For infectious diseases such as pertussis, susceptibility is determined by immunity, which is chronological age-dependent. We consider an age-structured epidemiological model that accounts for both passively acquired maternal antibodies that decay and active immunity that wanes, permitting reinfection. The model is a 6-dimensional system of partial differential equations (PDE). By assuming constant rates within each age-group, the PDE system can be reduced to an ordinary differential equation (ODE) system with aging from one age-group to the next. We derive formulae for the effective reproduction number ℛ and provide their biological interpretation in some special cases. We show that the disease-free equilibrium is stable when ℛ < 1 and unstable if ℛ > 1.

中文翻译：

---

在具有年龄和物质免疫力的年龄结构流行病模型中OF的计算。

对于诸如百日咳这样的传染性疾病，易感性是由免疫力决定的，免疫力是依年龄而定的。我们考虑一种年龄结构的流行病学模型，该模型既考虑了被动获得的母体抗体的衰变，又考虑了主动免疫力的减弱（允许再次感染）。该模型是6维偏微分方程（PDE）系统。通过在每个年龄组中假设恒定的比率，PDE系统可以简化为具有从一个年龄组到下一个年龄组的老化的常微分方程（ODE）系统。我们推导了有效繁殖数ℛ的公式，并在某些特殊情况下提供了它们的生物学解释。我们证明当ℛ<1时，无病平衡是稳定的；如果and> 1，则无病平衡是不稳定的。