show that the selectively targeted removal of sensitive cancer cells may cause the initially heterogeneous population to become a more resistant population.Moreover,the decline of immune recruitment is a stronger determinant of cancer escape from immune surveillance or targeted therapy than the decay in immune predation strength.Sensitivity analysis of model parameters provides insight into the roles of the immune system combined with targeted therapy in determining treatment outcomes.

Aggravation of Cancer,Heart Diseases and Diabetes Subsequent to COVID-19 Lockdown via Mathematical Modeling

The global populationhas beenandwill continue to be severely impacted by theCOVID-19 epidemic.The primary objective of this research is to demonstrate the future impact of COVID-19 on those who suffer from other fatal conditions such as cancer,heart disease,and diabetes.Here,using ordinary differential equations(ODEs),two mathematical models are developed to explain the association between COVID-19 and cancer and between COVID-19 and diabetes and heart disease.After that,we highlight the stability assessments that can be applied to these models.Sensitivity analysis is used to examine how changes in certain factors impact different aspects of disease.The sensitivity analysis showed that many people are still nervous about seeing a doctor due to COVID-19,which could result in a dramatic increase in the diagnosis of various ailments in the years to come.The correlation between diabetes and cardiovascular illness is also illustrated graphically.The effects of smoking and obesity are also found to be significant in disease compartments.Model fitting is also provided for interpreting the relationship between real data and the results of thiswork.Diabetic people,in particular,need tomonitor their health conditions closely and practice heart health maintenance.People with heart diseases should undergo regular checks so that they can protect themselves from diabetes and take some precautions including suitable diets.The main purpose of this study is to emphasize the importance of regular checks,to warn people about the effects of COVID-19(including avoiding healthcare centers and doctors because of the spread of infectious diseases)and to indicate the importance of family history of cancer,heart diseases and diabetes.The provision of the recommendations requires an increase in public consciousness.

Mathematical Modeling of Cell Polarity Establishment of Budding Yeast

The budding yeast Saccharomyces cerevisiae is a powerful model system for studying the cell polarity establishment.The cell polarization process is regulated by signaling molecules,which are initially distributed in the cytoplasm and then recruited to a proper location on the cell membrane in response to spatial cues or spontaneously.Polarization of these signaling molecules involves complex regulation,so the mathematical models become a useful tool to investigate the mechanism behind the process.In this review,we discuss how mathematical modeling has shed light on different regulations in the cell polarization.We also propose future applications for the mathematical modeling of cell polarization and morphogenesis.

Mathematical Modeling of the Co-Infection Dynamics of HIV and Tuberculosis Incorporating Inconsistency in HIV Treatment

A non-linear HIV-TB co-infection has been formulated and analyzed. The positivity and invariant region has been established. The disease free equilibrium and its stability has been determined. The local stability was determined and found to be stable under given conditions. The basic reproduction number was obtained and according to findings, co-infection diminishes when this number is less than unity, and persists when the number is greater than unity. The global stability of the endemic equilibrium was calculated. The impact of HIV on TB was established as well as the impact of TB on HIV. Numerical solution was also done and the findings indicate that when the rate of HIV treatment increases the latent TB increases while the co-infected population decreases. When the rate of HIV treatment decreases the latent TB population decreases and the co-infected population increases. Encouraging communities to prioritize the consistent treatment of HIV infected individuals must be emphasized in order to reduce the scourge of HIV-TB co-infection.

Mathematical Modeling of HIV Investigating the Effect of Inconsistent Treatment

HIV is a retrovirus that infects and impairs the cells and functions of the immune system. It has caused a great challenge to global public health systems and leads to Acquired Immunodeficiency Syndrome (AIDS), if not attended to in good time. Antiretroviral therapy is used for managing the virus in a patient’s lifetime. Some of the symptoms of the disease include lean body mass and many opportunistic infections. This study has developed a SIAT mathematical model to investigate the impact of inconsistency in treatment of the disease. The arising non-linear differential equations have been obtained and analyzed. The DFE and its stability have been obtained and the study found that it is locally asymptotically stable when the basic reproduction number is less than unity. The endemic equilibrium has been obtained and found to be globally asymptotically stable when the basic reproduction number is greater than unity. Numerical solutions have been obtained and analyzed to give the trends in the spread dynamics. The inconsistency in treatment uptake has been analyzed through the numerical solutions. The study found that when the treatment rate of those infected increases, it leads to an increase in treatment population, which slows down the spread of HIV and vice versa. An increase in the rate of treatment of those with AIDS leads to a decrease in the AIDS population, the reverse happens when this rate decreases. The study recommends that the community involvement in advocating for consistent treatment of HIV to curb the spread of the disease.

Refined mathematical model for the breaching of concrete-face sand-gravel dams due to overtopping failure

Overtopping is one of the main reasons for the breaching of concrete-face sand-gravel dams(CFSGDs).In this study,a refined mathematical model was established based on the characteristics of the overtopping breaching of CFSGDs.The model characteristics were as follows:(1)Based on the Renormailzation Group(RNG)k-εturbulence theory and volume of fluid(VOF)method,the turbulent characteristics of the dam-break flow were simulated,and the erosion surface of the water and soil was tracked;(2)In consideration of the influence of the change in the sediment content on the dam-break flow,the dam material transport equation,which could reflect the characteristics of particle settlement and entrainment motion,was used to simulate the erosion process of the sand gravels;(3)Based on the bending moment balance method,a failure equation of the concrete face slab under dead weight and water load was established.The proposed model was verified through a case study on the failure of the Gouhou CFSGD.The results showed that the proposed model could well simulate the erosion mode of the special vortex flow of the CFSGD scouring the support body of the concrete face slab inward and reflect the mutual coupling relationship between the dam-break flow,sand gravels,and concrete face slabs.Compared with the measured values,the relative errors of the peak discharge,final breach average width,dam breaching duration,and maximum failure length of the face slab calculated using the proposed model were all less than 12%,thus verifying the rationality of the model.The proposed model was demonstrated to perform better and provide more detailed results than three selected parametric models and three simplified mathematical models.The study results can aid in establishing the risk level and devising early warning strategies for CFSGDs.

Tight gas charging and accumulation mechanisms and mathematical model

The gas-water distribution and production heterogeneity of tight gas reservoirs have been summarized from experimental and geological observations, but the charging and accumulation mechanisms have not been examined quantitatively by mathematical model. The tight gas charging and accumulation mechanisms were revealed from a combination of physical simulation of nuclear magnetic resonance coupling displacement, numerical simulation considering material and mechanical equilibria, as well as actual geological observation. The results show that gas migrates into tight rocks to preferentially form a gas saturation stabilization zone near the source-reservoir interface. When the gas source is insufficient, gas saturation reduction zone and uncharged zone are formed in sequence from the source-reservoir interface. The better the source rock conditions with more gas expulsion volume and higher overpressure, the thicker the gas saturation stabilization and reduction zones, and the higher the overall gas saturation. When the source rock conditions are limited, the better the tight reservoir conditions with higher porosity and permeability as well as larger pore throat, the thinner the gas saturation stabilization and reduction zones, but the gas saturation is high. The sweet spot of tight gas is developed in the high-quality reservoir near the source rock, which often corresponds to the gas saturation stabilization zone. The numerical simulation results by mathematical model agree well with the physical simulation results by nuclear magnetic resonance coupling displacement, and reasonably explain the gas-water distribution and production pattern of deep reservoirs in the Xujiaweizi fault depression of the Songliao Basin and tight gas reservoirs in the Linxing-Huangfu area of the Ordos Basin.

Quantification of Ride Comfort Using Musculoskeletal Mathematical Model Considering Vehicle Behavior

This research aims to quantify driver ride comfort due to changes in damper characteristics between comfort mode and sport mode,considering the vehicle’s inertial behavior.The comfort of riding in an automobile has been evaluated in recent years on the basis of a subjective sensory evaluation given by the driver.However,reflecting driving sensations in design work to improve ride comfort is abstract in nature and difficult to express theoretically.Therefore,we evaluated the human body’s effects while driving scientifically by quantifying the driver’s behavior while operating the steering wheel and the behavior of the automobile while in motion using physical quantities.To this end,we collected driver and vehicle data using amotion capture system and vehicle CAN and IMU sensors.We also constructed a three-dimensional musculoskeletal mathematical model to simulate driver movements and calculate the power and amount of energy per unit of time used for driving the joints and muscles of the human body.Here,we used comfort mode and sport mode to compare damper characteristics in terms of hardness.In comfort mode,damper characteristics are soft and steering stability is mild,but vibration from the road is not easily transmitted to the driver making for a lighter load on the driver.In sport mode,on the other hand,damper characteristics are hard and steering stability is comparatively better.Still,vibration from the road is easily transmitted to the driver,whichmakes it easy for a load to be placed on the driver.As a result of this comparison,it was found that a load was most likely to be applied to the driver’s neck.This result in relation to the neck joint can therefore be treated as an objective measure for quantifying ride comfort.

Simulation Analysis of New Energy Vehicle Engine Cooling System Based on K-E Turbulent Flow Mathematical Model

New energy vehicles have better clean and environmental protection characteristics than traditional fuel vehicles.The new energy engine cooling technology is critical in the design of new energy vehicles.This paper used oneand three-way joint simulation methods to simulate the refrigeration system of new energy vehicles.Firstly,a k-εturbulent flow model for the cooling pump flow field is established based on the principle of computational fluid dynamics.Then,the CFD commercial fluid analysis software FLUENT is used to simulate the flow field of the cooling pump under different inlet flow conditions.This paper proposes an optimization scheme for new energy vehicle engines’“boiling”phenomenon under high temperatures and long-time climbing conditions.The simulation results show that changing the radiator’s structure and adjusting the thermostat’s parameters can solve the problem of a“boiling pot.”The optimized new energy vehicle engine can maintain a better operating temperature range.The algorithm model can reference each cryogenic system component hardware selection and control strategy in the new energy vehicle’s engine.

Problem-based Learning Combining Seminar Teaching Method for the Practice of Mathematical Modeling Course's Teaching Reform for Computer Discipline

Mathematical modeling course has been one of the fast development courses in China since 1992,which mainly trains students’innovation ability.However,the teaching of mathematical modeling course is quite difficult since it requires students to have a strong mathematical foundation,good ability to design algorithms,and programming skills.Besides,limited class hours and lack of interest in learning are the other reasons.To address these problems,according to the outcome-based education,we adopt the problem-based learning combined with a seminar mode in teaching.We customize cases related to computer and software engineering,start from simple problems in daily life,step by step deepen the difficulty,and finally refer to the professional application in computer and software engineering.Also,we focus on ability training rather than mathematical theory or programming language learning.Initially,we prepare the problem,related mathematic theory,and core code for students.Furtherly,we train them how to find the problem,and how to search the related mathematic theory and software tools by references for modeling and analysis.Moreover,we solve the problem of limited class hours by constructing an online resource learning library.After a semester of practical teaching,it has been shown that the interest and learning effectiveness of students have been increased and our reform plan has achieved good results.

Well-Posedness and a Finite Difference Approximation for a Mathematical Model of HPV-Induced Cervical Cancer

We present a first-order finite difference scheme for approximating solutions of a mathematical model of cervical cancer induced by the human papillomavirus (HPV), which consists of four nonlinear partial differential equations and a nonlinear first-order ordinary differential equation. The scheme is analyzed and used to provide an existence-uniqueness result. Numerical simulations are performed in order to demonstrate the first-order rate of convergence. A sensitivity analysis was done in order to compare the effects of two drug types, those that increase the death rate of HPV-infected cells, and those that increase the death rate of the precancerous cell population. The model predicts that treatments that affect the precancerous cell population by directly increasing the corresponding death rate are far more effective than those that increase the death rate of HPV-infected cells.

A Method for Constructing Mathematical Modeling of the Spread of a New Crown Pneumonia Epidemic Based on the Effect of Temperature

To better predict the spread of the COVID-19 outbreak, mathematical modeling and analysis of the spread of the COVID-19 outbreak is proposed based on data analysis and infectious disease theory. Firstly, the mathematical model indicators of the spread of the new coronavirus pneumonia epidemic are determined by combining the theory of infectious diseases, the basic assumptions of the spread model of the new coronavirus pneumonia epidemic are given based on the theory of data analysis model, the spread rate of the new coronavirus pneumonia epidemic is calculated by combining the results of the assumptions, and the spread rate of the epidemic is inverted to push back into the assumptions to complete the construction of the mathematical modeling of the diffusion. Relevant data at different times were collected and imported into the model to obtain the spread data of the new coronavirus pneumonia epidemic, and the results were analyzed and reflected. The model considers the disease spread rate as the dependent variable of temperature, and analyzes and verifies the spread of outbreaks over time under real temperature changes. Comparison with real results shows that the model developed in this paper is more in line with the real disease spreading situation under specific circumstances. It is hoped that the accurate prediction of the epidemic spread can provide relevant help for the effective containment of the epidemic spread.

Modellings of Infectious Diseases and Cancers under Wars and Pollution Impacts in Iraq with Reference to a Novel Mathematical Model and Literature Review

Microbial pathogens include bacteria, viruses, fungi, and parasites and together account for a significant percentage of acute and chronic human diseases. In addition to understanding the mechanisms by which various pathogens cause human disease, research in microbial pathogenesis also addresses mechanisms of antimicrobial resistance and the development of new antimicrobial agents and vaccines. Answering fundamental questions regarding host-microbe interactions requires an interdisciplinary approach, including microbiology, genomics, informatics, molecular and cellular biology, biochemistry, immunology, epidemiology, environment and interaction between host and microbe. Studies investigating the direct effects of pollutants on respiratory tract infections are very vast, but those interested in the role of a pre-existing disease and effects of the exposure on the response to secondary stresses are few. In an experimental study at concentrations of air pollutants found in urban environments, frank toxicological responses are rarely observed, however, exposure to secondary stress like the respiratory challenge with infectious bacteria can exacerbate the response of the experimental host. The models like experimental, mechanical, and mathematical are the most abstract, but they allow analysis and logical proofs in a way that other approaches do not permit. The present review is mostly concerned with these model representations particularly with a novel mathematical model explaining the interaction between pathogen and immunity including the equivalence point.

Mathematical models for foam-diverted acidizing and their applications

Foam diversion can effectively solve the problem of uneven distribution of acid in layers of different permeabilities during matrix acidizing. Based on gas trapping theory and the mass conservation equation, mathematical models were developed for foam-diverted acidizing, which can be achieved by a foam slug followed by acid injection or by continuous injection of foamed acid. The design method for foam-diverted acidizing was also given. The mathematical models were solved by a computer program. Computed results show that the total formation skin factor, wellhead pressure and bottomhole pressure increase with foam injection, but decrease with acid injection. Volume flow rate in a highpermeability layer decreases, while that in a low-permeability layer increases, thus diverting acid to the low-permeability layer from the high-permeability layer. Under the same formation conditions, for foamed acid treatment the operation was longer, and wellhead and bottomhole pressures are higher. Field application shows that foam slug can effectively block high permeability layers, and improve intake profile noticeably.

Thermodynamic Consistency of Plate and Shell Mathematical Models in the Context of Classical and Non-Classical Continuum Mechanics and a Thermodynamically Consistent New Thermoelastic Formulation

Inclusion of dissipation and memory mechanisms, non-classical elasticity and thermal effects in the currently used plate/shell mathematical models require that we establish if these mathematical models can be derived using the conservation and balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the mathematical models. Thermodynamic consistency ensures thermodynamic equilibrium during the evolution of the deformation. When the mathematical models are thermodynamically consistent, the second law of thermodynamics facilitates consistent derivations of constitutive theories in the presence of dissipation and memory mechanisms. This is the main motivation for the work presented in this paper. In the currently used mathematical models for plates/shells based on the assumed kinematic relations, energy functional is constructed over the volume consisting of kinetic energy, strain energy and the potential energy of the loads. The Euler’s equations derived from the first variation of the energy functional for arbitrary length when set to zero yield the mathematical model(s) for the deforming plates/shells. Alternatively, principle of virtual work can also be used to derive the same mathematical model(s). For linear elastic reversible deformation physics with small deformation and small strain, these two approaches, based on energy functional and the principle of virtual work, yield the same mathematical models. These mathematical models hold for reversible mechanical deformation. In this paper, we examine whether the currently used plate/shell mathematical models with the corresponding kinematic assumptions can be derived using the conservation and balance laws of classical or non-classical continuum mechanics. The mathematical models based on Kirchhoff hypothesis (classical plate theory, CPT) and first order shear deformation theory (FSDT) that are representative of most mathematical models for plates/shells are investigated in this paper for their thermodynamic consistency. This is followed by the details of a general and higher order thermodynamically consistent plate/shell thermoelastic mathematical model that is free of a priori consideration of kinematic assumptions and remains valid for very thin as well as thick plates/shells with comprehensive nonlinear constitutive theories based on integrity. Model problem studies are presented for small deformation behavior of linear elastic plates in the absence of thermal effects and the results are compared with CPT and FSDT mathematical models.

Consistency and Validity of the Mathematical Models and the Solution Methods for BVPs and IVPs Based on Energy Methods and Principle of Virtual Work for Homogeneous Isotropic and Non-Homogeneous Non-Isotropic Solid Continua

Energy methods and the principle of virtual work are commonly used for obtaining solutions of boundary value problems (BVPs) and initial value problems (IVPs) associated with homogeneous, isotropic and non-homogeneous, non-isotropic matter without using (or in the absence of) the mathematical models of the BVPs and the IVPs. These methods are also used for deriving mathematical models for BVPs and IVPs associated with isotropic, homogeneous as well as non-homogeneous, non-isotropic continuous matter. In energy methods when applied to IVPs, one constructs energy functional (I) consisting of kinetic energy, strain energy and the potential energy of loads. The first variation of this energy functional (δI) set to zero is a necessary condition for an extremum of I. In this approach one could use δI = 0 directly in constructing computational processes such as the finite element method or could derive Euler’s equations (differential or partial differential equations) from δI = 0, which is also satisfied by a solution obtained from δI = 0. The Euler’s equations obtained from δI = 0 indeed are the mathematical model associated with the energy functional I. In case of BVPs we follow the same approach except in this case, the energy functional I consists of strain energy and the potential energy of loads. In using the principle of virtual work for BVPs and the IVPs, we can also accomplish the same as described above using energy methods. In this paper we investigate consistency and validity of the mathematical models for isotropic, homogeneous and non-isotropic, non-homogeneous continuous matter for BVPs that are derived using energy functional consisting of strain energy and the potential energy of loads. Similar investigation is also presented for IVPs using energy functional consisting of kinetic energy, strain energy and the potential energy of loads. The computational approaches for BVPs and the IVPs designed using energy functional and principle of virtual work, their consistency and validity are also investigated. Classical continuum mechanics (CCM) principles i.e. conservation and balance laws of CCM with consistent constitutive theories and the elements of calculus of variations are employed in the investigations presented in this paper.

MATHEMATICAL MODELS ARISING FROM A SURVEY OF FERROMAGNETIC MATERIALS UNDER MAGNETISATION

This paper focuses on mathematical models describing the mechanical behavior of ferromagnetic materials under magnetization. Through combination of the electromagnetic field theory with the theory of elastic mechanics, several nonlinear systems of fourth order partial differential equations were deduced. By making further assumption, the first-order approximation of the above equations was established, of which the solutions are good enough for engineering application.

PHYSICAL AND MATHEMATICAL MODELS OF POLLUTION DISPERSION IN THE YANGTZE ESTUARY

The method of combining a physical model with a mathematical model is described to study the concentration profile of pollutant dispersion in the Yangtze Estuary. the Experiments are described regarding a jet in a tidal physical model and two-dimensional calculations of diffusion using momentum and mass conservation equations of unsteady flow. The feature of dispersion in the tidal flow, which is different from that in the steady flow such as rivers, is explained. Dilution and dispersion mainly depend on the volume of runoff and tidal range. The results of the measurement and calculation are presented, and it can be seen that they are in good agreement.

Comparison on Mathematical Models for Description of Flow Curves of Stable Austenitic Steels

The flow curves were measured for the stable austenitic steels 304L and 304LN by means of tensile test at room temperature,which are described by the models σ=K1εn1 + exp(K2 + n2ε), σ=Kεn1+n2lnε and σ=σ0+Kεn (where, K1, K2, n1 andn2; K, n1 and n2; σ0, K and n are constant). The comparison of the maximum deviations and the consideration of thevariation of the work hardening rate with true strain show that the flow curves for the austenitic steels 304L and 304LN canbe described by the model σ=Kεn1+n2 lnε at higher precision.The derivatives of the models σ=K1εn1 + exp(K2 + n2ε) and σ=Kεn1+n2lnε with respect to true strain, exhibit theextreme at low true strain. This inherent character indicates that both models are unsuitable to describe the part of the workhardening rate curve at low true strain.