On analytical approaches to epidemics on networks. (English) Zbl 1118.92055
Summary: One way to describe the spread of an infection on a network is by approximating the network by a random graph. However, the usual way of constructing a random graph does not give any control over the number of triangles in the graph, while these triangles will naturally arise in many networks (e.g., in social networks). In this paper, random graphs with a given degree distribution and a given expected number of triangles are constructed.
By using these random graphs we analyze the spread of two types of infection on a network: infections with a fixed infectious period and infections for which an infective individual will infect all of its susceptible neighbors or none. These two types of infection can be used to give upper and lower bounds for \(R_0\), the probability of extinction and other measures of dynamics of infections with more general infectious periods.
By using these random graphs we analyze the spread of two types of infection on a network: infections with a fixed infectious period and infections for which an infective individual will infect all of its susceptible neighbors or none. These two types of infection can be used to give upper and lower bounds for \(R_0\), the probability of extinction and other measures of dynamics of infections with more general infectious periods.
MSC:
| 92D30 | Epidemiology |
| 05C80 | Random graphs (graph-theoretic aspects) |
| 94C99 | Circuits, networks |
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
References:
| [1] | Albert, R.; Barabasi, A. L., Statistical mechanics of complex networks, Rev. Mod. Phys., 74, 47-96 (2002) · Zbl 1205.82086 |
| [2] | Andersson, H., Epidemic models and social networks, Math. Sci., 24, 2, 128-147 (1999) · Zbl 0951.92022 |
| [3] | Andersson, H., Britton, T., 2000. Stochastic epidemic models and their statistical analysis. Springer Lecture Notes in Statistics, vol. 151. Springer, New York.; Andersson, H., Britton, T., 2000. Stochastic epidemic models and their statistical analysis. Springer Lecture Notes in Statistics, vol. 151. Springer, New York. · Zbl 0951.92021 |
| [4] | Ball, F., A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models, Adv. Appl. Probab., 18, 289-310 (1986) · Zbl 0606.92018 |
| [5] | Ball, F.; Mollison, D.; Scalia-Tomba, G., Epidemics with two levels of mixing, Ann. Appl. Probab., 7, 46-89 (1997) · Zbl 0909.92028 |
| [6] | Bearman, P. S.; Moody, J.; Stovel, C., Chains of affection: the structure of adolescent romantic and sexual networks, Am. J. Sociol., 110, 1, 44-91 (2004) |
| [7] | Cox, J. T.; Durrett, R., Limit theorems for the spread of epidemics and forest fires, Stochastic Process. Appl., 30, 2, 171-191 (1988) · Zbl 0667.92016 |
| [8] | Diekmann, O.; Heesterbeek, J. A.P., Mathematical Epidemiology of Infectious Diseases (2000), Wiley: Wiley Chichester · Zbl 0997.92505 |
| [9] | Durrett, R., Random Graph Dynamics (2006), Cambridge University Press: Cambridge University Press Cambridge |
| [10] | Ferguson, N. M.; Donnelly, C. A.; Anderson, R. M., Transmission intensity and impact of control policies on the foot and mouth epidemic in Great-Britain, Nature, 413, 542-547 (2001) |
| [11] | Grimmett, G. R., Percolation (1999), Springer: Springer Berlin, Heidelberg, New York · Zbl 0926.60004 |
| [12] | van der Hofstad, R.; Hooghiemstra, G.; van Mieghem, P., Distances in random graphs with finite variance degrees, Random Struct. Algorithms, 26, 76-123 (2005) · Zbl 1074.05083 |
| [13] | van der Hofstad, R., Hooghiemstra, G., Znamenski, D., 2006. Distances in random graphs with finite mean and infinite variance degrees, preprint at arXiv:math.PR/0502581.; van der Hofstad, R., Hooghiemstra, G., Znamenski, D., 2006. Distances in random graphs with finite mean and infinite variance degrees, preprint at arXiv:math.PR/0502581. · Zbl 1126.05090 |
| [14] | Jagers, P., Branching Processes with Biological Applications (1975), Wiley: Wiley London · Zbl 0356.60039 |
| [15] | Keeling, M. J., The effects of local spatial structure on epidemiological invasions, Proc. R. Soc. London B, 266, 859-867 (1999) |
| [16] | Keeling, M. J., The implications of network structure for epidemic dynamics, Theor. Popul. Biol., 67, 1, 1-8 (2005) · Zbl 1072.92043 |
| [17] | Keeling, M. J.; Eames, K. T.D., Review. Networks and epidemic models, J. R. Soc. Interface, 2, 4, 295-307 (2005) |
| [18] | Kuulasmaa, K., The spatial general epidemic and locally dependent random graphs, J. Appl. Probab., 19, 745-758 (1982) · Zbl 0509.60094 |
| [19] | Liljeros, F.; Edling, C. R.; Amaral, L. A.N.; Stanley, H. E.; Aberg, Y., The web of human sexual contacts, Nature, 411, 907-908 (2001) |
| [20] | Meester, R., Trapman, J.P., 2006. Estimation in branching processes with restricted observations. Adv. Appl. Probab. 38, 1098-1115.; Meester, R., Trapman, J.P., 2006. Estimation in branching processes with restricted observations. Adv. Appl. Probab. 38, 1098-1115. · Zbl 1119.92058 |
| [21] | Mollison, D., Spatial contact models for ecological and epidemic spread, J. R. Stat. Soc. Ser. B, 39, 3, 283-326 (1977) · Zbl 0374.60110 |
| [22] | Murrell, D. J.; Dieckmann, U.; Law, R., On moment closures for population dynamics in continuous space, J. Theor. Biol., 229, 3, 421-432 (2004) · Zbl 1440.92057 |
| [23] | Newman, M. E.J., Spread of epidemic disease on networks, Phys. Rev. E, 66, 1 (2002), Art. No. 016128 · Zbl 1001.68931 |
| [24] | Newman, M. E.J., Properties of highly clustered networks, Phys. Rev. E, 68, 2 (2003), Art. No. 026121 · Zbl 1029.68010 |
| [25] | Newman, M. E.J., The structure and function of complex networks, SIAM Rev., 45, 2, 167-256 (2003) · Zbl 1029.68010 |
| [26] | Petermann, T.; De Los Rios, P., Cluster approximations for epidemic processes: a systematic description of correlations beyond the pair level, J. Theor. Biol., 229, 1, 1-11 (2004) · Zbl 1440.92065 |
| [27] | Rand, D. A., Correlation equations and pair approximations for spatial ecologies, (McGlade, J., Advanced Ecological Theory (1999), Blackwell Scientific Publishing: Blackwell Scientific Publishing London), 99-143 · Zbl 1001.92552 |
| [28] | Trapman, J.P., 2006. On stochastic models for the spread of infections. Ph.D. Thesis, Vrije Universiteit Amsterdam.; Trapman, J.P., 2006. On stochastic models for the spread of infections. Ph.D. Thesis, Vrije Universiteit Amsterdam. |
| [29] | Watts, D. J.; Strogatz, S. H., Collective dynamics of ‘small-world’ networks, Nature, 393, 440-442 (2002) · Zbl 1368.05139 |
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