Modeling and analysis of COVID-19 in India with treatment function through different phases of lockdown and unlock

India, unlike several other countries, has witnessed a greater challenge in overcoming the pandemic during the second wave crisis. The strategies on lockdown imposition during the first wave of the pandemic were well implemented due to which the year-long wave did not witness uncontrollable surges in the infections. In this study, we present a detailed study on the disease spread during the first and second wave of COVID-19, by performing numerical simulation in a phase-wise manner as per the lockdown and unlock phases implemented in India. The inclusion of a piecewise treatment function in the framed epidemiological model is a noteworthy aspect of the study since this function takes into consideration the availability of medical equipment, based upon the threshold number of infections. This function framed in this model first grows linearly reaching a peak, then it declines due to shortage of medical resources and finally gets saturated. Analysis of the impact of lockdown is presented for each phase of the pandemic in India. Though the data considered for the study is for a period that marked the beginning of the pandemic, major analysis and predictions are presented based on the second wave data in terms of sensitivity analysis and time series behavior. A comparison of deterministic and stochastic differential equations is presented with simulation results on certain parameter set to examine variation in treatment and recovery. The simulations are performed using MATLAB and R softwares. The work is validated with the real data and model fitting is done applying the Maximum Likelihood method. The study implies that, under accurate lockdown strategies and sufficient medical care, the peak in cases would be attained by 16 May 2021, after which a decline in the cases could be observed.