Objective:
In this lab we will investigate the "S-I-R" model for describing the spread of an epidemic through a population. Our first purpose is to study the significance of changes in the parameter values. Our second purpose is to compare the model with known data on the spread of the Hong Kong Flu epidemic through the population of New York City in 1968-69.
In the text, we developed three differential equations and three initial conditions to model the spread of an epidemic:
dS/dt = -a I(t) S(t) S(0) = 7,900,000
dI/dt = a I(t) S(t) - l I(t) I(0) = 10
dR/dt = l I(t) R(0) = 0.
Recall that a = b/N, where N is the 1969 population of New York City (7,900,000), and b is the number of contacts per day by an infected person that are sufficient to spread the disease. The constant l is the fraction of the number of infected individuals who recover in a given day.
In this lab, we will approximate solutions to this initial value problem by using Euler's Method.
We start with experimentally determined values for b and l:
b = 0.6
l = 0.34.
We will model the epidemic over a time period T, which is initially set at 91 days (13 weeks), and with a time step of a half-day.
This tells Mathcad to compute all
three up-dating equations at once.
The first line of this vector equation
represents the Euler's Method recursive
formula for S; the second represents
the formula for I; the third line represents the formula for R.
1. a) Keep l fixed at 0.34, and experiment with different values of b between 0.5 and 2.0.
Describe how these changes affect the graph of I(t). (Stay alert for automatic changes in
the vertical scale.)
b) Explain briefly why these changes are reasonable from an intutitive understanding of the
epidemic model.
2. a) Return b to 0.6, and experiment with different values of l between 0.1 and 0.65.
Describe the changes you see in the graph of I(t). Again, be alert for changes in the
vertical scale.
c) There is a change in the character of the graph of I(t) near the end of the range (0.1 to 0.65)
given for l. What is the change, and where does it occur? Why do you think it happens
there? (Hint: What does the constant b represent? What does the constant l represent? If your graph seems to disappear, try changing the upper limit on the vertical axis so you can see it.)
3. Return l to its original value of 0.34; b should still be 0.6. Using the same plot box (graph) as before, now graph Ri instead of Ii. Then graph Si. Explain the shapes of these graphs in terms of your intuitive understanding of the model.
4. We next investigate the extent to which the S-I-R model matches our data for the Hong Kong Flu epidemic. Recall our assumption that the number of new flu cases in a week was proportional to the number of deaths reported in a later week; we will test that assumption now.
a) First we have to determine what the model says about new cases each week.
Explain why that number can be computed as the number of susceptibles at the start
of the week minus the number of susceptibles at the end of the week.
b) We have already calculated the number of susceptibles at half-day time steps.
To use the idea in part (a) for finding the number of new cases for each week,
we need to use every fourteenth value of Sk. Explain carefully how the formula
below gives the numbers of new cases for each of the 13 weeks:
c) The number of "excess" deaths (deaths over and above the usual number) reported during the Hong Kong Flu epidemic is recorded in the table below:
How do the two graphs compare? Play around with values for b and l and see if you
can bring the model into closer agreement with the observed data.
Keep in mind:
i) 1/l can't vary too much from the known recovery time for flu
ii) there is no guarantee that flu actually fits the S-I-R model
iii) even if it does fit, the assumed proportionality between excess deaths and new cases may not be correct
iv) the comparison needs to be between deaths in a given week and new cases in an earlier week.
When you are satisfied that you have come as close as you can in modeling the data,
briefly discuss the assumptions that went into this model. Which ones seem most
sensible? Which ones are most questionable? Describe two additional features of real
epidemics that you might want to try to include in an improved model.
