I am doing a research on artificial (non-native) bilingualism, i.e. when a child is brought up a foreign language in a monolingual family environment, providing the language taught is not a mother tongue of any of the parents.
I would like to apply a sociolinguistic model in my research, such as the model of horizontal and vertical multilingualism, the model of social networks, or the model of ethnolinguistic vitality. The problem is that within this kind of bilingualism, it is difficult to determine sociological variables such as the number of speakers, the territory where it is used, the institutional support and so on.
Is it possible to apply any of these sociolinguistic models to this type of bilingualism? What kind of approach would be appropriate to study artificial bilingualism?