Stochastic epidemics in growing populations

Consider a uniformly mixing population which grows as a super-critical linear birth and death process. At some time an infectious disease (of SIR or SEIR type) is introduced by one individual being infected from outside. It is shown that three different scenarios may occur: 1) an epidemic never takes off, 2) an epidemic gets going and grows but at a slower rate than the community thus still being negligible in terms of population fractions, or 3) an epidemic takes off and grows quicker than the community eventually leading to an endemic equilibrium. Depending on the parameter values, either scenario 1 is the only possibility, both scenario 1 and 2 are possible, or scenario 1 and 3 are possible.