Mathematical Models for Society and Biology

Edward Beltrami

• Publisher: Academic Press
• Year: 2013
• ISBN:
  9780124046931
• Edition: 2
• Topic: Social Sciences
• Subtopic: Sociology
• Index: Social Sciences

Mathematical Models for Society and Biology, 2e, is a useful resource for researchers, graduate students, and post-docs in the applied mathematics and life science fields. Mathematical modeling is one of the major subfields of mathematical biology. A mathematical model may be used to help explain a system, to study the effects of different components, and to make predictions about behavior.

Mathematical Models for Society and Biology, 2e, draws on current issues to engagingly relate how to use mathematics to gain insight into problems in biology and contemporary society. For this new edition, author Edward Beltrami uses mathematical models that are simple, transparent, and verifiable. Also new to this edition is an introduction to mathematical notions that every quantitative scientist in the biological and social sciences should know. Additionally, each chapter now includes a detailed discussion on how to formulate a reasonable model to gain insight into the specific question that has been introduced.

• Offers 40% more content – 5 new chapters in addition to revisions to existing chapters
• Accessible for quick self study as well as a resource for courses in molecular biology, biochemistry, embryology and cell biology, medicine, ecology and evolution, bio-mathematics, and applied math in general
• Features expanded appendices with an extensive list of references, solutions to selected exercises in the book, and further discussion of various mathematical methods introduced in the book

Chapter 1

Crabs and Criminals

1.1 Background

A hand reaches into the still waters of the shallow lagoon and gently places a shell on the sandy bottom. We watch. A little later a tiny hermit crab scurries out of a nearby shell and takes possession of the larger one just put in. This sets off a chain reaction in which another crab moves out of its old quarters and scuttles over to the now-empty shell of the previous owner. Other crabs do the same, until at last some barely habitable shell is abandoned by its occupant for a better shelter, and it remains unused.

One day the president of a corporation decides to retire. After much fanfare and maneuvering within the firm, one of the vice presidents is promoted to the top job. This leaves a vacancy, which, after a lapse of a few weeks, is filled by another executive, whose position is now occupied by someone else in the corporate hierarchy. Some months pass, and the title of the last position to be vacated is merged with some currently held job title and the chain terminates.

A lovely country home is offered by a real estate agency when the owner dies and his widow decides to move into an apartment. An upwardly mobile young professional buys it and moves his family out of the split-level they currently own after selling it to another couple of moderate income. That couple sold their modest house in a less-than-desirable neighborhood to an entrepreneurial fellow who decides to make some needed repairs and rent it.

What do these examples have in common? In each case a single vacancy leaves in its wake a chain of opportunities that affect several individuals. One vacancy begets another as individuals move up the social ladder. Implicit here is the assumption that individuals want or need a resource unit (shells, houses, or jobs) that is somehow better (newer, bigger, more status) or, at least, no worse than the one they already possess. There are a limited number of such resources and many applicants. As units trickle down from prestigious to commonplace, individuals move in the opposite direction to fill the opening created in the social hierarchy.

A chain begins when an individual dies or retires or when a housing unit is newly built or a job created. The assumption is that each resource unit is reusable when it becomes available and that the trail of vacancies comes to an end when a unit is merged, destroyed, or abandoned, or because some new individual enters the system from the outside. For example, a rickety shell is abandoned by its last resident, and no other crab in the lagoon claims it, or else a less fortunate hermit crab, one who does not currently have a shell to protect its fragile body, eagerly snatches the last shell.

A mathematical model of movement in a vacancy chain is formulated in the next section and is based on two notions common to all the examples given. The first notion is that the resource units belong to a finite number, usually small, of categories that we refer to as states; the second notion is that transitions take place among states whenever a vacancy is created. The crabs acquire protective shells formerly occupied by snails that have died, and these snail shells come in various size categories. These are the states. Similarly, houses belong to varying price/prestige categories, while jobs in a corporate structure can be labeled by different salary/prestige classes.

Let’s now consider an apparently different situation. A crime is committed, and, in police jargon, the perpetrator is apprehended and brought to justice and sentenced to “serve time” in jail. Some crimes go unsolved, however, and of the criminals that get arrested only a few go to prison; most go free on probation or because charges are dropped. Moreover, even if a felon is incarcerated or is released after arrest or even if he was never caught to begin with, it is quite possible that the same person will become a recidivist, that is, a repeat offender. What this has in common with the mobility examples given earlier is that here, too, there are transitions between states, where in this case “state” means the status of an offender as someone who has just committed a crime, has just been arrested, has just been jailed, or, finally, has “gone straight,” never to repeat a crime again. This, too, is a kind of social mobility, and we will see that it fits the same mathematical framework that applies to the other examples.

One of the problems associated with models of social mobility is the difficult chore of obtaining data regarding the frequency of moves between states. If price, for example, measures the state of housing, then what dollar bracket constitutes a single state? Obviously the narrower we make a price category, the more homogeneous is the housing stock that lies within a given grouping. On the other hand, this homogeneity requires a large number of states, which exacerbates the data-gathering effort necessary to estimate the statistics of moves between states.

We chose to tell the crab story because it is a recent and well-documented study that serves as a parable for larger-scale problems in sociology connected with housing and labor. It is not beset by some of the technical issues that crop up in these other areas, such as questions of race that complicate moves within the housing and labor markets. By drastically simplifying the criminal justice system, we are also able to address some significant questions about the chain of moves of career criminals that curiously parallel those of crabs on the sandy sea bottom. These examples are discussed in Sections 1.3 through 1.5.

More recent work on crab mobility shows that, in contrast to the solitary crab behavior discussed earlier in which a single individual searches for a larger shell before vacating its existing home, the terrestrial hermit crab Coenobita clypeatus engages in a more aggregate behavior, in which a cluster of crabs piggyback each other in order to move together as a group. The crabs grasp the shell of another denizen from behind, with the leader dragging itself along traile...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Dedication
• Preface to the Second Edition
• Chapter 1. Crabs and Criminals
• Chapter 2. It Isn’t Fair
• Chapter 3. While the City Burns
• Chapter 4. Clean Streets
• Chapter 5. A Bayesian Take on Colorectal Screening, Baseball, Fund Managers, and a Murder
• Chapter 6. What Are the Odds of That?
• Chapter 7. It’s Normal Not to Be Normal
• Chapter 8. Boom and Bust
• Chapter 9. Viral Outbreaks and Blood Clots
• Chapter 10. Red Tides and Whatever Happened to the Red Squirrel?
• Chapter 11. Spatial Patterns: The Cat’s Tail and Spreading Slime
• Chapter 12. The Coil of Life
• Afterthoughts on Modeling
• Appendix A. The Normal Density
• Appendix B. Poisson Events
• Appendix C. Nonlinear Differential Equations and Oscillations
• Appendix D. Conditional Probability
• References
• Index