 ## Applications of Systems of Differential Equations

### Theoretical Introduction

Note: Before attempting this laboratory, you should first complete the Predator-Prey Laboratory.

Let's consider modeling a new situation, namely, the way in which a disease spreads through a population. The population could be made up of any life form, but let's keep things personal, and say that we're working with a human population. Also, let's talk about a rather familiar disease—influenza.

#### Choosing Variables

Before we can go very far in creating our model, we need to choose some variables to represent the quantities of interest. We'll need to talk about how many people have caught influenza already, how many people haven't caught it yet, and how many people have already recovered from the illness. Of course, we can think of each of these quantities as being a function of time. The following list of variable names should cover what we need:

• S(t) represents the number of people who have not yet caught the disease. The letter S is used because these people are susceptible to the illness.

• I(t) represents the number of people who currently have the disease. The letter I is used because these people are infected by the illness.

• R(t) represents the number of people who have recovered from the illness. In the model we are creating, we consider patients that have already had the disease as having developed an immunity to re-infection. This means that recovery from the illness does not return you to the susceptible group. (Note: the model would work in just the same way if the patients actually died from the illness. After all, once they die they are no longer susceptible or infected. Because of this fact, when some researchers work with this model they say that the R represents people that have been removed from the disease-interacting population, either by death or recovery. But let's be positive—they recovered!)

• t represents time.

#### Finding the Rates of Change

Now let's think about how these sub-groups interact with one another. We'll consider the rates of change, (derivatives,) of each of our functions in turn.

S′(t): the rate of change of the susceptible population. In the context of the spread of disease, this group is reduced by one person each time one of them becomes infected. So, the rate at which S is changing is the same as the rate at which infection is taking place. The underlying question, then, is "When does infection take place?"

Infection occurs when the disease is passed from an infected person to a susceptible person. This occurs during some of the times when susceptibles come in contact with infecteds. Now the number of susceptible-infected contacts will be proportional to the product of S and I, and of these contacts a proportion will catch the disease. It makes sense then to come up with the following formula for the rate of change of the susceptible population:

S′ = - a S I

where a is some positive constant of proportionality which takes into account the total number of contacts between the two groups, and how many of these contacts actually lead to infection. The negative sign is present because when infection occurs, the susceptible population is reduced.

I′(t): the rate of change of the infected population. We have already discussed why this population grows. They are constantly recruiting members from the susceptible group. We can even name the rate at which growth occurs due to this transformation—it's a S I. After all, the rate at which people are leaving the susceptible population is exactly the same as the rate at which they are joining the infected population. That's where they are going!

But we have another factor to consider! There are also people recovering from the disease. It seems reasonable to assume that the number of people moving from the infected group to the recovered group is some kind of fixed proportion of the infected groups size, e.g. we can see the sense in a statement such as "On any given day, an average of 15% of our influenza patients enter recovery and are released from our care." So, the rate at which the infected group is reduced by recoveries should be a quantity that looks like -b I, where b is the constant of proportionality representing the recovery rate. (For the example above, b = 0.15.)

Putting our infection and recovery rates together, we get a quantity representing the rate of change of the infected population:

I′ = a S I - b I

Whether the infected group is growing or shrinking in size would be determined by the relative sizes of these two terms.

R′(t): the rate of change of the recovered population. We already know exactly what rate this group is growing at. Their members come exclusively from the recoveries we mentioned above, at a rate of b I. The formula for the rate of change of the recovered population is therefore:

R′ =  b I

Note that this quantity is positive, since the recovered group is growing all of the time as people overcome the illness.

#### Putting it All Together

So far, our three rates of change have led to three differential equations. If we throw in three initial conditions to keep them company we should have a nicely formed initial value problem.

The initial conditions are obvious. We'll use the sizes of the three groups at whatever time we're choosing to call time zero—the beginning of our analysis. Let's call these sizes So, Io, and Ro respectively. The initial value problem becomes:

System Initial Conditions
S′ = - a S I
I′ = a S I - b I
R′ =  b I
S(0) = So
I(0) = Io
R(0) = Ro

Make sure you understand why each part of the model makes sense. It's now time to begin our computer analysis, so let's move on and see how we might implement these ideas in Mathematica. If you're lost, impatient, want an overview of this laboratory assignment, or maybe even all three, you can click on the compass button on the left to go to the table of contents for this laboratory assignment.

ODE Laboratories: A Sabbatical Project by Christopher A. Barker

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