With the development of modern medical technology, a large number of antibiotics have been developed to kill microorganisms that pose a threat to human health. However, the misuse of antibiotics has led to widespread antibiotic resistance, which is now a significant public issue [1]. Antibiotic resistance, also known as drug resistance, refers to the resistance of microorganisms, parasites and tumour cells to the effects of chemotherapy antibiotics [2, 3]. Antibiotic resistance can be divided into two types: acquired antibiotic resistance and natural antibiotic resistance. Pathogens in nature, such as a strain of bacteria, can also have natural resistance, regardless of increasing doses of the antibiotic [4]. Acquired resistance occurs due to specific changes in the DNA (mutation) or through contact with external sources such as plasmids, transposons or integrons [5]. At present, acquired drug resistance is considered as the main cause for the prevalence of antibiotic resistant. Bacterial cells are capable of transferring genes horizontally. When resistance genes are mobile, being carried on plasmids or phages, their spread can be greatly accelerated [6,7]. A plasmid is a small piece of extrachromosomal DNA that may be easily transferred between bacteria via horizontal gene transfer and, in particular, has been implicated in the acquisition of antibiotic resistance genes to many antibiotics [8]. In this paper, we investigate the problem of celltocell transmission of antibiotic resistance caused by plasmids carrying resistance genes.
When antibiotics are applied for a long time, the majority of the sensitive bacteria are killed continuously, and the antibiotic resistant bacteria will keep multiplying in large numbers, so that the resistance rate of the bacteria to the antibiotic keeps rising [2]. The experiments show that the use of antibiotics will promote the transmission of plasmids in the colonies [9]. In fact, the concentration of antibiotics in the environment is not static, but changes continuously during the absorption. We analyse the effect of the drug concentration in the numerical simulation section. The speed at which the bacteria gets wiped out depends on the duration of administration of the antibiotic [10]. Researchers find that the optimal dosing does not necessarily correspond to the continuous administration of the antibiotics [11].
A mathematical model that describes the population dynamics of bacteria exposed to multiple antibiotics simultaneously is formulated [12], assuming the acquisition of resistance is through mutations from antibiotics exposure. In general, the specific mutations that confer resistance to antibiotic control have a cost, which may manifest as a reduction in reproductive capacity or the ability to compete. In the absence of medications, the infection rate of resistant bacteria is less or equal than the infection rate of sensitive bacteria; nevertheless, the population of resistant bacteria could increase to far exceed the population of sensitive bacteria under treatment [13]. Under the condition of decreasing drug concentration, the development of antibiotic resistance may be reversible [14]. This has implications for the management of antibiotic resistance, which can not only be managed by reducing the development of resistance, but also requires the restoring of existing resistance.
Assuming that drug resistance is acquired through mutations and plasmids transmission, a deterministic model for the population dynamics of sensitive and resistant bacteria to multiple bactericidal and bacteriostatic antibiotics is formulated and analysed [15]. Svara and Rankin [16] established an ordinary differential equation model to investigate the dynamics of plasmidcarried antibiotic resistance. By numerical simulations, it is shown that the transmission of plasmids is the key factor influencing plasmidborne antibiotic resistance, and the dosage of antibiotic and the interval between treatments are also important. By employing qualitative analysis of ordinary differential equations, Xu et al. [17] investigate the collective resistance of the bacteria population with resistant horizontal gene transfer under sublethal bactericide pressure. The results show that the possible mechanism of variations in antibiotic susceptibility is the dominance of different bacterial genotypes under sublethal bactericide pressure, rather than persistence, tolerance or resistance. As a matter of fact, this can be seen as one of the innate characteristics of interaction between bacteria and bactericides.
In a systematic review, Quentin et al. [18] searched for mathematical modelling studies that focus on horizontal transfer of antimicrobial resistance (AMR) genes. Their findings highlight the existence of a research gap in the dynamics of transformation and transduction, and the overall public health implications of the horizontal transfer of AMR genes.
Because horizontal transfer of plasmids is an important mechanism, whereby resistance is spread through bacterial populations, we develop a mathematical model to describe this process quantitatively, using it to optimise antimicrobial dosage regimens to minimise resistance development. The transmission of antibiotic resistant plasmids in the coexisting environment of two types of bacteria is discussed in this paper.
In this article, we assume that cells are divided into normal cells and pathogenic bacteria. In the mathematical model we established, the letter ‘a’ represents normal cells and the letter ‘b’ represents pathogenic bacteria. We assume that when antibiotics are used, they only kill the pathogenic bacteria and have no effect on normal cells.
To simplify the model, we assume as follows:
The densitydependent death rates of the two types of cells are equal, and is given by. It means that whether a cell is antibiotic resistant or not has no effect on its natural growth rate or densitydependent death rate.
We assume an antibioticinduced fitness cost, that is, antibiotics can have two effects on a cell: they can either kill the bacteria or they can prevent their reproduction.
To keep our model simple, we assume that genes for resistance against antibiotics are carried by a plasmid rather than by the chromosome.
Due to the absence of resistance genes, the plasmidfree types of pathogenic bacteria are more sensitive to antibiotics. Mathematically, it is reasonable to assume that.
Details of parameters used in our model are given in Table 1.
Model variables and parameters
Density of plasmidfree type of normal cells  
Density of normal cells infected with a resistancecarrying plasmid  
Density of plasmidfree type of pathogenic bacteria  
Density of pathogenic bacteria infected with a resistancecarrying plasmid  
Intrinsic percapita growth rate of cells  
Extrinsic densitydependent death rate of cells  
Total density of cells 

Rate of horizontal transfer of resistance plasmid from infected normal cells  
Rate of horizontal transfer of resistance plasmid from infected pathogenic bacteria  
Rate of the resistance plasmid segregation from normal cells  
Rate of the resistance plasmid segregation from pathogenic bacteria  
Rate of the resistance plasmid segregation from pathogenic bacteria  
Cost of infected normal cells carried on plasmid  
Cost of antibiotic resistance when pathogenic bacteria is carried on resistance plasmid  
Concentration of antibiotic  
Death rate of plasmidfree type of pathogenic bacteria due to antibiotics  
Death rate of pathogenic bacteria with resistancecarrying plasmid due to antibiotics 
Assuming logistic growth, with a growth rate of
For the sake of simplicity, let
Boundary equilibria and conditions
Always exist  

Always exist 



It is too complicated to express 

It is too complicated to express 
The boundary equilibria are analysed using stability theory. Through analysis, we can draw the following conclusions.
The proofs of Theorems 1–3 are as follows.
For
For
For
We built numerical simulations to investigate how parameters affect the variation of system (1). To attain a more realistic model, we assume that the antibiotic degrade at an exponential rate [19]:
When using an antibiotic to treat the disease caused by pathogenic bacteria, the concentration of the drug must reach a certain level; otherwise, it cannot kill the pathogenic bacteria. In other words, if the concentration of the antibiotic is not high to some extent, it becomes ineffective at treating the disease. The drug concentration over time is marked by a blue dotted curve in each figure for clearer emphasis. The four other lines show the density variations of cells. As we know from Eq. (2), this means that the smaller the
Here, the plots are calculated by running the simulation for a number of parameter values for over 500 time steps. Parameters used are:
With the increase of
As for the parameter of cost, when cells are carrying plasmids, the corresponding number of cells infected with resistancecarrying plasmids decreases with the increase of cost value. In Figure 4(a), when
Figure 5 shows the role of parameters involved in the segregation rate of plasmids. We can see from Figure 5 that the ability of resistancecarried plasmids to spread is reduced with the increasing segregation rate value
The goals of sensitivity analysis with respect to random perturbations of the model parameters is as follows:
To show how robust the simplified uncomplicated bacteria resistance model is in relation to perturbed parameter values.
To explore which parameters the system is more sensitive to understand the key process and bacteria resistance system mechanisms.
The sensitivity of drug degradation rates is considered in this section. Figure 6 shows the effect of drug degradation rates on the population of bacteria. The values of
The sensitivity of the parameter
The sensitivity of the parameter
The sensitivity of segregation rate
In Figures 9 and 10, curves of the same colour show the variations of the proportion of the bacteria under the same set of parameters. As we can see from the figures, the effect of
It is shown that the concentration of antibiotic drugs must reach a certain level to kill the pathogenic bacteria. If the concentration of antibiotic is not high to some extent, it becomes an ineffective treatment. The rate of horizontal transfer of resistance plasmids is directly related to the spread of drug resistance. With the increase in the horizontal transfer of resistance plasmids, cells in the body gradually turn into cells with antibiotic resistance, which cause great difficulties in the treatment of diseases.
In addition, the risks associated with the development of antibiotic resistance in populations are often latent and undetectable. However, when people get sick, it will be much harder for them to heal because of antibiotic resistance. Therefore, the potential threat of drug resistance to people's health cannot be ignored.
Boundary equilibria and conditions
Always exist  

Always exist 



It is too complicated to express 

It is too complicated to express 
Model variables and parameters
Density of plasmidfree type of normal cells  
Density of normal cells infected with a resistancecarrying plasmid  
Density of plasmidfree type of pathogenic bacteria  
Density of pathogenic bacteria infected with a resistancecarrying plasmid  
Intrinsic percapita growth rate of cells  
Extrinsic densitydependent death rate of cells  
Total density of cells 

Rate of horizontal transfer of resistance plasmid from infected normal cells  
Rate of horizontal transfer of resistance plasmid from infected pathogenic bacteria  
Rate of the resistance plasmid segregation from normal cells  
Rate of the resistance plasmid segregation from pathogenic bacteria  
Rate of the resistance plasmid segregation from pathogenic bacteria  
Cost of infected normal cells carried on plasmid  
Cost of antibiotic resistance when pathogenic bacteria is carried on resistance plasmid  
Concentration of antibiotic  
Death rate of plasmidfree type of pathogenic bacteria due to antibiotics  
Death rate of pathogenic bacteria with resistancecarrying plasmid due to antibiotics 
Law of interest rate changes in financial markets based on the differential equation model of liquidity Research and implementation of smart city public information mining analysis system based on mobile edge model of game theory Basalt fibre continuous reinforcement composite pavement reinforcement design based on finite element model Design of college education evaluation based on accompanying data acquisition and mathematical analysis Industrial transfer and regional economy coordination based on multiple regression model Enterprise financial strategy and performance management analysis based on principal component analysis Study on the Dynamic Change of Regional Water Level and Climate Based on Forecast Equation Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Municipal Civil Engineering Construction Based on Finite Element Differential Equations Financial Risk Prevention Model of Financial Institutions Based on Linear Partial Differential Equation Analysis of the relationship between industrial agglomeration and regional economic growth based on the multiobjective optimisation model Constraint effect of enterprise productivity based on constrained form variational computing The impact of urban expansion in Beijing and Metropolitan Area urban heat Island from 1999 to 2019 Ultrasonic wave promoting ice melt in ice storage tank based on polynomial fitting calculation model Training Model of Basketball Offensive Route Based on Nonlinear Differential Equation Optimization Algorithm of New Media Hot Event Push Based on Nonlinear Differential Equation Mathematical Differential Equation in Calculating the Strength of Building Beam Structure Impact Resistance Stability of Building Structural Engineering Based on Fractional Differential Equations The Technical Research on the Assessment of Network Security Situation Based on DS Evidence Theory Computer big data modeling system based on finite element mathematical equation simulation Uniqueness of system integration scheme of artificial intelligence technology in fractional differential mathematical equation Uniqueness of system integration scheme of artificial intelligence technology in fractional differential mathematical equation The Composition System of Pop Music Movement Based on Finite Element Differential Equations The Structure and Influencing Factors of Innovation and Entrepreneurship Ability of Higher Vocational Students Based on Structural Equation Model Model System Study of Accordion Score Based on Fractional Differential Equations Data mining of Chain convenience stores location Parameter Id of Metal Hipressure State Equation PMatrix Reasoning and Information Intelligent Mining Research on Detection Model of Abnormal Data in Engineering Cost List Research on deformation monitoring of tunnel engineering based on 3D laser scanning Intelligent Recommendation System for English Vocabulary Learning – Based on Crowdsensing Application of Nonlinear Fractional Differential Equations in Computer Artificial Intelligence Algorithms Preservation and utilisation of historic buildings in old district of Guangzhou from the perspective of space syntax Application of calculus function and value chain analysis method in the quantification of financial flexibility management indicators Copyright protection of original online music products based on applied statistical mathematics – take music trade network as an example Optimal Research in Piano Shape Sound and Sound Propagation Model Based on Nonlinear Differential Equations Regarding new wave distributions of the nonlinear integropartial Ito differential and fifthorder integrable equations Badminton players’ trajectory under numerical calculation method Research on the influencing factors of agricultural product purchase willingness in social ecommerce situation Research on mathematical quantitative model of the incremental cost and income of green small towns based on the whole life cycle Mathematical methodology in the seismic resilience evaluation of the water supply system Prediction of surface quality in end milling based on modified convolutional recurrent neural network Research on sustainable design of sports centre based on algorithm verification Research on an early warning model of effectiveness evaluation in ideological and political teaching based on big data Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Numerical simulation of vortex vibration in main girder of cablestayed bridge based on bidirectional fluid–structure coupling Optimisation of construction mode of residential houses based on the genetic algorithm under BIM technology A study on the reform of college English education inspired by the cultural differences between China and the United States Research on innovative strategies of college students’ English teaching under the blessing of big data Institutional investor company social responsibility report and company performance Research on multidimensional optimisation design of user interface under Rhino/GH platform Mathematical analysis of China's birth rate and research on the urgency of deepening the reform of art education Firstprinciples calculations of magnetic and mechanical properties of Febased nanocrystalline alloy Fe_{80}Si_{10}Nb_{6}B_{2}Cu_{2} Has the belt and road initiative boosted the resident consumption in cities along the domestic route? – evidence from credit card consumption Attitude control for the rigid spacecraft with the improved extended state observer A long command subsequence algorithm for manufacturing industry recommendation systems with similarity connection technology Application of HigherOrder Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Informatisation of educational reform based on fractional differential equations Application research on piano teaching in colleges and universities based on remote wireless network communication Research on the control of quantitative economic management variables under the numerical method based on stochastic ordinary differential equations Network monitoring and processing accuracy of big data acquisition based on mathematical model of fractional differential equation System dynamics model of output of ball mill Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Design of Morlet wavelet neural network to solve the nonlinear influenza disease system Motion about equilibrium points in the JupiterEuropa system with oblateness Badminton players’ trajectory under numerical calculation method Ultrashortterm power forecast of photovoltaic power station based on VMD–LSTM model optimised by SSA Optimal preview repetitive control for impulsefree continuoustime descriptor systems Development of main functional modules for MVB and its application in rail transit Analysis of agricultural economic development and optimisation measures under the strategy of rural revitalisation Study on the impact of forest fire prevention policy on the health of forest resources Value Creation of Real Estate Company Spinoff Property Service Company Listing Selection by differential mortality rates Digital model creation and image meticulous processing based on variational partial differential equation The modelling and implementation of the virtual 3D animation scene based on the geometric centreofmass algorithm The policy efficiency evaluation of the Beijing–Tianjin–Hebei regional government guidance fund based on the entropy method The transfer of stylised artistic images in eye movement experiments based on fuzzy differential equations Research on behavioural differences in the processing of tenant listing information: An eyemovement experiment Innovation of Economic Management Risk Control in Retailer Supply Chain Based on Differential Equation Model A review of the treatment techniques of VOC Some classes of complete permutation polynomials in the form of ( x ^{pm} −x +δ )^{s} +ax ^{pm} +bx overF _{p2m}The consistency method of linguistic information and other four preference information in group decisionmaking Statistical Model of College Ideological and Political Learning Based on Fractional Differential Equations Research on the driving principle and guiding strategy of the public's collaborative supervision of the sharing economy in my country Research on the willingness of Forest Land’s Management Rights transfer under the Beijing Forestry Development Nonlinear Differential Equation in Antiaging Test of Polymer Nanomaterials A mathematical model of the fractional differential method for structural design dynamics simulation of lower limb force movement step structure based on Sanda movement Fractal structure of magnetic island in tokamak plasma Mechanics of Building Structural Materials Based on Lagrangian Mathematical Model Analysis Numerical calculation and study of differential equations of muscle movement velocity based on martial articulation body ligament tension Study on the maximum value of flight distance based on the fractional differential equation for calculating the best path of shot put The Mental Health Education Management of Higher Vocational Students Based on Fractional Differential Equations Sports intensity and energy consumption based on fractional linear regression equation Application of regression function model based on panel data in financial risk management of bank resource allocation University Ideological and Political Learning Model Based on Statistical Memory Curve Mathematical Equation Analysis of the properties of matrix rank and the relationship between matrix rank and matrix operations Research on Resonance Properties of Semantic Wave Fractal Fractals Based on Quantitative Analysis of English Corpus Research on urban landscape big data information processing system based on ordinary differential equations Modeling of fractional differential equation in cloud computing image fusion algorithm Study on Establishment and Improvement Strategy of Aviation Equipment Research on Financial Risk Early Warning of Listed Companies Based on Stochastic Effect Mode Nonlinear strategic human resource management based on organisational mathematical model Conventional Algorithms in Sports Training Based on Fractional Differential Equations Dynamic Nonlinear System Based on Complex System Theory in the Development of Vocational Education The Algorithm Accuracy of Mathematical Model to Improve the Transmission Speed of Ecommerce Platform The Security of Database Network Model Based on Fractional Differential Equations Electric Vehicle Mechanical Transmission System Based on Fractional Differential Equations The Practice System of Physics and Electronics Courses in Higher Vocational Colleges Based on Fractional Differential Equations The Economic Model of Rural Supply and Demand Under the Data Analysis Function Based on Ordered Probit 3D Modeling System of Indoor Environment Art Landscape Design under Statistical Equation Electronic Information Security Model of Nonlinear Differential Equations The Optimization Model of College Students' Physical Exercise Motivation and Selfcontrol Ability Based on the Mathematical Model of Probability Theory A new variety from Sichuan, China— Paris polyphylla var.nullopetalina (Trilliaceae)Impact of ASEANChina free trade area on fishery value chain based on differenceindifference method Spatial–temporal graph neural network based on node attention A contrastive study on the production of double vowels in Mandarin Mathematical analysis of civil litigation and empirical research of corporate governance Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multifractal detrend fluctuation analysis Application and risk assessment of the energy performance contracting model in energy conservation of public buildings Sensitivity analysis of design parameters of envelope enclosure performance in the dryhot and drycold areas TOPSIS missile target selection method supported by the posterior probability of target recognition The incentive contract of subject librarians in university library under the nonlinear task importance Research on indoor environment design of comprehensive commercial shopping center based on numerical simulation Empirical analysis of the economic absolute income hypothesis based on mathematical statistics Realization of Book Collaborative Filtering Personalized Recommendation System Based on Linear Regression Equation Study on Interactive Relations between Enterprise Social Media and Decision Style Based on a vector Autoregressive Model Research on Dynamics of Flexible Multibody System with Deployable Antenna Based on Static Lagrangian Function The Optimization of Mathematics Teaching Models in Colleges and Universities Based on Nonlinear Differential Equations Study on spatial planning and design of learning commons in university libraries based on fuzzy matrix model The Stability Model of Piano Tone Tuning Based on Ordinary Differential Equations Construction and application of automobile user portrait based on kmean clustering model The Homework Model of Screening English Teaching Courses Based on Fractional Differential Equations Research on a method of completeness index based on complex model Fake online review recognition algorithm and optimisation research based on deep learning Research on the sustainable development and renewal of Macao inner harbour under the background of digitisation Support design of main retracement passage in fully mechanised coal mining face based on numerical simulation Study on the crushing mechanism and parameters of the twoflow crusher Interaction design of financial insurance products under the Era of AIoT Modeling the pathway of breast cancer in the Middle East ARMA analysis of the green innovation technology of core enterprises under the ecosystem – Time series data Image design and interaction technology based on Fourier inverse transform Calculus Logic Function in Integrated Manufacturing Automation of Single Chip Microcomputer Football Offense Training Strategy Based on Fractional Differential Mathematical Modeling Research on China interregional industrial transformation slowdown and influencing factors of industrial transformation based on numerical simulation The medical health venture capital network community structure, information dissemination and the cognitive proximity The Mathematical Analysis Model of Educational System in Music Courses in Colleges and Universities Continuing Education Network Data Center Model Based on Fractional Differential Mathematical Equations A generative design method of building layout generated by path Analysis of the causes of the influence of the industrial economy on the social economy based on multiple linear regression equation Research of neural network for weld penetration control Computer Art Design Model Based on Nonlinear Fractional Differential Equations The Optimization Model of Public Space Design Teaching Reform Based on Fractional Differential Equations The Approximate Solution of Nonlinear Vibration of Tennis Based on Nonlinear Vibration Differential Equation Graphical Modular Power Technology of Distribution Network Based on Machine Learning Statistical Mathematical Equation Employment and Professional Education Training System of College Graduates Based on the Law of Large Numbers Economic Research on Multiple Linear Regression in Fruit Market inspection and Management Nonlinear Differential Equations in Preventing Financial Risks Lagrange’s Mathematical Equations in the Sports Training of College Students Simulation Research of Electrostatic Precipitator Power Supply Voltage Control System Based on Finite Element Differential Equation Research on the effect of generative adversarial network based on wavelet transform hidden Markov model on face creation and classification Research on Lightweight Injection Molding (CAE) and Numerical Simulation Calculate of New Energy Vehicle Power Flow Calculation in Smart Distribution Network Based on Power Machine Learning Based on Fractional Differential Equations Demonstration of application program of logistics public information management platform based on fuzzy constrained programming mathematical model Basketball Shooting Rate Based on Multiple Regression LogicalMathematical Algorithm The Optimal Application of Lagrangian Mathematical Equations in Computer Data Analysis Similarity Solutions of the Surface Waves Equation in (2+1) Dimensions and Bifurcation Research on predictive control of students’ performance in PE classes based on the mathematical model of multiple linear regression equation Beam control method for multiarray antennas based on improved genetic algorithm The influence of X fuzzy mathematical method on basketball tactics scoring Optimal decisions and channel coordination of a green supply chain with marketing effort and fairness concerns Application of regression function model based on panel data in bank resource allocation financial risk management Game theoretic model for low carbon supply chain under carbon emissions reduction sensitive random demand Research on aerobics training posture motion capture based on mathematical similarity matching statistical analysis Limit cycles of a generalised Mathieu differential system Application of SobolevVolterra projection and finite element numerical analysis of integral differential equations in modern art design Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Research on motion capture of dance training pose based on statistical analysis of mathematical similarity matching Application of data mining in basketball statistics Application of Btheory for numerical method of functional differential equations in the analysis of fair value in financial accounting Research on the influence of fuzzy mathematics simulation model in the development of Wushu market Study on audiovisual family restoration of children with mental disorders based on the mathematical model of fuzzy comprehensive evaluation of differential equation Differenceindifferences test for micro effect of technological finance cooperation pilot in China Application of multiattribute decisionmaking methods based on normal random variables in supply chain risk management Exploration on the collaborative relationship between government, industry, and university from the perspective of collaborative innovation The impact of financial repression on manufacturing upgrade based on fractional Fourier transform and probability AtanKA New SVM Kernel for Classification Validity and reliability analysis of the Chinese version of planned happenstance career inventory based on mathematical statistics Visual positioning system for marine industrial robot assembly based on complex variable function Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Study of a linearphysicalprogrammingbased approach for web service selection under uncertain service quality The Relationship Between College Students’ Taekwondo Courses and College Health Based on Mathematical Statistics Equations A mathematical model of plasmidcarried antibiotic resistance transmission in two types of cells Fractional Differential Equations in the Exploration of Geological and Mineral Construction AdaBoost Algorithm in Trustworthy Network for Anomaly Intrusion Detection Burnout of frontline city administrative lawenforcing personnel in new urban development areas: An empirical research in China Enterprise Financial Risk Early Warning System Based on Structural Equation Model A Study on the Application of Quantile Regression Equation in Forecasting Financial Value at Risk in Financial Markets Fractional Differential Equations in the Model of Vocational Education and Teaching Practice Environment Information transmission simulation of Internet of things communication nodes under collision free probability equation Image denoising model based on improved fractional calculus mathematical equation Random Fourier Approximation of the Kernel Function in Programmable Networks The Complexity of Virtual Reality Technology in the Simulation and Modeling of Civil Mathematical Models University Library Lending System Model Based on Fractional Differential Equations Calculation and Performance Evaluation of Text Similarity Based on Strong Classification Features Intelligent Matching System of Clauses in International Investment Arbitration Cases Based on Big Data Statistical Model Evaluation and Verification of Patent Value Based on Combination Forecasting Model Financial Institution Prevention Financial Risk Monitoring System Under the Fusion of Partial Differential Equations Prediction and Analysis of ChiNext Stock Price Based on Linear and Nonlinear Composite Model Calculus Logic Function in Tax Risk Avoidance in Different Stages of Enterprises The Psychological Memory Forgetting Model Based on the Analysis of Linear Differential Equations Optimization Simulation System of University Science Education Based on Finite Differential Equations The Law of Large Numbers in Children's Education Optimization System of Strength and Flexibility Training in Aerobics Course Based on Lagrangian Mathematical Equation Data structure simulation for the reform of the teaching process of university computer courses Calculating university education model based on finite element fractional differential equations and macrocontrol analysis Educational research on mathematics differential equation to simulate the model of children's mental health prevention and control system Analysis of enterprise management technology and innovation based on multilinear regression model Verifying the validity of the whole person model of mental health education activities in colleges based on differential equation RETRACTION NOTE Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Research on nonlinear visual matching model under inherent constraints of images Good congruences on weakly Uabundant semigroups Can policy coordination facilitate unimpeded trade? An empirical study on factors influencing smooth trade along the Belt and Road Research on the processing method of multisource heterogeneous data in the intelligent agriculture cloud platform Internal control index and enterprise growth: An empirical study of Chinese listedcompanies in the automobile manufacturing industry Determination of the minimum distance between vibration source and fibre under existing optical vibration signals: a study Nonlinear differential equations based on the BSM model in the pricing of derivatives in financial markets FedUserPro: A user profile construction method based on federated learning A multifactor Regression Equationbased Test of Fitness Maximal Aerobic Capacity in Athletes Garment Image Retrieval based on Grab Cut Auto Segmentation and Dominate Color Method Financial Risk Prediction and Analysis Based on Nonlinear Differential Equations Constructivist Learning Method of Ordinary Differential Equations in College Mathematics Teaching Multiple Effects Analysis of Hangzhou Issuing Digital Consumer Coupons Based on Simultaneous Equations of CDM Model Response Model of Teachers’ Psychological Education in Colleges and Universities Based on Nonlinear Finite Element Equations A Hybrid Computational Intelligence Method of Newton's Method and Genetic Algorithm for Solving Compatible Nonlinear Equations Pressure Image Recognition of Lying Positions Based on Multifeature value Regularized Extreme Learning Algorithm English Intelligent Question Answering System Based on elliptic fitting equation Precision Machining Technology of Jewelry on CNC Machine Tool Based on Mathematical Modeling Application Research of Mathematica Software in Calculus Teaching Computer Vision Communication Technology in Mathematical Modeling Skills of Music Creation Based on Homogeneous FirstOrder Linear Partial Differential Equations Mathematical Statistics Technology in the Educational Grading System of Preschool Students Music Recommendation Index Evaluation Based on Logistic Distribution Fitting Transition Probability Function Children's Educational Curriculum Evaluation Management System in Mathematical Equation Model Query Translation Optimization and Mathematical Modeling for EnglishChinese CrossLanguage Information Retrieval The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization research on prefabricated concrete frame buildings based on the dynamic equation of eccentric structure and horizontaltorsional coupling Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Green building considering image processing technology combined with CFD numerical simulation MCM of Student’s Physical Health Based on Mathematical Cone Sports health quantification method and system implementation based on multiple thermal physiology simulation Research on visual optimization design of machine–machine interface for mechanical industrial equipment based on nonlinear partial equations Research on identifying psychological health problems of college students by logistic regression model based on data mining Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching A mathematical model of PCNN for image fusion with nonsampled contourlet transform Nonlinear Differential Equations in ComputerAided Modeling of Big Data Technology The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Financial customer classification by combined model Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on Logarithmic Growth Function Model Design of Tennis Mobile Teaching Assistant System Based on Ordinary Differential Equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis High Simulation Reconstruction of Crowd Animation Based on Optical Flow Constraint Equation Construction of Intelligent Search Engine for Big Data Multimedia Resource Subjects Based on Partial Least Squares Structural Equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with DiamondSquare Algorithm Analysis of the Teaching Quality of Physical Education Class by Using the Method of Gradient Difference The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Animation VR scene mosaic modeling based on generalized Laplacian equation Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Crossborder Ecommerce Platform Operation Based on Fractional Differential Equations The influence of accounting computer information processing technology on enterprise internal control under panel data simultaneous equation Research on Stability of Timedelay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Construction of comprehensive evaluation index system of watersaving irrigation project integrating penman Montei the quation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of AirConditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression Analysis of technical statistical indexes of college tennis players under the winlose regression function equation Automatic extraction and discrimination of vocal main melody based on quadratic wave equation Analysis of wireless English multimedia communication based on spatial state model equation Optimization of Linear Algebra Core Function Framework on Multicore Processors Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Mathematical function data model analysis and synthesis system based on shortterm human movement Human gait modelling and tracking based on motion functionalisation The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Nonlinear Differential Equations in Crossborder Ecommerce Controlling Return Rate Differential equation model of financial market stability based on Internet big data 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Children’s cognitive function and mental health based on finite element nonlinear mathematical model Fractional Differential Equations in Electronic Information Models BIM Engineering Management Oriented to Curve Equation Model Leakage control of urban water supply network and mathematical analysis and location of leakage points based on machine learning Analysis of higher education management strategy based on entropy and dissipative structure theory Prediction of corporate financial distress based on digital signal processing and multiple regression analysis Mathematical Method to Construct the Linear Programming of Football Training Multimedia sensor image detection based on constrained underdetermined equation The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Application of Numerical Computation of Partial Differential Equations in Interactive Design of Virtual Reality Media Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Knowledge Analysis of Charged Particle Motion in Uniform Electromagnetic Field Based on Maxwell Equation Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Term structure of economic management rate based on parameter analysis of estimation model of ordinary differential equation Influence analysis of piano music immersion virtual reality cooperation based on mapping equation Chinese painting and calligraphy image recognition technology based on pseudo linear directional diffusion equation Label big data compression in Internet of things based on piecewise linear regression Animation character recognition and character intelligence analysis based on semantic ontology and Poisson equation Design of language assisted learning model and online learning system under the background of artificial intelligence Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering Action of Aut( G ) on the set of maximal subgroups ofp groupsThe internal mechanism of corporate social responsibility fulfillment affecting debt risk in China: analysis of intermediary transmission effect based on degree of debt concentration and product market competitive advantage Study on transmission characteristics in three kinds of deformed finlines based on edgebased finite element method Asymptotic stability problem of predator–prey system with linear diffusion Research on loyalty prediction of ecommerce customer based on data mining Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Research on threat assessment problems of island air defence system based on the leaderfollower model Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on ecommerce platform Study on inefficient land use determination method for cities and towns from a city examination perspective A sentiment analysis method based on bidirectional long shortterm memory networks Evaluation of ecosystem health in Futian mangrove wetland based on the PSRAHP model A study of local smoothnessinformed convolutional neural network models for image inpainting Towards more efficient control of the ironmaking blast furnace: modelling gaseous reduction of iron ores in H_{2}N_{2} atmosphere Algorithm of overfitting avoidance in CNN based on maximum pooled and weight decay Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Crossborder Ecommerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model ChineseEnglish Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model