Exponential Growth: Difference Equations

Let's start with a very simple example of disease spread. This example is somewhat artificial, and we'll be examining more realistic and detailed models in further topics. We'll use this simple model, however, to define and illustrate the fundamental concepts of mathematical models.

Suppose that at the beginning, a single index case is introduced into a community, and that the person will be infected for one week and then recover. At the end of the week, the person will infect, let's say, 2 new people. Each of the new people will then be infected for a week and each of them will infect two new people. What happens?

At the outset, you have only the index case; we'll say that at time zero, the number of infectives is 1, and we'll write this I(0)=1.

Now after one week has gone by, the original case recovers, but the index case has infected two new people, so there are two infectives. These are the two secondary cases caused by the original case, so now, at week 1, there are two infectives, and so I(1)=2.

What about after another week? By then, the two secondary cases have recovered too, but each of them has infected two more people. So there must now be 2 × 2 = 4 cases: I(2)=4.

Now, let's go back to the beginning again. We're still going to say that everyone is infective for one week, and each infective during that week infects two new people. But now, suppose there were three infective people introduced into the community at the beginning. We would then write I(0)=3. After one week, there are I(1)=6 infectives, after two weeks, there are I(2)=12 infectives, and so on.

The pattern is certainly clear: every week, there would be twice as many cases as the previous week.

Let's backtrack one more time and try to summarize all this. We could write the model this way: I(t+1)=2 × I(t) (for t=0, 1, 2, and so on). If you know the number of infectives at time t, which we're calling I(t), then using this equation, you can calculate I(t+1), the number of infectives at time t+1, one time unit later (remember, here a time unit is one week). For this very simple epidemic, the equation
I(t+1)=2 × I(t)
represents the dynamics. If you know the starting value I(0) (that is, I(t) at t=0), you can calculate the next value I(1), and using I(1), you can calculate I(2), and so on. The equation I(t+1)=2 × I(t) is called a difference equation; notice that although it expresses the entire epidemic as far out in time as you want to go, you can't calculate anything with it unless you know the value at the beginning, I(0). The value at the beginning, I(0), is called the initial condition of the difference equationI(t+1)=2 × I(t).

Now, let's come up with a formula for the number of infectives at any time. We know that I(t+1)=2 × I(t) is true for any time t that is 0, 1, 2, and so on. So it must be true that I(1)=2 × I(0). It is also true that I(2)= 2 × I(1). We can substitute and get I(2)=2 × 2 × I(0). Let's do it again: we know I(3)= 2 × I(2), so therefore I(3)=2 × 2 × 2 × I(0) = 2³ × I(0). The pattern is clear: I(t) = 2^t × I(0). This formula tells us the number of infectives at any time. Notice two things: it contains the initial condition I(0), and that we computed it using the difference equation; it is called the solution of the difference equation.

So now we have a difference equation I(t+1) = 2 × I(t) that represents the epidemiology (such as it is!) of this simple system; the difference equation tells us how the system gets from here to there so to speak. We also have the initial condition I(0) that tells us where we start. And we also have the formula I(t) = 2^t × I(0) for the actual number at any time t. So we have a difference equation, its initial condition, and its solution. The pattern is that we use knowledge about the process that takes place to write the difference equation. Together with an appropriate initial condition, we then calculate the solution of the difference equation, which tells us how the epidemic progresses.

Why is I(t) = 2^t × I(0) the solution of the difference equation? Because when you substitute it back in to the difference equation, you get an identity (something that is always true). Here's how it works. Remember, the difference equation says that I(t+1)=2 × I(t). The solution says I(t+1) has to be 2^t+1. The difference equation says this has to equal 2 × I(t), and the formula gives us I(t)=2^t. So now on the right hand side of the difference equation we have 2 × 2^t, and the difference equation winds up insisting that 2^t+1 has to be equal to 2 × 2^t. But this is true, for sure; it is an identity!

Why all this trouble with difference equations and so on? Why not just write down the solution and be done with it? It turns out that in many situations, writing a difference equation (or some other kind of dynamical equation) to tell us how the system we're studying changes over some time interval quantifies the process in a convenient or natural way. We can then solve the difference equation to get the solution (if we know the initial condition). And it may not be obvious, at all, what the solution ought to be!

Let's back up yet again. We've been assuming each infective causes two new infections at the end of the week. But what if this number is not two, but something else? What if it is 3? Or 4? Let's say that the number of infections at the end of the week is a. We could write the difference equation I(t+1)=a × I(t). Before, we were assuming a=2, but now we let it take any integer value. The solution is I(t)=a^tI(0); this tells us how many infectives at any time t. Here, we've represented the same epidemic process, but in a more general form; we can assume any number of new infections, and not just 2. We say that the value a is one of the parameters of the model.

So this very simple example illustrates the fundamental vocabulary of modeling: dynamical equations (in particular, difference equations), the initial condition, the solution of the dynamical equations, and the parameters that appear in them.

We'll next look at some simple examples.