AML 100: Introduction to Applied Mathematics for the Life and Social Sciences

[This course meets general studies MA requirement]

Fall 2009

Professor: Marco Janssen

Class number: 86536

Tuesday and Thursday: 9.00-10.15. Physical Sciences A 106

Course content
Throughout the history of civilizations, mathematics has been developed both to solve practical problems and for its aesthetic beauty. In modern times, many of puzzles in the life and social sciences can be addressed with the help of mathematics. In this course you will learn what mathematical tools are applied in the life and social sciences, what the basic principles are that underlie these mathematical methods, and how you can apply them to basic problems.
A model is a simplified representation of reality, and mathematical models are used to find solutions for practical problems. Students will learn the basic steps in developing a model, analyze it and to test it on actual data.
This course will provide a broad overview of different methods that applied mathematicians use to develop models and how they apply them to practical problems. For each method, basic principles are provided and you will learn the first steps concerning how to represent these methods using formal mathematical techniques including developing equation-based relationships, plotting graphs, and, solving equations using computer software.
 

Course format
Lectures,  computer lab, home work assignments,

Required book

Elizabeth S. Allman and John A. Rhodes (2006) Mathematical Models in Biology: An Introduction, Cambridge University Press

Software

We will use Maple 12 which is available via ASU MyApps.

Schedule:

1. Overview of course (August 25 & 27)

What can you expect in this course? Discussion of the syllabus and overview of the course. What are the expected competencies? What are the expectations with regard to homework and exams? Some practical examples will be discussed how mathematics has helped to solve important problems in life and social sciences..

2. The history of mathematics (September 1 & 3)

In this week we will discuss important discoveries in the history of mathematics and we will show that important mathematical innovations are the result of solving concrete problems. We will discuss examples from Mesopotamia, Ancient Greece, Egypt and China as well as more recent examples like the work of Isaac Newton who showed that the motion of objects on Earth and of celestial bodies can be explained by the same mathematical model.

3. Organization of Data (September 8 & 10)

When we use mathematics in the life and social sciences we typically use data to define the problem we would like to solve. What are the different types of data? How does one effectively describe data by tables and graphs? How does one calculate the average and variance of data?

Required readings: Chapter 2 of Johnson and Bhattacharyya (2006)

4. Graphical analysis of Data (September 15 & 17)

This week we will look at the different ways to graphically look at data, and discuss time plots, box plots, histograms, scatter plots, bar charts, etc.

Required readings: Chapter 2 of Johnson and Bhattacharyya (2006) (Reading material)

5. How to construct a model (September 22 & 24)

What is a mathematical model? What is a system? What are variables and parameters? How do we represent phenomena in life and social sciences in a mathematical way? We will discuss a number of examples of models in different application areas of the life and social sciences.

Required readings: Chapter 1 and 2 of Otto and Day (2007)

6. Difference equations (September 29 & October 1)

Systems are fundamentally defined by how they change over time. How do we represent change of a system? We introduce difference equations to define how state variables change between one time period and the next.. We illustrate this with a number of examples including population dynamics, the build up of CO2 in the atmosphere, and the changes of balances in bank accounts.

Required readings: Section 1.1 and 1.2 of Allman and Rhodes (2006)

7. Analyzing the Dynamics of Models (September 6 & October 8)

We continue with difference equation models, and illustrate some methods to analyze the dynamics graphically over time.

Required Readings Section 1.3-1.5 of Allman and Rhodes (2006)

8. Linear models and Matrices (October 13 & 15)

The matrix is an important tool in applied mathematics. We will discuss this week the basics of matrices and matrix algebra.

Required Readings: Section 2.1 of Allman and Rhodes (2006)

9. Projection Matrices (October 20 & 22)

This week we will discuss how matrices can be used to study populations that include subgroups, like freshman, sophomore, juniors, and seniors in a student population. Using such linear models, we can study the change in the composition of these groups over time. We can apply this technique to models forest growth, disease spreading, and demographics.

Required Readings: Section 2.2 of Allman and Rhodes (2006)

10. Models of Predators and Prey (October 27 & 29)

So far we have looked at single populations, but populations of different species interact, whether these are biological species or different social groups. Predator-prey models are general set of models that capture interactions between different populations.

Required Readings: Sections 3.1, 3.2, and 3.4 of Allman and Rhodes (2006)

11. Models of Infectious Diseases (November 3 & 5)

An important application of mathematics is the study of infectious diseases. How do different diseases – from the flu to AIDS – spread through a population, and can we find insights from mathematical models that may help us to eradicate these diseases?

Required Readings: Sections 7.1, 7.2, and 7.3 of Allman and Rhodes (2006)

12. Probability (November 10 & 12)

What is your chance of winning the lottery? Or getting infected by influenza? How do we characterize such phenomena? We introduce the concept of probability and discuss a number of applications in the life and social sciences.

Required Readings: Sections 4.2 and 4.3 of Allman and Rhodes (2006)

13. Fitting Models to Data (November 17 & 19)

How do we fit models to data? Suppose you have developed a mathematical model and want to test it on empirical data. How to change the parameters to get the best fit of the model to the data. An introduction to the methods of least squares is discussed..

Required reading: Chapter 8 of Allman and Rhodes (2006).

14 Networks (December 1 & 3)

There is an increasing use of networks to study biological and social systems. What are networks? How do we represent them graphically and using equations? What are the different types of networks? We will discuss examples on how networks are used to study the spread of diseases and the flow of information in the Blogosphere.