Modelling the COVID-19 Pandemic: Asymptomatic Patients, Lockdown and Herd Immunity

The SARS-Cov-2 is a type of coronavirus that has caused the COVID-19 pandemic. 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 viruses, there is a distinct "asymptomatic" group A, who do not show any symptoms, but can nevertheless infect others, at the same rate as infected patients. This situation is captured in a model known as SAIR (Susceptible, Asymptomatic, Infected, Removed), introduced in Robinson and Stilianakis (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 stabiilty of the SAIR model. Next, we present methods for estimating the parameters in the SAIR model. We apply these estimation methods to data from several countries including India, and show that the predicted trajectories of the disease closely match actual data. Copyright (C) 2020 The Authors.

Estimating the herd immunity threshold by accounting for the hidden asymptomatics using a COVID-19 specific model.

A quantitative COVID-19 model that incorporates hidden asymptomatic patients is developed, and an analytic solution in parametric form is given. The model incorporates the impact of lock-down and resulting spatial migration of population due to announcement of lock-down. A method is presented for estimating the model parameters from real-world data, and it is shown that the various phases in the observed epidemiological data are captured well. It is shown that increase of infections slows down and herd immunity is achieved when active symptomatic patients are 10-25% of the population for the four countries we studied. Finally, a method for estimating the number of asymptomatic patients, who have been the key hidden link in the spread of the infections, is presented.

Modelling the spread of SARS-CoV-2 pandemic - Impact of lockdowns & interventions.

BACKGROUND & OBJECTIVES: To handle the current COVID-19 pandemic in India, multiple strategies have been applied and implemented to slow down the virus transmission. These included clinical management of active cases, rapid development of treatment strategies, vaccines computational modelling and statistical tools to name a few. This article presents a mathematical model for a time series prediction and analyzes the impact of the lockdown. METHODS: Several existing mathematical models were not able to account for asymptomatic patients, with limited testing capability at onset and no data on serosurveillance. In this study, a new model was used which was developed on lines of susceptible-asymptomatic-infected-recovered (SAIR) to assess the impact of the lockdown and make predictions on its future course. Four parameters were used, namely ß, γ, η and ε. ß measures the likelihood of the susceptible person getting infected, and γ denotes recovery rate of patients. The ratio ß/γ is denoted by R0 (basic reproduction number). RESULTS: The disease spread was reduced due to initial lockdown. An increase in γ reflects healthcare and hospital services, medications and protocols put in place. In Delhi, the predictions from the model were corroborated with July and September serosurveys, which showed antibodies in 23.5 and 33 per cent population, respectively. INTERPRETATION & CONCLUSIONS: The SAIR model has helped understand the disease better. If the model is correct, we may have reached herd immunity with about 380 million people already infected. However, personal protective measures remain crucial. If there was no lockdown, the number of active infections would have peaked at close to 14.7 million, resulted in more than 2.6 million deaths, and the peak would have arrived by June 2020. The number of deaths with the current trends may be less than 0.2 million.

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

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.