The SARS-CoV-2 is a type of coronavirus that has caused the pandemic known as the Coronavirus Disease of 2019, or COVID-19. In traditional epidemiological models such as SEIR (Susceptible, Exposed, Infected, Removed), the exposed group E does not infect the susceptible group S. A distinguishing feature of COVID-19 is that, unlike with previous viral diseases, there is a distinct “asymptomatic” group A, which does not show any symptoms, but can nevertheless infect others, at the same rate as infected symptomatic patients. This situation is captured in a model known as SAIR (Susceptible, Asymptomatic, Infected, Removed), introduced in Robinson and Stillianakis (2013). The dynamical behavior of the SAIR model is quite different from that of the SEIR model. In this paper, we use Lyapunov theory to establish the global asymptotic stabililty of the SAIR model, both without and with vital dynamics. Then we develop compartmental SAIR models to cater to the migration of population across geographic regions, and once again establish global asymptotic stability.

Next, we go beyond long-term asymptotic analysis and present methods for estimating the parameters in the SAIR model. We apply these estimation methods to data from several countries including India, and demonstrate that the predicted trajectories of the disease closely match actual data. We show that “herd immunity” (defined as the time when the number of infected persons is maximum) can be achieved when the total of infected, symptomatic and asymptomatic persons is as low as 25% of the population. Previous estimates are typically 50% or higher. We also conclude that “lockdown” as a way of greatly reducing inter-personal contact has been very effective in checking the progress of the disease.

SARS-CoV-2是一种冠状病毒，已引起大流行，被称为2019年冠状病毒病或COVID-19。在传统的流行病学模型如SEIR（易感，曝光，感染，删除），则暴露组ë不会感染易感组小号。与以前的病毒性疾病不同，COVID-19的一个显着特征是有一个明显的“无症状” A组，它没有表现出任何症状，但是仍然可以以与感染症状患者相同的速度感染其他人。这种情况是在Robinson和Stillianakis（2013）中引入的称为SAIR（易感，无症状，感染，移除）的模型中捕获的。SAIR模型的动力学行为与SEIR模型的动力学行为完全不同。在本文中，我们使用李雅普诺夫理论建立了具有或没有生命动力学的SAIR模型的全局渐近稳定性。然后，我们开发分区SAIR模型，以适应跨地理区域的人口迁移，并再次建立全局渐近稳定性。

接下来，我们将不进行长期渐近分析，而是介绍用于估算SAIR模型中参数的方法。我们将这些估算方法应用于包括印度在内的多个国家/地区的数据，并证明该疾病的预测轨迹与实际数据非常匹配。我们证明，当被感染，有症状和无症状的人的总数低至总人口的25％时，就可以实现“畜群免疫”（即被感染人数最大的时间）。先前的估算通常为50％或更高。我们还得出结论，“封锁”作为一种大大减少人际交往的方式，在检查疾病的进展方面非常有效。