Now, the nice thing about thinking of the continuous time

version of the quantitative model is that there's a very straight forward,

somewhat elegant formula that tells you exactly how much

your money is growing to over a time period t.

So, if your money is growing at a nominal annual interest rate of R%,

I'm using the letter capital R there, then it turns out that the amount of money

you've got at time t, Pt is just equal to P0 your principal times e,

that's the exponential function coming in there, to the power RT.

Now, note that's a little r there because I've taken the interest rate, capital R,

and turned it into an out of a hundred.

I've divide it through by a hundred.

And so, for example, if you're interest rate,

the nominal interest rate, was 4%, that little r would be 0.04.

So there is a very nice formula for continuous compounding.

So that's an alternative way of modelling a growth or decline

process rather than doing it in discreet time, we could do it in continuous time.

And we end up with a very neat formula

that interestingly involved the exponential function.

That was one of the reasons why I said in the introductory

module that it was one of the functions you needed to know.

It comes up naturally here.

I'm going to do a quick example with continuous compounding,

show you how you would do a calculation.

The important thing to note, though, with continuous compounding,

is that the value t now can actually take on any value.

Remember when we were talking about discrete,

it could only take on specific values, the end of each year or the end of each month.

Now that we're in continuous time, t can take on any value inside an interval.

So let's have a look what happened if we would continuously compound

a $1000 at a nominal annual interest rate of 4%.

After one year, make the calculation easy, we'll put to t over one year.

Then, what you're going to end up with is a 1000 times e to the power 0.04.

Again, you'd do a calculation like e to the power 0.04 on your calculator,

or using a spreadsheet.

It turns out that if you do that calculation,

you'll end up with $1,040.80 after one year, and

notice that that's a little bit different from the $1,040 if you just compounded

at a single point in time, at the end of the year, 4% of 1,000 gives you 40.

But if we continuously compound, then we end up with $1,040 and 80 cents.

So it's a little bit different, the end result of continuously

compounding rather than discreetly compounding.

And I talk about a nominal annual interest rate of 4% because, of course,

at the end of the year, if it was continuously compounded,

you earned a little bit more than 4%.

So 4% is just called nominal.

You earn 4.08% to be more precise.

So, that's the effective interests rate.

So there's a little bit about continuous compounding.

Now, I'm going to apply this exponential growth model, now,

back to the epidemic we were talking about.

Sure, I introduced the continuous compounding in a investment context but

this exponential models that they give rise to are much more general than

just talking about money.

And, at least in the early stages of an epidemic,

it's not unreasonable to think of a exponential model as a starting model.

So, let's consider modelling the epidemic with an exponential function.

So, when we have this exponential models, here I'm writing Pt = P0,

that's a starting amount or starting number of infections, starting number of

cases times e to the power rt, we call that exponential growth or decay.

And if the letter r, the number in practice,

is greater than zero, then it's a growth process, and

if it's less than zero, if r is negative, then it's a decay process.

So, these models can capture growth or decay, increasing or decreasing functions.