Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics Number 127 Mathematical Biology Modeling and Analysis Avner Friedman with support from the 10.1090/cbms/127 Mathematical Biology Modeling and Analysis Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics Number 127 Mathematical Biology Modeling and Analysis Avner Friedman Published for the Conference Board of the Mathematical Sciences by the with support from the 2018 NSF-CBMS Regional Research Conferences in the Mathematical Sciences on “Mathematical Biology: Modeling and Analysis” hosted by Howard Univeristy in Washington, DC, May 21–25, 2018. The author acknowledges support from NSF grant number 1743144. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. 2010 Mathematics Subject Classification. Primary 35Q92, 35R35, 37N25, 49J10, 92C50, 92D25; Secondary 35B32, 35B35, 35B50. For additional information and updates on this book, visit www.ams.org/bookpages/cbms-127 Library of Congress Cataloging-in-Publication Data Names: Friedman, Avner, author. | National Science Foundation (U.S.) Title: Mathematical biology : modeling and analysis / Avner Friedman. Description: Providence, Rhode Island : published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, [2018] | Series: CBMS regional conference series in mathematics ; number 127 | “Partially supported by the National Science Foundation.” | Includes bibliographical references and index. Identifiers: LCCN 2018015203 | ISBN 9781470447151 (alk. paper) Subjects: LCSH: Biology–Mathematical models. | AMS: Partial differential equations – Equations of mathematical physics and other areas of application – PDEs in connection with biology and other natural sciences. msc | Partial differential equations – Miscellaneous topics – Free boundary problems. msc | Dynamical systems and ergodic theory – Applications – Dynamical systems in biology. msc | Calculus of variations and optimal control; optimization – Existence theories – Free problems in two or more independent variables. msc | Biology and other natural sciences – Physiological, cellular and medical topics – Medical applications (general). msc | Biology and other natural sciences – Genetics and population dynamics – Population dynamics (general). msc | Partial differential equations – Qualitative properties of solutions – Bifurcation. msc | Partial differential equations – Qualitative properties of solutions – Stability. msc | Partial differential equations – Qualitative properties of solutions – Maximum principles. msc Classification: LCC QH323.5 .F75 2018 | DDC 570.1/5118–dc23 LC record available at https://lccn.loc.gov/2018015203 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2018 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 232221201918 Contents Preface vii Chapter 1. Introductory biology 1 Chapter 2. Introduction to modeling 7 Chapter 3. Models of population dynamics 15 3.1. Chemostat 15 3.2. Infectious diseases 18 3.3. A cancer model 21 Chapter 4. Cancer and the immune system 25 Chapter 5. Parameters estimation 33 Chapter 6. Mathematical analysis inspired by cancer models 41 Chapter 7. Mathematical model of atherosclerosis: Risk of high cholesterol 51 Chapter 8. Mathematical analysis inspired by the atherosclerosis model 63 Chapter 9. Mathematical models of chronic wounds 71 Chapter 10. Mathematical analysis inspired by the chronic wound model 79 Appendix: Introduction to PDEs 85 A.1. Elliptic equations 85 A.2. Parabolic equations 88 A.3. Nonlinear equations and systems 89 A.4. Free boundary problems 91 Bibliography 95 Index 99 v Preface Mathematical biology is a fast growing field which is concerned with problems that arise in biology. The aim is to address biological questions using mathemat- ics. The mathematical models that are used to address these questions depend on the specific biological context. They include dynamical systems, probability, statistics, and discrete mathematics. The approach in addressing a biological ques- tion is to develop a mathematical model that represents the biological background needed in order to address the question, to show that simulations of the model are in agreement with known biological facts, and, finally, to provide a solution to the original question. This approach to mathematical biology was carried out in two recent books: Introduction to Mathematical Biology, by C.-S. Chou and A. Friedman (Springer, 2016) and Mathematical Modeling of Biological Processes by A. Friedman and C.-Y. Kao (Springer, 2014). Both of these books were based on a one-semester course (the first one for undergraduate students and the second for master’s students) taught over several years at The Ohio State University in Columbus, Ohio. Each of the books included MATLAB simulations and exercises. The present monograph considers biological processes that are described by systems of partial differential equations (PDEs). It focuses on modeling such processes, not on numerical methods and simulations. On the other hand it also includes results in mathematical analysis of the mathematical models, or of their simplified versions, as well as many open problems. The monograph is addressed primarily to students and researchers in the math- ematical sciences who do not necessarily have any background in biology, and who may have had little exposure to PDEs. We have included in an Appendix a “short course” in PDEs in order to familiarize the reader with the mathematical aspects of the models that appear in the book. The first chapter introduces the basic biology that will be used in the book. The second chapter introduces the basic blocks in building models, for example how to express the fact that a ligand activates an immune cell. The third chapter gives several simple examples of models on pop- ulation dynamics. The fourth chapter develops two models of cancer. The choice of parameters in the cancer models, as in all other PDE models, is critically im- portant if the models are to have a predictive value. In Chapter 5, we illustrate how to estimate the parameters of the first cancer models of Chapter 4 using both experimental data and some “reasonable” assumptions. Chapter 6 describes mathematical results inspired by cancer models, includ- ing stability of spherical tumors and symmetry-breaking bifurcations, and it also suggests many open problems. Chapter 7 addresses the question of the risk of atherosclerosis associated with cholesterol levels. The model develops a system of PDEs that describe the growth of a plaque in the artery. Chapter 8 describes mathematical results and open problems vii viii PREFACE for a simplified model of plaque growth. Chapters 9 and 10 follow the format of Chapters 7 and 8: Chapter 9 develops a model of wound healing, and Chapter 10 describes mathematical results and open problems associated with this model. Almost all the PDE models introduced in this book are free boundary problems, that is, the domain where each PDE system holds is unknown in advance, and its boundary has to be determined together with the solution to the PDE system. This book was written for the 2018 NSF-CBMS conference on “Mathematical Biology: Modeling and Analysis”, hosted by Howard University in Washington, DC, during May 21–25, 2018. It is our hope that this monograph will demonstrate to the reader the challenges, the excitement, and the opportunities for research at the interface of mathematics and biology. It is finally my pleasure to express my thanks and appreciation to Dr. Xiulan Lai for typing the manuscript and drawing all the figures. Avner Friedman Bibliography [1] H M Byrne, M A J Chaplain, D L Evans, and I Hopkinson, Mathematical modelling of angiogenesis in wound healing: comparison of theory and experiment, Computational and Mathematical Methods in Medicine 2 (2000), no. 3, 175–197. [2] Duan Chen, Julie M. Roda, Clay B. Marsh, Timothy D. Eubank, and Avner Friedman, Hypoxia inducible factors-mediated inhibition of cancer by GM-CSF: a mathematical model, Bull. Math. Biol. 74 (2012), no. 11, 2752–2777. MR2994755 [3] Xinfu Chen, Shangbin Cui, and Avner Friedman, A hyperbolic free boundary problem model- ing tumor growth: asymptotic behavior, Trans. Amer. Math. Soc. 357 (2005), no. 12, 4771– 4804, DOI 10.1090/S0002-9947-05-03784-0.