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 Note to faculty and students: I had whooping cough as I started to write this book. I was sick for 3 full months during which I didn’t once sleep through the night because of the coughing fits. Often, I didn’t sleep for more than 20 minutes at a time. I broke 2 ribs because it was so violent. Remember that your vaccine for that wears off in about 5 years, so go get a booster!

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? The chance that B is infected is 0.8, or 80%.

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? IF B was infected, then there is still a 0.8 chance it is passed to C. Emphasize this – students need to recognize the precondition matters. Overall, there is a 0.64 chance that C is infected – an 80% chance that B was infected, and in turn an 80% chance in that scenario that C is infected. 0.8 *0.8 = 0.64.

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? Talk about the 4 cases: C can catch it 1) just from A, 2) just from B, 3) from both, 4) from neither. In a 1-threshold, the first 3 cases result in infection. In a 2-threshold, only case 3 does. Compute all 4 probabilities and add them up for the 1-threshold case. Here are the probabilities: 0.8 * 0.2 (yes from A, no from B) = 0.16 2) 0.2 * 0.8 (no from A, yes from B) = 0.16 3) 0.8*0.8 (yes from both) = 0.64 4) 0.2*0.2 (no from both) = 0.04 Note that the sum of these 4 options is 1 – that means there is a 100% chance that one of these things happens Also explain how 1-p is the probability that a node does NOT pass on the infection.

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.

All the possibilities!

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. Some interesting simulations will be listed on the website.