  Mathematical modelling of the spread of epidemics has been an interesting
challenge in the field of epidemiology. The SIR Model proposed by Kermack and
McKendrick in 1927 is a prototypical model of epidemiology. However, it has its
limitations. In this paper, we show two independent ways of generalizing this
model, the first one if the vaccine isn't discovered or ready to use and the
next one, if the vaccine is discovered and ready to use. In the first part, we
have pointed out a major over-simplification, i.e., assumption of variation of
the time derivatives of the variables with the linear or quadratic powers of
the individual variables and introduce two new parameters to incorporate
further nonlinearity in the number of infected people in the model. As a result
of this, we show how this additional nonlinearity, in the newly introduced
parameters, can bring a significant shift in the peak time of infection, i.e.,
the time at which the infected population reaches maximum. We show that in
special cases, even we can get a transition from epidemic to a non-epidemic
stage of a particular infectious disease. We further study one such special
case and treat it as a problem of phase transition. Then, we investigate all
the necessary parameters of this phase transition, like the order parameter and
critical exponent. We observe that $O_p \sim (q_c-q)^{\beta}$. {\it As far as
we know the phase transition and its quantification in terms of the scaling
behaviour is not yet know in the context of pandemic}. In the second part, we
incorporate in the model, a consideration of artificial herd immunity and show
how we can decrease the peak time of infection with a subsequent decrease in
the maximum number of infected people.

Populations and Evolution;Physics and Society