3.2 Analysis Plan
Most epidemic and pandemic diseases grow exponentially in the initial stages of the outset in a country [53]. A popular modelling technique that demonstrates this is the Susceptible-Infectious-Recovered (SIR) model[54].If S denotes the fraction of susceptible individuals to a pandemic, I indicate the fraction of infectious people, R is the fraction of recovered patients, β indicates the transmission rate per infectious individual, and the recovery rate is denoted by γ, the infectious period is exponentially distributed with a mean of 1/ γ. This indicates that following.
\begin{equation} \frac{\text{dS}}{\text{dt}}=\ -\beta SI\nonumber \\ \end{equation}\begin{equation} \frac{\text{dI}}{\text{dt}}=\ \beta SI-\ \gamma I\nonumber \\ \end{equation}\begin{equation} \frac{\text{dR}}{\text{dt}}=\gamma I\nonumber \\ \end{equation}
Linearizing this about the disease-free equilibrium, we get the following.
\begin{equation} \frac{\text{dI}}{\text{dt}}\approx\left(\beta-\gamma\right)I\nonumber \\ \end{equation}
Hence from the above expression, if \(\beta-\gamma>0\), then the infection function I(t) grows exponentially about the disease-free equilibrium point. In addition to this, at the onset of the infection,\(S\approx 1\) and hence the incidence rate \(C=\beta SI\) also grows exponentially. Hence, modelling the initial stages on a pandemic like COVID-19 is both relevant and crucial in understanding the growth of the infection. Although sub-exponential and polynomial modelling have worked in cases of outbreaks like Ebola, HIV, and foot and mouth diseases [55], they generally work well with proceeding generations. For pandemics like COVID-19, the exponential growth model proves relevant and using Least-squares to do the modelling at the initial stages can give precise insights.
The Exponential growth model bears the expression\(y(t)=ae^{\text{bt}}\), where ‘a’ is a function of the initial cases reported and ‘b’ depends on the rate at which the infection spreads. This model is extremely sensitive to the initial few periods and analysis of the last few data points concerning the model itself can assist in understanding if the interventions and policy implementations by the government of a country are effective in terms of containing the infection or not. Other factors like infrastructure, availability of doctors, temperature and humidity of the country during the spread can also significantly affect the growth rates of the infection. However, the objective of doing the exponential growth model for this research is to understand if the actual infections are lower than the predicted infections for the last few infections thereby forming a base data to design a classifier with this as the dependent variable. Last seven time periods were considered for comparison with the predicted values of the corresponding model for a particular country. If these actual data points of a particular country are significantly lower than the predicted values with the exponential growth model, it indicates a presence of an initial sign of containment owing to several factors like policies, infrastructures, behavioural changes, actions etc.
Then, the sign of containment was used as a dependent variable and for machine learning models like logistic regression, support vector machines, decision trees, and random forest algorithms. The independent variables for the study included physicians per thousand individuals, hospitals per thousand individuals, percentages of lockdown days since the first contact, cases per million population, deaths per million population, days since the first case, serious cases per thousand infections, average temperature since the first infection, and average humidity since the first infection. A combination of infrastructure, infection, policies, and environmental-related variables were used to train the model. A comparative analysis of the accuracy and error metrics in terms of predicting the country’s ability to contain the infection for the corresponding algorithms are reported. Python was used to do all the analysis and learning model developments for this research. Figure 1 shows the analysis plan to achieve the objectives of the research.