This module is designed for use in a firstsemester differential calculus course to stimulate interest in the derivative as a tool for modeling rate of change.
About this Module and its Authors
Click on the button corresponding to your preferred computer algebra system (CAS). This will download a file which you may open with your CAS.
Ver. 6 or higher 
Ver. 3.0 or higher 
Ver. 5.1 or higher 
Published December, 2001
This module is designed for use in a firstsemester differential calculus course to stimulate interest in the derivative as a tool for modeling rate of change.
About this Module and its Authors
Click on the button corresponding to your preferred computer algebra system (CAS). This will download a file which you may open with your CAS.
Ver. 6 or higher 
Ver. 3.0 or higher 
Ver. 5.1 or higher 
Published December, 2001
During the winter of 19681969, the United States was swept by a virulent new strain of influenza, named Hong Kong flu for its place of discovery. At that time, no flu vaccine was available, so many more people were infected than would be the case today. We will study the spread of the disease through a single urban population, that of New York City. The data displayed in the following table are weekly totals of "excess" pneumoniainfluenza deaths, that is, the numbers of such deaths in excess of the average numbers to be expected from other sources. The graph displays the same data. (Source: Centers for Disease Control. Click here for a recent CDC view of influenza pandemics.)

Relatively few flu sufferers die from the disease or its complications, even without a vaccine. However, we may reasonably assume that the number of excess deaths in a week was proportional to the number of new cases of flu in some earlier week, say, three weeks earlier. Thus the figures in the table reflect (proportionally) the rise and subsequent decline in the number of new cases of Hong Kong flu. We will model the spread of such a disease so that we can predict what might happen with similar epidemics in the future.
At any given time during a flu epidemic, we want to know the number of people who are infected. We also want to know the number who have been infected and have recovered, because these people now have an immunity to the disease. (As a matter of convenience, we include in the recovered group the relative handful who do not recover but die  they too can no longer contract the disease.) The remainder of the population is still susceptible to the disease. (We assume that the same people are in New York during the weeks of the epidemic  that is, we ignore the relatively few who are born or die from nonflu causes or move into or out of the city during those weeks.)
Thus, at any time, the fixed total population (approximately 7,900,000 in the case of New York City in the late 1960's) may be divided into three distinct groups:
In the next part, we will investigate a simple model that accounts for a few of the features of the spread of a disease such as the Hong Kong Flu.
As the first step in the modeling process, we identify the independent and dependent variables. The independent variable is time t, measured in days. We consider two related sets of dependent variables.
The first set of dependent variables counts people in each of the groups, each as a function of time:
S = S(t)  is the number of susceptible individuals, 
I = I(t)  is the number of infected individuals, and 
R = R(t)  is the number of recovered individuals. 
The second set of dependent variables represents the fraction of the total population in each of the three categories. So, if N is the total population (7,900,000 in our example), we have
s(t) = S(t)/N,  the susceptible fraction of the population, 
i(t) = I(t)/N,  the infected fraction of the population, and 
r(t) = R(t)/N,  the recovered fraction of the population. 
It may seem more natural to work with population counts, but some of our calculations will be simpler if we use the fractions instead. The two sets of dependent variables are proportional to each other, so either set will give us the same information about the progress of the epidemic.
Next we make some assumptions about the rates of change of our dependent variables:
No one is added to the susceptible group, since we are ignoring births and immigration. The only way an individual leaves the susceptible group is by becoming infected. We assume that the timerate of change of S(t), the number of susceptibles,^{1} depends on the number already susceptible, the number of individuals already infected, and the amount of contact between susceptibles and infecteds. In particular, suppose that each infected individual has a fixed number b of contacts per day that are sufficient to spread the disease. Not all these contacts are with susceptible individuals. If we assume a homogeneous mixing of the population, the fraction of these contacts that are with susceptibles is s(t). Thus, on average, each infected individual generates b s(t) new infected individuals per day. [With a large susceptible population and a relatively small infected population, we can ignore tricky counting situations such as a single susceptible encountering more than one infected in a given day.]
Let's see what these assumptions tell us about derivatives of our dependent variables.
(1) 
(2) 
(3) 
(4) 
(5) 
Finally, we complete our model by giving each differential equation an initial condition. For this particular virus  Hong Kong flu in New York City in the late 1960's  hardly anyone was immune at the beginning of the epidemic, so almost everyone was susceptible. We will assume that there was a trace level of infection in the population, say, 10 people.^{2} Thus, our initial values for the population variables are
S(0) = 7,900,000 
I(0) = 10 
R(0) = 0 
In terms of the scaled variables, these initial conditions are
s(0) = 1 
i(0) = 1.27 x 10^{ 6} 
r(0) = 0 
(Note: The sum of our starting populations is not exactly N, nor is the sum of our fractions exactly 1. The trace level of infection is so small that this won't make any difference.) Our complete model is
(6) 
We don't know values for the parameters b and k yet, but we can estimate them, and then adjust them as necessary to fit the excess death data. We have already estimated the average period of infectiousness at three days, so that would suggest k = 1/3. If we guess that each infected would make a possibly infecting contact every two days, then b would be 1/2. We emphasize that this is just a guess. The following plot shows the solution curves for these choices of b and k.
In Part 3, we will see how solution curves can be computed even without formulas for the solution functions.
^{1} Note that we have turned the adjective "susceptible" into a noun. It is common usage in epidemiology to refer to "susceptibles," "infecteds," and "recovereds" rather than always use longer phrases such as "population of susceptible people" or even "the susceptible group."
^{2} While I(0) is normally small relative to N, we must have I(0) > 0 for an epidemic to develop. Equation (5) says, quite reasonably, that if I = 0 at time 0 (or any time), then dI/dt = 0 as well, and there can never be any increase from the 0 level of infection.
In Part 3, we displayed solutions of an SIR model without any hint of solution formulas. This suggests the use of a numerical solution method, such as Euler's Method, which we assume you have seen in the context of a single differential equation. Nevertheless, we review the basic idea here.
Recall the idea of Euler's Method: If we have a "slope formula," i.e., a way to calculate dy/dt at any point (t,y), then we can generate a sequence of yvalues,
y_{0}, y_{1}, y_{2}, y_{3}, ...
by starting from a given y_{0} and computing each rise as slope x run. That is,
y_{n} = y_{n1} + slope_{n1} Delta_t
where Delta_t is a suitably small step size in the time domain.
It really doesn't matter in this calculation if the slope formula happens to depend not just on t and y but on other variables, say x and z  as long as we know how x and z are related to t and y. If x and z happen to be other dependent variables in a system of differential equations, we can generate values of x and z in the same way.
Of course, for the SIR model, we want the dependent variable names to be s, i, and r. Thus we have three Euler formulas of the form
s_{n} = s_{n1} + sslope_{n1} Delta_t,
i_{n} = i_{n1} + islope_{n1} Delta_t,
r_{n} = R_{n1 }+ rslope_{n1} Delta_t,
More specifically, given the SIR equations,
the Euler formulas become
Of course, to calculate something from these formulas, we must have explicit values for b, k, s(0),
i(0), r(0), and Delta_t. In this part we explore the adequacy of these formulas for generating solutions of the SIR model. If your helper application has Euler's Method as an option, we will use that rather than construct the formulas from scratch.
The infectious period for Hong Kong Flu is known to average about three days, so our estimate of k = 1/3 is probably not far off. However, our estimate of b was nothing but a guess. Furthermore, a good estimate of the "mixing rate" of the population would surely depend on many characteristics of the population, such as density. In this part, we will experiment with the effects of these parameters on the solutions, and then try to find values that are in agreement with the excess deaths data from New York City. We focus our experimentation on the infectedfraction, i(t), since that function tells us about the progress of the epidemic.
In Part 5 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 

= 
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 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.
The differential equation in step 1 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.
Three features of this new differential equation are particularly worth noting:
There are two times when we know (or can estimate) the values of i and s  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 0 again, and s(infinity) has settled to its steady state value. If there has been good reporting of the numbers who have contracted the disease, then the steady state is observable as the fraction of the population that did not get the disease.
Each strain of flu is a disease that confers future immunity on its sufferers. For such a disease, if almost everyone has had it, then those who have not had it are protected from getting it  there are not enough susceptibles left in the population to allow an epidemic to get under way. This group protection is called herd immunity.
In Part 4 you experimented with the relative sizes of b and k, and you found that, if b is small enough relative to k, then no epidemic can develop. In the language of Part 5, if the contact number c = b/k is small enough, then there will be no epidemic. But another way to prevent an epidemic is to reduce the initial susceptible population artificially by inoculation.
The point of inoculation is to create herd immunity by stimulating in as many people as possible the antibodies that confer immunity  but without actually giving those people the disease. Thus inoculation creates a direct path from the susceptible group to the recovered group without passing through the infected group (see the diagram below). And a largescale inoculation program to head off an impending epidemic does this rapidly enough to lower the initial susceptible population to a safe level  safe enough that if a trace level of infection enters the population, a few people may get sick, but no epidemic will develop.
So, what fraction of the population must be inoculated to obtain herd immunity? Or, put another way, how small must s_{0} be to insure that an epidemic cannot get started? It depends on the contact number.