probability in propagation
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Probability in Propagation. 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. Example.

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transmission rates
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
example
Example
  • Assume node A is infected.
  • Let the transmission rate be p. In this example, p=0.8.
  • What is the chance that B is infected?
example1
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?
multiple neighbors
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?
a closer look at the possibilities
A closer look at the possibilities

Now let p=0.6. Let’s work out the possible scenarios from the previous slide.

a more extensive example
A more extensive example
        • A and B start out infected. Let p=0.6 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.
when we need simulation
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!
simulations
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.
simulation exercise
Simulation Exercise
  • SI model
  • 1-threshold
  • transmission rate = 0.7
  • Assume a susceptible node can be infected at each time step
  • Use a random number generator to get a number between 0 and 100
  • http://www.random.org/
  • If number <70, infect, otherwise do not.
simulation example
Simulation Example
  • A and B are infected, 50% chance D is infected
  • Does C become infected?
    • Random number to see if infection comes from A
    • If not from A, random number to see if infection comes from B
  • 50% chance D is infected
    • Random number to decideif D is actually infected
  • Does E become infected?
    • If C is infected, random numberto see if C infects
    • If D is infected, random number to see if D infects
now you try
Now you try
  • Initial infection
    • D (100% chance of infection)
    • H (80% chance of infection)
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