Exact solution methods for time-inhomogeneous household models

Develop general exact solution methods for time-inhomogeneous household epidemic models with realistic, combinatorially growing state spaces, beyond the numerical approximations provided by the Peano–Baker and Magnus expansions.

Background

The paper examines household epidemic models whose transition-rate matrix depends explicitly on time. Unlike time-homogeneous models, these systems generally cannot be solved using a simple matrix exponential. The authors discuss the Peano–Baker series and Magnus expansion as formally exact representations, but explain that both become computationally impractical or intractable as household size and the associated state space increase.

The unresolved issue is whether mathematically tractable exact solution techniques can be developed for realistic time-inhomogeneous household models. Such methods would preserve the analytical advantages of homogeneous continuous-time Markov-chain formulations while accommodating time-varying external infection rates.