The Most Important Diagram

The Most Simple Model(?)

)(rdtSdS

SSr

The Most Simple Model(?)

)(trSdt

dS

SSr

This model characterizes how S(t) is changing

The Most Simple Model(?)

)(trSdt

dS

SSr

How many people are susceptible at time = t?What is S(t) = ? There are a number of approaches…

Approaches

Discrete / Numerical

Analytical

Simulation of stochastic events

Bonus: scheduling events via a stochastic algorithm

The Most Simple Model(?)

S

)(~ trSt

S

dt

dS

)()( tSttSS We know that:

r

Approach #1

The Most Simple Model(?)

S

)()()(

trSt

tSttS

Sr

Approach #1

The Most Simple Model(?)

SSr

ttrStSttS )()()(

Approach #1

The Most Simple Model(?)

SSr

ttrStSttS )()()(

Discrete(aka “Numerical”)

Approach #1

The Most Simple Model(?)

)(trSdt

dS

SSr

Differential equation

This one is not that difficult to solve

Approach #2

The Most Simple Model(?)

)(trSdt

dS

SSr

rteStS 0)(Solving for S(t)yields:

Analytic

Approach #2

The Most Simple Model(?)

S

SI1

0T

In each time step, each individual “leaves” S with probability = r

A population of 10 susceptibleIndividuals at T0

I10

I2 I3

I9

I4

I8

I5

I7I6

Approach #3

The Most Simple Model(?)

0T

In each time step, each individual “leaves” S with probability = r

- Generate a random number, U, between0 and 1 for each individual

- If U > r, individual stays in S

- If U < r, individual leaves S

- Suppose r = 0.05

U ~ Uniform on the interval [0, 1]

S

SI1

I10

I2 I3

I9

I4

I8

I5

I7I6

Approach #3

The Most Simple Model(?)

0T

In each time step, each individual “leaves” S with probability = r

Individuali

Ui

1 0.871

2 0.600

3 0.290

4 0.335

5 0.421

6 0.180

7 0.033

8 0.663

9 0.338

10 0.246

S

SI1

I10

I2 I3

I9

I4

I8

I5

I7I6

Approach #3

The Most Simple Model(?)

0T

In each time step, each individual “leaves” S with probability = r

S

SI1

I10

I2 I3

I9

I4

I8

I5

I7I6

S

SI1

I10

I2 I3

I9

I4

I8

I5

I6

1T

Approach #3

The Most Simple Model(?)

0T

Using indicator variables to denote if an individual is still in “S”

S

S1

1

1 1

1

1

1

1

1 1

1T

S

S1

1

1 1

1

1

1

1

0 1

Approach #3

The Most Simple Model(?)

1T

Repeat for the desired number of time steps to determine S(t)

S

S1

1

1 1

1

1

1

1

0 1

tT

S

S?

?

? ?

?

?

?

?

0 ?

Approach #3

Stochastic

A Stochastic Algorithm

If events are happening at a constant rate, l, we can model the time to the next event using an exponential distribution

Recall that the mean of an exponential distribution is 1/l

The c.d.f. is: F = P(T < t) = 1 – e –lt

Solving for t gives: t = -ln(1 – P) / = l F-1

Lets see a visual….

Suppose l = 2

We can see, for example, that the probability that we’ll wait more than one unit of time is about 0.2. The probability that we’ll have to wait less than one unit is about 0.8.

P(T < t) = 1 – e –lt

Suppose l = 2

We can generate a exponentially distributed r. v.’s using the following algorithm: Generate a uniform r.v. (U) using a

random number generator (e.g. in R) Plug in U in for P in F-1 (U) = -ln(1 – U) / l The result is a exponentially distributed

waiting time Repeat

Suppose l = 2

F-1 = -ln(1 – P) / l

Mean = 1/2

Multiple events

More than one type of event can occur in most models (e.g. birth, death, infection, recovery)

We can add up the rates of each of these to find out the overall rate that events occur

Use our algorithm to determine the time to the next event (stochastically!)

Randomly select which event occurred (proportional to it’s rate)