Pairwise and Edge-based Models of Epidemic Dynamics on Correlated Weighted Networks

P. Rattana¹ ; J.C. Miller² ; I.Z. Kiss¹

¹ School of Mathematical and Physical Sciences, Department of Mathematics University of Sussex, Falmer, Brighton BN1 9QH, UK
² School of Mathematical Sciences, School of Biological Sciences, and the Monash Academy for Cross & Interdisciplinary Mathematics, Monash University, , VIC 800, Australia

Mathematical modelling of natural phenomena, Tome 9 (2014) no. 2, pp. 58-81

Résumé

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

Détail

DOI : 10.1051/mmnp/20149204

Bibliographie

[1] D.A. Rand CWI Quarterly. 1999 329 368

[2] D.T. Gillespie J. Phys. Chem. 1977 2340 2361

[3] F. Ball, P. Neal Math. Biosci. 2008 69 87

[4] J.C. Miller. Epidemics on networks with large initial conditions or changing structure. Available at http://arxiv.org/abs/1208.3438.

[5] J.C. Miller J. Roy. Soc. Interface. 2009 1121 1134

[6] J.C. Miller, A.C. Slim, E.M. Volz J. Roy. Soc. Interface. 2012 890 906

[7] J.C. Miller, E.M. Volz. Edge-based compartmental modeling with disease and population structure. Available at http://arxiv.org/abs/1106.6344.

[8] J. Joo, J.L. Lebowitz Phys. Rev. E. 2004 066105

[9] F.H. Willeboordse J. Math. Biol. 2011 143 164

[10] K.B. Athreya, P.E. Ney, Branching processes. Dover Publications, Inc., Mineola, New York, 2008.

[11] K.J. Sharkey, C. Fernandez, K.L. Morgan, E. Peeler, M. Thrush, J.F. Turnbull, R.G. Bowers J. Math. Biol. 2006 61 85

[12] K.T.D. Eames Theor. Popul. Biol. 2008 104 111

[13] K.T.D. Eames, J.M. Read, W.J. Edmunds Epidemics. 2009 70 76

[14] K.T.D. Eames, M.J. Keeling Proc. Natl. Acad. Sci. USA. 2002 13330 13335

[15] M. Deijfen Math. Biosci. 2011 57 65

[16] M.E.J. Newman Phys. Rev. E. 2002 016128

[17] M. Gilbert, A. Mitchell, D. Bourn, J. Mawdsley, R. Clifton-Hadley, W. Wint Nature. 2005 491 496

[18] M.J. Keeling Proc. R. Soc. Lond. B. 1999 859 867

[19] M. Molloy, B. Reed Random Struct Alg. 1995 161 180

[20] 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).

[21] R. Olinky, L. Stone Phys. Rev. E. 2004 030902(R)

[22] T. Britton, M. Deijfen, F. Liljeros J. Stat. Phys. 2011 1368 1384

[23] T. House, M.J. Keeling J. Roy. Soc. Interface. 2011 67 73

[24] A. Noël, L. Hébert-Dufresne, A. Allard, L.J. Dubé Phys. Rev. E. 2010 036116

[25] J. C. Miller, I. Z. Kiss Math. Model. Nat. Phenom. Vol. 2014 4 42