Local approximation of Markov chains in time and space

Area of Science:

• Epidemiology and Mathematical Biology
• Stochastic Processes and Computational Statistics

Background:

• Early disease emergence is characterized by low infectious numbers, making outcomes sensitive to environmental randomness.
• Continuous-time Markov chains (CTMCs) model this randomness, with extinction probability linked to hitting specific states.
• Discrete-time Markov chains (DTMCs) are often used to study these extinction probabilities.

Purpose of the Study:

• To develop and apply a novel method for approximating discrete-time Markov chains (DTMCs) on countably infinite state spaces.
• To utilize this approximation technique for calculating disease extinction probabilities in epidemic models.
• To evaluate the effectiveness of heterogeneous disease control strategies within a metapopulation framework.

Main Methods:

• The study introduces an approach to approximate a DTMC on a countably infinite state space with a DTMC on a finite state space.
• This approximation facilitates the computation of hitting probabilities, crucial for extinction analysis.
• The method is applied to a specific epidemic model and a metapopulation disease control scenario.

Main Results:

• The proposed method effectively approximates the hitting probabilities for DTMCs on infinite state spaces.
• Accurate estimations of disease extinction probabilities were obtained for the studied epidemic model.
• The approach allowed for the evaluation of a heterogeneous disease control strategy in a metapopulation.

Conclusions:

• The developed finite-state DTMC approximation offers a practical computational tool for analyzing extinction events in stochastic epidemic models.
• This methodology enhances the ability to assess disease spread dynamics and the impact of control interventions.
• The study provides a valuable framework for integrating environmental randomness into epidemiological predictions and public health strategy development.