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2016 | Book

Introduction Read chapter Read first chapter

Introduction to Mathematical Biology

Modeling, Analysis, and Simulations

Authors: Ching Shan Chou, Avner Friedman

Publisher: Springer International Publishing

Book Series : Springer Undergraduate Texts in Mathematics and Technology

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book is based on a one semester course that the authors have been teaching for several years, and includes two sets of case studies. The first includes chemostat models, predator-prey interaction, competition among species, the spread of infectious diseases, and oscillations arising from bifurcations. In developing these topics, readers will also be introduced to the basic theory of ordinary differential equations, and how to work with MATLAB without having any prior programming experience.


The second set of case studies were adapted from recent and current research papers to the level of the students. Topics have been selected based on public health interest. This includes the risk of atherosclerosis associated with high cholesterol levels, cancer and immune interactions, cancer therapy, and tuberculosis. Readers will experience how mathematical models and their numerical simulations can provide explanations that guide biological and biomedical research.


Considered to be the undergraduate companion to the more advanced book "Mathematical Modeling of Biological Processes" (A. Friedman, C.-Y. Kao, Springer – 2014), this book is geared towards undergraduate students with little background in mathematics and no biological background.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
The progress in the biological sciences over the last several decades has been revolutionary, and it is reasonable to expect that this pace of progress, facilitated by huge advances in technology, will continue in the following decades. Mathematics has historically contributed to, as well as benefited from, progress in the natural sciences, and it can play the same role in the biological sciences. For this reason we believe that it is important to introduce students very early, already at the freshman or sophomore level, with just basic knowledge in Calculus, to the interdisciplinary field of mathematical biology. A typical case study in mathematical biology consists of several steps. The initial step is a description of a biological process which gives rise to several biological questions where mathematics could be helpful in providing answers. The second step is to develop a mathematical model that represents the relevant biological process. The next step is to use mathematical theories and computational methods in order to derive mathematical predictions from the model. The final step is to check that the mathematical predictions provide answers to the biological question. One can then further explore related biological questions by using the mathematical model.
Ching-Shan Chou, Avner Friedman
Chapter 2. Bacterial Growth in Chemostat
Abstract
Each chapter should be preceded by an abstract (10–15 lines long) that summarizes the content. The abstract will appear online at www.SpringerLink.com and be available with unrestricted access. This allows unregistered users to read the abstract as a teaser for the complete chapter. As a general rule the abstracts will not appear in the printed version of your book unless it is the style of your particular book or that of the series to which your book belongs. Please use the ’starred’ version of the new Springer abstract command for typesetting the text of the online abstracts (cf. source file of this chapter template abstract) and include them with the source files of your manuscript. Use the plain abstract command if the abstract is also to appear in the printed version of the book.
Ching-Shan Chou, Avner Friedman
Chapter 3. System of Two Linear Differential Equations
Abstract
In Chapter 5 we shall model the interaction between predator y and prey x by a system of two differential equations: the differential equation for x will involve the predator y and the differential equation for y will involve the prey x. The functions f(x, y) and g(x, y) will generally be nonlinear functions. We shall develop the theory in two stages: The first stage to be taken up in this chapter deals with the special case where f and g are linear functions, and the second stage, to be taken up in Chapter 4, will extend the theory to nonlinear functions f and g. Before we start, with a linear system of two equations, however, it will be instructive to consider one linear differential equations of the second order.
Ching-Shan Chou, Avner Friedman
Chapter 4. Systems of Two Differential Equations
Abstract
In Chapter 3, we considered linear differential system of the form (3.10). In this chapter we study general systems of two differential equations of the first order,
Ching-Shan Chou, Avner Friedman
Chapter 5. Predator–Prey Models
Abstract
A predator is an organism that eats another organism. A prey is an organism that a predator eats. In ecology, a predation is a biological interaction where a predator feeds on a prey. Predation occurs in a wide variety of scenarios, for instance in wild life interactions (lions hunting zebras, foxes hunting rabbits), in herbivore–plant interactions (cows grazing), and in parasite–host interactions.
Ching-Shan Chou, Avner Friedman
Chapter 6. Two Competing Populations
Abstract
Competition is an interaction between organisms, or species, sharing resources that are in limited supply. This is an important topic in ecology. The ‘competitive exclusion principle’ asserts that species less suited to compete will either adapt or die out. In aggressive competition one species may attempt to kill the other. This situation occurs, for example, among some species of ants, and some species or yeast. When enough data is known about the history of a specific competition between two species, mathematics can then be used to predict whether both species will survive and coexist or whether one of them will die out.
Ching-Shan Chou, Avner Friedman
Chapter 7. General Systems of Differential Equations
Abstract
In this chapter, we develop a theory for a system of differential equations that will be used to study models with many species.
Ching-Shan Chou, Avner Friedman
Chapter 8. The Chemostat Model Revisited
Abstract
In Chapter 2 we considered the chemostat model and used mathematics to answer the question: How should we choose the outflow rate in order to harvest the maximum amount of bacteria. Our model however was incomplete because we assumed that the nutrient concentration in the growth chamber is constant in time, and hence our answer is questionable. In the present chapter we want to correct the answer, by basing it on a more complete mathematical model of the chemostat.
Ching-Shan Chou, Avner Friedman
Chapter 9. Spread of Disease
Abstract
Epidemiology is the study of patterns, causes, and effects of health and disease conditions in a population. It provides critical support for public health by identifying risk factors for disease and targets for preventive medicine. Epidemiology has helped develop methodology used in clinical research and public health studies. Major areas of epidemiological study include disease etiology, disease break, disease surveillance, and comparison of treatment effects such as in clinical trials.
Ching-Shan Chou, Avner Friedman
Chapter 10. Enzyme Dynamics
Abstract
Cells The online version of this chapter contains supplementary material, which is available to authorized users. are the basic units of life. A cell consists of a concentrated aqueous solution of molecules contained in a membrane, called plasma membrane. A cell is capable of replicating itself by growing and dividing. Cells that have a nucleus are called eukaryotes, and cells that do not have a nucleus are called prokaryotes. Bacteria are prokaryotes, while yeast and amoebas, as well as most cells in our body, are eukaryotes. The Deoxyribonucleic acid (DNA) are very long polymeric molecules, consisting of two strands of chains, having double helix configuration, with repeated nucleotide units A, C, G, and T. The DNA is packed in chromosomes, within the nucleus in eukaryotes. In humans, the number of chromosomes is 46, except in sperm and egg cells where the number is 23.
Ching-Shan Chou, Avner Friedman
Chapter 11. Bifurcation Theory
Abstract
Consider two populations, x and y, that are interacting either by competition, or as predator and prey. They may end up near a stable steady state, or possibly in seasonally oscillating states; this could depend on their proliferation rates, death rates, available resources, climate change, etc. In this chapter we wish to explore these varied possibilities using mathematics. To do that we begin by a short introduction to the theory of bifurcations. Bifurcation theory is concerned with the question of how the behavior of a system which depends on a parameter p changes with the parameter. It focuses on any critical value, p = p cr , where the behavior of the system undergoes radical change; such values are called bifurcation points.
Ching-Shan Chou, Avner Friedman
Chapter 12. Atherosclerosis: The Risk of High Cholesterol
Abstract
Arteries are blood vessels that carry oxygen-rich blood to the heart, brain, and other parts of the body. Atherosclerosis is a disease in which a plaque, a thick hard deposit of fatty material, builds up inside arteries. The plaque contains cholesterol, calcium, cells from the blood, and cells from the arterial wall. Over time the plaque grows, hardens, and narrows the artery. This reduces the flow of oxygen-rich blood, and also make it more likely to cause a blood clot, or thrombus, that will block the blood flow. A blockage formed in the coronary arteries may trigger a heart attack. A blockage formed in the carotid artery (located on each side of the neck, feeding oxygen to the brain) may cause a stroke. Atherosclerosis is the leading cause of death in the United States and worldwide, with annual deaths of 900,000 in the United States and 13 millions worldwide.
Ching-Shan Chou, Avner Friedman
Chapter 13. Cancer-Immune Interaction
Abstract
An abnormally new growth of tissue with cells that grow more rapidly than normal cells and has no physiological function is called a neoplasm or a tumor. The abnormally rapidly growing cells compete with normal cells for space and nutrients. When the new growth is localized, it is called a benign tumor. When a tumor in tissue has reached a size of several millimeters it requires a large supply of nutrients, for otherwise it can no longer grow. Until reaching this stage the tumor is called avascular. Avascular tumors that reached the stage where they require new supply of nutrients try to induce the formation of new blood vessels (angiogenesis) and direct their movement toward them. They do so by secreting vascular endothelial growth factor (VEGF) and, if successful, the tumors become vascular. As a tumor continues to grow some of its cells may break away and travel to other parts of the body through the bloodstream or the lymph system. Metastatic cancer is a tumor that spread from the original location where it started to other parts of the body. Metastatic cancer is also called malignant cancer, or, briefly, cancer, although people often use the words tumor and cancer interchangeably. Most cancer deaths are due to metastasized cancer.
Ching-Shan Chou, Avner Friedman
Chapter 14. Cancer Therapy
Abstract
There are many drugs that are used in the treatment of cancer; some drugs kill cancer cells directly while others change the cancer microenvironment to make it resistant to cancer cells growth. In Chapter 13, we considered a drug, TGF-β inhibitor, which changes the macrophage phenotype, thereby enabling the immune system to kill cancer cells more effectively. In this chapter we consider two entirely different kinds of anti-cancer drugs. The first one blocks the activity of vascular endothelial growth factor (VEGF), and the second one uses virus to kill cancer cells.
Ching-Shan Chou, Avner Friedman
Chapter 15. Tuberculosis
Abstract
Tuberculosis (TB) is an infective disease caused by Mycobacterium tuberculosis (Mtb). The bacteria is spread through the air when people who have active TB infection cough or sneeze. The bacteria attack the lungs, primarily, but can also spread and attack other parts of the body. The most common symptom of active TB infection is chronic cough with blood-tinged sputum. It is estimated that one-third of the world’s population are infected with Mtb, although only 13 million chronic cases are active, and 1.5 million associated deaths occur. Treatment of TB uses antibiotics to kill the bacteria, but the treatment is not entirely effective. Vaccination in children decreases significantly the risk of infection.
Ching-Shan Chou, Avner Friedman
Backmatter
Metadata
Title
Introduction to Mathematical Biology
Authors
Ching Shan Chou
Avner Friedman
Copyright Year
2016
Electronic ISBN
978-3-319-29638-8
Print ISBN
978-3-319-29636-4
DOI
https://doi.org/10.1007/978-3-319-29638-8

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