# The Accuracy of Mean-Field Approximation for Susceptible-Infected-Susceptible Epidemic Spreading with Heterogeneous Infection Rates

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## Abstract

The epidemic spreading over a network has been studied for years by applying the mean-field approach in both homogeneous case, where each node may get infected by an infected neighbor with the same rate, and heterogeneous case, where the infection rates between different pairs of nodes are also different. Researchers have discussed whether the mean-field approaches could accurately describe the epidemic spreading for the homogeneous cases but not for the heterogeneous cases. In this paper, we explore if and under what conditions the mean-field approach could perform well when the infection rates are heterogeneous. In particular, we employ the Susceptible-Infected-Susceptible (SIS) model and compare the average fraction of infected nodes in the metastable state, where the fraction of infected nodes remains stable for a long time, obtained by the continuous-time simulation and the mean-field approximation. We concentrate on an individual-based mean-field approximation called the N-intertwined Mean Field Approximation (NIMFA), which is an advanced approach considered the underlying network topology. Moreover, for the heterogeneity of the infection rates, we consider not only the independent and identically distributed (i.i.d.) infection rate but also the infection rate correlated with the degree of the two end nodes. We conclude that NIMFA is generally more accurate when the prevalence of the epidemic is higher. Given the same effective infection rate, NIMFA is less accurate when the variance of the i.i.d. infection rate or the correlation between the infection rate and the nodal degree leads to a lower prevalence. Moreover, given the same actual prevalence, NIMFA performs better in the cases: 1) when the variance of the i.i.d. infection rates is smaller (while the average is unchanged); 2) when the correlation between the infection rate and the nodal degree is positive. Our work suggests the conditions when the mean-field approach, in particular NIMFA, is more accurate in the approximation of the SIS epidemic with heterogeneous infection rates.

## Keywords

Infection Rate Metastable State Nodal Degree Average Fraction Epidemic Spreading## References

- 1.Albert, R., Jeong, H., Barabási, A.L.: Internet: Diameter of the world-wide web. Nature
**401**(6749), 130–131 (1999)Google Scholar - 2.Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science
**286**(5439), 509–512 (1999)Google Scholar - 3.Barrat, A., Barthelemy, M., Pastor-Satorras, R., Vespignani, A.: The architecture of complex weighted networks. Proceedings of the National Academy of Sciences of the United States of America
**101**(11), 3747–3752 (2004)Google Scholar - 4.Buono, C., Vazquez, F., Macri, P., Braunstein, L.: Slow epidemic extinction in populations with heterogeneous infection rates. Physical Review E
**88**(2), 022,813 (2013)Google Scholar - 5.Caldarelli, G., Marchetti, R., Pietronero, L.: The fractal properties of internet. EPL (Europhysics Letters)
**52**(4), 386 (2000)Google Scholar - 6.Cohen, R., Erez, K., ben Avraham, D., Havlin, S.: Resilience of the internet to random breakdowns. Physical Review Letters
**85**, 4626–4628 (2000). DOI 10.1103/PhysRevLett.85. 4626. URL http://link.aps.org/doi/10.1103/PhysRevLett.85.4626 - 7.Li, C., van de Bovenkamp, R., Van Mieghem, P.: Susceptible-infected-susceptible model: A comparison of n-intertwined and heterogeneous mean-field approximations. Phys. Rev. E
**86**(2), 026,116 (2012)Google Scholar - 8.Li, D., Qin, P., Wang, H., Liu, C., Jiang, Y.: Epidemics on interconnected lattices. EPL (Europhysics Letters)
**105**(6), 68,004 (2014). URL http://stacks.iop.org/0295-5075/105/i=6/a=68004 - 9.Li, W., Cai, X.: Statistical analysis of airport network of china. Physical Review E
**69**(4), 046,106 (2004)Google Scholar - 10.Liu, M., Li, D., Qin, P., Liu, C., Wang, H., Wang, F.: Epidemics in interconnected small-world networks. PloS one
**10**(3), e0120,701 (2015)Google Scholar - 11.Macdonald, P., Almaas, E., Barabási, A.L.: Minimum spanning trees of weighted scale-free networks. EPL (Europhysics Letters)
**72**(2), 308 (2005)Google Scholar - 12.Pastor-Satorras, R., Castellano, C., Van Mieghem, P., Vespignani, A.: Epidemic processes in complex networks. arXiv preprint arXiv:1408.2701 (2014)Google Scholar
- 13.Pastor-Satorras, R., Vespignani, A.: Epidemic dynamics and endemic states in complex networks. Physical Review E
**63**(6), 066,117 (2001)Google Scholar - 14.Pastor-Satorras, R., Vespignani, A.: Epidemic spreading in scale-free networks. Physical Review Letters
**86**(14), 3200 (2001)Google Scholar - 15.Qu, B., Wang, H.: SIS epidemic spreading with heterogeneous infection rates. arXiv preprint arXiv:1506.07293 (2015)Google Scholar
- 16.Qu, B., Wang, H.: The accuracy of mean-field approximation for susceptible-infectedsusceptible epidemic spreading. arXiv preprint arXiv:1609.01105 (2016)Google Scholar
- 17.Qu, B.,Wang, H.: SIS epidemic spreading with correlated heterogeneous infection rates. arXiv preprint arXiv:1608.07327 (2016)Google Scholar
- 18.Van Mieghem, P.: Performance analysis of communications networks and systems. Cambridge University Press (2014)Google Scholar
- 19.Van Mieghem, P., Omic, J., Kooij, R.: Virus spread in networks. IEEE/ACM Transactions on Networking
**17**(1), 1–14 (2009)Google Scholar - 20.Wang, H., Li, Q., D’Agostino, G., Havlin, S., Stanley, H.E., Van Mieghem, P.: Effect of the interconnected network structure on the epidemic threshold. Physical Review E
**88**(2), 022,801 (2013)Google Scholar - 21.Wang,W.,Wu, Z.,Wang, C., Hu, R.: Modelling the spreading rate of controlled communicable epidemics through an entropy-based thermodynamic model. Sci. Sin.-Phys. Mech. Astron.
**56**(11), 2143 (2013). DOI 10.1007/s11433-013-5321-0 - 22.Yang, Z., Zhou, T.: Epidemic spreading in weighted networks: an edge-based mean-field solution. Physical Review E
**85**(5), 056,106 (2012)Google Scholar