Mathematics and Life Sciences (eBook)

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Alexandra V. Antoniouk, Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine; Roderick V. N. Melnik, Wilfrid Laurier University, Waterloo, Ontario, Canada.

1 Introduction 13
1.1 Scientific Frontiers at the Interface of Mathematics and Life Sciences 15
1.1.1 Developing the Language of Science and Its Interdisciplinary Character 15
1.1.2 Challenges at the Interface: Mathematics and Life Sciences 17
1.1.3 What This Book Is About 22
1.1.4 Concluding Remarks 26
2 Mathematical and Statistical Modeling of Biological Systems 29
2.1 Ensemble Modeling of Biological Systems 31
2.1.1 Introduction 31
2.1.2 Background 33
2.1.3 Ensemble Model 37
2.1.4 Computational Techniques 39
2.1.5 Application to Viral Infection Dynamics 42
2.1.6 Ensemble Models in Biology 46
2.1.7 Conclusions 48
3 Probabilistic Models for Nonlinear Processes and Biological Dynamics 55
3.1 Nonlinear Lévy and Nonlinear Feller Processes: an Analytic Introduction 57
3.1.1 Introduction 57
3.1.2 Dual Propagators 61
3.1.3 Perturbation Theory for Weak Propagators 64
3.1.4 T-Products 66
3.1.5 Nonlinear Propagators 69
3.1.6 Linearized Evolution Around a Path of a Nonlinear Semigroup 72
3.1.7 Sensitivity Analysis for Nonlinear Propagators 76
3.1.8 Back to Nonlinear Markov Semigroups 78
3.1.9 Concluding Remarks 80
4 New Results in Mathematical Epidemiology and Modeling Dynamics of Infectious Diseases 83
4.1 Formal Solutions of Epidemic Equation 85
4.1.1 Introduction 85
4.1.2 Epidemic Models 87
4.1.3 Formal Solutions 88
4.1.4 Separation of Variables 91
4.1.5 Solvability of General Equations 92
4.1.6 Concluding Remarks 96
5 Mathematical Analysis of PDE-based Models and Applications in Cell Biology 99
5.1 Asymptotic Analysis of the Dirichlet Spectral Problems in Thin Perforated Domains with Rapidly Varying Thickness and Different Limit Dimensions 101
5.1.1 Introduction 101
5.1.2 Description of a Thin Perforated Domain with Quickly Oscillating Thickness and Statement of the Problem 102
5.1.3 Equivalent Problem 104
5.1.4 The Homogenized Theorem 106
5.1.5 Asymptotic Expansions for the Eigenvalues and Eigenfunctions 112
5.1.6 Conclusions 119
6 Axiomatic Modeling in Life Sciences with Case Studies for Virus-immune System and Oncolytic Virus Dynamics 123
6.1 Axiomatic Modeling in Life Sciences 125
6.1.1 Introduction 125
6.1.2 Boosting Immunity by Anti-viral Drug Therapy: Timing, Efficacy and Success 127
6.1.3 Predictive Modeling of Oncolytic Virus Dynamics 135
6.1.4 Conclusions 150
7 Theory, Applications, and Control of Nonlinear PDEs in Life Sciences 157
7.1 On One Semilinear Parabolic Equation of Normal Type 159
7.1.1 Introduction 159
7.1.2 Semilinear Parabolic Equation of Normal Type 160
7.1.3 The Structure of NPE Dynamics 165
7.1.4 Stabilization of Solution for NPE by Start Control 170
7.1.5 Concluding Remarks 171
7.2 On some Classes of Nonlinear Equations with L1 -Data 173
7.2.1 Nonlinear Elliptic Second-order Equations with L1-data 174
7.2.2 Nonlinear Fourth-order Equations with Strengthened Coercivity and L1-Data 190
7.2.3 Concluding Remarks 197
8 Mathematical Models of Pattern Formation and Their Applications in Developmental Biology 201
8.1 Reaction-Diffusion Models of Pattern Formation in Developmental Biology 203
8.1.1 Introduction 203
8.1.2 Mechanisms of Developmental Pattern Formation 205
8.1.3 Motivating Application: Pattern Control in Hydra 206
8.1.4 Diffusive Morphogens and Turing Patterns 209
8.1.5 Receptor-based Models 212
8.1.6 Multistability 218
8.1.7 Discussion 219
9 Modeling the Dynamics of Genetic Mechanism, Pattern Formation, and the Genetics of “Geometry” 225
9.1 Modeling the Positioning of Trichomes on the Leaves of Plants 227
9.1.1 Introduction 227
9.1.2 Activator-inhibitor Reaction-diffusion Modeling of the Trichome Positioning 230
9.1.3 Hexagonal Recursion 233
9.1.4 Conclusions 237
10 Statistical Modeling in Life Sciences and Direct Measurements 241
10.1 Error Estimation for Direct Measurements in May-June 1986 of 131I Radioactivity in Thyroid Gland of Children and Adolescents and Their Registration in Risk Analysis 243
10.1.1 Introduction 243
10.1.2 Materials and Methods 245
10.1.3 Conclusion and Discussion 251
10.1.4 Appendix. Approximation of Conditional Expectations 252
11 Design and Development of Experiments for Life Science Applications 257
11.1 Physiological Effects of Static Magnetic Field Exposure in an in vivo Acute Visceral Pain Model in Mice 259
11.1.1 Introduction 259
11.1.2 Methods 261
11.1.3 Results 268
11.1.4 Discussion 278
11.1.5 Conclusions 281
12 Mathematical Biomedicine and Modeling Avascular Tumor Growth 289
12.1 Continuum Models of Avascular Tumor Growth 291
12.1.1 Introduction 291
12.1.2 Diffusion-limited Models of Avascular Tumor Growth 293
12.1.3 Tumor Invasion 301
12.1.4 Multiphase Models of Avascular Tumor Growth 307
12.1.5 Conclusions 315
Index 325