  The Susceptible-Infected-Recovered (SIR) epidemic model as well as its
generalizations are extensively used for the study of the spread of infectious
diseases, and for the understanding of the dynamical evolution of epidemics.
From SIR type models only the model without vital dynamics has an exact
analytic solution, which can be obtained in an exact parametric form. The SIR
model with vital dynamics, the simplest extension of the basic SIR model, does
not admit a closed form representation of the solution. However, in order to
perform the comparison with the epidemiological data accurate representations
of the time evolution of the SIR model with vital dynamics would be very
useful. In the present paper, we obtain first the basic evolution equation of
the SIR model with vital dynamics, which is given by a strongly nonlinear
second order differential equation. Then we obtain a series representation of
the solution of the model, by using the Adomian and Laplace-Adomian
Decomposition Methods to solve the dynamical evolution equation of the model.
The solutions are expressed in the form of infinite series. The series
representations of the time evolution of the SIR model with vital dynamics are
compared with the exact numerical solutions of the model, and we find that, at
least for a specific range of parameters, there is a good agreement between the
Adomian and Laplace-Adomian semianalytical solutions, containing only a small
number of terms, and the numerical results.
