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EXACT ANALYTICAL EXPRESSIONS FOR THE FINAL EPIDEMIC SIZE OF AN SIR MODEL ON SMALL NETWORKS

  • K. MCCULLOCH (a1), M. G. ROBERTS (a1) and C. R. LAING (a1)
Abstract

We investigate the dynamics of a susceptible infected recovered (SIR) epidemic model on small networks with different topologies, as a stepping stone to determining how the structure of a contact network impacts the transmission of infection through a population. For an SIR model on a network of $N$ nodes, there are $3^{N}$ configurations that the network can be in. To simplify the analysis, we group the states together based on the number of nodes in each infection state and the symmetries of the network. We derive analytical expressions for the final epidemic size of an SIR model on small networks composed of three or four nodes with different topological structures. Differential equations which describe the transition of the network between states are also derived and solved numerically to confirm our analysis. A stochastic SIR model is numerically simulated on each of the small networks with the same initial conditions and infection parameters to confirm our results independently. We show that the structure of the network, degree of the initial infectious node, number of initial infectious nodes and the transmission rate all significantly impact the final epidemic size of an SIR model on small networks.

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[1] Ball, F. and Neal, P., “On methods for studying stochastic disease dynamics”, J. R. Soc. Interface 5 (2008) 171181; doi:10.1016/j.mbs.2008.01.001.
[2] Bansal, S., Grenfell, B. and Meyers, L., “When individual behaviour matters: homogeneous and network models in epidemiology”, J. R. Soc. Interface 4 (2007) 879891; doi:10.1098/rsif.2007.1100.
[3] Barabási, A. and Albert, R., “Emergence of scaling in random networks”, Science 286 (1999) 509512; doi:10.1126/science.286.5439.509.
[4] Danon, L., Ford, A., House, T., Jewell, C., Kelling, M., Roberts, G., Ross, J. and Vernon, M., “Networks and the epidemiology of infectious disease”, Interdiscip. Perspect. Infect. Dis. 2011 (2011) 128; doi:10.1155/2011/284909.
[5] Gleeson, J., “High-accuracy approximation of binary-state dynamics on networks”, Phys. Rev. Lett. 107 (2011) 068701; doi:10.1103/physrevlett.107.068701.
[6] House, T. and Keeling, M., “Epidemic prediction and control in clustered populations”, J. Theoret. Biol. 272 (2010) 17; doi:10.1016/j.jtbi.2010.12.009.
[7] Keeling, M., “The effects of local spatial structures on epidemiological invasions”, Proc. R. Soc. Lond. B 266 (1999) 859867; doi:10.1098/rspb.1999.0716.
[8] Keeling, M. and Eames, K., “Networks and epidemic models”, J. R. Soc. Interface 2 (2005) 295307; doi:10.1098/rsif.2005.0051.
[9] Keeling, M. and Rohani, P., Modeling infectious diseases in humans and animals (Princeton University Press, Princeton, NJ, 2008).
[10] Kiss, I., Morris, C., Selley, F., Simon, P. and Wilkinson, R., “Exact deterministic representation of Markovian SIR epidemics on networks with and without loops”, J. Math. Biol. 70 (2015) 437464; doi:10.1007/s00285-014-0772-0.
[11] Lindquist, J., Ma, J., van den Driessche, P. and Willeboordse, F., “Effective degree network disease models”, J. Math. Biol. 62 (2011) 143164; doi:10.1007/s00285-010-0331-2.
[12] Lloyd, A. and Valeika, S., “Network models in epidemiology: an overview”, in: World Scientific Lecture Notes in Complex Systems (eds Blasius, B., Kurths, J. and Stone, L.), (World Scientific, Singapore, 2007) 189214 ; doi:10.1142/9789812771582_0008.
[13] Ma, J., van den Driessche, P. and Willeboordse, F., “The importance of contact network topology for the success of vaccination strategies”, J. Theoret. Biol. 325 (2013) 1221; doi:10.1016/j.jtbi.2013.01.006.
[14] Marceau, V., Noel, P., Hebert-Dufresne, L., Allard, A. and Dube, L., “Adaptive networks: coevolution of disease and topology”, Phys. Rev. E 82 (2010) 19; doi:10.1103/physreve.82.036116.
[15] Miller, J., “A note on a paper by Erik Volze: SIR dynamics in random networks”, J. Math. Biol. 62 (2010) 349358; doi:10.1007/s00285-010-0337-9.
[16] Miller, J. and Kiss, I., “Epidemic spread in networks: existing methods and current challenges”, Math. Model. Nat. Phenom. 9 (2014) 442; doi:10.1051/mmnp/20149202.
[17] Miller, J., Slim, A. and Volz, E., “Edge-based compartmental modelling for infectious disease spread”, J. R. Soc. Interface 9 (2012) 890906; doi:10.1098/rsif.2011.0403.
[18] Miller, J. and Volz, E., “Incorporating disease and population structure into models of SIR disease in contact networks”, PLoS ONE 8 (2013) e69162; doi:10.1371/journal.pone.0069162.
[19] Newman, M., Networks: an introduction (Oxford University Press, New York, 2010).
[20] Pastor-Satorras, R. and Vespignani, A., “Epidemic dynamics and endemic states in complex networks”, Phys. Rev. E 63 (2001) 066117; doi:10.1103/physreve.63.066117.
[21] Pellis, L., Ball, F., Bansal, S., Eames, K., House, T., Isham, V. and Trapman, P., “Eight challenges for network epidemic models”, Epidemics 10 (2015) 5862; doi:10.1016/j.epidem.2014.07.003.
[22] Ritchie, M., Berthouze, L., House, T. and Kiss, I., “Higher-order structure and epidemic dynamics in clustered networks”, J. Theoret. Biol. 348 (2014) 2132; doi:10.1016/j.jtbi.2014.01.025.
[23] Simon, P., Taylor, M. and Kiss, I., “Exact epidemic models on graphs using graph-automorphism driven lumping”, J. Math. Biol. 62 (2010) 479508; doi:10.1007/s00285-010-0344-x.
[24] Taylor, M., Taylor, T. and Kiss, I., “Epidemic threshold and control in a dynamic network”, Phys. Rev. E 85 (2012) 016103; doi:10.1103/physreve.85.016103.
[25] Volz, E., “SIR dynamics in random networks with heterogenous connectivity”, J. Math. Biol. 56 (2008) 293310; doi:10.1007/s00285-007-0116-4.
[26] Watts, D. and Strogatz, S., “Collective dynamics of ‘small-world’ networks”, Nature 393 (1998) 440442; doi:10.1038/30918.
[27] Zhao, Y. and Roberts, M., “Simulating epidemics on networks”, Res. Lett. Inform. Math. Sci. 6 (2007) 101103; http://www.massey.ac.nz/massey/learning/colleges/college-of-sciences/research/natural-mathematical-sciences/mathematics-research/en/simulating-epidemics-on-networks.cfm.
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