Since you state that "R [is] the average number of people an infected person goes on to infect." have you not just made R_hospital = 0.7+0.2=0.9 <1 but R_community = 0.8 + 0.4 =1.2 > 1, we need the total per person, not split into components.
Otherwise we could, for instance, split R_teacher = 1.5 = 0.9 (infected at school) + 0.6 (infected in community) to get all component Rs less than 1.