EoN.Epi_Prob_cts_time

EoN.Epi_Prob_cts_time(Pk, tau, gamma, umin=0, umax=10, ucount=1001, number_its=100)

Encodes System (6.3) of Kiss, Miller, & Simon. Please cite the book if using this algorithm.

The equations are rescaled by setting $u=\gamma T$. Then it becomes

P = 1- int_0^infty \psi(alpha(u/\gamma)) e^{-u} du alpha_d(u/\gamma) = 1- p(u/\gamma)

• p(u/\gamma) int_0^infty

  (\psiPrime(alpha(hat{u}/\gamma))/<K>) e^{-u}du

where p(u/\gamma) = 1 - e^{-tau u/\gamma}

Define

hat{p}(u) = p(u/\gamma), and hat{alpha}(u) = alpha(u/\gamma)

and then drop hats to get

P = 1-int_0^infty \psi(alpha(u)) e^{-u} du alpha(u) = 1-p(u) + p(u)

int_0^infty

(\psiPrime(alpha(u))/<K>)e^{-u} du

with initial guess

alpha_1(u) = e^{-tau u/\gamma}

and

p(u) = 1-e^{-tau u/\gamma}

Arguments:

Pk dict

Pk[k] is probability a node has degree k.

tau float

transmission rate

gamma float

recovery rate

umin minimal value of \gamma T used in calculation umax maximum value of \gamma T used in calculation ucount number of points taken for integral.

So this integrates from umin to umax using simple Riemann sum.

number_its int

number of iterations before assumed converged. default value is 100

Returns:

PE float

Calculated Epidemic probability (assuming configuration model)