Deterministic and Stochastic Infection Dynamics in a Population Subject to Stress

Physiological stress fundamentally alters disease susceptibility in aquatic environments. In this paper, we develop a stress-structured epidemiological model where host vulnerability is dynamically driven by water quality. Analytically, we establish that the system exhibits a classic forward bifurcation at $R_0=1$ , confirming that the basic reproduction number remains a valid threshold for eradication. However, stochastic analysis reveals a critical asymmetry not captured by deterministic thresholds. We show that while $R_0$ predicts stability, the probability of an outbreak depends on the initial physiological state and, in time-varying environments, on the exact time $t_0$ of pathogen introduction. Introducing infection into a stressed sub-population leads to an immediate rapid growth of the disease, whereas introduction into the normal compartment occurs later, with the delay depending on the level of stress.