Abstract
Moment closure on general discrete structures often requires one of the following: (i) an absence of short-closed loops (zero clustering); (ii) existence of a spatial scale; (iii) ad hoc assumptions. Algebraic methods are presented to avoid the use of such assumptions for populations based on clumps and are applied to both SIR and macroparasite disease dynamics. One approach involves a series of approximations that can be derived systematically, and another is exact and based on Lie algebraic methods.
References
Addiss DG (2013) Epidemiologic models, key logs, and realizing the promise of WHA 54.19. PLoS Negl Trop Dis 7(2):e2092
Anderson RM (1993) Epidemiology, Chapter 4. In: Cox FEG (ed) Modern parasitology, 2nd edn. Blackwell Science, Oxford, pp 75–116
Anderson RM, May RM (1991) Infectious diseases of humans. Oxford University Press, Oxford
Anderson RM, Truscott JE, Pullan RL, Brooker SJ, Hollingsworth TD (2013) How effective is school-based deworming for the community-wide control of soil-transmitted helminths? PLoS Negl Trop Dis 7(2):e2027
Ball F, Neal P (2008) Network epidemic models with two levels of mixing. Math Biosci 212(1):69–87
Ball F, Mollison D, Scalia-Tomba G (1997) Epidemics with two levels of mixing. Ann Appl Probab 7(1):46–89
Blanes S, Casas F, Oteo JA, Ros J (1998) Magnus and Fer expansions for matrix differential equations: the convergence problem. J Phys A 31(1):259–268
Blanes S, Casas F, Ros J (2002) High order optimized geometric integrators for linear differential equations. BIT Numer Math 42(2):262–284
Blanes S, Casas F, Oteo JA, Ros J (2013) The Fer and Magnus expansions. In: Engquist B et al (eds) Encyclopedia of applied and computational mathematics. Springer, London
Cornell SJ, Ovaskainen O (2008) Exact asymptotic analysis for metapopulation dynamics on correlated dynamic landscapes. Theor Popul Biol 74(3):209–225
Danon L, Ford AP, House T, Jewell CP, Keeling MJ, Roberts GO, Ross JV, Vernon MC (2011) Networks and the epidemiology of infectious disease. Interdiscip Perspect Infect Dis 2011:1–28
Decreusefond L, Dhersin J-S, Moyal P, Tran VC (2012) Large graph limit for a SIR process in random network with heterogeneous connectivity. Ann Appl Probab 22(2):541–575
Dodd P, Ferguson N (2007) Approximate disease dynamics in household-structured populations. J R Soc Interface 4(17):1103–1106
Ferguson NM, Donnelly CA, Anderson RM (2001) The foot-and-mouth epidemic in Great Britain: pattern of spread and impact of interventions. Science 292(5519):1155–1160
Ghoshal G, Sander LM, Sokolov IM (2004) SIS epidemics with household structure: the self-consistent field method. Math Biosci 190(1):71–85
House T (2010) Exact epidemic dynamics for generally clustered, complex networks. arXiv:1006.3483
House T (2012) Lie algebra solution of population models based on time-inhomogeneous Markov chains. J Appl Probab 49(2):472–481
House T, Keeling MJ (2008) Deterministic epidemic models with explicit household structure. Math Biosci 213(1):29–39
House T, Keeling MJ (2009) UK household structure and infectious disease transmission. Epidemiol Infect 137(5):654–661
House T, Keeling MJ (2010) The impact of contact tracing in clustered populations. PLoS Comput Biol 6(3):e1000721
Iserles A, Nørsett SP (1999) On the solution of linear differential equations in Lie groups. Philos Trans R Soc 357(1754):983–1019
Isham V (1995) Stochastic models of host-macroparasite interaction. Ann Appl Probab 5(3):720–740
Karrer B, Newman MEJ (2010) Random graphs containing arbitrary distributions of subgraphs. Phys Rev E 82(6):066118
Keeling M, Ross J (2008) On methods for studying stochastic disease dynamics. J R Soc Interface 5(19):171–181
Keeling MJ (1999) The effects of local spatial structure on epidemiological invasions. Proc R Soc B 266(1421):859–867
Kirkwood JG, Boggs EM (1942) The radial distribution function in liquids. J Chem Phys 10(6):394–402
Kiss IZ, Simon PL (2012) New moment closures based on a priori distributions with applications to epidemic dynamics. Bull Math Biol 74(7):150–1515
Ma J, Driessche P, Willeboordse F (2013) Effective degree household network disease model. J Math Biol 66(1–2):75–94
Magnus W (1954) On the exponential solution of differential equations for a linear operator. Commun Pure Appl Math 7(4):649–673
Miller JC (2011) A note on a paper by Erik Volz: SIR dynamics in random networks. J Math Biol 62(3):349–358
Miller JC, Slim AC, Volz EM (2012) Edge-based compartmental modelling for infectious disease spread. J R Soc Interface 9(70):890–906
Ovaskainen O, Cornell SJ (2006) Space and stochasticity in population dynamics. Proc Natl Acad Sci 103(34):12781–12786
Rogers T (2011) Maximum-entropy moment-closure for stochastic systems on networks. J Stat Mech 2011(05):P05007
Ross JV (2012) On parameter estimation in population models III: time-inhomogeneous processes and observation error. Theor Popul Biol 82(1):1–17
Ross JV, House T, Keeling MJ (2010) Calculation of disease dynamics in a population of households. PLoS ONE 5:e9666
Shang Y (2012) A Lie algebra approach to susceptible-infected-susceptible epidemics. Electron J Differ Equ 2012(233):1–7
Sidje RB (1998) Expokit. A software package for computing matrix exponentials. ACM Trans Math Softw 24(1):130–156
Sumner JG (2013) Lie geometry of \(2\times 2\) Markov matrices. J Theor Biol 327:88–90
Taylor M, Simon PL, Green DM, House T, Kiss IZ (2012) From Markovian to pairwise epidemic models and the performance of moment closure approximations. J Math Biol 64(6):1021–1042
Volz EM (2008) SIR dynamics in random networks with heterogeneous connectivity. J Math Biol 56(3):293–310
Volz EM, Miller JC, Galvani A, Ancel Meyers L (2011) Effects of heterogeneous and clustered contact patterns on infectious disease dynamics. PLoS Comput Biol 7(6):e1002042
Wilcox RM (1967) Exponential operators and parameter differentiation in quantum physics. J Math Phys 8:962–982
Acknowledgments
Work funded by the UK Engineering and Physical Sciences Research Council (EPSRC). The author would like to thank Joshua Ross, the editors, and three anonymous reviewers for helpful comments on this manuscript.
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House, T. Algebraic Moment Closure for Population Dynamics on Discrete Structures. Bull Math Biol 77, 646–659 (2015). https://doi.org/10.1007/s11538-014-9981-3
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DOI: https://doi.org/10.1007/s11538-014-9981-3