Epidemic spreading on complex networks with general degree and weight distributions

Wei Wang, Ming Tang, Hai-Feng Zhang, Hui Gao, Younghae Do, and Zong-Hua Liu

Phys. Rev. E 90, 042803 – Published 6 October 2014

Abstract

The spread of disease on complex networks has attracted wide attention in the physics community. Recent works have demonstrated that heterogeneous degree and weight distributions have a significant influence on the epidemic dynamics. In this study, a novel edge-weight-based compartmental approach is developed to estimate the epidemic threshold and epidemic size (final infected density) on networks with general degree and weight distributions, and a remarkable agreement with numerics is obtained. Even in complex networks with the strong heterogeneous degree and weight distributions, this approach is used. We then propose an edge-weight-based removal strategy with different biases and find that such a strategy can effectively control the spread of epidemic when the highly weighted edges are preferentially removed, especially when the weight distribution of a network is extremely heterogenous. The theoretical results from the suggested method can accurately predict the above removal effectiveness.

Received 2 July 2014

DOI:https://doi.org/10.1103/PhysRevE.90.042803

Authors & Affiliations

Wei Wang¹, Ming Tang^1,2,*, Hai-Feng Zhang³, Hui Gao¹, Younghae Do⁴, and Zong-Hua Liu⁵

¹Web Sciences Center, University of Electronic Science and Technology of China, Chengdu 610054, China
²Center for Atmospheric Remote Sensing(CARE), Kyungpook National University, Daegu 702-701, South Korea
³School of Mathematical Science, Anhui University, Hefei 230039, China
⁴Department of Mathematics, Kyungpook National University, Daegu 702-701, South Korea
⁵Department of Physics, East China Normal University, Shanghai 200062, China

^*tangminghuang521@hotmail.com

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Vol. 90, Iss. 4 — October 2014

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Figure 1
The influence of degree and weight distributions on the epidemic threshold and epidemic size. (a) The relative epidemic threshold $β_{c} / β_{c}^{0}$ as a function of weight exponent $γ_{W}$ for degree exponents $γ_{D} = 2.1$ (gray squares), $γ_{D} = 2.5$ (red circles), and $γ_{D} = 4.0$ (blue up-triangles), where $β_{c}^{0} \approx 0.016$ is the theoretical threshold at $γ_{D} = 4.0, γ_{W} = 2.1$ . The black solid, red dashed, and blue dot-dashed lines are the numerical solutions from Eq. (18). (b) Epidemic size $R (\infty)$ versus unit infection probability $β$ for $γ_{D} = 2.1, γ_{W} = 2.1$ (gray squares), $γ_{D} = 2.1, γ_{W} = 4.0$ (red circles), $γ_{D} = 4.0, γ_{W} = 2.1$ (blue up-triangles), and $γ_{D} = 4.0, γ_{W} = 4.0$ (green down-triangles). The black solid, red dashed, blue dot-dashed, and green dot lines are the numerical solutions from Eqs. (11, 12, 13, 14). The inset of (b) shows the numerical solutions of $〈 λ (w) 〉$ from Eq. (16) as function of $β$ for three different values of $γ_{W}$ (i.e., 2.1, 2.5, and 4.0), corresponding to the black solid, red dashed, and blue dot-dashed lines.
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Figure 2
Epidemic size changes with the degree and weight exponents. Color-coded values of epidemic size from numerical simulations (a) and theoretical solutions (b) are shown on the $γ_{D}, γ_{W}$ plane. (c) The increment of epidemic size $Δ R (γ_{D}, γ_{W}^{0}, \infty)$ as a function of degree exponent $γ_{D}$ at weight exponents $γ_{W}^{0} = 2.1$ (gray squares), $γ_{W}^{0} = 2.5$ (red circles), and $γ_{W}^{0} = 4.0$ (blue up-triangles). (d) $Δ R (γ_{D}^{0}, γ_{W}, \infty)$ as a function of $γ_{W}$ at $γ_{D}^{0} = 2.1$ (gray squares), $γ_{D}^{0} = 2.5$ (red circles), and $γ_{D}^{0} = 4.0$ (blue up-triangles). The values in (b) and the lines in (c) and (d) are the numerical solutions of Eqs. (11, 12, 13, 14) in the limit $t \to \infty$ . The unit infection probability is set to $β = 0.04$ .
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Figure 3
The control effectiveness of the edge-weight-based removal strategy with different bias. The relative epidemic threshold $β_{c} / β_{c}^{0}$ (a) and epidemic size $R (\infty)$ (b) as a function of parameter $α$ on different networks with tunable parameters $γ_{D} = 2.1, γ_{W} = 2.1$ (gray squares), $γ_{D} = 2.1, γ_{W} = 4.0$ (red circles), $γ_{D} = 4.0, γ_{W} = 2.1$ (blue up-triangles), and $γ_{D} = 4.0, γ_{W} = 4.0$ (green down-triangles). The black solid, red dashed, blue dot-dashed, and green dot lines are the analytical predictions from Eq. (18) for the relative threshold and Eqs. (11, 12, 13, 14) for the epidemic size, with the degree and weight distributions according to Eqs. (20) and (25), respectively. In (a), $β_{c}^{0}$ is the theoretical threshold on the original network.
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Figure 4
The decrement of epidemic size $▵ R^{'} (\infty)$ as a function of $γ_{D}$ and $γ_{W}$ for different values of $α$ . (a) and (c) represent, respectively, the numerical simulations for $α = 0.0$ and $α = 5.0$ , and the theoretical predictions for $α = 0.0$ and $α = 5.0$ are shown in (b) and (d), respectively. The unit infection probability is $β = 0.04$ .
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Figure 5
The relative epidemic threshold as a function of removal proportion $1 - f$ for (a) $α = 0.0$ and (b) $α = 5.0$ on different networks, including the parameters $γ_{D} = 2.1, γ_{W} = 2.1$ (gray squares), $γ_{D} = 2.1, γ_{W} = 4.0$ (red circles), $γ_{D} = 4.0, γ_{W} = 2.1$ (blue up-triangles), and $γ_{D} = 4.0, γ_{W} = 4.0$ (green down-triangles). The black solid, red dashed, blue dot-dashed, and green dot lines are the analytical predictions from Eq. (18) with the degree and weight distributions according to Eqs. (20) and (25), respectively. The parameter $β_{c}^{0}$ is the theoretical threshold on the original network.
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