The purpose of this paper is to build a mathematical model for the study of the roles of lock-down, social distancing, vaccination, detection efficiency, and health care capacity planning of the COVID-19 pandemic taking into account the demographic topology of the State of Tamil Nadu, India. Two mathematical models are proposed for the evolution of the first and second wave of COVID-19 pandemic. The model for the first wave considers lock-down orders, social distancing measures, and detection efficiency. The model for the second wave considers more sub-populations and incorporates two more elements, vaccination and health care capacity. Daily reported data on the evolution of the COVID-19 pandemic are used to determine the parameter values. The dynamics produced by the mathematical model closely follow the evolution of COVID-19 in the State of Tamil Nadu. Numerical simulation shows that the lock-down effect is limited. Social distancing implementation and detection of positive cases are relatively ineffective compared with other big cities. Shortage of health care resources is one of the factors responsible for rapidly spreading in the second wave in Tamil Nadu.
Keywords: COVID-19; epidemic; mathematical modeling; numerical simulation.