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.