• Motivation
  • relatives sick
  • graph theory stuff is neato
  • Politicians and religious people got sick first
  • 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 nr 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]
    • refer to latex notes for info about number of newly infected of each type
• Proposition 0: what happens with identity mapping among distribution
  • Any point is a fixed point
  • will technically be minor variations
  • but will stick around the initial outbreak point
  • epidemic will depend purely on whether n_ii > 1/r?
• Proposition 1: fixed point and convergence
  • Note that, holding the parameters \eta_{..} constant, the system of equations describing each \Phi_{i}^{\prime} is a mapping from a probability distribution to itself, which doesn’t depend on the transmission rate r.
  • In other words, I’m splitting up two questions about how the contagion spreads: how does the distribution of types amongst the infected evolve?, and does the incidence of the infection grow or shrink?
  • Because it’s a continous mapping from a simplex to the same simplex, there must be at least one fixed point.
  • This fixed point is not necessarily unique.
  • If the connecton parameters \left{ \eta\right} are chosen such that \eta_{ij}\neq0 for any i\neq j , then the fixed point must be in the interior of the probability. simplex.
  • In the latter case, subsequent generations of infectious will evolve to have a distribution of types which converges to the fixed distribution.
  • Then the question of whether the contagion can spread to a large portion of the network depends on the average number of new tranmissions per person among this distribution of infectious people.
  • Then after the distribution evolves in a way determined by the structure of connections between types, the number of new transmissions per infectious person will be a weighted sum of the average number of transmissions per type of person \sum_{k}\left[\Phi_{k}\sum_{j}rn_{kj}\right]

-Continuous example with two types is in pdf

-exmaple with only two people

-note about endogenizing connections )pdf

-point about eigenvectors

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