Say influenza, if tomorrow, everyone with influenza spontaneously died, would the disease cease to exist?
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Generally, no, for two reasons. The first reason depends on the disease, because some infectious diseases that infect humans also infect other animals, and can jump from those animals to humans. Influenza, for example, also infects birds and pigs. So, even if you killed all people who had the flu, the virus would survive in its animal hosts and sooner or later it would infect a person again.
Of course, we could expand your definition of “everyone” to also include all animals who had the disease. If you did this, then you’d get a lot closer to eradicating the disease. But you still have a problem, which is that the virus can survive for short periods outside its host. So just after you press the “kill all the sick”-button, there will still be virus floating around in the air near the deceased animas and humans, and there will stil be virus on their skin, in their feces and phlegm, and on surfaces that they touched. It’s likely that this remaining virus would manage to infect at least a few more live animals, which would then continue the spread.
Of course, if this particular disease were already close to extinction, then your odds of success improve. If there are only two humans left in the world with a certain virus (and no animals), and you killed them off, then it only would only take a few lucky coin flips for their remaining virus not to infect any further people, and then that’s the end of the road for that virus. But then the key factor there is really the vastly reduced numbers of the virus *before* you hit the kill-switch.
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