Why is a lottery conducted for sick patients to be cured?

Because the vaccine's production was limited. As found on Wikipedia:

The vaccine's production is rapidly increased, but due to limited production, the CDC awards vaccinations by lottery based on birth date. Inoculations take place for one full year until every survivor is vaccinated. First responders, doctors and others designated by the government are declared exempt from the lottery.

As to why use a lottery to decide who gets the vaccine, according to this Bulletin on the World Health Organization about Ethical considerations for vaccination programmes in acute humanitarian emergencies:

A small supply of vaccine could be equitably distributed through a lottery, but prioritizing particularly susceptible groups and individuals, or those most likely to spread the disease, would not be possible. Different rules govern decision-making and priority-setting during acute crises. Resource distribution during a crisis is often suboptimal because those engaged in humanitarian assistance can only do the “best they can” in the context of imperfect information, exceptional circumstances and needs far outweighing the available resources.

So it was a realistic way to show they were trying the best they could to distribute the vaccine in the most fair way possible during such a crisis.

Where there is something that everyone (in the pertinent group) needs or should get, but this is not possible (usually because there are not enough)…

The obvious way to decide how to choose who gets the thing is according to merit. Perhaps, for instance, a much older person should lose out to a healthy young person on the grounds that the former has already lived a full life… or the CEO of a large company has more value to society than does an alcoholic who lives on the street.

If there is some crisis, then choosing people according to their value in fighting the crisis is justifiable — hence choosing (for instance) doctors or soldiers over everyone else.

Beyond that… choosing people according to merit is very much justifiable — at least at face value. However, there are two difficulties.

One difficulty is that the question of who actually qualifies as being of greater merit is a very complex and difficult question… over and above which is the fact that people can get very irrational about it if their lives are at stake.

The other difficulty is that, arguably, if the issue is human life, then apparently related measures such as {value to society} or {value to immediate family} are actually not relevant — that every individual has the attribute {life}, and that is all that is relevant.

It is arguable that the issue is correctly positioned as being about fairness… and that the fairest way to choose is randomly.

As for the lottery by birth date: this is a ready way to assign numbers to people, for the sake of the lottery.