These are referred to as Employee`s Health Insurance (kenkou hoken) and Japanese National Health Insurance (kokumin hoken). Under the Japanese National Health Insurance the patient pays 30% of the cost while the government pays the remaining 70%. The monthly insurance premium is paid per household and is scaled to annual income. The Japanese National Health Insurance (Kokumin Hoken) is great because it is convenient but costs a lot in the long run and you must live permanently in Japan.
I would suggest that you should have basic travel insurance as this will cover you almost 100% for medical and hospital treatment in Japan. I lived in Japan for over 10 years and found that initially basic travel insurance from home covered me more than adequately. You will need to pay for the cost yourself upfront but get reimbursed later for the costs. You just need to supply a doctor`s certificate and the original receipts from the doctor`s visit.
I would also advise that you visit a local clinic rather than a larger hospital. I found a local clinic just around the corner from my apartment and this was a lot more convenient and saved a lot of time waiting in a big hospital. The local clinic over a number of years treated me for different aliments including colds and virus. I found it a lot better as I got to know the doctors and staff a lot easier and save time. I hope this helps you with information about Medical Insurance and Health Care in Japan.
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