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The Most Important Diagram. The Most Simple Model(?). r. S. S. The Most Simple Model(?). r. S. S. This model characterizes how S(t) is changing. The Most Simple Model(?). r. S. S. How many people are susceptible at time = t?

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The Most Important Diagram

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The most important diagram

The Most Important Diagram


The most simple model

The Most Simple Model(?)

r

S

S


The most simple model1

The Most Simple Model(?)

r

S

S

This model characterizes how S(t) is changing


The most simple model2

The Most Simple Model(?)

r

S

S

How many people are susceptible at time = t?

What is S(t) = ? There are a number of approaches…


Approaches

Approaches

  • Discrete / Numerical

  • Analytical

  • Simulation of stochastic events

  • Bonus: scheduling events via a stochastic algorithm


The most simple model3

The Most Simple Model(?)

Approach #1

r

S

We know that:


The most simple model4

The Most Simple Model(?)

Approach #1

r

S

S


The most simple model5

The Most Simple Model(?)

Approach #1

r

S

S


The most simple model6

The Most Simple Model(?)

Approach #1

r

S

S

Discrete

(aka “Numerical”)


The most simple model7

The Most Simple Model(?)

Approach #2

r

S

S

Differential equation

This one is not that difficult to solve


The most simple model8

The Most Simple Model(?)

Approach #2

r

S

S

Solving for S(t)

yields:

Analytic


The most simple model9

The Most Simple Model(?)

Approach #3

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

S

S

I2

I3

I1

I4

I5

A population of 10 susceptible

Individuals at T0

I10

I6

I8

I9

I7


The most simple model10

The Most Simple Model(?)

Approach #3

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

S

  • - Generate a random number, U, between

  • 0 and 1 for each individual

  • If U > r, individual stays in S

  • If U < r, individual leaves S

  • Suppose r = 0.05

S

I2

I3

I1

I4

I5

I10

I6

I8

I9

I7

U ~ Uniform on the interval [0, 1]


The most simple model11

The Most Simple Model(?)

Approach #3

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

S

S

I2

I3

I1

I4

I5

I10

I6

I8

I9

I7


The most simple model12

The Most Simple Model(?)

Approach #3

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

S

S

S

S

I2

I2

I3

I3

I1

I1

I4

I4

I5

I5

I10

I10

I6

I6

I8

I8

I9

I7

I9


The most simple model13

The Most Simple Model(?)

Approach #3

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

S

S

S

S

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

0

1

1


The most simple model14

The Most Simple Model(?)

Approach #3

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

S

S

S

S

?

1

?

1

?

1

?

1

?

1

?

1

?

1

?

1

?

0

1

0

Stochastic


A stochastic algorithm

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

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 21

Suppose l = 2

  • We can generate a exponentially distributed r. v.’susing 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 22

Suppose l = 2

  • F-1 = -ln(1 – P) / l

Mean = 1/2


Multiple events

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)


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