[Paper Review] Coupling a branching process to an infinite dimensional epidemic process

A. D. Barbour|ArXiv.org|Oct 19, 2007

COVID-19 epidemiological studies10 references4 citations

TL;DR

This paper establishes a coupling between a continuous-time Markovian epidemic process (the BK-model for schistosomiasis) and a branching process, showing they coincide with high probability until $ M_N = o(N^{2/3}) $ infections occur. The key contribution is a rigorous pathwise coupling with asymptotically small total variation distance, extending prior results that were limited to $ o(N^{1/2}) $, using likelihood ratio analysis and martingale concentration inequalities to control deviations.

ABSTRACT

Branching process approximation to the initial stages of an epidemic process has been used since the 1950's as a technique for providing stochastic counterparts to deterministic epidemic threshold theorems. One way of describing the approximation is to construct both branching and epidemic processes on the same probability space, in such a way that their paths coincide for as long as possible. In this paper, it is shown, in the context of a Markovian model of parasitic infection, that coincidence can be achieved with asymptotically high probability until o(N^{2/3}) infections have occurred, where N denotes the total number of hosts.

Motivation & Objective

• To extend the range of pathwise coupling between epidemic and branching processes beyond the previously known $ o(N^{1/2}) $ threshold.
• To establish asymptotically small total variation distance between the epidemic and branching processes up to $ M_N = o(N^{2/3}) $ infections.
• To develop a rigorous coupling framework for continuous-time, infinite-dimensional epidemic models such as the BK-model for parasitic infections.
• To analyze the likelihood ratio of the epidemic and branching processes along paths to quantify their relative closeness.
• To apply martingale concentration inequalities to control deviations in the likelihood ratio and ensure high-probability path coincidence.

Proposed method

• Construct a coupling between the infinite-dimensional BK-model epidemic process and a branching process on the same probability space.
• Use the likelihood ratio of the two processes along paths of length $ M $ to assess their relative closeness.
• Apply a martingale concentration inequality (Lemma 4.1) to bound the tail probabilities of the log-likelihood ratio process.
• Define a stopping time $ t' $ to control the likelihood ratio up to $ M $ infections, ensuring path coincidence until that time.
• Use the condition $ ext{Pr}[ ilde{L}_M^N - 1 > 1] $ and $ \text{Pr}[|1 - ilde{L}_M^N| > \varepsilon_{M,N}^r / 2] $ to bound coupling failure probabilities.
• Establish relative closeness with tolerance $ \varepsilon_{M,N}^r = C_r \psi(M,N) \sqrt{\log(1/\psi(M,N))} $, where $ \psi(M,N) = S_M \sqrt{M}/N $.

Experimental results

Research questions

• RQ1 Can the range of pathwise coupling between epidemic and branching processes be extended beyond $ o(N^{1/2}) $ infections in continuous-time, infinite-dimensional models?
• RQ2 What conditions ensure that the likelihood ratio between the epidemic and branching processes remains close to 1 with high probability up to $ M_N $ infections?
• RQ3 How can martingale concentration inequalities be applied to control the deviation of the likelihood ratio in a coupling framework?
• RQ4 What is the optimal rate of convergence for the total variation distance between the epidemic and branching processes in the BK-model?
• RQ5 Under what conditions is the branching process a valid approximation to the epidemic process up to $ o(N^{2/3}) $ infections?

Key findings

• The epidemic and branching processes can be coupled such that their paths coincide with asymptotically high probability until $ M_N = o(N^{2/3}) $ infections occur.
• The total variation distance between the path distributions of the epidemic and branching processes is asymptotically small under the same condition.
• The likelihood ratio of the two processes is tightly controlled via martingale concentration, with tail bounds derived from Lemma 4.1.
• The coupling error is bounded by $ \varepsilon_{M,N}^r = C_r \psi(M,N) \sqrt{\log(1/\psi(M,N))} $, where $ \psi(M,N) = S_M \sqrt{M}/N $, ensuring relative closeness.
• The bound holds provided $ M \geq (1/5)C_r^2 \log N $ and $ \varepsilon_{M,N}^r \leq 1 $, ensuring the approximation remains valid in the large population limit.
• The result improves upon prior work by Ball and Donnelly (1995), who had shown coupling only up to $ o(N^{1/2}) $, by extending the range to $ o(N^{2/3}) $.