Modeling Vaccination#

In the previous chapter I presented the Kermack-McKendrick (KM) model of infectious disease and used it to model the Freshman Plague at Olin. In this chapter we’ll consider metrics intended to quantify the effects of the disease and interventions intended to reduce those effects.

We’ll use some of the functions from the previous chapter: make_system, update_func, and the last version of run_simulation, which returns the results in a DataFrame object.


Models like this are useful for testing “what if?” scenarios. As an example, we’ll consider the effect of immunization.

Suppose there is a vaccine that causes a student to become immune to the Freshman Plague without being infected. How might you modify the model to capture this effect?

One option is to treat immunization as a shortcut from susceptible to recovered without going through infectious. We can implement this feature like this:

def add_immunization(system, fraction):
    system.init.s -= fraction
    system.init.r += fraction

add_immunization moves the given fraction of the population from S to R.

As a basis for comparison, I’ll run the model with the same parameters as in the previous chapter, with no immunization.

tc = 3             # time between contacts in days 
tr = 4             # recovery time in days

beta = 1 / tc      # contact rate in per day
gamma = 1 / tr     # recovery rate in per day

system = make_system(beta, gamma)
results = run_simulation(system, update_func)

Now let’s see what happens if 10% of students are immune. I’ll make another System object with the same parameters, then run add_immunization to modify the initial conditions.

system2 = make_system(beta, gamma)
add_immunization(system2, 0.1)

Now we can run the simulation like this:

results2 = run_simulation(system2, update_func)

The following figure shows s as a function of time, with and without immunization.

results.s.plot(style='--', label='No immunization')
results2.s.plot(label='10% immunization')

decorate(xlabel='Time (days)',
         ylabel='Fraction of population')

With immunization, there is a smaller change in s; that is, fewer people are infected. In the next section we’ll compute this change and use it to quantify the effect of immunization.


When we plot a time series, we get a view of everything that happened when the model ran, but often we want to boil it down to a few numbers that summarize the outcome. These summary statistics are called metrics.

In the KM model, we might want to know the time until the peak of the outbreak, the number of people who are sick at the peak, the number of students who will still be sick at the end of the semester, or the total number of students who get sick at any point.

As an example, I will focus on the last one — the total number of sick students — and we will consider interventions intended to minimize it.

We can get the total number of infections by computing the difference in s at the beginning and the end of the simulation.

def calc_total_infected(results, system):
    s_0 = results.s[0]
    s_end = results.s[system.t_end]
    return s_0 - s_end

And here are the results from the two simulations.

calc_total_infected(results, system)
calc_total_infected(results2, system2)

Without immunization, almost 47% of the population gets infected at some point. With 10% immunization, only 31% get infected. That’s pretty good.

Sweeping Immunization#

Now let’s see what happens if we administer more vaccines. This following function sweeps a range of immunization rates:

def sweep_immunity(fraction_array):
    sweep = SweepSeries()

    for fraction in fraction_array:
        system = make_system(beta, gamma)
        add_immunization(system, fraction)
        results = run_simulation(system, update_func)
        sweep[fraction] = calc_total_infected(results, system)

    return sweep

The parameter of sweep_immunity is an array of immunization rates. The result is a SweepSeries object that maps from each immunization rate to the resulting fraction of students ever infected.

We can call it like this:

fraction_array = linspace(0, 1, 21)
infected_sweep = sweep_immunity(fraction_array)

The following figure plots the SweepSeries. Notice that the \(x\)-axis is the immunization rate, not time.


decorate(xlabel='Fraction immunized',
         ylabel='Total fraction infected',
         title='Fraction infected vs. immunization rate')

As the immunization rate increases, the number of infections drops steeply. If 40% of the students are immunized, fewer than 4% get sick. That’s because immunization has two effects: it protects the people who get immunized (of course) but it also protects the rest of the population.

Reducing the number of “susceptibles” and increasing the number of “resistants” makes it harder for the disease to spread, because some fraction of contacts are wasted on people who cannot be infected. This phenomenon is called herd immunity, and it is an important element of public health (see

The steepness of the curve is a blessing and a curse. It’s a blessing because it means we don’t have to immunize everyone, and vaccines can protect the “herd” even if they are not 100% effective.

But it’s a curse because a small decrease in immunization can cause a big increase in infections. In this example, if we drop from 80% immunization to 60%, that might not be too bad. But if we drop from 40% to 20%, that would trigger a major outbreak, affecting more than 15% of the population. For a serious disease like measles, just to name one, that would be a public health catastrophe.


In general, models are used to predict, explain, and design. In this chapter, we use an SIR model to predict the effect of immunization and to explain the phenomenon of herd immunity.

In the repository for this book, you will find a file called plague.ipynb that uses this model for design, that is, for making public health decisions intended to achieve a goal.

In the next chapter, we’ll explore the SIR model further by sweeping the parameters.

But first you might want to work on this exercise.


This chapter is available as a Jupyter notebook where you can read the text, run the code, and work on the exercises. You can access the notebooks at

Exercise 1#

Suppose we have the option to quarantine infected students. For example, a student who feels ill might be moved to an infirmary or a private dorm room until they are no longer infectious.

How might you incorporate the effect of quarantine in the SIR model?

Hide code cell content
# Solution

"""There is no unique best answer to this question,
but one simple option is to model quarantine as an
effective reduction in gamma, on the assumption that
quarantine reduces the number of infectious contacts
per infected student.

Another option would be to add a fourth compartment
to the model to track the fraction of the population
in quarantine at each point in time.  This approach
is more complex and might not be substantially better.

The following function could be used, like 
`add_immunization`, to adjust the
parameters in order to model various interventions.

In this example, `high` is the highest duration of
the infection period, with no quarantine.  `low` is
the lowest duration, on the assumption that it takes
some time to identify infectious students.

`fraction` is the fraction of infected students who 
are quarantined as soon as they are identified.

def add_quarantine(system, fraction):
    """Model the effect of quarantine by adjusting gamma.
    system: System object
    fraction: fraction of students quarantined
    # `low` represents the number of days a student 
    # is infectious if quarantined.
    # `high` is the number of days they are infectious
    # if not quarantined
    low = 1
    high = 4
    tr = high - fraction * (high-low)
    system.gamma = 1 / tr
Hide code cell content
# Solution

def sweep_quarantine(fraction_array):
    sweep = SweepSeries()

    for fraction in fraction_array:
        system = make_system(beta, gamma)
        add_quarantine(system, fraction)
        results = run_simulation(system, update_func)
        sweep[fraction] = calc_total_infected(results, system)

    return sweep
Hide code cell content
# Solution

fraction_array = linspace(0, 1, 21)
infected_sweep2 = sweep_quarantine(fraction_array)
Hide code cell content
# Solution


decorate(xlabel='Fraction quarantined',
         ylabel='Total fraction infected',
         title='Fraction infected vs. quarantine rate')