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Calculating numbers of week for a specific time period

Hi I have this data frame which contains the start and end dates of healthcare services. It looks like this:

id <- c("A", "B", "B", "C", "D")
start_date <- c("2014-08-18", "2014-08-04", "2016-10-10", "2016-04-01", "2015-12-31")
end_date <- c("2018-09-01", "2019-09-22", "2016-10-18", "2016-05-01", "2016-04-08")

df <- data.frame(id, start_date, end_date)

  id    start_date     end_date
1  A    2014-08-18    2018-09-01
2  B    2014-08-04    2019-09-22
3  B    2016-10-10    2016-10-18
4  C    2016-04-01    2016-05-01
5  D    2015-12-31    2016-04-08

The IDs are not unique because people may had multiple services simultaneously. These services may spanned across many years. However, I would like to calculate numbers of weeks for each service within the financial year 2016, which starts at 2016-04-01 and ends at 2017-03-31 . The data frame that I would like to have is:

  id    start_date    end_date       wks_FY16
1  A    2014-08-18    2018-09-01        52
2  B    2014-08-04    2019-09-22        52
3  B    2016-10-10    2016-10-18         1
4  C    2016-04-01    2016-05-01         4
5  D    2015-12-31    2016-04-08         1

where "wks_FY16" represents financial year 2016.

I think difftime should be useful, but I don't know how to specify the period that I need. I will be really grateful for your help on this.

We get the difference in 'weeks' using difftime after making corrections for the '2016' year

library(dplyr)
df %>% 
   mutate(across(start_date:end_date, as.Date), 
   wks_FY16 = as.integer(difftime(pmin(end_date, 
    as.Date('2017-03-31')), pmax(as.Date('2016-04-01'),
     start_date), units = "weeks")))

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