An extensively studied model in epidemiology
The key thing is allows nodes to change behaviors back and forth over time
Model of catching some recurring diseases, who to vote for, etc
Nodes are infected or susceptible
Probability that get infected is proportional to number of infected neighbors with rate v>0, plus spontaneous $\epsilon$
get well randomly in any period at rate $\delta > 0$
Let $\rho$ be the percent of the population that's infected any point in time.
Start wih benchmark where all players mix with even probabilities
Randomly meet an individual each period
Large Markov chain
Steady state mean-field: $d\rho/dt = 0$
$d\rho/dt = (1-\rho)(v\rho + \epsilon) - \rho \delta = 0$
solve the above expression, then we get the solution:
$\rho = [(v - \delta - \epsilon) + ((v -\delta -\epsilon)^2 + 4\epsilon v)^{1/2}] / 2v$
$d\rho/dt = (1-\rho)v\rho - \rho \delta = 0$
Two solutions:
$\rho = 1 - \delta/v$ ($if >0 ==> \delta < v$)
$\rho = 0$
so far uniformly random interaction
missing heterogeneity in degree
missing local patterns
we can at least address the first concern...
random matching with $d_i$ matches for node i
$\rho(d)$: fraction of nodes of degree d infected
$\theta$: fraction of randomly chosen neighbors infected
Now what's going to be important, is the fraction of people over all that might have something in their population, is not going to be the same as the fraction of people I meet. Because I'm more likely to meet people who are meeting lots of people. So, some people have lots of interactions. Those are the people I'm more likely to meet. Those are also the people who are more likely to be infected. Okay? So, so that's the process that's going on.
P(d): fraction of nodes that have d meetings, i.e. degree distribution
More likely to meet someone who has high d
likelihood of meeting node of degree d is:
$\theta = \sum_d \rho(d)P(d)d/E[d]$ fraction of infected neighbors/random partners
Steady state: for each d
Steady state: for each d
$\rho(d) = \frac {\lambda \theta d}{((\lambda \theta d + 1)}$ where $\lambda = \frac{v}{\delta}$
$\theta = \sum_d \frac{\rho(d) P(d) d}{E[d]} = \sum_d \frac{P(d) \lambda \theta d^2}{[(\lambda \theta d + 1)E[d]]}$
Steady state infection rate of people you meet is the solution to
What can we say about how this depends on the "network structure"?
How does infection rate $\theta$ depend on P(d), E(d)?