Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-01T07:42:05.726Z Has data issue: false hasContentIssue false
  • English
  • Français

Pairwise and Edge-based Models of Epidemic Dynamics onCorrelated Weighted Networks

Published online by Cambridge University Press:  24 April 2014

P. Rattana ,
J.C. Miller  and
I.Z. Kiss
P. Rattana
Affiliation:
School of Mathematical and Physical Sciences, Department of Mathematics University of Sussex, Falmer, Brighton BN1 9QH, UK
J.C. Miller
Affiliation:
School of Mathematical Sciences, School of Biological Sciences, and the Monash Academy for Cross & Interdisciplinary Mathematics, Monash University, , VIC 800, Australia
I.Z. Kiss*
Affiliation:
School of Mathematical and Physical Sciences, Department of Mathematics University of Sussex, Falmer, Brighton BN1 9QH, UK
*
⋆Corresponding author. E-mail: [email protected]
Get access

Abstract

In this paper we explore the potential of the pairwise-type modelling approach to beextended to weighted networks where nodal degree and weights are not independent. As abaseline or null model for weighted networks, we consider undirected, heterogenousnetworks where edge weights are randomly distributed. We show that the pairwise modelsuccessfully captures the extra complexity of the network, but does this at the cost oflimited analytical tractability due the high number of equations. To circumvent thisproblem, we employ the edge-based modelling approach to derive models corresponding to twodifferent cases, namely for degree-dependent and randomly distributed weights. Thesemodels are more amenable to compute important epidemic descriptors, such as early growthrate and final epidemic size, and produce similarly excellent agreement with simulation.Using a branching process approach we compute the basic reproductive ratio for both modelsand discuss the implication of random and correlated weight distributions on this as wellas on the time evolution and final outcome of epidemics. Finally, we illustrate that thetwo seemingly different modelling approaches, pairwise and edge-based, operate on similarassumptions and it is possible to formally link the two.

Keywords

weighted networkpairwise modeledge-based compartmental modellingprobability generating function
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 2: Epidemics models on networks , 2014 , pp. 58 - 81
Copyright
© EDP Sciences, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Rand, D.A.. Correlation equations and pair approximations for spatial ecologies. CWI Quarterly., 12 (1999), 329–368. Google Scholar
Gillespie, D.T.. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem., 81 (1977), 2340–2361. CrossRefGoogle Scholar
Ball, F., Neal, P.. Network epidemic models with two levels of mixing. Math. Biosci., 212 (2008), 69–87. CrossRefGoogle Scholar
J.C. Miller. Epidemics on networks with large initial conditions or changing structure. Available at http://arxiv.org/abs/1208.3438.
Miller, J.C.. Spread of infectious disease through clustered populations. J. Roy. Soc. Interface., 6 (2009), 1121–1134. CrossRefGoogle ScholarPubMed
Miller, J.C., Slim, A.C., Volz, E.M.. Edge-based compartmental modelling for infectious disease spread. J. Roy. Soc. Interface., 9 (2012), 890–906. CrossRefGoogle ScholarPubMed
J.C. Miller, E.M. Volz. Edge-based compartmental modeling with disease and population structure. Available at http://arxiv.org/abs/1106.6344.
Joo, J., Lebowitz, J.L.. Behavior of susceptible-infected-susceptible epidemics on heterogeneous networks with saturation. Phys. Rev. E., 69 (2004), 066105. CrossRefGoogle ScholarPubMed
J. Lindquist, J. Ma, P. Van den Driessche, Willeboordse, F.H.. Effective degree network disease models. J. Math. Biol., 62 (2011), 143–164. Google Scholar
K.B. Athreya, P.E. Ney, Branching processes. Dover Publications, Inc., Mineola, New York, 2008.
Sharkey, K.J., Fernandez, C., Morgan, K.L., Peeler, E., Thrush, M., Turnbull, J.F., Bowers, R.G.. Pair-level approximations to the spatio-temporal dynamics of epidemics on asymmetric contact networks. J. Math. Biol., 53 (2006), 61–85. CrossRefGoogle ScholarPubMed
Eames, K.T.D.. Modelling disease spread through random and regular contacts in clustered populations. Theor. Popul. Biol., 73 (2008), 104–111. CrossRefGoogle Scholar
Eames, K.T.D., Read, J.M., Edmunds, W.J., Epidemic prediction and control in weighted networks. Epidemics., 1 (2009), 70–76. CrossRefGoogle ScholarPubMed
Eames, K.T.D., Keeling, M.J.. Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases. Proc. Natl. Acad. Sci. USA., 99 (2002), 13330–13335. CrossRefGoogle ScholarPubMed
Deijfen, M.. Epidemics and vaccination on weighted graphs. Math. Biosci., 232 (2011), 57–65. CrossRefGoogle ScholarPubMed
Newman, M.E.J.. Spread of epidemic disease on networks. Phys. Rev. E., 66 (2002), 016128. CrossRefGoogle ScholarPubMed
Gilbert, M., Mitchell, A., Bourn, D., Mawdsley, J., Clifton-Hadley, R., Wint, W.. Cattle movements and bovine tuberculosis in Great Britain. Nature., 435 (2005), 491–496. CrossRefGoogle Scholar
Keeling, M.J.. The effects of local spatial structure on epidemiological invasions. Proc. R. Soc. Lond. B., 266 (1999), 859–867. CrossRefGoogle ScholarPubMed
Molloy, M., Reed, B.. A critical point for random graphs with a given degree sequence. Random Struct Alg., 6 (1995), 161–180. CrossRefGoogle Scholar
P. Rattana, K.B. Blyuss, K.T.D. Eames, I.Z. Kiss. A class of pairwise models for epidemic dynamics on weighted networks. Accepted for publication in Bull. Math. Biol., (2012).
Olinky, R., Stone, L.. Unexpected epidemic thresholds in heterogeneous networks: The role of disease transmission. Phys. Rev. E., 70 (2004), 030902(R). CrossRefGoogle ScholarPubMed
Britton, T., Deijfen, M., Liljeros, F.. A weighted configuration model and inhomogeneous epidemics. J. Stat. Phys., 145 (2011), 1368-1384. CrossRefGoogle Scholar
House, T., Keeling, M.J.. Insights from unifying modern approximations to infections on networks. J. Roy. Soc. Interface., 8 (2011), 67–73. CrossRefGoogle ScholarPubMed
V. Marceau, P-Noël, A., Hébert-Dufresne, L., Allard, A., Dubé, L.J.. Adaptive networks: coevolution of disease and topology. Phys. Rev. E., 82 (2010), 036116. Google Scholar
Miller, J. C., Kiss, I. Z.. Epidemic Spread in Networks: Existing Methods and Current Challenges, Math. Model. Nat. Phenom. Vol. 9, No. 2, (2014), 4–42. CrossRefGoogle ScholarPubMed