Estimating the epidemic threshold on networks by deterministic connections

Abstract

For many epidemic networks some connections between nodes are treated as deterministic, while the remainder are random and have different connection probabilities. By applying spectral analysis to several constructed models, we find that one can estimate the epidemic thresholds of these networks by investigating information from only the deterministic connections. Nonetheless, in these models, generic nonuniform stochastic connections and heterogeneous community structure are also considered. The estimation of epidemic thresholds is achieved via inequalities with upper and lower bounds, which are found to be in very good agreement with numerical simulations. Since these deterministic connections are easier to detect than those stochastic connections, this work provides a feasible and effective method to estimate the epidemic thresholds in real epidemic networks.

FIG. 1.

Example of an epidemic network with two unattached sub-networks: deterministic network G and stochastic network G^c.

FIG. 2.

Comparisons between the epidemic threshold β_c and upper-lower bound estimation under different network sizes n and uniform stochastic probability α = 0.01 in (a), (c), and α = 0.001 in (b), (d).

FIG. 3.

Comparisons between the epidemic threshold β_c and upper-lower bound estimation under different network sizes n and parameter η = 0.5 in (a), (c), and η = 0.2 in (b), (d).

FIG. 4.

Comparisons between the epidemic threshold β_c and upper-lower bound estimation under different network sizes n and parameters μ = 0.1, σ²= 0.001 in (a), (c), and μ = 0.2, σ²= 0.001 in (b), (d).

FIG. 5.

Comparisons between the epidemic threshold β_c and upper-lower bound estimation under different community sizes m and n.

FIG. 6.

Comparisons between the epidemic threshold β_c, upper bound, and optimal upper bound estimation under different community sizes m and n.