# American Institute of Mathematical Sciences

July  2018, 23(5): 2005-2020. doi: 10.3934/dcdsb.2018192

## On bounding exact models of epidemic spread on networks

 1 Institute of Mathematics, Eötvös Loránd University Budapest, Hungary 2 Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences, Hungary 3 School of Mathematical and Physical Sciences, Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, UK

* Corresponding author: Péter L. Simon

Received  March 2017 Revised  October 2017 Published  July 2018 Early access  May 2018

Fund Project: The first author is supported by Hungarian Scientific Research Fund, OTKA, (grant no. 115926).

In this paper we use comparison theorems from classical ODE theory in order to rigorously show that closures or approximations at individual or node level lead to mean-field models that bound the exact stochastic process from above. This will be done in the context of modelling epidemic spread on networks and the proof of the result relies on the observation that the epidemic process is negatively correlated (in the sense that the probability of an edge being in the susceptible-infected state is smaller than the product of the probabilities of the nodes being in the susceptible and infected states, respectively). The results in the paper hold for Markovian epidemics and arbitrary weighted and directed networks. Furthermore, we cast the results in a more general framework where alternative closures, other than that assuming the independence of nodes connected by an edge, are possible and provide a succinct summary of the stability analysis of the resulting more general mean-field models. While deterministic initial conditions are key to obtain the negative correlation result we show that this condition can be relaxed as long as extra conditions on the edge weights are imposed.

Citation: Péter L. Simon, Istvan Z. Kiss. On bounding exact models of epidemic spread on networks. Discrete and Continuous Dynamical Systems - B, 2018, 23 (5) : 2005-2020. doi: 10.3934/dcdsb.2018192
##### References:
 [1] B. Armbruster and E. Beck, Elementary proof of convergence to the mean-field model for the SIR process, Journal of Mathematical Biology, 75 (2017), 327-339.  doi: 10.1007/s00285-016-1086-1. [2] B. Armbruster and E. Beck, An elementary proof of convergence to the mean-field equations for an epidemic model, IMA Journal of Applied Mathematics, 82 (2017), 152-157.  doi: 10.1093/imamat/hxw010. [3] B. Armbruster, A. Besenyei and P. L. Simon, Bounds for the expected value of one-step processes, Commun. Math. Sci., 14 (2016), 1911-1923.  doi: 10.4310/CMS.2016.v14.n7.a6. [4] M. van Baalen, Pair approximations for different spatial geometries, chapter Pair approximations for different spatial geometries, Cambridge University Press, (2000), 359-387. [5] A. Bátkai, I. Z. Kiss, E. Sikolya and P. L. Simon, Differential equation approximations of stochastic network processes: An operator semigroup approach, Netw. Heter. Media, 7 (2012), 43-58.  doi: 10.3934/nhm.2012.7.43. [6] M. Boguná and R. Pastor-Satorras, Epidemic spreading in correlated complex networks, Physical Review E, 66 (2002), 047104. [7] E. Cator and P. Van Mieghem, Nodal infection in Markovian susceptible-infected-susceptible and susceptible-infected-removed epidemics on networks are non-negatively correlated, Physical Review E, 89 (2014), 052802.  doi: 10.1103/PhysRevE.89.052802. [8] T. E. Harris, Additive set-valued markov processes and graphical methods, The Annals of Probability, 6 (1978), 355-378.  doi: 10.1214/aop/1176995523. [9] M. W. Hirsch and H. Smith, Monotone dynamical systems, in Handbook of Differential Equations: Ordinary Differential Equations (eds. A. Canada, P. Drábek and A. Fonda), Elsevier BV Amsterdam, 2 (2005), 239–357. [10] E. Kamke, Zur theorie der systeme gewöhnlicher differentialgleichungen. Ⅱ, Acta Mathematica, 58 (1932), 57-85.  doi: 10.1007/BF02547774. [11] M. J. Keeling, The effects of local spatial structure on epidemiological invasions, Proceedings of the Royal Society of London. Series B: Biological Sciences, (2011), 480-488.  doi: 10.1515/9781400841356.480. [12] I. Z. Kiss, J. C. Miller and P. L. Simon, Mathematics of Network Epidemics: From Exact to Approximate Models, Springer-Verlag, New York, 2017. doi: 10.1007/978-3-319-50806-1. [13] I. Z. Kiss, C. G. Morris, F. Sélley, P. L. Simon and R. R. Wilkinson, Exact deterministic representation of markovian SIR epidemics on networks with and without loops, Journal of Mathematical Biology, 70 (2015), 437-464.  doi: 10.1007/s00285-014-0772-0. [14] I. Z. Kiss, G. Röst and Z. Vizi, Generalization of pairwise models to non-Markovian epidemics on networks Physical Review Letters, 115 (2015), 078701. doi: 10.1103/PhysRevLett.115.078701. [15] D. Kunszenti-Kovács and P. L. Simon, Mean-field approximation of counting processes from a differential equation perspective, Electronic Journal of Qualitative Theory of Differential Equations, 2016 (2016), 1-17. [16] A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Mathematical Biosciences, 28 (1976), 221-236.  doi: 10.1016/0025-5564(76)90125-5. [17] J. Lindquist, J. Ma, P. van den Driessche and F. H. Willeboordse, Effective degree network disease models, Journal of Mathematical Biology, 62 (2011), 143-164.  doi: 10.1007/s00285-010-0331-2. [18] 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: 10.1007/s00285-012-0572-3. [19] J. C. Miller and E. M. Volz, Model hierarchies in edge-based compartmental modeling for infectious disease spread, Journal of Mathematical Biology, 67 (2013), 869-899.  doi: 10.1007/s00285-012-0572-3. [20] M. Molloy and B. Reed, A critical point for random graphs with a given degree sequence, Random Structures & Algorithms, 6 (1995), 161-179.  doi: 10.1002/rsa.3240060204. [21] M. Müller, Über das fundamentaltheorem in der theorie der gewöhnlichen differentialgleichungen, Mathematische Zeitschrift, 26 (1927), 619-645.  doi: 10.1007/BF01475477. [22] R. Pastor-Satorras, C. Castellano, P. Van Mieghem and A. Vespignani, Epidemic processes in complex networks, Rev. Mod. Phys., 87 (2015), 925-979.  doi: 10.1103/RevModPhys.87.925. [23] R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Physical Review Letters, 86 (2001), 3200-3203.  doi: 10.1515/9781400841356.493. [24] D. A. Rand, Correlation equations and pair approximations for spatial ecologies, in Advanced ecological theory: principles and applications, Oxford: Blackwell Science, (1999), 100–142. [25] K. J. Sharkey, Deterministic epidemic models on contact networks: Correlations and unbiological terms, Theoretical Population Biology, 79 (2011), 115-129.  doi: 10.1016/j.tpb.2011.01.004. [26] K. J. Sharkey, I. Z. Kiss, R. R. Wilkinson and P. L. Simon, Exact equations for SIR epidemics on tree graphs, Bulletin of Mathematical Biology, 77 (2015), 614-645.  doi: 10.1007/s11538-013-9923-5. [27] P. L. Simon, M. Taylor and I. Z. Kiss, Exact epidemic models on graphs using graph-automorphism driven lumping, Journal of Mathematical Biology, 62 (2011), 479-508.  doi: 10.1007/s00285-010-0344-x. [28] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995. [29] J. Szarski, Differential Inequalities, Instytut Matematyczny Polskiej Akademi Nauk (Warszawa), 1965. [30] M. Taylor, P. L. Simon, D. M. Green, T. House and I. Z. Kiss, From Markovian to pairwise epidemic models and the performance of moment closure approximations, Journal of Mathematical Biology, 64 (2012), 1021-1042.  doi: 10.1007/s00285-011-0443-3. [31] P. Van Mieghem, The n-intertwined SIS epidemic network model, Computing, 93 (2011), 147-169.  doi: 10.1007/s00607-011-0155-y. [32] P. Van Mieghem, J. Omic and R. Kooij, Virus spread in networks, Networking, IEEE/ACM Transactions, 17 (2009), 1-14.  doi: 10.1109/TNET.2008.925623.

show all references

##### References:
 [1] B. Armbruster and E. Beck, Elementary proof of convergence to the mean-field model for the SIR process, Journal of Mathematical Biology, 75 (2017), 327-339.  doi: 10.1007/s00285-016-1086-1. [2] B. Armbruster and E. Beck, An elementary proof of convergence to the mean-field equations for an epidemic model, IMA Journal of Applied Mathematics, 82 (2017), 152-157.  doi: 10.1093/imamat/hxw010. [3] B. Armbruster, A. Besenyei and P. L. Simon, Bounds for the expected value of one-step processes, Commun. Math. Sci., 14 (2016), 1911-1923.  doi: 10.4310/CMS.2016.v14.n7.a6. [4] M. van Baalen, Pair approximations for different spatial geometries, chapter Pair approximations for different spatial geometries, Cambridge University Press, (2000), 359-387. [5] A. Bátkai, I. Z. Kiss, E. Sikolya and P. L. Simon, Differential equation approximations of stochastic network processes: An operator semigroup approach, Netw. Heter. Media, 7 (2012), 43-58.  doi: 10.3934/nhm.2012.7.43. [6] M. Boguná and R. Pastor-Satorras, Epidemic spreading in correlated complex networks, Physical Review E, 66 (2002), 047104. [7] E. Cator and P. Van Mieghem, Nodal infection in Markovian susceptible-infected-susceptible and susceptible-infected-removed epidemics on networks are non-negatively correlated, Physical Review E, 89 (2014), 052802.  doi: 10.1103/PhysRevE.89.052802. [8] T. E. Harris, Additive set-valued markov processes and graphical methods, The Annals of Probability, 6 (1978), 355-378.  doi: 10.1214/aop/1176995523. [9] M. W. Hirsch and H. Smith, Monotone dynamical systems, in Handbook of Differential Equations: Ordinary Differential Equations (eds. A. Canada, P. Drábek and A. Fonda), Elsevier BV Amsterdam, 2 (2005), 239–357. [10] E. Kamke, Zur theorie der systeme gewöhnlicher differentialgleichungen. Ⅱ, Acta Mathematica, 58 (1932), 57-85.  doi: 10.1007/BF02547774. [11] M. J. Keeling, The effects of local spatial structure on epidemiological invasions, Proceedings of the Royal Society of London. Series B: Biological Sciences, (2011), 480-488.  doi: 10.1515/9781400841356.480. [12] I. Z. Kiss, J. C. Miller and P. L. Simon, Mathematics of Network Epidemics: From Exact to Approximate Models, Springer-Verlag, New York, 2017. doi: 10.1007/978-3-319-50806-1. [13] I. Z. Kiss, C. G. Morris, F. Sélley, P. L. Simon and R. R. Wilkinson, Exact deterministic representation of markovian SIR epidemics on networks with and without loops, Journal of Mathematical Biology, 70 (2015), 437-464.  doi: 10.1007/s00285-014-0772-0. [14] I. Z. Kiss, G. Röst and Z. Vizi, Generalization of pairwise models to non-Markovian epidemics on networks Physical Review Letters, 115 (2015), 078701. doi: 10.1103/PhysRevLett.115.078701. [15] D. Kunszenti-Kovács and P. L. Simon, Mean-field approximation of counting processes from a differential equation perspective, Electronic Journal of Qualitative Theory of Differential Equations, 2016 (2016), 1-17. [16] A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Mathematical Biosciences, 28 (1976), 221-236.  doi: 10.1016/0025-5564(76)90125-5. [17] J. Lindquist, J. Ma, P. van den Driessche and F. H. Willeboordse, Effective degree network disease models, Journal of Mathematical Biology, 62 (2011), 143-164.  doi: 10.1007/s00285-010-0331-2. [18] 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: 10.1007/s00285-012-0572-3. [19] J. C. Miller and E. M. Volz, Model hierarchies in edge-based compartmental modeling for infectious disease spread, Journal of Mathematical Biology, 67 (2013), 869-899.  doi: 10.1007/s00285-012-0572-3. [20] M. Molloy and B. Reed, A critical point for random graphs with a given degree sequence, Random Structures & Algorithms, 6 (1995), 161-179.  doi: 10.1002/rsa.3240060204. [21] M. Müller, Über das fundamentaltheorem in der theorie der gewöhnlichen differentialgleichungen, Mathematische Zeitschrift, 26 (1927), 619-645.  doi: 10.1007/BF01475477. [22] R. Pastor-Satorras, C. Castellano, P. Van Mieghem and A. Vespignani, Epidemic processes in complex networks, Rev. Mod. Phys., 87 (2015), 925-979.  doi: 10.1103/RevModPhys.87.925. [23] R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Physical Review Letters, 86 (2001), 3200-3203.  doi: 10.1515/9781400841356.493. [24] D. A. Rand, Correlation equations and pair approximations for spatial ecologies, in Advanced ecological theory: principles and applications, Oxford: Blackwell Science, (1999), 100–142. [25] K. J. Sharkey, Deterministic epidemic models on contact networks: Correlations and unbiological terms, Theoretical Population Biology, 79 (2011), 115-129.  doi: 10.1016/j.tpb.2011.01.004. [26] K. J. Sharkey, I. Z. Kiss, R. R. Wilkinson and P. L. Simon, Exact equations for SIR epidemics on tree graphs, Bulletin of Mathematical Biology, 77 (2015), 614-645.  doi: 10.1007/s11538-013-9923-5. [27] P. L. Simon, M. Taylor and I. Z. Kiss, Exact epidemic models on graphs using graph-automorphism driven lumping, Journal of Mathematical Biology, 62 (2011), 479-508.  doi: 10.1007/s00285-010-0344-x. [28] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995. [29] J. Szarski, Differential Inequalities, Instytut Matematyczny Polskiej Akademi Nauk (Warszawa), 1965. [30] M. Taylor, P. L. Simon, D. M. Green, T. House and I. Z. Kiss, From Markovian to pairwise epidemic models and the performance of moment closure approximations, Journal of Mathematical Biology, 64 (2012), 1021-1042.  doi: 10.1007/s00285-011-0443-3. [31] P. Van Mieghem, The n-intertwined SIS epidemic network model, Computing, 93 (2011), 147-169.  doi: 10.1007/s00607-011-0155-y. [32] P. Van Mieghem, J. Omic and R. Kooij, Virus spread in networks, Networking, IEEE/ACM Transactions, 17 (2009), 1-14.  doi: 10.1109/TNET.2008.925623.
The relation of the joint and marginal probabilities.
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 $\left\langle {I_i} \right\rangle$ $\left\langle {S_i} \right\rangle$ $\left\langle {I_j} \right\rangle$ a b q $\left\langle {S_j} \right\rangle$ c d 1-q p 1-p
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