Modelling COVID-19 in India
Mathematical Model
SIR Model
This paper employed the SIR (Susceptible, Infective, and Recovery) model known as the Kermack-McKendrick model. The SIR model is extremely beneficial for forecasting the end and peak of epidemic illnesses, as well as other outbreak-related behaviour. The goal is to develop a mathematical model that monitors the spread of Covid-19 across the Indian population and predicts the maximum growth of the COVID-19 outbreak in the country. Three ordinary differential equations were developed using the SIR and solved using Euler’s method.
Assumptions
The population of India remains constant regarding the study COVID-19 outbreak in India.
The constant function represents the unchanging susceptible population used for the model, with a value of 1,357,413.
In the paper, it was estimated that the incubation time for the COVID-19 outbreak in India is approximately 14 days.
Another simplification made is that R(t) is the sum of the recovered population, the dead population as a result of the disease as well as the population naturally immune to the outbreak.
Lastly, it was assumed that the Recovered population is immune to the disease.
Parameters
r is the rate of the infection. It was obtained from the ratio of the infected population to the susceptible population.
a is the rate of recovery. It was based on the assumption that the average time of the outbreak is 14 days.
Equations
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S(t): the number of susceptible populations at time t, which is the total COVID-2019 tested population until April 7, 2020.
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I(t): represents the number of infected people at time t, which is the number of people infected with COVID-2019 in India until April 7, 2020.
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R(t): represents the number of recovered populations at time t, which is the number of recovered, died, or naturally immune COVID-2019 populations in India until 7 April 2020.
The Kermack-McKendrick SIR model was used in this study. For the sake of simplicity, R(t) was taken as the recovered population plus the died population from the COVID-19 outbreak in India on 7 April 2020 in the paper, as shown below.
The parameters 'a' and 'r' of the above differential equations are known as the infection rate and recovery/removal rate of COVID-19 in India, respectively. The average time of the COVID-19 outbreak in India was taken as approximately 14 days. The resulting equations are shown below as inflow-outflow.
The population size of susceptible cases in the COVID-19 outbreak in India was taken into account here as K. This is roughly equivalent to India's initial population S0. In this section, a small number of pathogens were introduced into the population. Therefore,
where Rn represent the reproductive number of the population
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Numerical methods such as Runge Kutta and Euler methods were proposed to solve the SIR model. However, the Euler method was used to solve the differential equation based on the SIR model.
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In addition, MATLAB software was used in the study to solve the differential equation using the initial conditions values of S0, I0, R0, a, and r.
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Euler's method is a numerical solution method for first-order differential equations. The obtained SIR model was a system of first-order differential equations. As a result, Euler's method is better suited to solving the proposed SIR models.
Pertinent Information to Note
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COVID-19 data set from India on 7 May 2020 was used for this analysis.
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The total initial number of infectives population was taken as Io
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The total number of recovered/removed cases was also taken as R0 at the initial level for analyzing the COVID-19 outbreak in India on 7 May 2020.
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The three initial populations are S0, I0, and R0.
Finally, the parameters and the initial conditions obtained were used in Matlab to generate a graph for virtual comparison.