Temporal Network Epidemiology pp 105-128 | Cite as

# Mean Field at Distance One

## Abstract

To be able to understand how infectious diseases spread on networks, it is important to understand the network structure itself in the absence of infection. In this text we consider dynamic network models that are inspired by the (static) configuration network. The networks are described by population-level averages such as the fraction of the population with *k* partners, *k* = 0, 1, 2, *…* This means that the bookkeeping contains information about individuals and their partners, but no information about partners of partners. Can we average over the population to obtain information about partners of partners? The answer is ‘it depends’, and this is where the mean field at distance one assumption comes into play. In this text we explain that, yes, we may average over the population (in the right way) in the static network. Moreover, we provide evidence in support of a positive answer for the network model that is dynamic due to partnership changes. If, however, we additionally allow for demographic changes, dependencies between partners arise. In earlier work we used the slogan ‘mean field at distance one’ as a justification of simply ignoring the dependencies. Here we discuss the subtleties that come with the mean field at distance one assumption, especially when demography is involved. Particular attention is given to the accuracy of the approximation in the setting with demography. Next, the mean field at distance one assumption is discussed in the context of an infection superimposed on the network. We end with the conjecture that an extension of the bookkeeping leads to an exact description of the network structure.

## Notes

### Acknowledgements

We would like to thank Pieter Trapman for opening our eyes during the Infectious Disease Dynamics meeting at the Isaac Newton Institute in Cambridge in 2013 as well as the members of the infectious disease dynamics journal clubs in Utrecht and Stockholm, and two anonymous reviewers for helpful comments.

K.Y. Leung is supported by the Netherlands Organisation for Scientific Research (NWO) [grant Mozaïek 017.009.082] and the Swedish Research Council [grant number 2015-05015_3].

## References

- 1.Leung, K.Y., Diekmann, O.: Dangerous connections: on binding site models of infectious disease dynamics. J. Math. Biol.
**74**, 619–671 (2017)MathSciNetCrossRefMATHGoogle Scholar - 2.Barbour, A.D., Reinert, G.: Approximating the epidemic curve. Electron. J. Probab.
**18**(54), 1–30 (2013)MathSciNetMATHGoogle Scholar - 3.Volz, E.M.: IR dynamics in random networks with heterogeneous connectivity. J. Math. Biol.
**56**, 293–310 (2008)MathSciNetCrossRefMATHGoogle Scholar - 4.Kiss, I.Z., Miller, J.C., Simon, O.: Mathematics of Epidemics on Networks: From Exact to Approximate Models. Springer, Cham (2017)CrossRefMATHGoogle Scholar
- 5.Durrett, R.: Random Graph Dynamics. Cambridge University Press, Cambridge (2006)CrossRefMATHGoogle Scholar
- 6.Van der Hofstad, R.: Random Graphs and Complex Networks, vol. I. Cambridge University Press, Cambridge (2016)Google Scholar
- 7.Leung, K.Y., Kretzschmar, M.E.E., Diekmann, O.: Dynamic concurrent partnership networks incorporating demography. Theor. Popul. Biol.
**82**, 229–239 (2012)CrossRefMATHGoogle Scholar - 8.Britton, T., Lindholm, M.: Dynamic random networks in dynamic populations. J. Stat. Phys.
**139**, 518–535 (2010)MathSciNetCrossRefMATHGoogle Scholar - 9.Britton, T., Lindholm, M., Turova, T.: A dynamic network in a dynamic population: asymptotic properties. J. Appl. Prob.
**48**, 1163–1178 (2011)MathSciNetCrossRefMATHGoogle Scholar - 10.Lashari, A.A., Trapman, P.: Branching process approach for epidemics in dynamic partnership network. J. Math. Biol. (2017). doi:10.1007/s00285-017-1147-0Google Scholar
- 11.Leung, K.Y., Kretzschmar, M.E.E., Diekmann, O.:
*SI*infection of a dynamic partnership network: characterization of*R*_{0}. J. Math. Biol.**71**, 1–56 (2015)MathSciNetCrossRefMATHGoogle Scholar - 12.Newman, M.E.J.: Assortative mixing in networks. Phys. Rev. Lett.
**89**(20), 208701 (2002)CrossRefGoogle Scholar - 13.Newman, M.E.J.: The structure and function of complex networks. SIAM Rev.
**45**(2), 167–256 (2003)MathSciNetCrossRefMATHGoogle Scholar - 14.Ball, F., Britton, T., Sirl, D.: A network with tunable clustering, degree correlation and degree distribution, and an epidemic thereon. J. Math. Biol.
**66**, 979–1019 (2013)MathSciNetCrossRefMATHGoogle Scholar - 15.Decreusefond, L., Dhersin, J.-S., Moyal, P., Tran, V.C.: Large graph limit for an SIR process in random network with heterogeneous connectivity. Ann. Appl. Probab.
**22**, 541–575 (2012)MathSciNetCrossRefMATHGoogle Scholar - 16.Janson, S., Luczak, M., Windridge, P.: Law of large numbers for the SIR epidemic on a random graph with given degrees. Random Struct. Algor.
**45**(4), 724–761 (2014)MathSciNetCrossRefMATHGoogle Scholar - 17.Diekmann, O., Heesterbeek, J.A.P., Britton, T.: Mathematical Tools for Understanding Infectious Disease Dynamics. Princeton University Press, Princeton (2013)MATHGoogle Scholar