I am doing a binomial not neg binomial but I would think this section from the documentation can help. I have site and individuals and ended up trying to nest but it looks like it gives the same effect as doing (1 |site) + (1|AnimalID).

"specify a model for the random effects, in the notation that is common

to the nlme and lme4 packages. Random effects are specified as x|g,

where x is an effect and g is a grouping factor (which must be a factor

variable, or a nesting of/interaction among factor variables). For example,

the formula would be 1|block for a random-intercept model or

time|block for a model with random variation in slopes through time

across groups specified by block. A model of nested random effects

(block within site) could be 1|site/block if block labels are reused

across multiple sites, or (1|site)+ (1|block) if the nesting structure

is explicit in the data and each level of block only occurs within one

site. A model of crossed random effects (block and year) would be

(1|block)+(1|year)."