AI 中文总结
该研究针对年龄结构分支种群，分析仅移除驱动对其拟平稳平衡的控制极限，揭示可达衰减率受执行器形状而非大小主导，并通过口蹄疫模型验证结果。
AI 中文摘要
亚临界年龄结构分支种群几乎必然灭绝。在存活的条件下，它收敛到Yaglom极限，这是一个拟平稳平衡，我们将其作为控制的操作点。我们将预防性移除（扑杀）建模为一个依赖年龄的执行器，它会提高死亡率但不改变后代法则，并且我们证明，由于结构原因，它对该平衡的控制能力是有限的。有两个事实驱动该结果：第一，输入与致死率匹配，但与Foster-Lyapunov漂移不匹配，因此仅由繁殖设定的传输壁垒$\bar{L}$在这种驱动下是不变的；第二，且这并不单纯由不变性得出，可达衰减率的上确界为$\bar{L}+\nu^\boldsymbol{\u2605}$，其中$\nu^\boldsymbol{\u2605}\u22640$是谱系在从未绝嗣条件下的马尔萨斯参数；该间隙$|\nu^\boldsymbol{\u2605}|$以闭式形式给出，且当没有个体拥有两个或更多后代时完全消失。因此，此类移除法则均无法达到该壁垒，且在可容许边界上，可达衰减率由执行器的形状而非其大小决定。我们在一个针对2001年坎布里亚口蹄疫疫情校准的模型上说明了这些结果。
英文摘要
A subcritical age-structured branching population dies out almost surely. Conditioned on survival, it converges to its Yaglom limit, a quasi-stationary equilibrium that we take as the operating point for control. We model preventive removal (culling) as an age-dependent actuator that raises the mortality rate and leaves the offspring law untouched, and we show that its authority over this equilibrium is bounded for structural reasons. Two facts drive the result. First, the input is matched to the killing rate but unmatched with respect to the Foster--Lyapunov drift, so the transmission barrier $\Lbar$ set by reproduction alone is invariant under such actuation. Second, and this does not follow from invariance alone,} the supremum of the reachable decay rates is $\Lbar+ν^\star$, where $ν^\star\le0$ is the Malthusian parameter of the lineage conditioned never to die childless; the gap $|ν^\star|$ is given in closed form and vanishes exactly when no individual has two or more offspring. Consequently no removal law of this class reaches the barrier, and along the admissibility boundary the achievable decay rate is governed by the shape of the actuator rather than by its size. We illustrate these results on a model calibrated to the 2001 Cumbrian foot-and-mouth outbreak.
CommentsThe authors gratefully acknowledge the support of the IUT Henri Poincar{é} de Longwy, Universit{é} de Lorraine. This work was initiated while O. Arezki was a visiting professor hosted by CRAN (CNRS UMR 7039) and the IUT Henri Poincar{é} de Longwy.A short version of this paper, restricted to Sections III--IV, has been submitted to the IEEE Control Systems Letters (L-CSS)
