# Monte Carlo simulation and analytic approximation of epidemic processes on large networks

Open Mathematics (2013)

• Volume: 11, Issue: 4, page 800-815
• ISSN: 2391-5455

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## Abstract

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Low dimensional ODE approximations that capture the main characteristics of SIS-type epidemic propagation along a cycle graph are derived. Three different methods are shown that can accurately predict the expected number of infected nodes in the graph. The first method is based on the derivation of a master equation for the number of infected nodes. This uses the average number of SI edges for a given number of the infected nodes. The second approach is based on the observation that the epidemic spreads along the cycle graph as a front. We introduce a continuous time Markov chain describing the evolution of the front. The third method we apply is the subsystem approximation using the edges as subsystems. Finally, we compare the steady state value of the number of infected nodes obtained in different ways.

## How to cite

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Noémi Nagy, and Péter Simon. "Monte Carlo simulation and analytic approximation of epidemic processes on large networks." Open Mathematics 11.4 (2013): 800-815. <http://eudml.org/doc/269800>.

@article{NoémiNagy2013,
abstract = {Low dimensional ODE approximations that capture the main characteristics of SIS-type epidemic propagation along a cycle graph are derived. Three different methods are shown that can accurately predict the expected number of infected nodes in the graph. The first method is based on the derivation of a master equation for the number of infected nodes. This uses the average number of SI edges for a given number of the infected nodes. The second approach is based on the observation that the epidemic spreads along the cycle graph as a front. We introduce a continuous time Markov chain describing the evolution of the front. The third method we apply is the subsystem approximation using the edges as subsystems. Finally, we compare the steady state value of the number of infected nodes obtained in different ways.},
author = {Noémi Nagy, Péter Simon},
journal = {Open Mathematics},
keywords = {SIS epidemic; ODE approximation; Network process; network process},
language = {eng},
number = {4},
pages = {800-815},
title = {Monte Carlo simulation and analytic approximation of epidemic processes on large networks},
url = {http://eudml.org/doc/269800},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Noémi Nagy
AU - Péter Simon
TI - Monte Carlo simulation and analytic approximation of epidemic processes on large networks
JO - Open Mathematics
PY - 2013
VL - 11
IS - 4
SP - 800
EP - 815
AB - Low dimensional ODE approximations that capture the main characteristics of SIS-type epidemic propagation along a cycle graph are derived. Three different methods are shown that can accurately predict the expected number of infected nodes in the graph. The first method is based on the derivation of a master equation for the number of infected nodes. This uses the average number of SI edges for a given number of the infected nodes. The second approach is based on the observation that the epidemic spreads along the cycle graph as a front. We introduce a continuous time Markov chain describing the evolution of the front. The third method we apply is the subsystem approximation using the edges as subsystems. Finally, we compare the steady state value of the number of infected nodes obtained in different ways.
LA - eng
KW - SIS epidemic; ODE approximation; Network process; network process
UR - http://eudml.org/doc/269800
ER -

## References

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1. [1] Barrat A., Barthélemy M., Vespignani A., Dynamical Processes on Complex Networks, Cambridge University Press, Cambridge, 2008 http://dx.doi.org/10.1017/CBO9780511791383 Zbl1198.90005
2. [2] Bollobás B., Random Graphs, 2nd ed., Cambridge Stud. Adv. Math., 73, Cambridge University Press, Cambridge, 2001 http://dx.doi.org/10.1017/CBO9780511814068
3. [3] Brauer F., van den Driessche P., Wu J. (Eds.), Mathematical Epidemiology, Lecture Notes in Math., 1945, Math. Biosci. Subser., Springer, Berlin-Heidelberg, 2008
4. [4] Danon L., Ford A.P., House T., Jewell C.P., Keeling M.J., Roberts G.O., Ross J.V., Vernon M.C., Networks and the epidemiology of infectious disease, Interdisciplinary Perspectives on Infectious Diseases, 2011, #284909
5. [5] Gleeson J.P., High-accuracy approximation of binary-state dynamics on networks, Phys. Rev. Lett., 2011, 107(6), #068701 http://dx.doi.org/10.1103/PhysRevLett.107.068701
6. [6] House T., Keeling M.J., Insights from unifying modern approximations to infections on networks, Journal of the Royal Society Interface, 2011, 8(54), 67–73 http://dx.doi.org/10.1098/rsif.2010.0179
7. [7] Keeling M.J., Eames K.T.D., Networks and epidemic models, Journal of the Royal Society Interface, 2005, 2(4), 295–307 http://dx.doi.org/10.1098/rsif.2005.0051
8. [8] Nåsell I., The quasi-stationary distribution of the closed endemic SIS model, Adv. in Appl. Probab., 1996, 28(3), 895–932 http://dx.doi.org/10.2307/1428186 Zbl0854.92020
9. [9] Sharkey K.J., Deterministic epidemic models on contact networks: Correlations and unbiological terms, Theoretical Population Biology, 2011, 79(4), 115–129 http://dx.doi.org/10.1016/j.tpb.2011.01.004
10. [10] Simon P.L., Taylor M., Kiss I.Z., Exact epidemic models on graphs using graph-automorphism driven lumping, J. Math. Biol., 2010, 62(4), 479–508 http://dx.doi.org/10.1007/s00285-010-0344-x Zbl1232.92068
11. [11] Taylor M., Simon P.L., Green D.M., House T., Kiss I.Z., From Markovian to pairwise epidemic models and the performance of moment closure approximations, J. Math. Biol., 2012, 646(6), 1021–1042 http://dx.doi.org/10.1007/s00285-011-0443-3 Zbl1252.92051

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