Stochasticity and Practical Identifiability in Epidemic Models: A Monte Carlo Perspective

Assessing the practical identifiability of epidemic models is essential for determining whether parameters can be meaningfully estimated from observed data. Monte Carlo (MC) methods provide an accessible and intuitive framework; however, their standard implementation - perturbing deterministic trajectories with independent Gaussian noise - rests on assumptions poorly suited to epidemic processes, which are inherently stochastic, temporally correlated, and highly variable, especially in small populations or under slow transmission. In this study, we investigate the structure of stochastic variability in the classic Susceptible-Infected-Recovered (SIR) model across a range of epidemiological regimes, and assess whether it can be represented within the independent Gaussian noise framework. We show that continuous-time Markov chain (CTMC) trajectories consistently exhibit super-Poissonian variability and strong temporal dependence. Through coverage analysis, we further demonstrate that independent Gaussian noise systematically underestimates the variability of the underlying stochastic process, leading to overly optimistic conclusions about parameter identifiability. In addition, we propose a hybrid simulation approach that introduces time- and amplitude-dependent variability into deterministic ODE trajectories, preserving computational efficiency while capturing key features of epidemic stochasticity. Our findings highlight the limitations of the standard MC algorithm and provide a pathway for incorporating more realistic noise structures into epidemic inference.