I am confused about how to decide whether to treat time as continuous or discrete in survival analysis. Specifically, I want to use survival analysis to identify child- and household-level variables that have the largest discrepancy in their impact on boys' versus girls' survival (up to age 5). I have a dataset of child ages (in months) along with an indicator for whether the child is alive, the age at death (in months), and other child- and household-level variables.
Since time is recorded in months and all children are under age 5, there are many tied survival times (often at half-year intervals: 0mos, 6mos, 12mos, etc). Based on what I have read about survival analysis, having many tied survival times makes me think I should be treating time as discrete. However, I have read several other studies where survival time is in, for example, person-years (and so surely there are tied survival times) and continuous-time methods like Cox proportional hazards are used.
What are the criteria I should use to decide whether to treat time as continuous or discrete? For my data and question, using some continuous-time model (Cox, Weibull, etc) makes intuitive sense to me, but the discrete nature of my data and the amount of tied survival times seem to suggest otherwise.