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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Probability in Propagation

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Probability in propagation

Probability in Propagation


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.


  • All the possibilities

    All the possibilities!


    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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