Estimating the Cost of Epidemic Outbreaks in Stochastic SIR models

Abstract In this paper we develop a probabilistic framework for quantifying the economic impact of epidemics under prevalence-based control policies. The epidemic dynamics are described by a novel continuous-time Markovian SIR model in which the infection rates and the cost structure switch when the number of infectious cases crosses a critical threshold. On top of the stochastic dynamics, we consider a state-dependent cost framework that distinguishes (i) time-in-state sojourn costs, (ii) per-infection costs and (iii) activation/deactivation expenses associated with implementing and lifting control measures. Within this setting, we define the total epidemic cost and the cost of a tagged infectious individual as random variables and derive recursive expressions for their Laplace–Stieltjes transforms (LSTs) and moments. Using matrix differential calculus and Kronecker products, we further obtain recursive formulas for the sensitivities and elasticities of the most representative moments, with respect to the main cost parameters. Numerical experiments illustrate how different cost components drive the mean and standard deviation of total and individual costs across a range of thresholds and epidemic scenarios. The proposed methodology thus combines stochastic epidemic modeling and perturbation analysis to provide distribution-aware, parameter-sensitive assessments of epidemic costs, with direct relevance for statistical evaluation and operations research methods in policy design.