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Who Spread the Flu?

Case Facts:

The week after Spring Break all 500 students and teachers at Contagious High School in Contagious, Maryland went home sick with influenza (the flu). It has been discovered that three students came back after Spring Break, each had a different infectious flu. The rates at which each of the flus can infect another person (Infection Rate) and the rates at which a person with the flu gets better (Recovery Rate) are commonly known for each of these strains. You have been asked to determine which of the three originally infected students was the source of the flu that was so virulent it affected nearly everyone in the school.

Student Name	Flu Strain	Infection Rate	Recovery Rate
Bobby Getwels	Ick Flu	.002	.5
Amess Besick	Plu Flu	.001	.5
Majorly Ruddy	Gacky Flu	.08	.2

Which student had the flu that caused everyone to get sick?

Approximately how many students would be infected by Bobby Getwels if he was the only source of infection? by Amess Besick?

What is the relationship between the recovery rate and the shape of the line for 'recovered'?

What is the relationship between the infection fate and the shape of the line for 'infected'?

Can you predict which flu will infect more people: one with an Infection rate of .001 or .002?

Parallel Example

Before building the model for this case, consider this case study:

Source: Mathematical Biology, J.D. Murray, Springer-Verlag, 1989.

Background:

In 1978, a study was conducted and reported in the British Medical Journal (4 March 1978) of an outbreak of the influenza virus in a boys boarding school. The school had a population of 763 boys. Of these 512 were confined to bed during the epidemic, which lasted from 22 January until 4 February. It seems that one infected boy initiated the epidemic. At the outbreak of the epidemic, none of the boys had previously had influenza, so no resistance to the infection was present.

Modeling Task

Build a computational model of this epidemic, looking to answer these two questions:

• On what day (with Day 0 being the beginning of the epidemic) did the majority of the boys have the flu?
• At what time did most or all of the boys recover from the illness?

To build this model, you should use the 1927 Kermack-McKendrick model known as the SIR algorithm. This algorithm looks at the change in three populations: susceptibles (S), infecteds (I), and recovereds (R). It assumes that once recovered, immunity from recontamination exists and therefore a patient cannot re-enter the susceptible or infected group. The algorithm is shown below in differential equation form:

dS/dt = -rSI

dI/dt = rSI - aI

dR/dt = aI

where S is the total susceptible population, I is the total infected population, R is the total recovered population, r is the infection rate and a is the removal rate of infectives (the recovery rate). We are, of course, only interested in non-negative solutions for S, I, and R. This is a very primitive algorithm, but provides relatively good results, especially in looking for the epidemic "hump". This algorithm assumes a constant population, that is, no immigration or emigration of potential players. This also assumes, naturally, that no one dies of the illness! In this model 1/a is the average infectious period. In this model, the infection rate is about 0.00218 per day, and the removal rate is approximately 1/2 per day -- an average infectious period of about two days.

Once you have built your model, show the three populations on your graph. You might also consider using a "Total Population" converter to ensure that all 763 boys are accounted for at all times.

Extension:

What would happen if some of the boys had been vaccinated prior to the beginning of the epidemic? Let's assume that 10% of the susceptible boys are vaccinated each day -- some of them getting the shot while the epidemic is happening in order not to get sick (this assumes, falsely, that instant immunity is realized once you've received the shot). Experiment with the 10% vaccination rate to determine how it changes the intensity and duration of the epidemic.

The completed model should look like this:

Equations are:

Infected(t) = Infected(t - dt) + (get_sick - get_better) * dt
INIT Infected = 1

INFLOWS:
get_sick = infection_probability*Susceptible*Infected

OUTFLOWS:
get_better = recovery_rate*Infected
Recovered(t) = Recovered(t - dt) + (get_better) * dt
INIT Recovered = 0

INFLOWS:
get_better = recovery_rate*Infected
Susceptible(t) = Susceptible(t - dt) + (- get_sick) * dt
INIT Susceptible = 762

OUTFLOWS:
get_sick = infection_probability*Susceptible*Infected
infection_probability = 0.00218
recovery_rate = 1/2

Results graph:

Online Flu Model

(Note: In addition to changing the infection and recovery rates in the model, you may also need to change the number of infected and susceptible people.)