 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  Contagion rate for “information” is hard to predict, but we make assumptions

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.

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.

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   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 decide if D is actually infected  Does E become infected?  If C is infected, random number to 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)