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, p=0.8. • 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 p=0.6. Let’s work out the possible scenarios from the previous slide.

7. 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.

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.

11. 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.

12. 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

13. Now you try • Initial infection • D (100% chance of infection) • H (80% chance of infection)