 Propagated outbreak

A propagated outbreak spreads through transmission from infected hosts to further hosts, producing successive infection generations. The transmission route can involve direct contact, a shared vehicle or a vector. Separated peaks in an epidemic curve can be suggestive, but overlapping transmission generations need not give visibly separated waves.^[ref]

For an elementary branching approximation, let $I_k$ count infections in generation $k$ and suppose each infectious host generates an average $R$ new infections while susceptibility remains effectively unchanged. Then $\mathbb{E}[I_{k+1}\mid I_k]=R{I}_k$ and $\mathbb{E}{I}_k=R^k{I}_0$. Starting from one infection with $R=2$ gives expected generation sizes $1,2,4,8$. This is a mean calculation, not a guarantee that every introduction survives: random transmission can produce extinction.

The generation interval controls the timing between infection generations. Depletion of susceptible hosts, changing contacts or interventions can change the effective reproduction number, invalidating the constant-$R$ approximation. A common-source outbreak can seed this mechanism, so distinguishing the initial source from later transmission is part of reconstructing the outbreak.