### The SIR Model for Spread of Disease

Part 5. The Contact Number

In Part 4 we took it for granted that the parameters  b  and  k  could be estimated somehow, and therefore it would be possible to generate numerical solutions of the differential equations. In fact, as we have seen, the fraction  k  of infecteds recovering in a given day can be estimated from observation of infected individuals. Specifically,  k  is roughly the reciprocal of the number of days an individual is sick enough to infect others. For many contagious diseases, the infectious time is approximately the same for most infecteds and is known by observation.

There is no direct way to observe  b,  but there is an indirect way. Consider the ratio of  b  to  k:

 b/k = b x 1/k = the number of close contacts per day per infected   x the number of days infected = the number of close contacts per infected individual.

We call this ratio the contact number, and we write  c = b/k.  The contact number  c  is a combined characteristic of the population and of the disease. In similar populations, it measures the relative contagiousness of the disease, because it tells us indirectly how many of the contacts are close enough to actually spread the disease. We now use calculus to show that  c  can be estimated after the epidemic has run its course. Then  b  can be calculated as the product of  c k.

Here again are our differential equations for  s  and  i:

We observe about these two equations that the most complicated term in both would cancel and leave something simpler if we were to divide the second equation by the first -- provided we can figure out what it means to divide the derivatives on the left.

1. Use the Chain Rule to explain why

2. Now show that

The differential equation in step 2 determines (except for dependence on an initial condition) the infected fraction  i  as a function of the susceptible fraction  s.  We will use solutions of this differential equation for two special initial conditions to describe a method for determining the contact number.

There are several points we need to note about this differential equation.

• The equation is independent of time. In other words, no matter when s achieves a certain value, it makes sense to discuss the rate of change of i with respect to s, and this rate of change is always given by -1 + 1/(cs)

• In this equation, we are considering s to be an independent variable. Moreover, the right-hand side does not involve the dependent variable i. So, we will be able to find the solutions by integrating with respect to s.

• Note that c is the only parameter in the differential equation. It is this fact that we will exploit to obtain a method for determining c.

1. Show that  i(s)  must have the form

where q is a constant.

2. Explain why the quantity

must be independent of time.

There are two times when we know (or can estimate) the values of   and   -- at  t = 0  and  t = infinity.  For a disease such as the Hong Kong flu,  i(0)  is approximately  0  and  s(0)  is approximately  1.  A long time after the onset of the epidemic, we have  i(infinity)  approximately   again, and  s(infinity)  has settled to its steady state value. If there has been good reporting of the numbers of individuals who have contracted the disease, then the steady state is observable as the fraction of the population that did not get the disease. (Look again at the graph of s(t) and note how it levels off to a constant value. That constant value is s(infinity).)

1. For such an epidemic, explain why

[Hint: Use the fact that the quantity in step 3 is the same at  t = 0  and at  t = infinity.]

2. Use one of your numerical solutions in Part 4 to estimate the value of  s(infinity).  Use this value to calculate the contact number  c  for the Hong Kong flu. Compare your calculated value with the one you get by direct calculation from the definition,  c = b/k.

Note that a value for b was used in the determination of the graph of s(t). Thus, what we are doing is making a consistency check on our work. We need an independent determination of s(infinity) to obtain an independent determination of c and, hence, of b.

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