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