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

Abstract

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.

Keywords: COVID-19; Herd immunity; Lockdown; Lyapunov stability; SAIR model; SARS-CoV-2.

Fig. 1

Flowchart of the SIR model without vital dynamics.

Fig. 2

Dependence of $I_{\max }$ on I₀ for various σ.

Fig. 3

Flowchart of the SIR model with vital dynamics.

Fig. 4

Flowchart of the SEIR model without vital dynamics.

Fig. 5

Flowchart of the SEIR model with vital dynamics.

Fig. 6

Flowchart of the SAIR model without vital dynamics.

Fig. 7

Flowchart of the simplified SAIR model without vital dynamics.

Fig. 8

Flowchart of the simplified SAIR model without vital dynamics.

Fig. 9

Removal (recovery+death) frequency γ for various countries.

Fig. 10

Delay difference plots for : France(left) and Switzerland(right), in order to find the correct γ .

Fig. 11

Time $t=0$ is 14th February 2020. Extracting δ (using the decaying exponential in the analytical solution 10–30 days post lockdown).

Fig. 12

Time $t=0$ is 14th February 2020. Extracting β (using the analytical solution at early time).

Fig. 13

Time $t=0$ is 14th February 2020. Numerical solution to the SAIR model using the estimated paramters β, δ and γ.

Fig. 14

Estimates and Predictions for Delhi: (a) Estimating γ. (b) Estimating β. (c) Reconstruction of past trajectory. (d) Prediction of future trajectory.