<p>Supplementary Results SEIR-Model with Dense Data.</p>

Global pandemics, such as the recent COVID-19 crisis, highlight the need for stochastic epidemic models that can capture the randomness inherent in the spread of disease. Such models must be accompanied by methods for estimating parameters in order to generate fast nowcasts and short-term forecasts that can inform public health decisions. This paper presents a comparison of two advanced Bayesian inference methods: 1) pseudo-marginal particle Markov chain Monte Carlo, using an unbiased likelihood estimate obtained by Particle Filter (PF), and 2) Conditional Normalizing Flows (CNF). We investigate their performance on three commonly used compartmental models: A classical Susceptible-Infected-Susceptible (SIS), a Susceptible-Infected-Recovered (SIR) model and a two-variant Susceptible-Exposed-Infected-Recovered (SEIR) model, complemented by an observation model that maps latent trajectories to empirical data. Addressing the challenges of intractable likelihoods for parameter inference in stochastic settings, our analysis highlights how particle-filter-based likelihood estimation and flow-based posterior approximation can provide accurate and robust inference capabilities. The results of our simulation study further underscore the effectiveness of these approaches in capturing the stochastic dynamics of epidemics, providing prediction capabilities for the control of epidemic outbreaks. Results on an Ethiopian cohort study demonstrate operational robustness under real-world noise and irregular data sampling. To facilitate reuse and to enable building pipelines that ultimately contribute to better informed decision making in public health, we make code and synthetic datasets publicly available.