Effects of void nodes on epidemic spreads in networks

Abstract

We present the pair approximation models for susceptible-infected-recovered (SIR) epidemic dynamics in a sparse network based on a regular network. Two processes are considered, namely, a Markovian process with a constant recovery rate and a non-Markovian process with a fixed recovery time. We derive the implicit analytical expression for the final epidemic size and explicitly show the epidemic threshold in both Markovian and non-Markovian processes. As the connection rate decreases from the original network connection, the epidemic threshold in which epidemic phase transits from disease-free to endemic increases, and the final epidemic size decreases. Additionally, for comparison with sparse and heterogeneous networks, the pair approximation models were applied to a heterogeneous network with a degree distribution. The obtained phase diagram reveals that, upon increasing the degree of the original random regular networks and decreasing the effective connections by introducing void nodes accordingly, the final epidemic size of the sparse network is close to that of the random network with average degree of 4. Thus, introducing the void nodes in the network leads to more heterogeneous network and reduces the final epidemic size.

Figure 1

Final epidemic size corresponding to the connection coefficient α and infection parameters: (a) Markovian process with basic reproduction number R₀ = 1/r = β/γ as the infection parameter; (b) non-Markovian process with basic reproduction number R₀ = βσ as the infection parameter. Additionally, the difference of final epidemic size between Markovian and non-Markovian process is shown in (c). The original network is assumed to be a regular random graph with Q = 8. The solid line means the critical curve for Markovian process from Eq. (26) and the dot line means the critical curve for non-Markovian process from Eq. (41).

Figure 2

Final epidemic size according to the fraction of void node x and the basic reproduction number R₀. (1) the case where the void node is homogeneously distributed on the network (α = 1 – x), and (2) the case where the void node is distributed so that the connection of [BB] is zero ($\alpha =\frac{1-2x}{1-x}$). (a) the Markovian process, (b) non-Markovian process, and (c) difference of final epidemic size between Markovian and non-Markovian processes. The original network is assumed a regular random graph with Q = 8. The solid line means the critical curve for Markovian process from combination of Eq. (26) and connection coefficient α and the dot line means the critical curve for non-Markovian process from Eq. (41).

Figure 3

Final epidemic size as a function of (a) the inverse of the effective recovery rate (1/r) for the Markovian process and (b) βσ for the non-Markovian process.