Temporal Network Epidemiology by Naoki Masuda & Petter Holme

Author:Naoki Masuda & Petter Holme
Language: eng
Format: epub
Publisher: Springer Singapore, Singapore

(7.3)

which is now a system of ODEs that is closed in terms of , where ⊤ is the transposition.

The IBA leads to β c cont = γ∕α max, where α max is the largest eigenvalue of A [5, 7, 31]. This result is obtained by linearising Eq. (7.3) around the disease-free state given by x i (t) = 0 (i = 1, …, N), which yields

(7.4)

where I is the identity matrix. The largest eigenvalue of βA −γI equals 0 at β = β c cont. Equation (7.4) indicates that exponentially decays in time if β < β c cont. If β > β c cont, exponentially grows in time as long as the initial condition is not orthogonal to the eigenvector corresponding to the largest eigenvalue of A. Therefore, one obtains β c cont = γ∕α max. The epidemic threshold β c cont depends on α max, which is determined by the structure of the network.

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