Probability in Propagation

1
Probability in Propagation
2
Transmission Rates

• Models discussed so far assume a 100
  transmission rate to susceptible individuals
  (e.g. Firefighter problem)
• Almost no diseases are this contagious
• Whooping cough 90 transmission rate
• HIV 2 transmission rate

3
Example

• Assume node A is infected.
• Let the transmission rate be p. In this example,
  p0.8.
• What is the chance that B is infected?

4
Example

• If B was infected by A, what is the chance that C
  is infected by B?
• What is the overall chance that C is infected?

5
Multiple Neighbors

• Both A and B are infected.
• What is the chance that C is infected in a
  1-threshold model?
• What about a 2-threshold model?

6
A closer look at the possibilities
Now let p0.6. Lets work out the possible
scenarios from the previous slide.
7
A more extensive example

• A and B start out infected. Let p0.6 as in the
  previous slide.
• What is the chance that C is infected in a
  1-threshold model?
• Let the probability that D is infected be 0.7.
  What is the probability that E gets infected?
• Repeat for a 2-threshold model.

8
All the possibilities!
9
When we need simulation

• A and B start infected. They can infect C and/or
  D
• If one node, say C, is uninfected, in the next
  time step it could be infected by A or B again,
  but it could also be infected by D.
• If we change to an SIS or SIR or SIRS model, all
  these calculations change.
• The way the disease propagates at each time step
  changes
• Too much to calculate by hand, especially in big
  nets!

10
Simulations

• Take a network. Set some nodes as I and others as
  S.
• When there is a probability, make a decision
  (infect or not). Repeat for as long as the
  simulation runs. Get results.
• Repeat the simulation, making decisions that may
  go the other way (e.g. a 60 transmission rate
  may lead to infection in one simulation and no
  infection in another)
• Do the simulation a lot of times, and look at the
  average result.