We study an SEIQR (Susceptible–Exposed–Infectious–Quarantined–Recovered) model due to Young et al. (2019) for an infectious disease, with time delays for latency and an asymptomatic phase. For fast pandemics where nobody has prior immunity and everyone has immunity after recovery, the SEIQR model decouples into two nonlinear delay differential equations (DDEs) with five parameters. One parameter is set to unity by scaling time. The simple subcase of perfect quarantining and zero self-recovery before quarantine, with two free parameters, is examined first. The method of multiple scales yields a hyperbolic tangent solution; and a long-wave (short delay) approximation yields a first order ordinary differential equation (ODE). With imperfect quarantining and nonzero self-recovery, the long-wave approximation is a second order ODE. These three approximations each capture the full outbreak, from infinitesimal initiation to final saturation. Low-dimensional dynamics in the DDEs is demonstrated using a six state non-delayed reduced order model obtained by Galerkin projection. Numerical solutions from the reduced order model match the DDE over a range of parameter choices and initial conditions. Finally, stability analysis and numerics show how a well executed temporary phase of social distancing can reduce the total number of people affected. The reduction can be by as much as half for a weak pandemic, and is smaller but still substantial for stronger pandemics. An explicit formula for the greatest possible reduction is given.

我们研究了 Young 等人提出的 SEIQR（易感 - 暴露 - 感染 - 隔离 - 恢复）模型。 （2019）针对传染病，具有潜伏期和无症状期的时间延迟。对于没有人事先具有免疫力并且每个人在恢复后都具有免疫力的快速流行病，SEIQR 模型解耦为两个具有五个参数的非线性延迟微分方程 (DDE)。通过缩放时间将一个参数设置为统一。首先检查具有两个自由参数的完美隔离和隔离前零自我恢复的简单子案例。多尺度方法产生双曲正切解；长波（短延迟）近似产生一阶常微分方程 (ODE)。由于不完美隔离和非零自恢复，长波近似是二阶 ODE。这三个近似值各自捕获了从无穷小的起始到最终饱和的完整爆发。使用由伽辽金投影获得的六态非延迟降阶模型演示了 DDE 中的低维动力学。降阶模型的数值解在一系列参数选择和初始条件上与 DDE 相匹配。最后，稳定性分析和数字表明，执行良好的临时社交距离阶段如何能够减少受影响的总人数。对于较弱的流行病，减少幅度可能高达一半；对于较严重的流行病，减少幅度较小，但仍然很大。给出了最大可能减少的明确公式。