Estimating Time-Inhomogeneous Transition Probabilities from District Level Daily COVID-19 Transmission Data in Sierra Leone

Compartmental epidemic models conventionally treat the probability of moving between disease states as fixed over time, an assumption that sits uneasily with the reality of a pandemic in which lockdowns, mask mandates, vaccination roll-out, and the arrival of new variants continually reshape transmission. This paper develops a time-inhomogeneous Markov chain framework for the Susceptible–Exposed–Infectious–Removed (SEIR) process, in which each transition probability p _ab ( t ) is allowed to vary with calendar time while respecting the structural zeros implied by the SEIR compartmental flow. We derive the constrained maximum-likelihood estimator of p _ab ( t ) under these structural constraints, establish its finite sample efficiency, asymptotic normality, and Wilson score confidence intervals, and construct a like-lihood ratio test of the null hypothesis that a compartment’s exit probability is constant over time. We further propose a stochastic machine learning hybrid extension in which the raw, kernel smoothed transition probabilities are regressed on policy and mobility covariates using both a logistic generalized linear model and a random forest, allowing the framework to attribute time-inhomogeneity to observable interventions. The methodology is applied to a compiled daily, district level COVID-19 surveillance panel for Sierra Leone spanning March 2020 to December 2023 (16 districts, 1,401 days). The likelihood ratio test rejects time-homogeneity of the exposed to infectious transition in 15 of 16 districts and of the infectious-to-removed transition in 8 of 16 districts ( α = 0.05), and the covariate augmented logistic model achieves an out of sample Brier score roughly 76 times smaller than a time-homogeneous pooled baseline, with healthcare capacity and the time trend emerging as the most influential predictors in the random-forest component. These results provide statistical evidence that time-inhomogeneous, covariate informed Markov models offer a materially better description of district-level COVID-19 transmission in Sierra Leone than classical time-homogeneous compartmental models, with implications for sub-national outbreak monitoring in resource constrained settings.