SEIRS epidemics in growing populations

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 $\mathit{\text{number}}$ of infected individuals grows exponentially, but the $\mathit{\text{fraction}}$ of infected individuals remains negligible; iii) the epidemic takes off, the $\mathit{\text{number}}$ 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 $\mathit{\text{number}}$ of infected individuals grows initially quicker than the population, the diseases fatalities turn the exponential growth of the population to an exponential decay.