# SIR Model – The Flue Season – Dynamic Programming

May 14, 2013
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(This article was first published on Econometrics by Simulation, and kindly contributed to R-bloggers)

# The SIR Model (susceptible, infected, and recovered) model is a common and useful tool in epidemiological modelling.

# In this post and in future posts I hope to explore how this basic model can be enriched by including different population groups or disease vectors.

# Simulation Population Parameters:
# Proportion Susceptible
Sp = .9

# Proportion Infected
Ip = .1

# Population
N = 1000

# Number of periods
r = 200

# Number of pieces in each time period.
# A dynamic model can be simulated by dividing each dynamic period into a sufficient number of discrete pieces.
# As the number of pieces approaches infinity then the differences between the simulated outcome and the outcome achieved by solving the dynamic equations approaches zero.
np = 1

# Model - Dynamic Change
DS = function() -B*C*S*I/N
DI = function() (B*C*S*I/N) - v*I
DZ = function() v*I
# I is the number of people infected, N the number of people in total, S is the number of people susceptible for infection, and Z is the number of people immune to the infection (from already recovering from the infection).

# Model Parameters:
# Transmition rate from contact with an infected individual.
B = .2
# Contact rate.  The number of people that someone becomes in contact with sufficiently to recieve transmition.
C = .5
# Recovery rate. Meaning the average person will recover in 20 days (3 weeks).
# This would have to be a particularly virolent form of the flu (not impossible at all).
v = .05

# Initial populations:

# Sesceptible population, Sv is a vector while S is the population values as the current period
Sv = S = Sp*N

# Infected, Iv is a vector while I is the population values as the current period
Iv = I = Ip*N

# Initial immunity.
Zv = Z = 0

# Now let's how the model works.
# Loop through periods
for (p in 1:r) {
# Loop through parts of periods
for (pp in 1:np) {

# Calculate the change values
ds = DS()/np
di = DI()/np
dz = DZ()/np

# Change the total populations
S = S + ds
I = I + di
Z = Z + dz

# Save the changes in vector form
Sv = c(Sv, S)
Iv = c(Iv, I)
Zv = c(Zv, Z)
}
}

# ggplot2 generates easily high quality graphics
require(ggplot2)

# Save the data to a data frame for easy manipulation with ggplot
mydata = data.frame(Period=rep((1:length(Sv))/np,3), Population = c(Sv, Iv, Zv), Indicator=rep(c("Uninfected", "Infected", "Recovered"), each=length(Sv)))

# This sets up the plot but does not actually graph anything yet.

p = ggplot(mydata, aes(x=Period, y=Population, group=Indicator))

# This graphs the first plot just by the use of the p command.
# Adding the geom_line plots the lines changing the color or the plot for each indicator (population group)
p + geom_line(aes(colour = Indicator)) + ggtitle("Flu Season")

# Save initial graph:
ggsave(file="2013-05-14flu.png")

# Let's do some back of the envelope cost calculations.
# Let's say the cost of being infected with the flu is about $10 a day (a low estimate) in terms of lost productivity as well as expenses on treatment. # This amounts to: sum(Iv/np)*10 # Which is a cost of$165,663.40 over an entire flu season for the thousand people in our simulated sample.
# Or about $165 per person. # Imagine if we could now do a public service intervention. # Telling people to wash their hands, practice social distancing, and avoid touching their noses and eyes, and staying at home when ill. # Let's say people take up these practices and it reduces the number of potential exposure periods per contact by half. C = .25 # .... p + geom_line(aes(colour = Indicator)) + ggtitle("Flu Season with Prevention") # Save initial graph: ggsave(file="2013-05-14flu2.png") # .... sum(Iv/np)*10 # Which is a cost of$76,331.58 over an entire flu season  for the thousand people in our simulated sample or about 76 dollars per person.

# The difference in costs is about 89 thousand dollars for the whole population or on average 89 per person.  The argument is therefore, so long as a public service intervention that reduces personal contact costs less than 89 thousand dollars for those 1000 people, then it is an efficient intervention (at least by the made up parameters I have here).