SIR Model: The Boarding School Example - SAMSI

SIR Model:
The Boarding School Example
Cammey Cole Manning
Meredith College
May 17, 2011
SIR Model: The Boarding School Example – p. 1/8

SIR Model: Equations
SI
N
S I R
I
dS
dt
dI
dt
dR
dt
= − βSI
N
= βSI
N − γI
= γI
N = S + I + R
SIR Model: The Boarding School Example – p. 2/8

Boarding School Example:
The Data
Time
Number Infected
1 3
2 6
3 25
4 73
5 222
6 294
7 258
8 237
9 191
10 125
11 69
12 27
13 11
14 4
From Influenza in a boarding school, British Medical Journal, March 4, 1978, p. 587.
SIR Model: The Boarding School Example – p. 3/8

Experiment with Values of β and γ
time=[1:1:14];
Infected=[3 6 25 73 222 294 258 237 191 125 69 27 11 4];
beta= % Choose values for beta and gamma
gamma =
N=763;
YInit=[760 3 0];
options=[ ];
[t,PredY]=ode15s(@SIRode, time, YInit, options, beta, gamma, N);
PredI=PredY(:,2);
figure
plot(time, Infected, ’*’, t, PredI, ’-’)
xlabel(’Time (in Days)’)
ylabel(’Number Infected’)
legend(’Data’, ’Model’)
SIR Model: The Boarding School Example – p. 4/8

Least Squares:
Sum of Squares of Residuals
Minimize the sum of the squares of the residuals between
the measured y (the data) and the y calculated in the linear
model.
S r =
=
=
n∑
i=1
e 2 i
n∑
(y i,measured − y i,model ) 2
i=1
n∑
(DataI i − PredI i ) 2
i=1
This criterion has a number of advantages:
• Positive differences do not cancel negative differences.
• Small differences become smaller, and large
differences are magnified.
SIR Model: The Boarding School Example – p. 5/8

Boarding School Example:
MATLAB Cost Code
function [Cost]=SIRCost(param, t,Infected, YInit, N)
% param: 2x1 vector of beta and gamma
% t: time data % Infected: infected data
% Cost: value of sum of squared errors
beta=param(1);
gamma=param(2);
options=[];
[t,PredY]=ode15s(@SIRode, t, YInit, options, beta, gamma, N);
PredInfected=PredY(:,2);
DiffY=PredInfected’-Infected;
DiffYsq=(DiffY).ˆ2;
Cost=sum(DiffYsq);
SIR Model: The Boarding School Example – p. 6/8

Boarding School Example:
Script Using Fminsearch and ODE
time=[1:1:14];
Infected=[3 6 25 73 222 294 258 237 191 125 69 27 11 4];
N=763;
%Initial Conditions: Susceptible 760, Infected: 3, Recovered: 0
YInit=[760 3 0];
options=[ ];
ParamInit=[0.01 0.1]; % Choose initial guesses for β and γ
[ParamOpt] = fminsearch(@SIRCost,ParamInit,options, time,Infected, YInit, N)
betaOpt=ParamOpt(1);
gammaOpt=ParamOpt(2);
[t,PredY]=ode15s(@SIRode, time, YInit, options, betaOpt, gammaOpt, N);
PredInfected=PredY(:,2);
figure
plot(time,Infected, ’*’, t, PredInfected, ’-’)
xlabel(’Time (in Days)’)
ylabel(’Number Infected’)
legend(’Data’, ’Model’)
SIR Model: The Boarding School Example – p. 7/8

Boarding School Example: MATLAB Plot
β = 1.6923, γ = 0.4485
300
Data
Model
250
200
Number Infected
150
100
50
0
0 2 4 6 8 10 12 14
Time (in Days)
SIR Model: The Boarding School Example – p. 8/8