Mathematical Modeling the Life Sciences: Numerical Recipes in Python and MATLAB® (Textbooks in Mathematics)

Description

The purpose of this unique textbook is to bridge the gap between the need for numerical solutions to modeling techniques through computer simulations to develop skill in employing sensitivity analysis to biological and life sciences applications. The underpinning mathematics is minimalized. The focus is on the consequences, implementation, and application. Historical context motivates the models. An understanding of the earliest models provides insight into more complicated ones. While the text avoids getting mired in the details of numerical analysis, it demonstrates how to use numerical methods and provides core codes that can be readily altered to fit a variety of situations. Numerical scripts in both Python and MATLAB® are included. Python is compiled in Jupyter Notebook to aid classroom use. Additionally, codes are organized and available online. One of the most important skills requiring the use of computer simulations is sensitivity analysis. Sensitivity analysis is increasingly used in biomathematics. There are numerous pitfalls to using sensitivity analysis and therefore a need for exposure to worked examples in order to successfully transfer their use from mathematicians to biologists. The interconnections between mathematics and the life sciences have an extensive history. This book offers a new approach to using mathematics to model applications using computers, to employ numerical methods, and takes students a step further into the realm of sensitivity analysis. With some guidance and practice, the reader will have a new and incredibly powerful tool to use. Cover Half Title Series Page Title Page Copyright Page Dedication Contents Foreword 1. Introduction 1.1. What Is a Model? 1.2. Projectile Motion 1.3. Problems 2. Mathematical Background 2.1. Mathematical Preliminaries 2.1.1. Linear 2.1.2. Nonlinear Equations 2.2. Linearization 2.3. Qualitative Analysis 2.4. Problems 2.5. Appendix: Planar Example 3. Introduction to the Numerical Methods 3.1. Introduction 3.2. Best Practices in Coding 3.2.1. Folder Structure 3.2.2. Naming Conventions 3.2.3. Code Structure 3.2.4. Comments 3.3. Getting the Programs Running 3.3.1. Python 3.3.2. MATLAB® 3.4. Initial Programs 3.4.1. Differential Equations in Python 3.4.2. Differential Equations in MATLAB® 3.5. Problems 3.6. Appendix: Sample Scripts 3.6.1. Python 3.6.2. MATLAB® 4. Ecology 4.1. Historical Background 4.2. Single Species Models 4.2.1. The Exponential Model 4.2.2. The Logistic Model 4.2.3. Analysis 4.2.4. Predator/Prey – Lotka-Volterra 4.2.5. Analysis 4.2.6. Sensitivity: One at a Time, Scatterplots 4.3. Competitive Exclusion 4.3.1. Model 4.3.2. Analysis 4.3.3. Sensitivity: Linear Regression 4.4. State of the Art and Caveats 4.5. Problems 5. Within-host Disease Models 5.1. Historical Background 5.2. Pathological: Tumor 5.2.1. Model 5.2.2. Analysis 5.2.3. Sensitivity: Direct Estimation 5.3. Viral: Acute Infection 5.3.1. Model 5.3.2. Analysis 5.3.3. Sensitivity Analysis: Feature Sensitivity 5.4. Chronic: Tuberculosis 5.4.1. Model 5.4.2. Analysis 5.4.3. Sensitivity: Relative Change 5.5. Problems 5.6. Appendix 6. Between Host-Disease Models 6.1. Historical Background 6.2. Two Compartment Models 6.2.1. Model 6.2.2. Analysis 6.2.3. Sensitivity Analysis: Spider Plot 6.3. Classical SIR 6.3.1. Model 6.3.2. Analysis 6.3.3. Sensitivity Analysis: Tornado Plots 6.4. Waning Antigens 6.4.1. Model: SIRS 6.4.2. Analysis 6.4.3. Sensitivity Analysis: Cobweb Diagrams 6.5. Caveats and State of the Art 6.6. Problems 7. Microbiology 7.1. Historical Background 7.2. Bacterial Growth: Chemostat 7.2.1. Model 7.2.2. Analysis 7.2.3. Sensitivity Analysis: Correlation Coefficient, Pearson’s Moment Correlation 7.3. Multiple State Models: Free/attached 7.3.1. Model: Freter 7.3.2. Analysis 7.3.3. Sensitivity Analysis: Correlation Coefficient, Spearman 7.4. Cooperators, Cheaters, and Competitions 7.4.1. Model 7.4.2. Analysis 7.4.3. Sensitivity Analysis: Sensitivity in Time and Partial Rank Correlation Coefficient (PRCC) 7.5. State of the Art and Caveats 7.6. Problems 8. Circulation and Cardiac Physiology 8.1. Historical Background 8.2. Blood Circulation Models 8.2.1. Model: Algebraic 8.2.2. Analysis 8.2.3. Sensitivity Analysis: Sampling Methods 8.3. Cardiac Physiology 8.3.1. Model: Noble 8.3.2. Analysis 8.3.3. Sensitivity Analysis: Morris Screening 8.4. State of the Art and Caveats 8.5. Problems 9. Neuroscience 9.1. Historical Background 9.2. Action Potential 9.2.1. Model: Hodgkin-Huxley 9.2.2. Analysis 9.2.3. Sensitivity Analysis: ANOVA – Sobol’ 9.3. Fitzhugh-Nagumo 9.3.1. Model 9.3.2. Analysis 9.3.3. Sensitivity: Moment Independent 9.4. State of the Art and Caveats 9.5. Problems 10. Genetics 10.1. Historical Background 10.2. Heredity 10.2.1. Mathematics 10.2.2. Analysis 10.2.3. Sensitivity: Factorial Design 10.3. State of the Art and Caveats 10.4. Problems Bibliography Index