Journal of Mathematical Biology

, Volume 62, Issue 4, pp 479–508

# Exact epidemic models on graphs using graph-automorphism driven lumping

• Péter L. Simon
• Michael Taylor
• Istvan Z. Kiss
Article

## Abstract

The dynamics of disease transmission strongly depends on the properties of the population contact network. Pair-approximation models and individual-based network simulation have been used extensively to model contact networks with non-trivial properties. In this paper, using a continuous time Markov chain, we start from the exact formulation of a simple epidemic model on an arbitrary contact network and rigorously derive and prove some known results that were previously mainly justified based on some biological hypotheses. The main result of the paper is the illustration of the link between graph automorphisms and the process of lumping whereby the number of equations in a system of linear differential equations can be significantly reduced. The main advantage of lumping is that the simplified lumped system is not an approximation of the original system but rather an exact version of this. For a special class of graphs, we show how the lumped system can be obtained by using graph automorphisms. Finally, we discuss the advantages and possible applications of exact epidemic models and lumping.

## Keywords

Network Epidemic Markov chain Lumping Graph automorphism

## Mathematics Subject Classification (2000)

92D25 92D30 92D40 00A71 00A72

## Preview

Unable to display preview. Download preview PDF.

## References

1. Anderson RM, May RM (1991) Infectious diseases of humans: dynamics and control. Oxford University PressGoogle Scholar
2. Andersson H, Djehich B (1997) A threshold limit theorem for the stochastic logistic epidemic. J Appl Prob 35: 662–670Google Scholar
3. Ball F, Mollison D, Salia-Tomba G (1997) Epidemics with two levels of mixing. Ann Appl Prob 7: 46–89
4. Brandes U, Erlebach T (2005) Network analysis: methodological foundations. Springer-Verlag, Berlin
5. Brauer F, van den Driessche P, Wu J (2008) In: (eds) Mathematical epidemiology. In: Lecture notes in mathematics. Springer-Verlag, BerlinGoogle Scholar
6. Broom M, Rychtář J (2008) An analysis of the fixation probability of a mutant on special classes of non-directed graphs. Proc R Soc A 464: 2609–2627
7. Daley DJ, Gani J (1999) Epidemic modelling: an intoduction. Cambridge University PressGoogle Scholar
8. Dent JE, Kao RR, Kiss IZ, Hyder K, Arnold M (2008) Contact structures in the poultry industry in Great Britain: exploring transmission routes for a potential avian influenza virus epidemic. BMC Vet Res 4: 27
9. Diekmann O, Heesterbeek JAP (2000) Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. Wiley, ChichesterGoogle Scholar
10. Diekmann O, De Jong MCM, Metz JAJ (1998) A deterministic epidemic model taking account of repeated contacts between the same individuals. J Appl Prob 35: 448–462
11. Ferguson NM, Donnelly CA, Anderson RM (2001) The foot-and mouth epidemic in Great Britain: pattern of spread and impact of interventions. Science 292: 1155–1160
12. Filliger L, Hongler M-O (2008) Lumping complex networks. Madeira Math Encounters XXXV. http://ccm.uma.pt/mme35/files/LumpingNetworks-Filliger.pdf
13. Green DM, Kiss IZ, Kao RR (2006) Modelling the initial spread of foot-and-mouth disease through animal movements. Proc R Soc B 273: 2729–2735
14. Gross JL, Yellen J (2003) Handbook of graph theory. CRC Press, Boca RatonGoogle Scholar
15. House T, Davies G, Danon L, Keeling MJ (2009) A motif-based approach to network epidemics. Bull Math Biol 71: 1693–1706
16. Hufnagel L, Brockmann D, Geisel T (2004) Forecast and control of epidemics in a globalized world. Proc Natl Acad Sci USA 101: 15124–15129
17. Hundsdorfer W, Verwer JG (2003) Numerical solution of time-dependent advection-diffusion-reactions equations. Springer, New YorkGoogle Scholar
18. Jacobi MN, Görnerup O (2009) A spectral method for aggregating variables in linear dynamical systems with application to cellular automata renormalization. Adv Complex Syst 12: 1–25
19. Kao RR, Danon L, Green DM, Kiss IZ (2006) Demographic structure and pathogen dynamics on the network of livestock movements in Great Britain. Proc R Soc B 273: 1999–2007
20. Keeling MJ (1999) The effects of local spatial structure on epidemiological invasions. Proc R Soc Lond B 266: 859–867
21. Keeling MJ (2005) The implications of network structure for epidemic dynamics. Theor Popul Biol 67: 1–8
22. Keeling MJ, Eames KTD (2005) Networks and epidemic models. J R Soc Interface 2: 295–307
23. Keeling MJ, Ross JV (2008) On methods for studying stochastic disease dynamics. J R Soc Interface 5: 171–181
24. Keeling MJ, Rand DA, Morris AJ (1997) Correlation models for childhood epidemics. Proc R Soc B 264: 1149–1156
25. Kemeny JG, Snell JL (1976) Finite Markov chains, 2nd edn. Springer, New York
26. Kenah E, Robins JM (2007) Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing. J Theor Biol 249: 706–722
27. Kermack WO, McKendrick AG (1927) A contribution to the mathematical study of epidemics. Proc R Soc Lond Ser A 115: 700–721
28. Kiss IZ, Green DM, Kao RR (2005) Disease contact tracing in random and clustered networks. Proc R Soc B 272: 1407–1414
29. Kiss IZ, Green DM, Kao RR (2006a) The network of sheep movements within Great Britain: network properties and their implications for infectious disease spread. J R Soc Interface 3: 669–677
30. Kiss IZ, Green DM, Kao RR (2006b) The effect of network heterogeneity and multiple routes of transmission on final epidemic size. Math Biosci 203: 124–136
31. Kiss IZ, Green DM, Kao RR (2008) The effect of network mixing patterns on epidemic dynamics and the efficacy of disease contact tracing. J R Soc Interface 5: 791–799
32. Kurtz TG (1970) Solutions of ordinary differential equations as limits of pure jump Markov processes. J Appl Prob 7: 49–58
33. Kurtz TG (1971) Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J Appl Prob 8: 344–356
34. Lipsitch M et al (2003) Transmission dynamics and control of severe acute respiratory syndrome. Science 300: 1966–1970
35. May RM, Lloyd AL (2001) Infection dynamics on scale-free networks. Phys Rev E 64: 066112
36. Meyers LA, Pourbohloul B, Newman MEJ, Skowronski DM, Brunham RC (2005) Network theory and SARS: predicting outbreak diversity. J Theor Biol 232: 71–81
37. Nåsell I (1996) The quasi-stationary distribution of the closed endemic SIS model. Adv Appl Probab 28: 895–932
38. Newman MEJ (2002) The spread of epidemic disease on networks. Phys Rev E 66: 016128
39. Picard P (1965) Sur les modèles stochastique logistiques en démographie. Ann Inst Henri Poincaré B II: 151–172
40. Rand DA (1999) Correlation equations for spatial ecologies. In: McGlade J (eds) Advanced ecological theory. Blackwell, Oxford, pp 100–142
41. Sato K, Matsuda H, Sasaki A (1994) Pathogen invasion and host extinction in lattice structured populations. J Math Biol 32: 251–268
42. Sharkey KJ (2008) Deterministic epidemiological models at the individual level. J Math Biol 57: 311–331
43. Smith GJ et al (2009) Origins and evolutionary genomics of the 2009 swine-origin H1N1 influenza A epidemic. Nature 459: 1122–1125
44. van Baalen M (2000) Pair approximations for different spatial geometries. In: Dieckmann U, Law R, Metz JAJ (eds) The geometry of ecological interactions: simplifying complexity. Cambridge University Press, pp 359–387Google Scholar
45. Yap HP (1986) Some topics in graph theory. In: London Mathematical Society, Lecture notes series 108. Cambridge University Press, CambridgeGoogle Scholar