كتاب Mathematical Modeling the Life Sciences - Numerical Recipes in Python and MATLAB
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منتدى هندسة الإنتاج والتصميم الميكانيكى
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 كتاب Mathematical Modeling the Life Sciences - Numerical Recipes in Python and MATLAB

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Mathematical Modeling the Life Sciences - Numerical Recipes in Python and MATLAB
N. G. Cogan

كتاب Mathematical Modeling the Life Sciences - Numerical Recipes in Python and MATLAB  M_m_i_11
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Contents
Foreword xi
1 Introduction 1
1.1 What Is a Model? . 1
1.2 Projectile Motion . 2
1.3 Problems . 8
2 Mathematical Background 9
2.1 Mathematical Preliminaries . 9
2.1.1 Linear . 11
2.1.2 Nonlinear Equations . 13
2.2 Linearization . 15
2.3 Qualitative Analysis . 18
2.4 Problems . 20
2.5 Appendix: Planar Example 23
3 Introduction to the Numerical Methods 27
3.1 Introduction 27
3.2 Best Practices in Coding . 29
3.2.1 Folder Structure 29
3.2.2 Naming Conventions . 30
3.2.3 Code Structure
3.2.4 Comments . 32
3.3 Getting the Programs Running 32
3.3.1 Python . 32
3.3.2 MATLAB® 34
3.4 Initial Programs 34
3.4.1 Differential Equations in Python . 37
3.4.2 Differential Equations in MATLAB® . 38
3.5 Problems . 40
vii
30viii Contents
3.6 Appendix: Sample Scripts 44
3.6.1 Python . 44
3.6.2 MATLAB® 45
4 Ecology 47
4.1 Historical Background 47
4.2 Single Species Models 50
4.2.1 The Exponential Model 51
4.2.2 The Logistic Model 53
4.2.3 Analysis 53
4.2.4 Predator/Prey – Lotka-Volterra 54
4.2.5 Analysis 56
4.2.6 Sensitivity: One at a Time, Scatterplots . 59
4.3 Competitive Exclusion 62
4.3.1 Model . 63
4.3.2 Analysis 65
4.3.3 Sensitivity: Linear Regression 68
4.4 State of the Art and Caveats . 71
4.5 Problems . 73
5 Within-host Disease Models 77
5.1 Historical Background 77
5.2 Pathological: Tumor . 81
5.2.1 Model . 82
5.2.2 Analysis 84
5.2.3 Sensitivity: Direct Estimation . 87
5.3 Viral: Acute Infection . 92
5.3.1 Model . 92
5.3.2 Analysis 94
5.3.3 Sensitivity Analysis: Feature Sensitivity . 96
5.4 Chronic: Tuberculosis 99
5.4.1 Model . 99
5.4.2 Analysis 101
5.4.3 Sensitivity: Relative Change . 103
5.5 Problems . 104
5.6 Appendix . 108Contents ix
6 Between Host-Disease Models 111
6.1 Historical Background 111
6.2 Two Compartment Models 115
6.2.1 Model . 115
6.2.2 Analysis 117
6.2.3 Sensitivity Analysis: Spider Plot . 118
6.3 Classical SIR . 121
6.3.1 Model . 121
6.3.2 Analysis 122
6.3.3 Sensitivity Analysis: Tornado Plots 125
6.4 Waning Antigens . 126
6.4.1 Model: SIRS . 127
6.4.2 Analysis 129
6.4.3 Sensitivity Analysis: Cobweb Diagrams . 129
6.5 Caveats and State of the Art . 131
6.6 Problems . 132
7 Microbiology 135
7.1 Historical Background 135
7.2 Bacterial Growth: Chemostat . 138
7.2.1 Model . 140
7.2.2 Analysis 141
7.2.3 Sensitivity Analysis: Correlation Coefficient,
Pearson’s Moment Correlation 142
7.3 Multiple State Models: Free/attached 146
7.3.1 Model: Freter . 148
7.3.2 Analysis 150
7.3.3 Sensitivity Analysis: Correlation Coefficient,
Spearman . 151
7.4 Cooperators, Cheaters, and Competitions 154
7.4.1 Model . 155
7.4.2 Analysis 156
7.4.3 Sensitivity Analysis: Sensitivity in Time and
Partial Rank Correlation Coefficient (PRCC) 157
7.5 State of the Art and Caveats . 161
7.6 Problems . 162x Contents
8 Circulation and Cardiac Physiology 167
8.1 Historical Background 167
8.2 Blood Circulation Models 173
8.2.1 Model: Algebraic . 174
8.2.2 Analysis 176
8.2.3 Sensitivity Analysis: Sampling Methods . 176
8.3 Cardiac Physiology 177
8.3.1 Model: Noble . 178
8.3.2 Analysis 181
8.3.3 Sensitivity Analysis: Morris Screening 183
8.4 State of the Art and Caveats . 186
8.5 Problems . 187
9 Neuroscience 189
9.1 Historical Background 189
9.2 Action Potential 192
9.2.1 Model: Hodgkin-Huxley . 194
9.2.2 Analysis 196
9.2.3 Sensitivity Analysis: ANOVA – Sobol’ 196
9.3 Fitzhugh-Nagumo 201
9.3.1 Model . 202
9.3.2 Analysis 203
9.3.3 Sensitivity: Moment Independent . 203
9.4 State of the Art and Caveats . 208
9.5 Problems . 209
10 Genetics 211
10.1 Historical Background 211
10.2 Heredity 215
10.2.1 Mathematics . 216
10.2.2 Analysis 217
10.2.3 Sensitivity: Factorial Design . 218
10.3 State of the Art and Caveats . 219
10.4 Problems . 220
Bibliography 223
Index 22
Index
R0, 112, 124, 127, 128
Linnaeus, 48
action potential, 178, 182, 190,
192
activation, 179
acute infection, 92
Anaconda, 32
ANOVA, 196, 197
antigen, 126
asymptotic, 158
axon, 190, 191, 193
bacteria, 77–79, 81, 102, 135–142
Bellman, 170
biofilm, 137, 147
bistable, 204
blood, 136, 167, 168, 174–176
cancer, 77, 79, 81
capacitance, 178, 191
cardiac, 168, 175, 177, 178, 183
cell, 79–81, 92, 93, 95, 101, 136,
139, 168, 177, 178, 189,
192
cellular, 80
cheater/cooperator, 155
chemostat, 138, 140–142, 146
circulation, 174
cobweb diagram, 129
Cole and Curtis, 190
competitive exclusion, 62, 66, 72
correlation coefficent
PRCC, 157
correlation coefficient, 145
Pearson, 158
PRCC, 159
Spearman, 158
Pearson, 142, 144
PRCC, 171
Spearman, 151, 153
covariance, 143, 160
COVID, 112, 123
curse of large dimension, 169
derivative, 3, 9–11, 13, 15, 52, 53,
87, 88, 97, 107, 118, 130,
134, 193
differential sensitivity, 87
disease, 79, 125
don’t panic
Starbix, 39
ecology, 47, 55, 63, 135
effector cell, 82, 83, 85, 91, 104
eigenvalues, 12, 13, 22, 84, 86,
95, 105, 124, 146, 158
endemic, 112, 128–130
epidemics, 118, 122, 131
epidemiology, 111, 116, 128
excitability, 182, 191, 192, 196,
201, 203
229230 Index
FAST, 201
feature sensitivity, 96
Fisher, 197
Fitzhugh Nagumo, 192
Fitzhugh-Nagumo, 201, 203, 206
Fourier Transform, 182
Freter, 148, 149
Galen, 168
gating variables, 194
Gause, 63
genetics, 211–214
half saturation constant, 119, 139
Hardy, 212
Harvey, 168
histogram, 102
Hodgkin-Huxley, 189, 192, 194,
201
Humboldt, 48
immune system, 79, 81, 99
inactivation, 179
infected, 93, 98, 115
infection, 78, 80, 92, 95, 96, 99,
101, 102, 115, 121–124,
126, 128, 130
infectious, 79, 112–114, 124, 126
initial value problem, 10
Jacobian, 18, 57, 73, 84–86, 89,
98, 104, 107
Kermack-McKendrick, 114
linear regression, 68, 71
linearization, 15, 18, 23, 54, 56,
74, 94
logistic, 53, 63
Lotka-Volterra, 54, 56, 63
Malthus, 52
mass action, 5, 54, 56, 83, 93, 94,
116
mass action kinetics, 94
MATLAB®, 34
membrane, 177, 182, 190, 191,
194
Mendel, 211
Michaelis-Menton, 139
microbiology, 135
moment independent, 203
Monod, 139, 149
Monte Carlo, 201
Morris Screening, 183
nerve, 190
neuron, 189, 190
Noble, 178, 181, 192
nullcine, 57
nullcline, 65
nullclines, 20, 75, 86
period, 58–60, 74, 164, 182
periodic, 58, 59, 61, 74, 220
phase-plane, 18, 58
population growth, 52
PRCC, 196
propagate, 115, 118, 125, 128
propagation, 126
Punnett, 212
Purkinje, 178
Purkinje fibers, 178
qualitative analysis, 18, 53
rank transformation, 152
recovered, 115, 127Index 231
refractory period, 196
regression, 68–71, 143, 146, 152,
159, 196
sampling, 153, 169
Latin Hyper Cube, 144, 170,
176
Monte Carlo, 144, 173, 176
Sobol’ sequence, 173
uniform, 144
scatter plot, 158
scatterplot, 59, 60, 146, 171
sensitivity in time, 89, 157, 160
sigmoidal curve, 119, 139
SIR, 114, 116, 121, 128
Sobol’, 196, 199, 201, 203, 204
spider plot, 118
steady state, 16, 53, 58, 65, 66,
85, 98, 101, 122
steady-state, 129, 141, 142
susceptible, 115
Taylors’ Theorem, 13, 15, 17, 24
tonic firing, 203
tornado plot, 125
tradgedy of the commons, 154
trajectories, 20, 90, 106, 107,
197, 203
Tuberculosis, 99
tumor, 81–83, 85, 86, 90, 91, 104,
106
vaccination, 79, 111, 126, 134,
137
vaccine, 81, 111, 126, 127, 137
van der Pol oscillator, 202
variance, 70, 143, 200
Verhulst, 53, 63
viral dynamics, 92
virus, 93
voltage trace, 190
within-host, 79, 9

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