Herd immunity as a public good

   Most economics students at high-school level should have encountered the concept "public good" already. Usually, they will be given examples like national defense, lighthouse, streetlights, fresh air, etc. I recently read some articles in relation to the current pandemic. They discuss herd immunity as a public good. It's timely that we use this example to explain the concept here.
   The "herd immunity" mentioned by the articles refers to vaccination that prevents widespread infections of a disease. It does not refer to an idea that is now proved to be a disaster -- allow more people to get infected by a disease (covid-19) so that a large part of the people in a community gets immunity and will not infect the rest of the society.
   Herd immunity in relation to vaccination is a much better idea than taking the risk to get infected first. But it is still not free of controversy. Some people (including influential people) believe that vaccination is unnatural, will cause severe side-effects, and does more harms than benefits. These accusations are not completely wrong though may be exaggerated in some aspects. Of course, these arguments were formulated before the covid-19 outbreak. They are more about flu, and covid-19 is more dangerous than flu. But these arguments cannot be simply ignored. We need some arguments for vaccination. The articles I read argue for the moral duty of accepting vaccines. They are not economics articles but try to use some economics concepts like public good.
   If many people have already taken vaccines, they are immune to an infectious disease. The disease cannot be easily spread in such a society. This is said to be a public good by the articles. Why? Recall the definition of public good. The good is non-rivalrous and non-excludable. A good is non-rivalrous if someone's taking it will NOT diminish its availability for others (such as streetlights). A good is non-excludable if there is no (easy) way to prevent one from taking or enjoying it (like fresh air) within a scope (say, in a city) when the good is already there. Now, herd immunity is both non-rivalrous and non-excludable. If most people are immune to the disease already, it is not easy for you to get infected. Your low risk of being infected will not make others more risky. Others still can enjoy the low infection risk. Also, you cannot prevent others from enjoying this low risk.
   If herd immunity is a public good, so what? To these article writers, they argue for the moral duty to take the vaccine so that immunity can be established at the community level. If few people take the vaccine, there is no significant effect at the social level though the few who have taken vaccine can protect themselves. But most people, without taking vaccines, will not run on street with a sense of safety. So, people should take vaccines. They should be told to do something good for the community.
   To economists, perhaps we tend to argue for policy supports. From an economic viewpoint, voluntary contribution to a public good will normally lead to under-supply of the good. If taking vaccines involve bearing a private cost (the side-effect of taking vaccines as well as monetary cost), and if others choose to take vaccines already, you are still safe without taking vaccines yourself. You are said to be a "free-rider" in economics. When many want to free-ride on others' providing the good (waiting for others to take the vaccines), the supply of the good will be too low. Therefore, some policy supports are justified, for example, subsidizing the vaccine costs. Of course, like what the article writers do, moral persuasion may also be done.
   Some public goods may be, or may be turned, excludable. For example, people living near Disneyland may free-ride on the fireworks displays sent from the theme park. They can watch the displays remotely but have never paid the theme park entrance fee. The good is non-excludable. But is it? In fact, before Disney builds a theme park, it will find site farther away from residential area and often requires the local government not to build new houses near the theme park. This effectively "excludes" those free riders from enjoying the good without paying. If a public good is excludable, we have a stronger case for voluntary contribution (like what Disney does). But in the case of herd immunity, it is not. So, voluntary contribution is less likely.