Abstract
Deterministic epidemic models are attractive due to their compact nature, allowing substantial complexity with computational efficiency. This partly explains their dominance in epidemic modelling. However, the small numbers of infectious individuals at early and late stages of an epidemic, in combination with the stochastic nature of transmission and recovery events, are critically important to understanding disease dynamics. This motivates the use of a stochastic model, with continuous-time Markov chains being a popular choice. Unfortunately, even the simplest Markovian S–I–R model—the so-called general stochastic epidemic—has a state space of order \(N^2\), where N is the number of individuals in the population, and hence computational limits are quickly reached. Here we introduce a hybrid Markov chain epidemic model, which maintains the stochastic and discrete dynamics of the Markov chain in regions of the state space where they are of most importance, and uses an approximate model—namely a deterministic or a diffusion model—in the remainder of the state space. We discuss the evaluation, efficiency and accuracy of this hybrid model when approximating the distribution of the duration of the epidemic and the distribution of the final size of the epidemic. We demonstrate that the computational complexity is \({\mathcal {O}}(N)\) and that under suitable conditions our approximations are highly accurate.
Similar content being viewed by others
References
Andersson H, Britton T (2000) Stochastic epidemic models and their statistical analysis, lecture notes in statistics, vol 151, 1st edn. Springer, New York. doi:10.1007/978-1-4612-1158-7
Bailey NTJ (1950) A simple stochastic epidemic. Biometrika. doi:10.1093/biomet/37.3-4.193
Bailey NTJ (1957) The mathematical theory of epidemics. Griffin, London
Ball F, Donnelly P (1995) Strong approximations for epidemic models. Stoch Proc Appl. doi:10.1016/0304-4149(94)00034-Q
Ball F, Neal P (2010) Applications of branching processes to the final size of SIR epidemics. In: Workshop on branching processes and their applications, lecture notes in statistics, vol 197. Springer, Berlin, pp 207–223. doi:10.1007/978-3-642-11156-3_15
Barbour A (1975) The duration of the closed stochastic epidemic. Biometrika. doi:10.1093/biomet/62.2.477
Barbour AD (1974) On a functional central limit theorem for Markov population processes. Adv Appl Prob. doi:10.2307/1426205
Barbour AD (1976) Quasi-stationary distributions in Markov population processes. Adv Appl Prob. doi:10.2307/1425906
Barbour AD (1980a) Density dependent Markov population processes. In: Biological growth and spread, lecture notes in biomathematics, vol 38, Springer, Berlin, pp 36–49. doi:10.1007/978-3-642-61850-5_4
Barbour AD (1980b) Equilibrium distributions of Markov population processes. Adv Appl Prob. doi:10.2307/1426422
Bartlett MS (1949) Some evolutionary stochastic processes. J R Stat Soc 11:211–229
Bartlett MS (1956) Deterministic and stochastic models for recurrent epidemics. In: Proceedings of the third Berkeley symposium on mathematical statistics and probability, University of California Press, Berkeley, vol 4
Black AJ, Ross J (2015) Computation of epidemic final size distributions. J Theor Biol. doi:10.1016/j.jtbi.2014.11.029
Coulson T, Rohani P, Pascual M (2004) Skeletons, noise and population growth: the end of an old debate? Trends Ecol Evol. doi:10.1016/j.tree.2004.05.008
Ethier SN, Kurtz TG (2008) Markov processes: characterisation and convergence. Wiley, Hoboken. doi:10.1002/9780470316658
Fox GA (1993) Life history evolution and demographic stochasticity. Evol Ecol. doi:10.1007/BF01237731
Grenfell BT, Wilson K, Finkenstadt BF, Coulson TN, Murray S, Albon SD, Pemberton JM, Clutton-Brock TH, Crawley MJ (1998) Noise and determinism in synchronized sheep dynamics. Nature. doi:10.1038/29291
Jenkinson G, Goutsias J (2012) Numerical integration of the master equation in some models of stochastic epidemiology. PLoS One. doi:10.1371/journal.pone.0036160
Keeling MJ, Wilson HB, Pacala SW (2000) Reinterpreting space, time lags, and functional responses in ecological models. Science. doi:10.1126/science.290.5497.1758
Kendall DG (1965) Mathematical models of the spread of infection. In: Mathematics and Computer Science in Biology and Medicine. H.M.S.O, London, pp 213–225
Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proc R Soc Lond A. doi:10.1098/rspa.1927.0118
Kurtz TG (1970) Solutions of ordinary differential equations as limits of pure jump Markov processes. J Appl Prob. doi:10.2307/3212147
Kurtz TG (1971) Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J Appl Prob. doi:10.2307/3211904
Lefèvre C (1990) Stochastic epidemic models for SIR infectious diseases: a brief survey of the recent general theory. In: Stochastic Processes in epidemic theory, lecture notes in biomathematics, vol 86. Springer, Berlin, pp 1–12. doi:10.1007/978-3-662-10067-7_1
Nagaev AV, Startsev AN (1970) The asymptotic analysis of a stochastic model of an epidemic. Theory Prob Appl 15(1):98–107. doi:10.1137/1115007
Rand DA, Wilson HB (1991) Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics. Proc R Soc Lond B. doi:10.1098/rspb.1991.0142
Safta C, Sargsyan K, Debusschere B, Najm HN (2015) Hybrid discrete/continuum algorithms for stochastic reaction networks. J Comput Phys. doi:10.1016/j.jcp.2014.10.026
Sazonov I, Kelbert M, Gravenor MB (2011) A two-stage model for the SIR outbreak: Accounting for the discrete and stochastic nature of the epidemic at the initial contamination stage. Math Biosci. doi:10.1016/j.mbs.2011.09.002
Scalia-Tomba G (1985) Asymptotic final-size distribution for some chain-binomial processes. Adv Appl Prob. doi:10.2307/1427116
Spagnolo B, Fiasconaro A, Valenti D (2003) Noise induced phenomena in Lotka-Volterra systems. Fluct Noise Lett. doi:10.1142/S0219477503001245
Watson R (1980) A useful random time-scale transformation for the standard epidemic model. J Appl Prob. doi:10.2307/3213022
Watson R (1981) An application of a martingale central limit theorem to the standard epidemic model. Stoc Proc Appl. doi:10.1016/0304-4149(81)90023-5
Waugh WAO (1958) Conditioned Markov processes. Biometrika. doi:10.1093/biomet/45.1-2.241
Yi S, Ulsoy A (2006) Solution of a system of linear delay differential equations using the matrix Lambert function. In: Proc. of the 25th American control conference, Minneapolis, pp 2433–2438. doi:10.1109/ACC.2006.1656585
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rebuli, N.P., Bean, N.G. & Ross, J.V. Hybrid Markov chain models of S–I–R disease dynamics. J. Math. Biol. 75, 521–541 (2017). https://doi.org/10.1007/s00285-016-1085-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-016-1085-2