Deterministic and stochastic infection dynamics in a population subject to stress

TL;DR

This work couples environmental stress, modulated by water quality, to host susceptibility in a stress-structured epidemiological model for aquatic populations. It develops a deterministic two-class model (normal and stressed) and proves a forward bifurcation at $\mathcal{R}_0=1$, with $\mathcal{R}_0 = \mathcal{R}_{01}+\mathcal{R}_{02}$ and explicit housing of stress-driven transmission, alongside a stochastic CTMC and branching-process analysis that yields extinction probabilities and first-introduction timing. A time-varying stress extension yields a bounded $\mathcal{R}(t)$ whose extremes depend on $\alpha_{min}$ and $\alpha_{max}$, highlighting scenarios where invasion is possible or suppressed under fluctuating DO. The results reveal a critical asymmetry: outbreaks depend on which physiological class is initially seeded, with stressed index cases driving rapid epidemics while normal-index introductions face a stochastic barrier, underscoring the importance of water-quality management in disease control.

Abstract

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 $\mathcal{R}_0=1$, 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 $\mathcal{R}_0$ predicts stability, the probability of an outbreak depends on the initial physiological state. Introducing infection into a stressed sub-population leads to immediate rapid growth of the disease, whereas introduction into the normal class faces a stochastic barrier that significantly delays the epidemic peak.

Figures (6)

• Figure 1: Flow diagram of the deterministic stress–infection model: Normal susceptible ($S_N$), Stressed susceptible ($S_S$), Normal infected ($I_N$), Stressed infected ($I_S$) and Recovered ($R$).
• Figure 2: Dissolved oxygen profiles $W(t)$ (left panel) and corresponding stress rates $\alpha(t)$ (right panel) under four illustrative water-quality scenarios: a well-oxygenated "wet" regime (8.5 mg L$^{-1}$), a borderline "medium" regime at the critical threshold ($W_{\mathrm{crit}}=6$ mg L$^{-1}$), a low-oxygen "dry" regime (4.5 mg L$^{-1}$) pfister2009assessing and a "seasonal" regime in which $W(t)$ oscillates between approximately 4 and 8 mg L$^{-1}$ over one year. The remain parameter values are given in Table \ref{['tab:parameters']}.
• Figure 3: Time-dependent reproduction number $\mathcal{R}(t)$ under bounded stress. Each coloured curve represents $\mathcal{R}(t)$ for a different stress profile $\alpha(t)$ satisfying $\alpha_{\min} \le \alpha(t) \le \alpha_{\max}$. The two horizontal lines correspond to the autonomous thresholds $\mathcal{R}_0(\alpha_{\min})$ and $\mathcal{R}_0(\alpha_{\max})$.
• Figure 4: Deterministic trajectories for the stress–infection model with time-varying stress rate $\alpha(t)$ with different basic reproduction numbers in Figure \ref{['fig:Rt-bounds']}. The left panel shows the number of non-stressed infected fish $I_N(t)$, the right panel shows the number of stressed infected fish $I_S(t)$, under four water-stress scenarios (dry, medium, seasonal, and wet).
• Figure 5: Distribution of extinction probability $\mathbb{P}_{\text{ext}}$ over initial conditions, for four stress levels $\alpha$ (wet, medium, seasonal, dry) and two seeding classes. Left: the epidemic is seeded in $I_N$; right: it is seeded in $I_S$.

Theorems & Definitions (13)

• Lemma 1
• proof
• Lemma 2
• proof
• Proposition 3
• Lemma 4
• proof
• Lemma 5
• Lemma 6
• proof