1. Predicting the Future To Predict the Future, “all we have to have is a knowledge of how things are and an understanding of the rules that govern the changes that will occur. From Calculus we know that change is measured by the derivative, and using the derivative to describe how a quantity changes is what the subject of Differential Equations is all about.” ---From Blanchard, Devaney, Hall, 2E

2. The Course of an Epidemic Consider a disease such as the measles: People get sick, recover, and become immune in the process. What sorts of things might we want to know about a measles epidemic? How many people are sick? How many people are going to get sick? How long will the epidemic last? How many sick people will there be at any given time? If we can describe the way that people get sick and recover, we can answer these questions.

3. The Measles What are some basic features of a Measles epidemic? There are sick people and there are well people The well people are either immune or susceptible to the disease. So there are three mutually exclusive populations: Susceptible Infected Recovered At a given time t, let S(t) be the number of susceptible people, I(t) be the number of infected people, and R(t) be the number of recovered people.

4. The Measles S I R We can picture the epidemic using a “compartment model.” Individuals in the population move from one compartment to another as shown in the diagram. But how does this happen? And how do we describe the process mathematically? Assumption: The population is fixed in size. So S + I +R is a constant.

5. How do people get sick? Interactions! SusceptibleInfected

6. Interactions The number of possible interactions is SI. Of course only a small portion a of these lead to an infection---a S I.

7. The SIR Model People getting well also affects the size of the infected population. How might we model this?

8. The SIR Model The numbers a and b are called “parameters.” Parameters are fixed values that depend on the particular situation being modeled. For instance, we will have one set of values for measles and another for whooping cough. Assumption: S+I+R is a constant also implies no one dies from the disease.

9. What will the graphs look like? S I R