Application of the theory of linear algebra on mathematical modeling practice

The application level of mathematics in each science discipline signs the level of development of this science. With the advancement of science and technology, especially the rapid development of computer technology, mathematics has permeated from natural scientific technology to agricultural construction, from economic activities to all areas of social life. Generally, when the actual problem requires us to provide quantitative results of analysis, forecasting, decision making, control and other aspects for real object under study, we are often inseparable from the application of mathematics. Mathematical modeling is the key to this process, whose purpose is to make mathematics applied to social and social services, and using mathematics to solve practical problems is through mathematical models. When using mathematical methods to solve some practical problems, we usually first transfer practical problems into mathematical language, and then abstract them into a mathematical model.

Application of differential equations in mathematical modeling

This paper introduces the main methods and steps of modeling principle by ordinary differential equations, and is used to explore the differential equation model to solve some practical problems, some features of the related problems. With the development of science and technology and production practice, differential equation is more closely connected with other subjects, and a mathematical model for some practical problems of good.

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.

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.

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.

Mathematical Modeling and Effect of Various Hot-Air Drying on Mushroom(Lentinus edodes)

An experimental study was performed to determine the characteristics and drying process of mushroom （Lentinus edodes） by 6 different hot-air drying methods namely isothermal drying, uniform raise drying, non-uniform raise drying, uniform intermittent drying, non-uniform intermittent drying and combined drying. The chemical composition （dry matter, ash, crude protein, crude fat, total sugars, dietary fiber, and energy）, color parameters （L, a＊, b＊, c＊, and h~） and rehydration capacities were determined. Among all the experiments, non-uniform intermittent drying reached a better comprehensive results due to the higher chemical composition, better color quality associated with high bright （26.381＋5.842）, high color tone （73.670＋2.975）, low chroma （13.349a：3.456） as well as the highest rehydration （453.76% weigh of dried body）. Nine kinds of classical mathematical model were used to obtained moisture data and the Midili-kucuk model can be described by the drying process with the coefficient （R2 ranged from 0.99790 to 0.99967）, chi-square （X2 ranged from 0.00003 to 0.00019） and root mean square error （RMSE ranged from 0.000486 to 0.0012367）.

Nonlinear Mathematical Modeling and Sensitivity Analysis of Hydraulic Drive Unit

The previous sensitivity analysis researches are not accurate enough and also have the limited reference value, because those mathematical models are relatively simple and the change of the load and the initial displacement changes of the piston are ignored, even experiment verification is not conducted. Therefore, in view of deficiencies above, a nonlinear mathematical model is established in this paper, including dynamic characteristics of servo valve, nonlinear characteristics of pressure-flow, initial displacement of servo cylinder piston and friction nonlinearity. The transfer function block diagram is built for the hydraulic drive unit closed loop position control, as well as the state equations. Through deriving the time-varying coefficient items matrix and time-varying free items matrix of sensitivity equations respectively, the expression of sensitivity equations based on the nonlinear mathematical model are obtained. According to structure parameters of hydraulic drive unit, working parameters, fluid transmission characteristics and measured friction-velocity curves, the simulation analysis of hydraulic drive unit is completed on the MATLAB/Simulink simulation platform with the displacement step 2 mm, 5 mm and 10 mm, respectively. The simulation results indicate that the developed nonlinear mathematical model is sufficient by comparing the characteristic curves of experimental step response and simulation step response under different constant load. Then, the sensitivity function time-history curves of seventeen parameters are obtained, basing on each state vector time-history curve of step response characteristic. The maximum value of displacement variation percentage and the sum of displacement variation absolute values in the sampling time are both taken as sensitivity indexes. The sensitivity indexes values above are calculated and shown visually in histograms under different working conditions, and change rules are analyzed. Then the sensitivity indexes values of four measurable parameters, such as supply pressure, proportional gain, initial position of servo cylinder piston and load force, are verified experimentally on test platform of hydraulic drive unit, and the experimental research shows that the sensitivity analysis results obtained through simulation are approximate to the test results. This research indicates each parameter sensitivity characteristics of hydraulic drive unit, the performance-affected main parameters and secondary parameters are got under different working conditions, which will provide the theoretical foundation for the control compensation and structure optimization of hydraulic drive unit.

Physical and Mathematical Modeling of the Argon-Oxygen Decarburization Refining Process of Stainless Steel

The available studies in the literature on physical and mathematical modeling of the argon oxygen decarburization (AOD) process of stainless steel have briefly been reviewed. The latest advances made by the author with his research group have been summarized. Water modeling was used to investigate the fluid flow and mixing characteristics in the bath of an 18 t AOD vessel, as well as the 'back attack' action of gas jets and its effects on the erosion and wear of the refractory lining, with sufficiently full kinematic similarity. The non rotating and rotating gas jets blown through two annular tuyeres, respectively of straight tube and spiral flat tube type, were employed in the experiments. The geometric similarity ratio between the model and its prototype (including the straight tube type tuyeres) was 1:3. The influences of the gas flow rate, the angle included between the two tuyeres and other operating parameters, and the suitability of the spiral tuyere as a practical application, were examined. These latest studies have clearly and successfully brought to light the fluid flow and mixing characteristics in the bath and the overall features of the back attack phenomena of gas jets during the blowing, and have offered a better understanding of the refining process. Besides, mathematical modeling for the refining process of stainless steel was carried out and a new mathematical model of the process was proposed and developed. The model performs the rate calculations of the refining and the mass and heat balances of the system. Also, the effects of the operating factors, including adding the slag materials, crop ends, and scrap, and alloy agents; the non isothermal conditions; the changes in the amounts of metal and slag during the refining; and other factors were all considered. The model was used to deal with and analyze the austenitic stainless steel making (including ultra low carbon steel) and was tested on data of 32 heats obtained in producing 304 grade steel in an 18 t AOD vessel. The changes in the bath composition and temperature during the refining process with time can be accurately predicted using this model. The model can provide some very useful information and a reliable basis for optimizing the process practice of the refining of stainless steel and control of the process in real time and online.

Mathematical modeling of tornadoes and squall storms

Recent advances in modeling of tornadoes and twisters consist of significant achievements in mathematical calculation of occurrence and evolution of a violent F5-class tornado on the Fujita scale, and four-dimensional mathematical modeling of a tornado with the fourth coordinate time multiplied by its characteristic velocity. Such a tornado can arise in a thunderstorm supercell filled with turbulent whirlwinds. A theory of the squall storms is proposed. The squall storm is modeled by running pertur- bation of the temperature inversion on the lower boundary of cloudiness. This perturbation is induced by the action of strong, hurricane winds in the upper and middle troposphere, and looks like a running solitary wave （soliton）; which is developed also in a field of pressure and velocity of a wind. If a soliton of a squall storm gets into the thunderstorm supercell then this soliton is captured by supercell. It leads to additional pressure fall of air inside a storm supercell and stimulate amplification of wind velocity here. As a result, a cyclostrophic balance inside a storm supercell generates a tornado. Comparison of the radial distribution of wind velocity inside a tornado calculated by using the new formulas and equations with radar observations of the wind velocity inside Texas Tornado Dummit in 1995 and inside the 3 May 1999 Oklahoma City Tornado shows good correspondence.

Mathematical modeling for dynamic stability of sandwich beam with variable mechanical properties of core

The paper is devoted to mathematical modelling of static and dynamic stability of a simply supported three-layered beam with a metal foam core. Mechanical properties of the core vary along the vertical direction. The field of displacements is for- mulated using the classical broken line hypothesis and the proposed nonlinear hypothesis that generalizes the classical one. Using both hypotheses, the strains are determined as well as the stresses of each layer. The kinetic energy, the elastic strain energy, and the work of load are also determined. The system of equations of motion is derived using Hamilton＇s principle. Finally, the system of three equations is reduced to one equation of motion, in particular, the Mathieu equation. The Bubnov-Galerkin method is used to solve the system of equations of motion, and the Runge-Kutta method is used to solve the second-order differential equation. Numerical calculations are done for the chosen family of beams. The critical loads, unstable regions, angular frequencies of the beam, and the static and dynamic equilibrium paths are calculated analytically and verified numerically. The results of this study are presented in the forms of figures and tables.

Mathematical modeling of elastic inverted pendulum control system

Inverted pendulums are important objects of theoretical investigation and experiment in the area of control theory and engineering. The researches concentrate on the rigid finite dimensional models which are described by ordinary differential equations (ODEs) .Complete rigidity is the approximation of practical models ; Elasticity should be introduced into mathematical models in the analysis of system dynamics and integration of highly precise controller. A new kind of inverted pendulum, elastic inverted pendulum was proposed, and elasticity was considered. Mathematical model was derived from Hamiltonian principle and variational methods, which were formulated by the coupling of partial differential equations (PDE) and ODE. Because of infinite dimensional, system analysis and control of elastic inverted pendulum is more sophisticated than the rigid one.

Mathematical Modeling of the Oxidation of Polyunsaturated Fatty Acids in Emulsions with Stirring and Limited Oxygen Compensation

The oxidation of polyunsaturated fatty acids （PUFA） in emulsion with stirring and limited oxygen compensation was studied. A mathematical model of diffusion-oxidation was developed considering the mass transfer resistance of a gas-liquid boundary, the resistance of the boundary layer from the emulsifier membrane, and the autocatalytic-type autoxidation reaction of PUFA. The dynamic mass transfer coefficient of the emulsifier membrane, k0, was introduced. The model was verified by comparing the predictions of the model with the experi- mental data. The results indicated that the model was in good agreement with the oxygen diffusion and linoleic acid oxidation in the emulsion, and showed good applicability in the prediction of the effect of the emulsifier type on the oxidation of PUFA in the emulsion. It indicated that the oxidation of PUFA in emulsions, with stirring and limited oxygen compensation from the atmosphere, was controlled mostly by mass transfer resistance from the emulsifier membrane.

MATHEMATICAL AND PHYSICAL MODELING OF INTERFACIALPHENOMENA IN CONTINOUS CASTING MOULD WITH ARGONINJECTION THROUGH SUBMERGED ENTRV NOZZLE

The study on the fluid flow, meniscus oscillation, slag entrapment in continuous casting mould was conducted mathematically and experimentally. The results show that the injection of argon into submerged nozzle enhances the meniscus oscillation, thus increases the probability of slag entrapment, and the critical argon blowing flow rate, which will give rise to slag entrapment, is around 10l/min. The trajectory of bubble is affected by the bubble diameter and the molten steel flow, and the bubble diameter is dominant. The bubble with diameter 1.4mm floats fastest with 0.47m/s terminal velocity.

Mathematical Modeling of Carcinogenesis Based on Chromosome Aberration Data

Objective： The progression of human cancer is characterized by the accumulation of genetic instability. An increasing number of experimental genetic molecular techniques have been used to detect chromosome aberrations. Previous studies on chromosome abnormalities often focused on identifying the frequent loci of chromosome alterations, but rarely addressed the issue of interrelationship of chromosomal abnormalities. In the last few years, several mathematical models have been employed to construct models of carcinogenesis, in an attempt to identify the time order and cause-and-effect relationship of chromosome aberrations. The principles and applications of these models are reviewed and compared in this paper. Mathematical modeling of carcinogenesis can contribute to our understanding of the molecular genetics of tumor development, and identification of cancer related genes, thus leading to improved clinical practice of cancer.

Germanium transport across supported liquid membrane with Cyanex 923: Mathematical modeling

A mathematical model was developed to monitor the facilitated transport of germanium(IV) from oxalic acid solutions through a flat sheet supported liquid membrane(FSSLM) containing four trialkylphosphine oxides(Cyanex 923). The FSSLM modeling was based on the extraction constant(Kext) calculated from the liquid-liquid extraction(LLX) modeling. The LLX model presented a reliable calculation of the extraction constant(Kex= 2.057×103 L/mol4). The FSSLM model was solved using Matlab■ software according to extraction constant, Fick’s law, and diffusional principles. The model predicts the overall mass transfer coefficient(Korg) to be 3.84 cm/s. Using this value, diffusion coefficients(Dm) for various Cyanex 923 concentrations of 0.126, 0.252, 0.378, 0.505, 0.631 and 0.757 mol/L are found to be 8.50×10^-4, 4.30×10^-4, 1.87×10^-4, 5.87×10^-5, 2.57×10^-5, 2.09×10^-5 cm2/s, respectively. The results show that the diffusion rate of the current study is approximately more than that of similar FSSLM systems containing Cyanex 923 used to transport various metals. The modeling values are in good agreement with the experimental data, showing the good reliability of the mathematical model.

Impact of pertinent parameters on foam behavior in the entrance region of porous media:mathematical modeling

Foam injection is a promising solution for control of mobility in oil and gas field exploration and development,including enhanced oil recovery,matrix-acidization treatments,contaminated-aquifer remediation and gas leakage prevention.This study presents a numerical investigation of foam behavior in a porous medium.Fractional flow method is applied to describe steady-state foam displacement in the entrance region.In this model,foam flow for the cases of excluding and including capillary pressure and for two types of gas,nitrogen(N2)and carbon dioxide(CO2)are investigated.Effects of pertinent parameters are also verified.Results indicate that the foam texture strongly governs foam flow in porous media.Required entrance region may be quite different for foam texture to accede local equilibrium,depending on the case and physical properties that are used.According to the fact that the aim of foaming of injected gas is to reduce gas mobility,results show that CO2 is a more proper injecting gas than N2.There are also some ideas presented here on improvement in foam displacement process.This study will provide an insight into future laboratory research and development of full-field foam flow in a porous medium.