zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Analysis of a stochastic SIR epidemic on a random network incorporating household structure. (English) Zbl 1192.92037
Summary: This paper is concerned with a stochastic SIR (susceptible infective removed) model for the spread of an epidemic amongst a population of individuals, with a random network of social contacts, that is also partitioned into households. The behaviour of the model as the population size tends to infinity in an appropriate fashion is investigated. A threshold parameter which determines whether or not an epidemic with few initial infectives can become established and lead to a major outbreak is obtained, as are the probability that a major outbreak occurs and the expected proportion of the population that are ultimately infected by such an outbreak, together with methods for calculating these quantities. Monte Carlo simulations demonstrate that these asymptotic quantities accurately reflect the behaviour of finite populations, even for only moderately sized finite populations. The model is compared and contrasted with related models previously studied in the literature. The effects of the amount of clustering present in the overall population structure and the infectious period distribution on the outcomes of the model are also explored.
MSC:
92D30Epidemiology
60J85Applications of branching processes
65C05Monte Carlo methods
References:
[1]Bartoszyński, R.: On a certain model of an epidemic, Zastos. mat. 13, 139 (1972/1973)
[2]Becker, N. G.; Dietz, K.: The effect of household distribution on transmission and control of highly infectious diseases, Math. biosci. 127, 207 (1995) · Zbl 0824.92025 · doi:10.1016/0025-5564(94)00055-5
[3]Ball, F. G.; Mollison, D.; Scalia-Tomba, G.: Epidemics with two levels of mixing, Ann. appl. Probab. 7, No. 1, 46 (1997) · Zbl 0909.92028 · doi:10.1214/aoap/1034625252
[4]Andersson, H.: Epidemic models and social networks, Math. sci. 24, No. 2, 128 (1999) · Zbl 0951.92022
[5]Newman, M. E. J.: Spread of epidemic disease on networks, Phys. rev. E 66, No. 1, 016128 (2002)
[6]Kenah, E.; Robins, J. M.: Second look at the spread of epidemics on networks, Phys. rev. E 76, No. 3, 036113 (2007)
[7]Meyers, L. A.; Pourbohloul, B.; Newman, M. E. J.; Skowronski, D. M.; Brunham, R. C.: Network theory and SARS: predicting outbreak diversity, J. theoret. Biol. 232, No. 1, 71 (2005)
[8]Kiss, I. Z.; Green, D. M.; Kao, R. R.: The effect of contact heterogeneity and multiple routes of transmission on final epidemic size, Math. biosci. 203, No. 1, 124 (2006) · Zbl 1099.92063 · doi:10.1016/j.mbs.2006.03.002
[9]Ball, F. G.; Neal, P. J.: Network epidemic models with two levels of mixing, Math. biosci. 212, No. 1, 69 (2008) · Zbl 1132.92020 · doi:10.1016/j.mbs.2008.01.001
[10]Britton, T.; Deijfen, M.; Lagerås, A. N.; Lindholm, M.: Epidemics on random graphs with tunable clustering, J. appl. Probab. 45, No. 3, 743 (2008) · Zbl 1147.92034 · doi:10.1239/jap/1222441827
[11]Ball, F. G.; Sirl, D. J.; Trapman, P.: Threshold behaviour and final outcome of an epidemic on a random network with household structure, Adv. appl. Probab. 41, No. 3, 765 (2009) · Zbl 1176.92042 · doi:10.1239/aap/1253281063
[12]Berchenko, Y.; Artzy-Randrup, Y.; Teicher, M.; Stone, L.: Emergence and size of the giant component in clustered random graphs with a given degree distribution, Phys. rev. Lett. 102, 138701 (2009)
[13]B. Bollobás, S. Janson, O. Riordan, Sparse random graphs with clustering, arXiv:0807.2040v1, 2008.
[14]Gleeson, J. P.: Bond percolation on a class of clustered random networks, Phys. rev. E 80, No. 3, 036107 (2009)
[15]Gleeson, J. P.; Melnik, S.: Analytical results for Bond percolation and k-core sizes on clustered networks, Phys. rev. E 80, No. 4, 046121 (2009)
[16]House, T.; Davies, G.; Danon, L.; Keeling, M. J.: A motif-based approach to network epidemics, Bull. math. Biol. 71, 1693 (2009) · Zbl 1173.92028 · doi:10.1007/s11538-009-9420-z
[17]Miller, J. C.: Bounding the size and probability of epidemics on networks, J. appl. Probab. 45, No. 2, 498 (2008) · Zbl 1145.92029 · doi:10.1239/jap/1214950363
[18]Miller, J. C.: Percolation and epidemics in random clustered networks, Phys. rev. E 80, No. 2, 020901 (2009)
[19]Miller, J. C.: Spread of infectious disease through clustered populations, J.R. soc. Interface 6, 1121 (2009)
[20]Newman, M. E. J.: Random graphs with clustering, Phys. rev. Lett. 103, 058701 (2009)
[21]Smieszek, T.; Fiebig, L.; Scholz, R. W.: Models of epidemics: when contact repetition and clustering should be included, Theor. biol. Med. model. 6, 11 (2009)
[22]Vazquez, A.: Spreading dynamics on small-world networks with connectivity fluctuations and correlations, Phys. rev. E 74, No. 5, 056101 (2006)
[23]Trapman, P.: On analytical approaches to epidemics on networks, Theor. popul. Biol. 71, 160 (2007) · Zbl 1118.92055 · doi:10.1016/j.tpb.2006.11.002
[24]Bender, E. A.; Canfield, E. R.: The asymptotic number of labeled graphs with given degree sequences, J. combinatorial theory (A) 24, No. 3, 296 (1978) · Zbl 0402.05042 · doi:10.1016/0097-3165(78)90059-6
[25]Molloy, M.; Reed, B.: A critical point for random graphs with a given degree sequence, Rand. struct. Alg. 6, No. 2 – 3, 161 (1995) · Zbl 0823.05050 · doi:10.1002/rsa.3240060204
[26]Pellis, L.; Ferguson, N. M.; Fraser, C.: The relationship between real-time and discrete-generation models of epidemic spread, Math. biosci. 216, No. 1, 63 (2008) · Zbl 1151.92029 · doi:10.1016/j.mbs.2008.08.009
[27]Durrett, R.: Random graph dynamics, Cambridge series in statistical and probabilistic mathematics, (2006)
[28]Ludwig, D.: Final size distributions for epidemics, Math. biosci. 23, 33 (1975) · Zbl 0318.92025 · doi:10.1016/0025-5564(75)90119-4
[29]N.G. Becker, K. Dietz, Reproduction numbers and critical immunity levels for epidemics in a community of households, in: Athens Conference on Applied Probability and Time Series Analysis, vol. I (1995), Lecture Notes in Statistics, vol. 114, Springer, New York, 1996, pp. 267 – 276. · Zbl 0854.92016
[30]Goldstein, E.; Paur, K.; Fraser, C.; Kenah, E.; Wallinga, J.; Lipsitch, M.: Reproductive numbers, epidemic spread and control in a community of households, Math. biosci. 221, 11 (2009) · Zbl 1172.92030 · doi:10.1016/j.mbs.2009.06.002
[31]H.E. Daniels, The distribution of the total size of an epidemic, in: Proceedings of the Fifth Berkeley Symposium in Mathematical Statistics and Probability, vol. IV, University of California, Berkeley, CA, 1967, pp. 281 – 293.
[32]Lefèvre, C.; Picard, P.: A nonstandard family of polynomials and the final size distribution of Reed – frost epidemic processes, Adv. appl. Probab. 22, No. 1, 25 (1990) · Zbl 0709.92020 · doi:10.2307/1427595
[33]Ball, F. G.: A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models, Adv. appl. Probab. 18, No. 2, 289 (1986) · Zbl 0606.92018 · doi:10.2307/1427301
[34]Ball, F. G.; O’neill, P. D.: The distribution of general final state random variables for stochastic epidemic models, J. appl. Probab. 36, No. 2, 473 (1999) · Zbl 0940.92021 · doi:10.1239/jap/1032374466
[35]Andersson, H.; Britton, T.: Stochastic epidemic models and their statistical analysis, Lecture notes in statistics 151 (2000)
[36]Ball, F. G.; Lyne, O. D.: Stochastic multitype SIR epidemics among a population partitioned into households, Adv. appl. Probab. 33, No. 1, 99 (2001) · Zbl 0978.92025 · doi:10.1239/aap/999187899
[37]Ball, F. G.; Neal, P. J.: A general model for stochastic SIR epidemics with two levels of mixing, Math. biosci. 180, 73 (2002) · Zbl 1015.92034 · doi:10.1016/S0025-5564(02)00125-6
[38]Gertsbakh, I. B.: Epidemic process on a random graph: some preliminary results, J. appl. Probab. 14, No. 3, 427 (1977) · Zbl 0373.92032 · doi:10.2307/3213446
[39]Kenah, E.; Robins, J. M.: Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing, J. theoret. Biol. 249, 706 (2007)
[40]F.G. Ball, Susceptibility sets and the final outcome of stochastic SIR epidemic models, Research Report 00-09, Division of Statistics, School of Mathematical Sciences, University of Nottingham, 2000.
[41]R. Meester, P. Trapman, Bounding basic characteristics of spatial epidemics with a new percolation model, arXiv:0812.4353v1, 2009.
[42]Durrett, R.: Probability: theory and examples, (2004)
[43]Grossman, Z.: Oscillatory phenomena in a model of infectious diseases, Theor. popul. Biol. 18, No. 2, 204 (1980) · Zbl 0457.92020 · doi:10.1016/0040-5809(80)90050-7
[44]Keeling, M. J.; Grenfell, B. T.: Effect of variability in infection period on the persistence and spatial spread of infectious diseases, Math. biosci. 147, No. 2, 207 (1998) · Zbl 0887.92028 · doi:10.1016/S0025-5564(97)00101-6
[45]Kuulasmaa, K.: The spatial general epidemic and locally dependent random graphs, J. appl. Probab. 19, No. 4, 745 (1982) · Zbl 0509.60094 · doi:10.2307/3213827
[46]Feller, W.: Second ed.an introduction to probability theory and its applications, An introduction to probability theory and its applications 2 (1971) · Zbl 0219.60003
[47]Martin-Löf, A.: Symmetric sampling, procedures, general epidemic processes and their threshold limit theorems, J. appl. Probab. 23, No. 2, 265 (1986) · Zbl 0605.92009 · doi:10.2307/3214172