Without any data, I suspect the incidence rate of PPD is unaffected by social factors but having a support system definitely helps mitigate the effects on the family (i.e. more hands to help with the baby). Fortunately my family has never had to deal with it but it's definitely not some minor illness for people.
Quickly looking at the research, seems you are right. Like most diseases, it's interplay between genetics and environment.
Social support is less important risk factor, while factors like marital satisfaction and partner's depression are more important. For women, adverse events such as loss of a loved one, husband's loss of job or serious economical issues seem to be important too (Escriba-Aguir et al. 2010).
This is also quite concordant with the fact that PPD is more common in less developed countries, as mentioned in another comment.
