# SIR Model – The Flue Season – Dynamic Programming

**Econometrics by Simulation**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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# 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).

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