  An SEIRS epidemic with disease fatalities is introduced in a growing
population (modelled as a super-critical linear birth and death process). The
study of the initial phase of the epidemic is stochastic, while the analysis of
the major outbreaks is deterministic. Depending on the values of the
parameters, the following scenarios are possible. i) The disease dies out
quickly, only infecting few; ii) the epidemic takes off, the \textit{number} of
infected individuals grows exponentially, but the \textit{fraction} of infected
individuals remains negligible; iii) the epidemic takes off, the
\textit{number} of infected grows initially quicker than the population, the
disease fatalities diminish the growth rate of the population, but it remains
super critical, and the \emph{fraction} of infected go to an endemic
equilibrium; iv) the epidemic takes off, the \textit{number} of infected
individuals grows initially quicker than the population, the diseases
fatalities turn the exponential growth of the population to an exponential
decay.
