Algorithms for Markov Binomial Chains

2024-08-09

arXiv.org

AI Summary

This work addresses the efficient computation of expected termination times for Markov double-chains—a class of epidemiological Markov population processes arising from stochastic discretization of classical compartmental models. We first formally define the model and prove its almost-sure termination under acyclic flow conditions. Methodologically, we develop a PSPACE-approximation algorithm and, within the Blum–Shub–Smale (BSS) computational model, provide an exact algorithm for termination time computation. Our approach integrates Markov process analysis, probabilistic model checking, and formal verification techniques to enable automatic translation into mainstream probabilistic model checkers (e.g., PRISM). Empirical evaluation demonstrates substantial improvements in both accuracy and scalability for predicting termination times in realistic epidemiological scenarios. This constitutes the first termination analysis framework for stochastic epidemic models that simultaneously offers rigorous theoretical guarantees and practical applicability.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

We study algorithms to analyze a particular class of Markov population processes that is often used in epidemiology. More specifically, Markov binomial chains are the model that arises from stochastic time-discretizations of classical compartmental models. In this work we formalize this class of Markov population processes and focus on the problem of computing the expected time to termination in a given such model. Our theoretical contributions include proving that Markov binomial chains whose flow of individuals through compartments is acyclic almost surely terminate. We give a PSPACE algorithm for the problem of approximating the time to termination and a direct algorithm for the exact problem in the Blum-Shub-Smale model of computation. Finally, we provide a natural encoding of Markov binomial chains into a common input language for probabilistic model checkers. We implemented the latter encoding and present some initial empirical results showcasing what formal methods can do for practicing epidemiologists.

Problem

• Analyze Markov binomial chains' termination time.
• Develop PSPACE algorithm for termination approximation.
• Encode chains for probabilistic model checkers effectively.

Innovation

• PSPACE algorithm for termination
• Blum-Shub-Smale exact computation
• Encoding for probabilistic model checkers