A partition method for bounding continuous-time Markov chain models of general reaction network

Area of Science:

• Stochastic modeling
• Chemical kinetics
• Computational biology

Background:

• Stochastic reaction networks are often modeled by multi-dimensional continuous-time Markov chains.
• Analyzing these complex systems, especially their behavior at infinity and numerical approximations, is challenging.

Purpose of the Study:

• To present a general method for establishing properties of multi-dimensional continuous-time Markov chains.
• To develop analytically and numerically tractable bounding processes for complex Markov chains.
• To derive explicit formulas for constructing optimal bounding processes.

Main Methods:

• State space partitioning
• Construction of one-dimensional birth and death processes
• Coupling arguments and transport theory
• Analysis of stationary distributions and truncation errors

Main Results:

• Developed a method to construct one-dimensional birth and death processes that bound the original multi-dimensional Markov chain.
• Derived explicit formulas for optimal bounding process construction.
• Demonstrated the method's utility in analyzing stationary distributions and state space truncation errors.
• Illustrated the approach with two chemical reaction network examples.

Conclusions:

• The proposed method provides a general and practical approach to analyze complex Markov chains in stochastic reaction networks.
• The bounding processes simplify the analysis of system properties, including long-term behavior and numerical accuracy.
• The choice of state space partition is crucial for obtaining relevant and accurate results.