= 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.<Principles of Epidemiology in Public Health Practice by the Centers for Disease Control and Prevention>[Lesson 1, section 11, Epidemic Patterns: propagated and mixed outbreaks]{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 $\mathbb E[I_{k+1}\mid I_k]=RI_k$ and $\mathbb E I_k=R^kI_0$. 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.
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