Modelling COVID-19 in India
SAIR Model
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) (Ansumali et al., 2020).
Below is a diagram showing the simplified SAIR model without vital dynamics such as the death rate from each compartment.
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The rate of infections due to Susceptible and Asymptomatic interactions is the same as that due to Susceptible and Infected interactions, hence the same constant β is used.
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All infected persons enter only group A, with a rate constant of β
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All asymptomatic patients recover, but some infected patients may recover whilst dome die. Hence, R represents the recovered population, either through recovery or death.
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The recovered population is no longer susceptible to infection.
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The removal rate from the I population is the sum of the recovery rate and death rate.
From the assumptions, the following set of first-order differential equations were developed:
Assumptions
The parameters used in solving the SAIR model differ from that of the SIR model as the data used in the estimation was obtained from the 7th of February, 2020. The parameter values are as follows:
Compared to the SIR graph, there’s a decrease in the maximum number of infected people in the population, which can be explained by the presence of the asymptomatic population, which can easily recover without moving into the infected population. Herd immunity occurs when the infected population reaches about 25% of the total population.
Model Figure
Parameter Sensitivity
SIR Model per the research paper
For the first parameter sensitivity test, the rate at which the susceptible are infected was decreased to mimic lockdown and other preventative measures such as social distancing and face masking methods put in place by the Indian government. The figure shows a shift in herd immunity, defined as the time when the number of infected persons is maximum. This occurs around the 38th day, which is a much longer time than when there are no measures put in place, with herd immunity occurring around the 12th day.
There is also a decrease in the value of herd immunity from 8 to 3. These results prove the efficacy of measures put in place to curb the spread of the disease.
Modified SIR Model with a decrease in rate of infection
A similar trend in the shift of herd immunity can be seen in the modification of the SAIR model, in which β, the rate of infection, is decreased from 0.02 to 0.01. We conclude that the rate of infection by both the Asymptomatic population as well as the Infected population is a key parameter in the analysis of COVID-19 and hence an important point of focus for healthcare workers and policymakers.
Modified SAIR Model with a decrease in rate of infection
For this modification, the rate at which the asymptomatic population moves to the infected population decreases. This models the situation whereby vaccination attempts have been employed early during the beginning of disease progression. This modification tracks a decrease in herd immunity, however, the population will still be infected since the asymptomatic population can also cause infection. Thus, this parameter is extremely significant in the model.
Modified SAIR Model with a decrease in the rate at which asymptomatic patients become infectious
Modified SIR Model for Death Rate
For this modification, the SIR model employed by the paper was modified to include a separate death rate from the infected population. The rate of removal thus no longer represented the sum of the recovered and dead but only the recovered. As predicted, there was a decrease in the number of the recovered population. This decrease was not as significant as expected, which we believed was attributed to the fact that the death rate changed across the waves of COVID-19 in India and was not a constant value as used in this modification. For this model, the death rate had a low significance to the model. However, it would be important to track COVID-19 statistics and analysing the overall progression of the disease.