Epidemic branching processes with and without vaccination
  Here we treat the transmission of disease through a population as a standard Galton-Watson branching process, modified to take the presence of vaccination into account. Vaccination reduces the number of secondary infections produced per infected individual. We show that introducing vaccination in a population therefore reduces the expected time to extinction of the infection. We also prove results relating the distribution of number of secondery infections with and without vaccinations. 
Arni S.R. Srinivasa Rao and Chris T. Bauch