Journal of Applied Probability

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

Thomas House

Abstract

Many natural populations are well modelled through time-inhomogeneous stochastic processes. Such processes have been analysed in the physical sciences using a method based on Lie algebras, but this methodology is not widely used for models with ecological, medical, and social applications. In this paper we present the Lie algebraic method, and apply it to three biologically well-motivated examples. The result of this is a solution form that is often highly computationally advantageous.

Article information

Source
J. Appl. Probab., Volume 49, Number 2 (2012), 472-481.

Dates
First available in Project Euclid: 16 June 2012

https://projecteuclid.org/euclid.jap/1339878799

Digital Object Identifier
doi:10.1239/jap/1339878799

Mathematical Reviews number (MathSciNet)
MR2977808

Zentralblatt MATH identifier
1319.92042

Citation

House, Thomas. Lie algebra solution of population models based on time-inhomogeneous Markov chains. J. Appl. Probab. 49 (2012), no. 2, 472--481. doi:10.1239/jap/1339878799. https://projecteuclid.org/euclid.jap/1339878799

References

• Danon, L., House, T., Keeling, M. J. and Read, J. M. (2012). Social encounter networks: collective properties and disease transmission. Submitted.
• 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.
• Gilks, W. R., Richardson, S. and Spiegelhalter, D. J. (1995). Markov Chain Monte Carlo in Practice. Chapman and Hall/CRC, Boca Raton, FL.
• Grimmett, G. R. and Stirzaker, D. R. (2001). Probability and Random Processes, 3rd edn. Oxford University Press, New York.
• 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.
• Johnson, J. E. (1985). Markov-type Lie groups in $\text{GL}(n,{\mathbb{R}})$. J. Math. Phys. 26, 252–257.
• Keeling, M. J. and Rohani, P. (2008). Modeling Infectious Diseases in Humans and Animals. Princeton University Press.
• Keeling, M. J. and Ross, J. V. (2008). On methods for studying stochastic disease dynamics. J. R. Soc. Interface 5, 171–181.
• Mourad, B. (2004). On a Lie-theoretic approach to generalized doubly stochastic matrices and applications. Linear Multilinear Algebra 52, 99–113.
• Ross, J. V. (2010). Computationally exact methods for stochastic periodic dynamics spatiotemporal dispersal and temporally forced transmission. J. Theoret. Biol. 262, 14–22.
• Sidje, R. B. (1998). Expokit: a software package for computing matrix exponentials. ACM Trans. Math. Software 24, 130–156.
• Sumner, J., Fernandez-Sanchez, J. and Jarvis, P. (2012). Lie Markov models. J. Theoret. Biol. 298, 16–32.
• 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.
• Wei, J. and Norman, E. (1963). Lie algebraic solution of linear differential equations. J. Math. Phys. 4, 575–581.
• Wilcox, R. M. (1967). Exponential operators and parameter differentiation in quantum physics. J. Math. Phys. 8, 962–982.