- Motivation - relatives sick - graph theory stuff is neato - Transmission through networks - Percolation threshold - On random graphs and on graphs with a specific structure - Tells us whether an "outbreak" will become an "epidemic" and spread to an infinite number of nodes - Not looking at: - the time path of how a disease spreads - (Sort of time invariant approach) - The mortality or costs involved. - Model – Tranmission - always happens at rate r - If you have n outneighbors, its expected that your case will cause an additional n*r new cases - If there is a total population *I* of your type, then together you'll create Inr new cases. - If nr > 1, then on average cases will grow. - population structure - Split into types indexed by *i* - types are defined by the number of connections is has to each other type. - infinite population of each type, so I'm not worrying about susceptibility and the like - parameters n_ij tell us the number of outdegrees from a type i node to a type j node - Suppose that each of these groups is very large and that the connections are chosen at random, save for the fact about which group they connect. - Obvious consequences - let \Phi_{i} be the portion of infectious people who are of type i. So $$\Phi_{i}\equiv\frac{I_{i}}{\sum_{k}I_{k}}$$. - expected number of new transmission events is \[\sum_{k}\left[I_{k}\sum_{j}rn_{kj}\right]\] - Proposition 1 [ ] What do checkmarks look like?