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

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

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 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)