zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Bounding the size and probability of epidemics on networks. (English) Zbl 1145.92029
Summary: We consider an infectious disease spreading along the edges of a network which may have significant clustering. The individuals in the population have heterogeneous infectiousness and/or susceptibility. We define the out-transmissibility of a node to be the marginal probability that it would infect a randomly chosen neighbor given its infectiousness and the distribution of susceptibility. For a given distribution of out-transmissibility, we find conditions which give the upper (or lower) bounds on the size and probability of an epidemic, under weak assumptions on the transmission properties, but very general assumptions on the network. We find similar bounds for a given distribution of in-transmissibility (the marginal probability of being infected by a neighbor). We also find conditions giving global upper bounds on the size and probability. The distributions leading to these bounds are network independent. In the special case of networks with high girth (locally tree-like), we are able to prove stronger results. In general, the probability and size of epidemics are maximal when the population is homogeneous and minimal when the variance of in- or out-transmissibility is maximal.

60K35Interacting random processes; statistical mechanics type models; percolation theory
Full Text: DOI
[1] Abbey, H. (1952). An examination of the Reed--Frost theory of epidemics. Human Biol. 24, 201--233.
[2] Anderson, R. M. and May, R. M. (1991). Infectious Diseases of Humans . Oxford University Press.
[3] Andersson, H. (1999). Epidemic models and social networks. Math. Scientist 24, 128--147. · Zbl 0951.92022
[4] Ball, F. (1985). Deterministic and stochastic epidemics with several kinds of susceptibles. Adv. Appl. Prob. 17, 1--22. JSTOR: · Zbl 0571.92019 · doi:10.2307/1427049 · http://links.jstor.org/sici?sici=0001-8678%28198503%2917%3A1%3C1%3ADASEWS%3E2.0.CO%3B2-%23&origin=euclid
[5] Ball, F. and O’Neill, P. (1999). The distribution of general final state random variables for stochastic epidemic models. J. Appl. Prob. 36, 473--491. · Zbl 0940.92021 · doi:10.1239/jap/1032374466
[6] Broder, A. et al. (2000). Graph structure in the web. Comput. Networks 33, 309--320.
[7] Del Valle, S. Y., Hyman, J. M., Hethcote, H. W. and Eubank, S. G. (2007). Mixing patterns between age groups in social networks. Social Networks 29, 539--554.
[8] Eubank, S. et al. (2004). Modelling disease outbreaks in realistic urban social networks. Nature 429, 180--184.
[9] Hastings, M. B. (2006). Systematic series expansions for processes on networks. Phys. Rev. Lett. 96, 148701.
[10] Keeling, M. J. (2005). The implications of network structure for epidemic dynamics. Theoret. Pop. Biol. 67, 1--8. · Zbl 1072.92043 · doi:10.1016/j.tpb.2004.08.002
[11] Keeling, M. J. and Eames, K. T. D. (2005). Networks and epidemic models. J. R. Soc. Interf. 2, 295--307.
[12] Kenah, E. and Robins, J. M. (2007). Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing. J. Theoret. Biol. 249, 706--722.
[13] Kenah, E. and Robins, J. M. (2007). Second look at the spread of epidemics on networks. Phys. Rev. E 76, 036113. · doi:10.1103/PhysRevE.76.036113
[14] Kermack, W. O. and McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proc. R. Soc. London Ser. A 115, 700--721. · Zbl 53.0517.01
[15] Kingman, J. F. C. (1978). Uses of exchangeability. Ann. Prob. 6, 183--197. JSTOR: · Zbl 0374.60064 · doi:10.1214/aop/1176995566 · http://links.jstor.org/sici?sici=0091-1798%28197804%296%3A2%3C183%3AUOE%3E2.0.CO%3B2-S&origin=euclid
[16] Kuulasmaa, K. (1982). The spatial general epidemic and locally dependent random graphs. J. Appl. Prob. 19, 745--758. JSTOR: · Zbl 0509.60094 · doi:10.2307/3213827 · http://links.jstor.org/sici?sici=0021-9002%28198212%2919%3A4%3C745%3ATSGEAL%3E2.0.CO%3B2-W&origin=euclid
[17] Kuulasmaa, K. and Zachary, S. (1984). On spatial general epidemics and bond percolation processes. J. Appl. Prob. 21, 911--914. JSTOR: · Zbl 0553.60095 · doi:10.2307/3213706 · http://links.jstor.org/sici?sici=0021-9002%28198412%2921%3A4%3C911%3AOSGSAB%3E2.0.CO%3B2-Z&origin=euclid
[18] Madar, N. et al. (2004). Immunization and epidemic dynamics in complex networks. Europ. Phys. J. B 38, 269--276.
[19] Meyers, L. A. (2007). Contact network epidemiology: bond percolation applied to infectious disease prediction and control. Bull. Amer. Math. Soc. 44, 63--86. · Zbl 1106.92061 · doi:10.1090/S0273-0979-06-01148-7
[20] Meyers, L. A., Newman, M. and Pourbohloul, B. (2006). Predicting epidemics on directed contact networks. J. Theoret. Biol. 240, 400--418. · doi:10.1016/j.jtbi.2005.10.004
[21] Miller, J. C. (2007). Epidemic size and probability in populations with heterogeneous infectivity and susceptibility. Phys. Rev. E 76, 010101.
[22] Mollison, D. (1977). Spatial contact models for ecological and epidemic spread. J. R. Statist. Soc. Ser. B 39, 283--326. JSTOR: · Zbl 0374.60110 · http://links.jstor.org/sici?sici=0035-9246%281977%2939%3A3%3C283%3ASCMFEA%3E2.0.CO%3B2-G&origin=euclid
[23] Molloy, M. and Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures Algorithms 6, 161--179. · Zbl 0823.05050 · doi:10.1002/rsa.3240060204
[24] Neal, P. (2007). Copuling of two SIR epidemic models with variable susceptibilities and infectivities. J. Appl. Prob. 44, 41--57. · Zbl 1130.92047 · doi:10.1239/jap/1175267162
[25] Newman, M. E. J. (2002). Spread of epidemic disease on networks. Phys. Rev. E 66, 16128.
[26] Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45, 167--256. · Zbl 1029.68010 · doi:10.1137/S003614450342480
[27] Pastor-Satorras, R. and Vespignani, A. (2001). Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200--3203.
[28] Serrano, M. and Boguñá, M. (2006). Percolation and epidemic thresholds in clustered networks. Phys. Rev. Lett. 97, 088701.
[29] Trapman, P. (2007). On analytical approaches to epidemics on networks. Theoret. Pop. Biol. 71, 160--173. · Zbl 1118.92055 · doi:10.1016/j.tpb.2006.11.002
[30] Van den Berg, J., Grimmett, G. R. and Schinazi, R. B. (1998). Dependent random graphs and spatial epidemics. Ann. Appl. Prob. 8, 317--336. · Zbl 0946.92028 · doi:10.1214/aoap/1028903529