  When an infectious disease propagates throughout society, the incidence
function may rise at first due to an increase in pathogenicity and then
decrease due to inhibitory effects until it reaches saturation. Effective
vaccination and treatment are very helpful for controlling the effects of such
infectious diseases. To analyze the impacts of these diseases, we proposed a
new compartmental model with a generalized non-linear incidence function,
vaccination function, and treatment function, along with time delays in the
respective functions, which show how its monotonic features influence the
stability of the model. Fundamental properties of a model, such as positivity,
boundedness, and the existence of equilibria, are examined in this work. The
basic reproduction number has been computed, and correlative studies for local
stability in view of the basic reproduction number have been examined at the
disease-free and endemic equilibrium points. A delay-independent global
stability result has been established, and to be more precise, we explicitly
derived the result on global stability by restricting delay parameters within a
very specific range. Furthermore, numerical simulations and some examples based
on COVID-19 real-time data are pointed out to emphasize the significance of how
the disease's dynamical behavior is characterized by various functions for
controlling the spread of disease in a population and to justify the
mathematical conclusions.
