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Modelling a pandemic with asymptomatic patients, impact of lockdown and herd immunity, with applications to SARS-CoV-2

  • Ansumali, Santosh1
  • Kaushal, Shaurya1
  • Kumar, Aloke2
  • Prakash, Meher K.1
  • Vidyasagar, M.3
  • 1 Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore, India
  • 2 Indian Institute of Science, Bangalore, India
  • 3 Indian Institute of Technology Hyderabad India
Published Article
Annual Reviews in Control
Publication Date
Oct 09, 2020
DOI: 10.1016/j.arcontrol.2020.10.003
PMID: 33071595
PMCID: PMC7546280
PubMed Central


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.

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