Lie algebra solution of population models based on time-inhomogeneous Markov chains

Journal of Applied Probability

Volumn 49, Issue 2, 2012, Pages 472-481

  House, Thomas ^a  

^a UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

Birth death process; Epidemic; Lie algebra; Markov chain; Time inhomogeneous

Indexed keywords

EID: 84871704277     PISSN: 00219002     EISSN: None     Source Type: Journal    
DOI: 10.1239/jap/1339878799     Document Type: Article

References (15)

1. Danon, L., House, T., Keeling, M. J. and Read, J.M. (2012). Social encounter networks: collective properties and disease transmission. Submitted.
2. Dodd, P. J. and Ferguson, N.M. (2009).A many-body field theory approach to stochastic models in population biology. PLoS ONE 4, e6855, 9pp.
3. Gilks, W. R., Richardson, S. and Spiegelhalter, D. J. (1995). Markov Chain Monte Carlo in Practice. Chapman and Hall/CRC, Boca Raton, FL.
4. Grimmett, G. R. and Stirzaker, D. R. (2001). Probability and Random Processes, 3rd edn. Oxford University Press, NewYork.
5. Jarvis, P. D., Bashford, J. D. and Sumner, J. G. (2005). Path integral formulation and Feynman rules for phylogenetic branching models. J. Phys. A 38, 9621-9647.
6. Johnson, J. E. (1985). Markov-type Lie groups in GL(n, R). J. Math. Phys. 26, 252-257.
7. Keeling, M. J. and Rohani, P. (2008). Modeling Infectious Diseases in Humans and Animals. Princeton University Press.
8. Keeling, M. J. and Ross, J.V. (2008). On methods for studying stochastic disease dynamics. J. R. Soc. Interface 5, 171-181.
9. Mourad, B. (2004). On a Lie-theoretic approach to generalized doubly stochastic matrices and applications. Linear Multilinear Algebra 52, 99-113.
10. Ross, J. V. (2010). Computationally exact methods for stochastic periodic dynamics spatiotemporal dispersal and temporally forced transmission. J. Theoret. Biol. 262, 14-22.
11. Sidje, R. B. (1998). Expokit: a software package for computing matrix exponentials. ACMTrans. Math. Software 24, 130-156.
12. Sumner, J., Fernandez-Sanchez, J. and Jarvis, P. (2012). Lie Markov models. J. Theoret. Biol. 298, 16-32.
13. Sumner, J. G., Holland, B. R. and Jarvis, P. D. (2011). The algebra of the general Markov model on phylogenetic trees and networks. Bull. Math. Biol. 17pp.
14. Wei, J. and Norman, E. (1963). Lie algebraic solution of linear differential equations. J. Math. Phys. 4, 575-581.
15. Wilcox, R. M. (1967). Exponential operators and parameter differentiation in quantum physics. J. Math. Phys. 8, 962-982.