Not sure if I understood your domain correctly. My understanding is that you have a lot of paths (aka referrals) between two arbitrary hospitals and want to sum them up. In this case use
START hospital1 = Node:Hospitals("*:*")
MATCH (hospital1)-[:CHILD_DOCTOR]->(doctor1)-[referral:REFERRED]-(doctor2)<-[:CHILD_DOCTOR]-(hospital2)
WHERE hospital1 <> hospital2
RETURN hospital1, hospital2, sum(referral.count)
N.B. didn't try the query myself. For further discussion please create a sample dataset on http://console.neo4j.org.