Delay differential equations and applications to biology:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
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Singapore
Springer
[2021]
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| Series: | Forum for Interdisciplinary Mathematics
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xvii, 286 Seiten Illustrationen, Diagramme |
| ISBN: | 9789811606250 9789811606281 |
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| adam_text | Contents Part I 1 Qualitative and Quantitative Features of Delay Differential Equations Qualitative Features of Delay Differential Equations ........................... Introduction ..................................................................................... Delay Models in Population Dynamics ......................................... 1.2.1 Logistic Equation with Discrete Delay ........................... 1.2.2 Logistic Equation with Distributed Delay ...................... 1.2.3 Delayed Lotka-Volterra System ....................................... 1.3 Stability of DDEs ............................................................................. 1.3.1 Stability of Linear Constant Coefficient DDEs .............. 1.3.2 Asymptotical Stability Region for Linear DDEs ............ 1.3.3 Stability of Linear NDDEs ........... 1.3.4 Asymptotic Stability Region for Linear NDDEs ............ 1.4 Stability of Non-linear DDEs and Contractivity Conditions......... 1.5 Stability of DDEs in Lyapunov Method............... ........................... 1.5.1 Lyapunov-Krasovskii Sense ............................................. 1.5.2 Lyapunov-Razumikhin Sense........................................... 1.5.3 Stability of Linear Systems with Discrete Delays .......... 1.6 Concluding Remarks ............................................................... References ................................................................................................... 1.1 1.2 2 Numerical Solutions of Delay Differential Equations........................... 2.1 2.2 2.3 2.4 Propagation and
Location of Discontinuities in DDEs ................. Method of Steps for DDEs............................................................... Existence and Uniqueness Solution of DDEs ............................... Numerical Approach for DDEs................................ 2.4.1 General Approach ............................................................. 2.4.2 Θ-Methods for DDEs ....................................................... 2.4.3 Continuous One-Step Runge-Kutta Methods for ODE.............................................................................. 2.4.4 Runge-Kutta Method for DDEs ....................................... 3 3 5 5 7 7 8 9 10 10 11 12 17 18 18 19 20 21 23 25 26 28 28 28 30 31 34 xi
Contents xii More General Classes of DDEs....................................................... 2.5.1 Neutral Delay Differential Equations (NDDEs) ............. 2.5.2 Equations with State-Dependent Lags............................. 2.5.3 Equation with a Small or Vanishing Lag ......................... 2.6 Stiffness Problems................................... 2.7 Software Aspects............................................................................... 2.7.1 Discretization Error ............ 2.7.2 Location of Jump Discontinuities .................................... 2.7.3 Stepsize Control ................................................................. 2.7.4 Interpolation tö y(t)........................................................... 2.7.5 DDE Solvers and Available Software .............................. 2.8 Concluding Remarks .............. References ................................................................................................... 2.5 3 Stability Concepts of Numerical Solutions of Delay Differential Equations ................................. Introduction ..................................................................... Stability of Numerical Methods for DDEs ........................... 3.2.1 Stability Regions for DDEs: P-stability and GP-stability .................................................. 53 3.2.2 Stability Regions for Linear NDDEs............................... 3.3 Contractivity Concepts and GPN-Stability .................................. 3.3.1 Contractivity Concepts and GRN-Stability .................... 3.4 Concluding Remarks
...................................................................... References ................................................................................................... 3.1 3.2 4 51 51 51 57 61 64 66 67 Numerical Solutions of Volterra Delay Integro-Differential Equations ............................................................................................... 69 Introduction ......................... Analytical Stability .......................................................................... Continuous Mono-Implicit RK Scheme for DDEs........................ Numerical Treatment of VDIDEs ................................................... 4.4.1 CMIRK Scheme for VDIDEs........................................... 4.4.2 Numerical Integration Formula and Boole’s Quadrature Rule .................................................. 77 4.4.3 MIDDÉ Software Aspects ............................................... 4.5 Numerical Stability............................... 4.6 Numerical Results and Simulations ................................................ 4.7 Concluding Remarks ....................................................................... References ................................................................................................... 4.1 4.2 4.3 4.4 5 36 37 37 39 40 43 43 45 45 46 47 47 48 Parameter Estimation with Delay Differential Equations................... 5.1 5.2 Introduction ...................................................................................... Parameter Estimation with DDEs
.................................................... 5.2.1 Non-linearity of Model Predictions.................................. 69 70 73 75 76 78 79 81 85 85 87 87 87 88
Contents Computation of Estimates ............................................................... 5.3.1 Basic Iteration.................................................................... 5.3.2 Acceptability...................................................................... 5.3.3 Convergence ...................................................................... 5.4 Discontinuities Associated with Delay........................................... 5.5 Solving the Minimization Problem.................................................. 5.6 Models and Goodness of Fit for Cell Growth ................................ 5.6.1 Problem 1: Fitting DDEs with Growth of Fission Yeast . .................................................................. 96 5.6.2 Fitting DDEs with Growthof Tetrahymena Pyriformis ......................................................................... 5.7 Concluding Remarks ...................................................................... References .................................................................................................. 5.3 6 Sensitivity Analysis of Delay Differential Equations ............................ 6.1 6.2 Introduction ...................................................................................... Sensitivity Functions......................................................................... 6.2.1 Adjoint Equations ............................................................ 6.3 Variational Approach ....................................................................... 6.4 Direct Approach
............................................................................... 6.5 Sensitivity of Optimum Parameter p to Data ................................ 6.5.1 Standard Deviation of Parameter Estimates .................... 6.5.2 Non-linearity and Indications of Bias ............................. 6.6 Numerical Results............................................................................. 6.7 Concluding Remarks ....................................................................... References ............................ 7 Stochastic Delay Differential Equations .................................................. Introduction ...................................................................................... 7.1.1 Preliminaries...................................................................... 7.2 Existence and Uniqueness of Solutions for SDDEs ....................... 7.3 Stability Criteria for SDDEs ............................................................ 7.4 Numerical Scheme for Autonomous SDDEs ................................ 7.4.1 Convergence and Consistency .......................................... 7.5 Numerical Schemes for Non-autonomous SDDE.................... 7.5.1 Taylor Approximation ........................................................ 7.5.2 Implicit Strong Approximations........................................ 7.6 Milstein Scheme for SDDEs ........................................................... 7.6.1 Convergence and Mean-Square Stability of the Milstein Scheme ...................................... 138 7.7 Concluding Remarks
...................................................................... References ................................................................................................... 7.1 xiii 89 90 91 91 92 94 95 98 101 101 103 103 104 105 106 109 112 113 114 114 120 122 123 123 124 127 130 133 134 135 136 136 137 140 140
Contents xiv Part II Applications of Delay Differential Equations in Biosciences 8 Delay Differential Equations with Infectious Diseases........................ 8.1 Introduction ............................................. ·....................................... 8.2 Time-Delay in Epidemic Models ................... 8.2.1 Development of SIR Model (8.1)..................................... 8.3 Delay Differential Models with Viral Infection ............................ 8.3.1 DDEs with НГѴ Infection of CD4+T-cells .................... 8.3.2 Steady States........................................................................ 8.3.3 Stability Analysis of Infected Steady State...................... 8.3.4 Existence of Hopf Bifurcation .......................................... 8.4 Physiology ............................. 8.5 Concluding Remarks ....................................................................... References ’...................... . ............. ;.......................................................... 9 Delay Differential Equations of Tumor-Immune System with Treatment and Control.......................................... 167 9.1 Introduction ..................... 9.2 Description of the Model ................................................................ 9.2.1 Non-negativity and Boundedness Solutions of Model (9.3)...................................................... 170 9.2.2 Model with Chemotherapy ................................................ 9.3 Drug-Free Steady States and Their Stability .................................. 9.3.1 Stability of Tumor-
Free SteadyState ................................ 9.3.2 Stability of Lethal Steady States ...................................... 9.3.3 Stability of Coexisting Steady States................................ 9.4 Optimal Control Problem Governed by DDEs .............................. 9.5 Existence of Optimal Solution.................... 9.6 Optimality Conditions ..................................................................... 9.7 Immuno-Chemotherapy ................................................................... 9.8 Numerical Simulations of Optimal Control System...................... 9.9 Concluding Remarks ....................................................................... References ................................................................................................... 10 Delay Differential Equations of Ecological Systems with Allee Effect ........................................................................................................... 10.1 Introduction ...................................... 10.2 Delay Differential Model of Two-Prey One-Predator System .................................................................................. 194 10.2.1 Positivity and Boundedness of the Solution ................... 10.3 Local Stability and Hopf Bifurcation .............................................. 10.3.1 Existence of Interior Equilibrium Points.......................... 10.3.2 Stability and Bifurcation Analysis of the Interior Equilibrium.......................................................... 199 10.4 Global Stability of Interior Steady State
£*.................................... 145 145 146 147 148 150 154 156 157 160 163 164 167 168 172 173 174 175 175 177 179 181 183 184 188 188 191 191 196 197 198 203
Contents 10.5 Sensitivity to Allee Effect ............................................................. 10.6 Numerical Simulations................................................................... 10.7 Concluding Remarks .................................................................... References .................................................................................................. 11 XV 205 206 209 209 Fractional-Order Delay Differential Equations with Predator-Prey Systems.............................................................. 211 Introduction .................................................................................... 11.1.1 Preliminaries...................................................................... 11.2 Fractional Delayed Predator-Prey Model...................................... 11.3 Local Stability Analysis and Hopf Bifurcation ............................ 11.3.1 Trivial and Semi-trivial Equilibria and Their Stabilities............................................................. 217 11.3.2 Interior Equilibrium and Its Stability............................... 11.4 Global Stability Analysis............................................................... 11.5 Implicit Euler’s Scheme for FODDEs .......................................... 11.5.1 Stability and Convergence of Implicit Scheme for FODDEs ................................... 225 11.5.2 Numerical Simulations .................................................... 11.6 Concluding Remarks ....................................... References ............................. 11.1 211 212
215 216 218 220 222 227 228 230 12 Fractional-Order Delay Differential Equations of Hepatitis C Virus......................................................................................................... 233 12.1 Introduction ..................................................................................... 12.2 Mathematical Model of HCV ........................................................ 12.3 Local Stability of Infection-Free and Infected Steady States .... 12.4 Global Stability of Infection-Free Steady State £q ........................ 12.5 Numerical Simulations and Validity of Model ............................ 12.5.1 Parameter Estimation andValidity of Model ................... 12.6 Concluding Remarks .......................................... References ................................................................................................... 13 233 235 237 242 243 245 250 251 Stochastic Delay Differential Model for Coronavirus Infection COVID-19............................................................................................... 253 13.1 Introduction .................................................................................... 13.2 Stochastic SIRC Epidemic Model.................................................. 13.3 Existence and Uniqueness of Positive Solution .......................... 13.4 Existence of Ergodic Stationary Distribution .............................. 13.5 Extinction........................................................................................ 13.6 Numerical Simulations and Discussions
...................................... 13.7 Concluding Remarks ..................................................................... References ................................................................................................... 253 255 257 259 266 268 273 274 Remarks and Current Challenges............................................................. 277 Appendix A: Fifth-Order Dormand and Prince RK Method .................... 281 14
xvi Appendix В: Adams-Bashforth-Moulton Method for Fractional-Order Delay Differential Equations ........................................................... Contents 283 Appendix C: Matlab Program for Stochastic Delay Differential Equations Using Milstein Scheme........ ....... 285
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| adam_txt |
Contents Part I 1 Qualitative and Quantitative Features of Delay Differential Equations Qualitative Features of Delay Differential Equations . Introduction . Delay Models in Population Dynamics . 1.2.1 Logistic Equation with Discrete Delay . 1.2.2 Logistic Equation with Distributed Delay . 1.2.3 Delayed Lotka-Volterra System . 1.3 Stability of DDEs . 1.3.1 Stability of Linear Constant Coefficient DDEs . 1.3.2 Asymptotical Stability Region for Linear DDEs . 1.3.3 Stability of Linear NDDEs . 1.3.4 Asymptotic Stability Region for Linear NDDEs . 1.4 Stability of Non-linear DDEs and Contractivity Conditions. 1.5 Stability of DDEs in Lyapunov Method. . 1.5.1 Lyapunov-Krasovskii Sense . 1.5.2 Lyapunov-Razumikhin Sense. 1.5.3 Stability of Linear Systems with Discrete Delays . 1.6 Concluding Remarks . References . 1.1 1.2 2 Numerical Solutions of Delay Differential Equations. 2.1 2.2 2.3 2.4 Propagation and
Location of Discontinuities in DDEs . Method of Steps for DDEs. Existence and Uniqueness Solution of DDEs . Numerical Approach for DDEs. 2.4.1 General Approach . 2.4.2 Θ-Methods for DDEs . 2.4.3 Continuous One-Step Runge-Kutta Methods for ODE. 2.4.4 Runge-Kutta Method for DDEs . 3 3 5 5 7 7 8 9 10 10 11 12 17 18 18 19 20 21 23 25 26 28 28 28 30 31 34 xi
Contents xii More General Classes of DDEs. 2.5.1 Neutral Delay Differential Equations (NDDEs) . 2.5.2 Equations with State-Dependent Lags. 2.5.3 Equation with a Small or Vanishing Lag . 2.6 Stiffness Problems. 2.7 Software Aspects. 2.7.1 Discretization Error . 2.7.2 Location of Jump Discontinuities . 2.7.3 Stepsize Control . 2.7.4 Interpolation tö y(t). 2.7.5 DDE Solvers and Available Software . 2.8 Concluding Remarks . References . 2.5 3 Stability Concepts of Numerical Solutions of Delay Differential Equations . Introduction . Stability of Numerical Methods for DDEs . 3.2.1 Stability Regions for DDEs: P-stability and GP-stability . 53 3.2.2 Stability Regions for Linear NDDEs. 3.3 Contractivity Concepts and GPN-Stability . 3.3.1 Contractivity Concepts and GRN-Stability . 3.4 Concluding Remarks
. References . 3.1 3.2 4 51 51 51 57 61 64 66 67 Numerical Solutions of Volterra Delay Integro-Differential Equations . 69 Introduction . Analytical Stability . Continuous Mono-Implicit RK Scheme for DDEs. Numerical Treatment of VDIDEs . 4.4.1 CMIRK Scheme for VDIDEs. 4.4.2 Numerical Integration Formula and Boole’s Quadrature Rule . 77 4.4.3 MIDDÉ Software Aspects . 4.5 Numerical Stability. 4.6 Numerical Results and Simulations . 4.7 Concluding Remarks . References . 4.1 4.2 4.3 4.4 5 36 37 37 39 40 43 43 45 45 46 47 47 48 Parameter Estimation with Delay Differential Equations. 5.1 5.2 Introduction . Parameter Estimation with DDEs
. 5.2.1 Non-linearity of Model Predictions. 69 70 73 75 76 78 79 81 85 85 87 87 87 88
Contents Computation of Estimates . 5.3.1 Basic Iteration. 5.3.2 Acceptability. 5.3.3 Convergence . 5.4 Discontinuities Associated with Delay. 5.5 Solving the Minimization Problem. 5.6 Models and Goodness of Fit for Cell Growth . 5.6.1 Problem 1: Fitting DDEs with Growth of Fission Yeast . . 96 5.6.2 Fitting DDEs with Growthof Tetrahymena Pyriformis . 5.7 Concluding Remarks . References . 5.3 6 Sensitivity Analysis of Delay Differential Equations . 6.1 6.2 Introduction . Sensitivity Functions. 6.2.1 Adjoint Equations . 6.3 Variational Approach . 6.4 Direct Approach
. 6.5 Sensitivity of Optimum Parameter p to Data . 6.5.1 Standard Deviation of Parameter Estimates . 6.5.2 Non-linearity and Indications of Bias . 6.6 Numerical Results. 6.7 Concluding Remarks . References . 7 Stochastic Delay Differential Equations . Introduction . 7.1.1 Preliminaries. 7.2 Existence and Uniqueness of Solutions for SDDEs . 7.3 Stability Criteria for SDDEs . 7.4 Numerical Scheme for Autonomous SDDEs . 7.4.1 Convergence and Consistency . 7.5 Numerical Schemes for Non-autonomous SDDE. 7.5.1 Taylor Approximation . 7.5.2 Implicit Strong Approximations. 7.6 Milstein Scheme for SDDEs . 7.6.1 Convergence and Mean-Square Stability of the Milstein Scheme . 138 7.7 Concluding Remarks
. References . 7.1 xiii 89 90 91 91 92 94 95 98 101 101 103 103 104 105 106 109 112 113 114 114 120 122 123 123 124 127 130 133 134 135 136 136 137 140 140
Contents xiv Part II Applications of Delay Differential Equations in Biosciences 8 Delay Differential Equations with Infectious Diseases. 8.1 Introduction . ·. 8.2 Time-Delay in Epidemic Models . 8.2.1 Development of SIR Model (8.1). 8.3 Delay Differential Models with Viral Infection . 8.3.1 DDEs with НГѴ Infection of CD4+T-cells . 8.3.2 Steady States. 8.3.3 Stability Analysis of Infected Steady State. 8.3.4 Existence of Hopf Bifurcation . 8.4 Physiology . 8.5 Concluding Remarks . References ’. . . ;. 9 Delay Differential Equations of Tumor-Immune System with Treatment and Control. 167 9.1 Introduction . 9.2 Description of the Model . 9.2.1 Non-negativity and Boundedness Solutions of Model (9.3). 170 9.2.2 Model with Chemotherapy . 9.3 Drug-Free Steady States and Their Stability . 9.3.1 Stability of Tumor-
Free SteadyState . 9.3.2 Stability of Lethal Steady States . 9.3.3 Stability of Coexisting Steady States. 9.4 Optimal Control Problem Governed by DDEs . 9.5 Existence of Optimal Solution. 9.6 Optimality Conditions . 9.7 Immuno-Chemotherapy . 9.8 Numerical Simulations of Optimal Control System. 9.9 Concluding Remarks . References . 10 Delay Differential Equations of Ecological Systems with Allee Effect . 10.1 Introduction . 10.2 Delay Differential Model of Two-Prey One-Predator System . 194 10.2.1 Positivity and Boundedness of the Solution . 10.3 Local Stability and Hopf Bifurcation . 10.3.1 Existence of Interior Equilibrium Points. 10.3.2 Stability and Bifurcation Analysis of the Interior Equilibrium. 199 10.4 Global Stability of Interior Steady State
£*. 145 145 146 147 148 150 154 156 157 160 163 164 167 168 172 173 174 175 175 177 179 181 183 184 188 188 191 191 196 197 198 203
Contents 10.5 Sensitivity to Allee Effect . 10.6 Numerical Simulations. 10.7 Concluding Remarks . References . 11 XV 205 206 209 209 Fractional-Order Delay Differential Equations with Predator-Prey Systems. 211 Introduction . 11.1.1 Preliminaries. 11.2 Fractional Delayed Predator-Prey Model. 11.3 Local Stability Analysis and Hopf Bifurcation . 11.3.1 Trivial and Semi-trivial Equilibria and Their Stabilities. 217 11.3.2 Interior Equilibrium and Its Stability. 11.4 Global Stability Analysis. 11.5 Implicit Euler’s Scheme for FODDEs . 11.5.1 Stability and Convergence of Implicit Scheme for FODDEs . 225 11.5.2 Numerical Simulations . 11.6 Concluding Remarks . References . 11.1 211 212
215 216 218 220 222 227 228 230 12 Fractional-Order Delay Differential Equations of Hepatitis C Virus. 233 12.1 Introduction . 12.2 Mathematical Model of HCV . 12.3 Local Stability of Infection-Free and Infected Steady States . 12.4 Global Stability of Infection-Free Steady State £q . 12.5 Numerical Simulations and Validity of Model . 12.5.1 Parameter Estimation andValidity of Model . 12.6 Concluding Remarks . References . 13 233 235 237 242 243 245 250 251 Stochastic Delay Differential Model for Coronavirus Infection COVID-19. 253 13.1 Introduction . 13.2 Stochastic SIRC Epidemic Model. 13.3 Existence and Uniqueness of Positive Solution . 13.4 Existence of Ergodic Stationary Distribution . 13.5 Extinction. 13.6 Numerical Simulations and Discussions
. 13.7 Concluding Remarks . References . 253 255 257 259 266 268 273 274 Remarks and Current Challenges. 277 Appendix A: Fifth-Order Dormand and Prince RK Method . 281 14
xvi Appendix В: Adams-Bashforth-Moulton Method for Fractional-Order Delay Differential Equations . Contents 283 Appendix C: Matlab Program for Stochastic Delay Differential Equations Using Milstein Scheme. . 285 |
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| author | Rihan, Fathalla A. |
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| id | DE-604.BV048857192 |
| illustrated | Illustrated |
| index_date | 2024-07-03T21:41:21Z |
| indexdate | 2024-07-10T09:48:00Z |
| institution | BVB |
| isbn | 9789811606250 9789811606281 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-034122371 |
| oclc_num | 1376414570 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR |
| owner_facet | DE-355 DE-BY-UBR |
| physical | xvii, 286 Seiten Illustrationen, Diagramme |
| publishDate | 2021 |
| publishDateSearch | 2021 |
| publishDateSort | 2021 |
| publisher | Springer |
| record_format | marc |
| series2 | Forum for Interdisciplinary Mathematics |
| spelling | Rihan, Fathalla A. Verfasser (DE-588)1139646699 aut Delay differential equations and applications to biology Fathalla A. Rihan Singapore Springer [2021] xvii, 286 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Forum for Interdisciplinary Mathematics Analysis Integral Equations Mathematical analysis Analysis (Mathematics) Integral equations Biowissenschaften (DE-588)4129772-6 gnd rswk-swf Differentialgleichungssystem (DE-588)4121137-6 gnd rswk-swf Differentialgleichungssystem (DE-588)4121137-6 s Biowissenschaften (DE-588)4129772-6 s DE-604 Erscheint auch als Online-Ausgabe 978-981-16-0626-7 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034122371&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rihan, Fathalla A. Delay differential equations and applications to biology Analysis Integral Equations Mathematical analysis Analysis (Mathematics) Integral equations Biowissenschaften (DE-588)4129772-6 gnd Differentialgleichungssystem (DE-588)4121137-6 gnd |
| subject_GND | (DE-588)4129772-6 (DE-588)4121137-6 |
| title | Delay differential equations and applications to biology |
| title_auth | Delay differential equations and applications to biology |
| title_exact_search | Delay differential equations and applications to biology |
| title_exact_search_txtP | Delay differential equations and applications to biology |
| title_full | Delay differential equations and applications to biology Fathalla A. Rihan |
| title_fullStr | Delay differential equations and applications to biology Fathalla A. Rihan |
| title_full_unstemmed | Delay differential equations and applications to biology Fathalla A. Rihan |
| title_short | Delay differential equations and applications to biology |
| title_sort | delay differential equations and applications to biology |
| topic | Analysis Integral Equations Mathematical analysis Analysis (Mathematics) Integral equations Biowissenschaften (DE-588)4129772-6 gnd Differentialgleichungssystem (DE-588)4121137-6 gnd |
| topic_facet | Analysis Integral Equations Mathematical analysis Analysis (Mathematics) Integral equations Biowissenschaften Differentialgleichungssystem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034122371&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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