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The SIR Model for Spread of Disease

Part 2: The Differential Equation Model

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.

  1. Under the assumptions we have made, describe in words how do you think  s(t)  should vary with time? How should  r(t)  vary with time? How should  i(t)  vary with time?

  2. Sketch on a piece of paper what you think the graph of each of these functions looks like.

  3. Explain why, at each time  ts(t) + i(t) + r(t) = 1.

Next we make some assumptions about the rates of change of our dependent variables:

Let's see what these assumptions tell us about derivatives of our dependent variables.

  1. The Susceptible Equation.

    (a) Explain carefully how each component of the differential equation


    follows from the text preceding this step. In particular,

    (b) Now explain how this equation leads to the following differential equation for  s(t).

  2. The Recovered Equation. Explain how the corresponding differential equation for  r(t),


    follows from one of the assumptions preceding Step 4.

  3. The Infected Equation.

    (a) Explain why


    What assumption about the model does this reflect?

    (b) Now explain carefully how each component of the equation


    follows from what you have done thus far. In particular,

    • Why are there two terms?

    • Why is it reasonable that the rate of flow from the infected population to the recovered population should depend only on  i(t) ?

    • Where did the minus sign come from?

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. 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


We don't know values for the parameters  b  and   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

Solutions of the SIR Model

  1. In steps 1 and 2, you recorded your ideas about what the solution functions should look like. How do those ideas compare with the figure above? In particular,

    • What do you think about the relatively low level of infection at the peak of the epidemic?

    • Can you see how a low peak level of infection can nevertheless lead to more than half the population getting sick? Explain.

Because of the standard choice of symbols for the dependent variables, this model is called the SIR Model. In Part 3, we will see how solution curves can be computed even without formulas for the solution functions. [Note: If you have already worked through the Systems of Differential Equations module, you may skip Part 3 of this module and go straight to Part 4.]


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