Maybe it can take a month to kill, so the area where it started with have an initially higher death rate as things lag.
Also, the sicker you are, the less likely you are to move/travel, so a local higher death rate makes some sense.
Spanish Flu was so bad, in part, because the sickest WW1 soldiers, those carrying the worst strain, would be transferred across continents, circumventing a natural check on transmission.