A network epidemic model is studied. The underlying social network has two different types of group structures, households and workplaces, such that each individual belongs to exactly one household and one workplace. The random network is constructed such that a parameter $\theta$ controls the degree of overlap between the two group structures: $\theta=0$, corresponding to all household members belonging to the same workplace, and $\theta=1$, corresponding to all household members belonging to distinct workplaces. On the network a stochastic SIR epidemic is defined, having an arbitrary but specified infectious period distribution, with global (community), household, and workplace infectious contacts. The stochastic epidemic model is analysed as the population size $n\to\infty$, with the asymptotic probability, and size, of a major outbreak obtained. In addition to allowing for a tunable overlap $0 \leq \theta \leq 1$ between the two group structures, we also extend the model by allowing for a non-constant infectious period, the presence or absence of global infection, and potentially asymptotically infinite local outbreaks.

We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate θ, new individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have independent and identically distributed lifetime durations (nonnecessarily exponential) during which they give birth independently at a constant rate b. First, using spine decomposition, we relax previously known assumptions required for almost-sure convergence of the total population size. Then, we consider three models of structured populations: either all immigrants have a different type, or types are drawn in a discrete spectrum or in a continuous spectrum. In each model, the vector (P 1, P 2,…) of relative abundances of surviving families converges almost surely. In the first model, the limit is the GEM distribution with parameter θ / b.