EE365: Epidemic Example
1
Monte Carlo simulation
to approximate
e = E f(x0, . . . , xT ) =∑
s0,...,sT∈X
f(x0, . . . , xT )ds0Ps0s1 · · ·PsT−1sT
(a sum with nT+1 terms)
I simulate N trajectories x(i)t , and let
e =1
N
N∑i=1
f(x(i)0 , . . . , x
(i)T )
I e is an unbiased estimate of e
I works for any function f
2
Example: Epidemic models
I undirected graph
I k vertices, each represents an individual
I each may spread infection to neighbors
I each individual is either susceptible,infected, or removed
I called the S, I,R model
I transition probabilities for each individualdepends on infection state of neighbors
I X = {S, I,R}k, so |X | = 3k
I for modest k, we cannot store a distribution on X
I one approach is to use Monte Carlo methods to estimate useful quantities
3
Example: Epidemic models
I graph is 30× 30 grid
I transition probabilities for an individual, with states ordered (S, I,R) 0.6 0.4 00.2 0.6 0.20 0 1
with an infected neighbor
1 0 00.2 0.6 0.20 0 1
with no infected neighbors
I simulate Markov chain to see spread of infection
4
Susceptible
Infected
Removed
t = 0
Example: Epidemic models
5
Susceptible
Infected
Removed
t = 1
Example: Epidemic models
6
Susceptible
Infected
Removed
t = 2
Example: Epidemic models
7
Susceptible
Infected
Removed
t = 3
Example: Epidemic models
8
Susceptible
Infected
Removed
t = 4
Example: Epidemic models
9
Susceptible
Infected
Removed
t = 5
Example: Epidemic models
10
Susceptible
Infected
Removed
t = 6
Example: Epidemic models
11
Susceptible
Infected
Removed
t = 7
Example: Epidemic models
12
Susceptible
Infected
Removed
t = 8
Example: Epidemic models
13
Susceptible
Infected
Removed
t = 9
Example: Epidemic models
14
Susceptible
Infected
Removed
t = 10
Example: Epidemic models
15
Susceptible
Infected
Removed
t = 15
Example: Epidemic models
16
Susceptible
Infected
Removed
t = 20
Example: Epidemic models
17
Susceptible
Infected
Removed
t = 25
Example: Epidemic models
18
Susceptible
Infected
Removed
t = 30
Example: Epidemic models
19
Susceptible
Infected
Removed
t = 35
Example: Epidemic models
20
Susceptible
Infected
Removed
t = 40
Example: Epidemic models
21
Susceptible
Infected
Removed
t = 45
Example: Epidemic models
22
Susceptible
Infected
Removed
t = 50
Example: Epidemic models
23
Susceptible
Infected
Removed
t = 55
Example: Epidemic models
24
Susceptible
Infected
Removed
t = 60
Example: Epidemic models
25
Susceptible
Infected
Removed
t = 65
Example: Epidemic models
26
Susceptible
Infected
Removed
t = 70
Example: Epidemic models
27
Susceptible
Infected
Removed
t = 75
Example: Epidemic models
28
Susceptible
Infected
Removed
t = 80
Example: Epidemic models
29
Example: Epidemic models
disease spread over time
10 20 30 40 50 60 70 80 90 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
percentageofpopulation
30
Example: Epidemic models
I histogram of fraction of population that is removed after 100 time steps
I in 4% of runs, less than 5% is removed
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
fraction of population removed
fractionofruns
31