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January  2017, 14(1): 67-77. doi: 10.3934/mbe.2017005

## Network-based analysis of a small Ebola outbreak

 1 College of Public Health, The Ohio State University, Columbus, OH 43210, USA 2 Mathematical Biosciences Institute, The Ohio State University, Columbus, OH 43210, USA 3 Department of Mathematics and Mathematical Biosciences Institute, The Ohio State University, Columbus, OH 43210, USA 4 College of Public Health and Mathematical Biosciences Institute, The Ohio State University, Columbus, OH 43210, USA

Received  November 06, 2015 Accepted  April 15, 2016 Published  October 2016

Fund Project: This research has been supported in part by the Mathematical Biosciences Institute and the National Science Foundation under grants RAPID DMS-1513489 and DMS-1440386.

We present a method for estimating epidemic parameters in network-based stochastic epidemic models when the total number of infections is assumed to be small. We illustrate the method by reanalyzing the data from the 2014 Democratic Republic of the Congo (DRC) Ebola outbreak described in Maganga et al. (2014).

Citation: Mark G. Burch, Karly A. Jacobsen, Joseph H. Tien, Grzegorz A. Rempała. Network-based analysis of a small Ebola outbreak. Mathematical Biosciences & Engineering, 2017, 14 (1) : 67-77. doi: 10.3934/mbe.2017005
##### References:
 [1] L. J. Allen, An introduction to stochastic epidemic models, Mathematical Epidemiology,Springer, 1945 (2008), 81-130. doi: 10.1007/978-3-540-78911-6_3. [2] M. Altmann, The deterministic limit of infectious disease models with dynamic partners, Mathematical Biosciences, 150 (1998), 153-175.  doi: 10.1016/S0025-5564(98)00012-1. [3] H. Andersson and T. Britton, Stochastic Epidemic Models and Their Statistical Analysis vol. 4, Springer New York, 2000. doi: 10.1007/978-1-4612-1158-7. [4] A.D. Barbour and G. Reinert, Approximating the epidemic curve, Electronic Journal of Probability, 18 (2013), 1-30.  doi: 10.1214/EJP.v18-2557. [5] M. Boguñá, C. Castellano and R. Pastor-Satorras, Nature of the epidemic threshold for the susceptible-infected-susceptible dynamics in networks, Physical Review Letters, 111 (2013), 068701. [6] T. Bohman and M. Picollelli, SIR epidemics on random graphs with a fixed degree sequence, Random Structures & Algorithms, 41 (2012), 179-214.  doi: 10.1002/rsa.20401. [7] T. Britton and P.D. O'Neill, Bayesian inference for stochastic epidemics in populations with random social structure, Scandinavian Journal of Statistics, 29 (2002), 375-390.  doi: 10.1111/1467-9469.00296. [8] B. Choi and G.A. Rempala, Inference for discretely observed stochastic kinetic networks with applications to epidemic modeling, Biostatistics, 13 (2012), 153-165.  doi: 10.1093/biostatistics/kxr019. [9] G. Chowell and H. Nishiura, Transmission dynamics and control of Ebola virus disease (EVD): A review, BMC Medicine, 12 (2014), p196. doi: 10.1186/s12916-014-0196-0. [10] L. Decreusefond, J.-S. Dhersin, P. Moyal and V.C. Tran, Large graph limit for an SIR process in random network with heterogeneous connectivity, The Annals of Applied Probability, 22 (2012), 541-575.  doi: 10.1214/11-AAP773. [11] C. Groendyke, D. Welch and D.R. Hunter, Bayesian inference for contact networks given epidemic data, Scandinavian Journal of Statistics, 38 (2011), 600-616.  doi: 10.1111/j.1467-9469.2010.00721.x. [12] C. Groendyke, D. Welch and D.R. Hunter, A network-based analysis of the 1861 Hagelloch measles data, Biometrics, 68 (2012), 755-765.  doi: 10.1111/j.1541-0420.2012.01748.x. [13] T. House and M.J. Keeling, Insights from unifying modern approximations to infections on networks, Journal of The Royal Society Interface, 8 (2011), 67-73.  doi: 10.1098/rsif.2010.0179. [14] S. Janson, M. Luczak and P. Windridge, Law of large numbers for the SIR epidemic on a random graph with given degrees, Random Structures & Algorithms, 45 (2014), 726-763.  doi: 10.1002/rsa.20575. [15] J. Janssen, Semi-Markov Models: Theory and Applications, New York, Plenum Press, 1986.  doi: 10.1007/978-1-4899-0574-1. [16] M. Keeling, The implications of network structure for epidemic dynamics, Theoretical Population Biology, 67 (2005), 1-8.  doi: 10.1016/j.tpb.2004.08.002. [17] M.J. Keeling, The effects of local spatial structure on epidemiological invasions, Proceedings of the Royal Society of London B: Biological Sciences, 266 (1999), 859-867.  doi: 10.1098/rspb.1999.0716. [18] M. Keeling, Correlation equations for endemic diseases: externally imposed and internally generated heterogeneity, Proceedings of the Royal Society of London B: Biological Sciences, 266 (1999), 953-960.  doi: 10.1098/rspb.1999.0729. [19] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 115, The Royal Society, 1927,700-721. [20] T.G. Kurtz, Limit theorems for sequences of jump markov processes approximating ordinary differential processes, Journal of Applied Probability, 8 (1971), 344-356.  doi: 10.1017/S002190020003535X. [21] J. Legrand, R. Grais, P. Boelle, A. Valleron and A. Flahault, Understanding the dynamics of Ebola epidemics, Epidemiology and Infection, 135 (2007), 610-621. [22] G.D. Maganga, J. Kapetshi, N. Berthet, B. Kebela Ilunga, F. Kabange, P. Mbala Kingebeni, V. Mondonge, J.-J. T. Muyembe, E. Bertherat and S. Briand, Ebola virus disease in the democratic republic of congo, New England Journal of Medicine, 371 (2014), 2083-2091.  doi: 10.1056/NEJMoa1411099. [23] S. Meloni, N. Perra, A. Arenas, S. Gómez, Y. Moreno and A. Vespignani, Modeling human mobility responses to the large-scale spreading of infectious diseases, Scientific Reports, 1 (2011), 1-7.  doi: 10.1038/srep00062. [24] J.C. Miller, A.C. Slim and E.M. Volz, Edge-based compartmental modelling for infectious disease spread, Journal of the Royal Society Interface, 9 (2012), 890-906.  doi: 10.1098/rsif.2011.0403. [25] J.C. Miller and E.M. Volz, Incorporating disease and population structure into models of SIR disease in contact networks, PloS One, 8 (2013), e69162.  doi: 10.1371/journal.pone.0069162. [26] M. Newman, Networks: An Introduction, Oxford University Press, 2010.  doi: 10.1093/acprof:oso/9780199206650.001.0001. [27] L. Pellis, F. Ball, S. Bansal, K. Eames, T. House, V. Isham and P. Trapman, Eight challenges for network epidemic models, Epidemics, 10 (2015), 58-62.  doi: 10.1016/j.epidem.2014.07.003. [28] D. Rand, Correlation equations and pair approximations for spatial ecologies, Advanced Ecological Theory: Principles and Applications, Oxford Blackwell Science, 1999. [29] E. J. Schwartz, B. Choi and G. A. Rempala, Estimating epidemic parameters: Application to H1N1 pandemic data, Math. Biosciences, 270 (2015), 198-203 (e-pub, ahead of print). doi: 10.1016/j.mbs.2015.03.007. [30] W.E.R. Team, Ebola virus disease in west africa-the first 9 months of the epidemic and forward projections, N Engl J Med, 371 (2014), 1481-1495. [31] E. Volz, SIR dynamics in random networks with heterogeneous connectivity, Journal of Mathematical Biology, 56 (2008), 293-310.  doi: 10.1007/s00285-007-0116-4. [32] D. Welch, S. Bansal and D.R. Hunter, Statistical inference to advance network models in epidemiology, Epidemics, 3 (2011), 38-45.  doi: 10.1016/j.epidem.2011.01.002.

show all references

##### References:
 [1] L. J. Allen, An introduction to stochastic epidemic models, Mathematical Epidemiology,Springer, 1945 (2008), 81-130. doi: 10.1007/978-3-540-78911-6_3. [2] M. Altmann, The deterministic limit of infectious disease models with dynamic partners, Mathematical Biosciences, 150 (1998), 153-175.  doi: 10.1016/S0025-5564(98)00012-1. [3] H. Andersson and T. Britton, Stochastic Epidemic Models and Their Statistical Analysis vol. 4, Springer New York, 2000. doi: 10.1007/978-1-4612-1158-7. [4] A.D. Barbour and G. Reinert, Approximating the epidemic curve, Electronic Journal of Probability, 18 (2013), 1-30.  doi: 10.1214/EJP.v18-2557. [5] M. Boguñá, C. Castellano and R. Pastor-Satorras, Nature of the epidemic threshold for the susceptible-infected-susceptible dynamics in networks, Physical Review Letters, 111 (2013), 068701. [6] T. Bohman and M. Picollelli, SIR epidemics on random graphs with a fixed degree sequence, Random Structures & Algorithms, 41 (2012), 179-214.  doi: 10.1002/rsa.20401. [7] T. Britton and P.D. O'Neill, Bayesian inference for stochastic epidemics in populations with random social structure, Scandinavian Journal of Statistics, 29 (2002), 375-390.  doi: 10.1111/1467-9469.00296. [8] B. Choi and G.A. Rempala, Inference for discretely observed stochastic kinetic networks with applications to epidemic modeling, Biostatistics, 13 (2012), 153-165.  doi: 10.1093/biostatistics/kxr019. [9] G. Chowell and H. Nishiura, Transmission dynamics and control of Ebola virus disease (EVD): A review, BMC Medicine, 12 (2014), p196. doi: 10.1186/s12916-014-0196-0. [10] L. Decreusefond, J.-S. Dhersin, P. Moyal and V.C. Tran, Large graph limit for an SIR process in random network with heterogeneous connectivity, The Annals of Applied Probability, 22 (2012), 541-575.  doi: 10.1214/11-AAP773. [11] C. Groendyke, D. Welch and D.R. Hunter, Bayesian inference for contact networks given epidemic data, Scandinavian Journal of Statistics, 38 (2011), 600-616.  doi: 10.1111/j.1467-9469.2010.00721.x. [12] C. Groendyke, D. Welch and D.R. Hunter, A network-based analysis of the 1861 Hagelloch measles data, Biometrics, 68 (2012), 755-765.  doi: 10.1111/j.1541-0420.2012.01748.x. [13] T. House and M.J. Keeling, Insights from unifying modern approximations to infections on networks, Journal of The Royal Society Interface, 8 (2011), 67-73.  doi: 10.1098/rsif.2010.0179. [14] S. Janson, M. Luczak and P. Windridge, Law of large numbers for the SIR epidemic on a random graph with given degrees, Random Structures & Algorithms, 45 (2014), 726-763.  doi: 10.1002/rsa.20575. [15] J. Janssen, Semi-Markov Models: Theory and Applications, New York, Plenum Press, 1986.  doi: 10.1007/978-1-4899-0574-1. [16] M. Keeling, The implications of network structure for epidemic dynamics, Theoretical Population Biology, 67 (2005), 1-8.  doi: 10.1016/j.tpb.2004.08.002. [17] M.J. Keeling, The effects of local spatial structure on epidemiological invasions, Proceedings of the Royal Society of London B: Biological Sciences, 266 (1999), 859-867.  doi: 10.1098/rspb.1999.0716. [18] M. Keeling, Correlation equations for endemic diseases: externally imposed and internally generated heterogeneity, Proceedings of the Royal Society of London B: Biological Sciences, 266 (1999), 953-960.  doi: 10.1098/rspb.1999.0729. [19] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 115, The Royal Society, 1927,700-721. [20] T.G. Kurtz, Limit theorems for sequences of jump markov processes approximating ordinary differential processes, Journal of Applied Probability, 8 (1971), 344-356.  doi: 10.1017/S002190020003535X. [21] J. Legrand, R. Grais, P. Boelle, A. Valleron and A. Flahault, Understanding the dynamics of Ebola epidemics, Epidemiology and Infection, 135 (2007), 610-621. [22] G.D. Maganga, J. Kapetshi, N. Berthet, B. Kebela Ilunga, F. Kabange, P. Mbala Kingebeni, V. Mondonge, J.-J. T. Muyembe, E. Bertherat and S. Briand, Ebola virus disease in the democratic republic of congo, New England Journal of Medicine, 371 (2014), 2083-2091.  doi: 10.1056/NEJMoa1411099. [23] S. Meloni, N. Perra, A. Arenas, S. Gómez, Y. Moreno and A. Vespignani, Modeling human mobility responses to the large-scale spreading of infectious diseases, Scientific Reports, 1 (2011), 1-7.  doi: 10.1038/srep00062. [24] J.C. Miller, A.C. Slim and E.M. Volz, Edge-based compartmental modelling for infectious disease spread, Journal of the Royal Society Interface, 9 (2012), 890-906.  doi: 10.1098/rsif.2011.0403. [25] J.C. Miller and E.M. Volz, Incorporating disease and population structure into models of SIR disease in contact networks, PloS One, 8 (2013), e69162.  doi: 10.1371/journal.pone.0069162. [26] M. Newman, Networks: An Introduction, Oxford University Press, 2010.  doi: 10.1093/acprof:oso/9780199206650.001.0001. [27] L. Pellis, F. Ball, S. Bansal, K. Eames, T. House, V. Isham and P. Trapman, Eight challenges for network epidemic models, Epidemics, 10 (2015), 58-62.  doi: 10.1016/j.epidem.2014.07.003. [28] D. Rand, Correlation equations and pair approximations for spatial ecologies, Advanced Ecological Theory: Principles and Applications, Oxford Blackwell Science, 1999. [29] E. J. Schwartz, B. Choi and G. A. Rempala, Estimating epidemic parameters: Application to H1N1 pandemic data, Math. Biosciences, 270 (2015), 198-203 (e-pub, ahead of print). doi: 10.1016/j.mbs.2015.03.007. [30] W.E.R. Team, Ebola virus disease in west africa-the first 9 months of the epidemic and forward projections, N Engl J Med, 371 (2014), 1481-1495. [31] E. Volz, SIR dynamics in random networks with heterogeneous connectivity, Journal of Mathematical Biology, 56 (2008), 293-310.  doi: 10.1007/s00285-007-0116-4. [32] D. Welch, S. Bansal and D.R. Hunter, Statistical inference to advance network models in epidemiology, Epidemics, 3 (2011), 38-45.  doi: 10.1016/j.epidem.2011.01.002.
The empirical secondary case distribution in the DRC outbreak dataset (neglecting the index case), as given by Maganga et al.[22]
Estimated posterior densities for the parameters of interest for the non-Markovian MCMC sampler. Green line denotes the posterior mean
Final outbreak size distribution based on 20,000 simulations of the branching processes from the posterior parameter distribution. The actual outbreak size of 69 based on the DRC dataset (black line) is shown for comparison
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