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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 equation*I(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) = 2 ^{3}
× I(0)*. The pattern is clear:

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

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

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 ^{t}I(0)*; this tells us how
many infectives at any time

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.

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