Differential Equations: The Epic Journey: Let's talk about diseases: 8.1

Chapter 8 is entitled “An Introduction to Systems.”

Section 8.1 is entitled “Definitions and Examples.”

I’m going to make the observation here and now that any [chapter].1 section will be entirely dedicated to exposition for the succeeding sections.

In other words, [chapter].1 summaries will be particularly dull. I try my best (considering the people who pretend to read/care/enjoy this blog are not emotionally attached to the blog or the class whatsoever, besides my teacher, of course) but for the most part, I don’t have a lot of math jokes/puns to throw at you. Instead, I have math numbers and math words.

Anyway, ready for chapter 8? And the five sections the weekly schedule says we’re covering?

Let’s start off this summary with talking about sick people.

Let’s say there’s a population of people, N. There’s also there’s this disease that’s going around. We have three assumptions about this disease:

1. The disease lasts a short amount of time and doesn’t kill people (not often, anyway).

2. The disease spreads through physical contact.

3. After someone recovers from the disease, he/she will be immune.

Knowing these things, we can split our population N into three groups: people who have never had the disease, people who currently have the disease, and people who have recovered from the disease. We’ll call the first group the susceptible, S(t); the second group will be called the infected, I(t); the third group will be called the recovered, R(t). This means N = S + I + R.

Our first assumption means we can ignore births and deaths. This is nice because N is then constant. Let’s also think about how S(t) will change; susceptible people will catch the disease from infected people and then change groups. This means the rate of change for these two groups is proportional to the number of contacts made. The number of contacts will then be proportional to the product SI of the two populations. This will take the form

In this case, a is a positive constant.

Thinking about how I(t) will change, we think about two different ways. Susceptible people get sick, and infected people get better. We figured out the “susceptible people get sick” part, with our product above. With the “infected people get better part,” this means there will be a rate of recoveries (with some positive constant b). Thus the form will be

Thinking about how R(t) will change, infected people will get better. Thus we already have the form for the rate of change of R:

If we want to wrap everything up in a nice bow, we’ll have

This is called the SIR model. Notice it is nonlinear and autonomous.

The last equation isn’t really needed. The first two equations form a planar and autonomous system. We’ll get the system

This is also referred to as the SIR model.

Notice the SIR model only involves first-order derivatives of the system. It is called a first-order system. Thus we can conclude the order of a system is determined by the highest order derivative in that system. Generally, a first-order system of two equations will have the form

In this case, f and g are functions. A solution to this system would have the form

As always, this will be on some interval of t.

The number of equations should always be equal to the number of unknowns. This will be called the dimension of the system. Thus the system above has dimension 2, and a general system of n equations with n unknowns will have dimension n. A dimension 2 system will be called a planar system.

Notice that our first form of the SIR model is dimension 3, but the second form is a planar system.

If we have a general system

We can write this system in terms of vectors. We can set

One last thing for this section: “…there is a system of first-order equations that is equivalent to any system of higher-order equations, in the sense that a solution to one leads easily to a solution to the other” (335). This means that any application that involves higher-order equations will have an equivalent model with a first-order system. This is nice for the book because we can spend a lot more time studying first-order equations. This is also nice because numerical solvers (remember the two sections from chapter 6?) are more geared toward first-order equations.

Here’s an example of how you can find the first-order system of an equation: