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## PowerPoint Slideshow about 'Probability in Propagation' - gunda

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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

- 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?

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

- 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

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

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

- 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

- 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

- 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

- 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

- Initial infection
- D (100% chance of infection)
- H (80% chance of infection)

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