The Outer Limits: "Patient Zero"

I really enjoyed this episode of the Outer Limits because I found it very interesting and the way of time travel was pretty cool. Basically, there is a group of people from a time period of the plague, a deadly disease that is killing everyone. This group of people decides to send a man back in time to kill 'patient zero' so that the plague never starts. They send the man back in, what you could call a time machine. It is surrounded by coils and when he is send back it looks almost as if he is electrocuted and there are flashes of light. He seems to go through some pain that goes away quickly. The man, who is sent back, has lost his wife and child to the plague and is very eager to kill patient zero. When he gets to the past, he is run over by a car, and patient zero, who is a girl, comes to his rescue. He is taken to her house where he eats dinner there and sleeps. He slowly begins to learn more about her and realizes she is a good person and can't seem to kill her. When he does not go back to the future when expected, another man is sent back to finish the deed. The first man then finds a way to stop the plague without having to kill anyone, and has the rest of the day to do it. The doctor that patient zero has to come into contact with to start the plague, is beat up by him and told not to see patient zero ever again. This would stop the plague from happening. But the second man coming from the future then tells the first that his plan was right, but in doing it, he became patient zero. To become patient zero, you had to come into contact with the bodily fluids of three people, all of which he did. The episode ended with the first man being killed. A question that I had was, if he died in the past, then wouldn’t his child never exist?  I thought that this story demonstrated a good way to use time travel, which is to stop a worldwide epidemic.