  The global pandemic due to the outbreak of COVID-19 ravages the whole world
for more than two years in which all the countries are suffering a lot since
December 2019. In order to control this ongoing waves of epidemiological
infections, attempts have been made to understand the dynamics of this pandemic
in deterministic approach with the help of several mathematical models. In this
article characteristics of a multi-wave SIR model have been studied which
successfully explains the features of this pandemic waves in India. Stability
of this model has been studied by identifying the equilibrium points as well as
by finding the eigen values of the corresponding Jacobian matrices. Complex
eigen values are found which ultimately give rise to the oscillatory solutions
for the three categories of populations, say, susceptible, infected and
removed. In this model, a finite probability of the recovered people for
becoming susceptible again is introduced which eventually lead to the
oscillatory solution in other words. The set of differential equations has been
solved numerically in order to obtain the variation for numbers of susceptible,
infected and removed people with time. In this phenomenological study, finally
an additional modification is made in order to explain the aperiodic
oscillation which is found necessary to capture the feature of epidemiological
waves particularly in India.
