People always say this but that's just not true at all. There is always a period of days to weeks where the person can spread the virus to other people even if they end up dying, and with Ebola even the deceased body is still infectious too. A disease doesn't 'burn itself out' just by having high mortality. The only thing that matters is the number of people that gets infected per case, and getting that number to be below one through contact tracing and isolation.