Applying offset correctly in an Interrupted Time Series Analysis using segmented regression in GLM in R [closed]

I would like to apply an offset to allow for the population within segmented regression using GLM on count data in R and need to ensure that the formula I am using is correct. Adding offset in 2 different ways to the formula below seems to operate as expected but I just wanted to check if they are right.

model <- glm(outcome ~  1 + Intv + Time * (1 - Intv) + Time * (Intv) * offset(log(Pop_Total)),
data = data,
family = poisson)

model4 <- glm(outcome ~ 1 + Intv + Time * (1 - Intv) + Time *  (Intv),
data = data,
family = poisson,
offset = log(Pop_Total))


closed as off-topic by jbowman, Michael Chernick, kjetil b halvorsen, Peter Flom♦Feb 3 '18 at 14:26

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Why do you have Time * (1 - Intv) + Time * (Intv) in formula? It yields the same as outcome ~ Time * (Intv). Also, I guess you intend to have ... + offset(log(Pop_Total)) and not ... * offset(log(Pop_Total)). Otherwise, I am not sure that model and model4 gives the same. See the examples in help("glm").