Comparing virus incubation time in SIRC models: Deterministic versus stochastic approaches.

Time delays are a fundamental feature in modeling stochastic epidemic systems, as they capture the incubation period and other physiological lags inherent in disease transmission. In this work, we investigate a stochastic SIRC (Susceptible-Infectious-Recovered-Cross-immune) epidemic model where the delay is incorporated into the transmission term to reflect the incubation period. To account for environmental variability, we examine two stochastic formulations: the classical approach, which adds independent white noise to each compartment, and a probabilistic, event-driven model in which stochasticity arises directly from transition probabilities. A key focus of our study is the comparison between different delay formulations in the transmission term, specifically contrasting the standard approach-where the delay acts only on the infected compartment-with alternative formulations that distribute the delay across both susceptible and infected populations. Through systematic numerical simulations, we find that the choice of delay formulation strongly influences the timing and magnitude of the initial epidemic peak, while the long-term (asymptotic) behavior is more robust but remains sensitive to the underlying stochastic framework. The probabilistic model, in particular, offers a more faithful depiction of correlated fluctuations and extinction phenomena, capturing the biological complexity of epidemic processes more accurately than the classical approach. These results underscore the importance of both the delay representation and the stochastic modeling strategy in shaping the qualitative and quantitative features of epidemic dynamics.