Herd immunity is all in the news but in many cases the news media is not really describing the concept clearly. While technically the concept of herd immunity derives from mathematical models of disease transmission, when you drill down, it is just generalizing common-sense thinking about disease transmission.<\/p>\n\n\n\n

Let\u2019s start by imagining only one person still has a disease and everyone else is immune. Then,  after you cure that person, assuming there is no animal reservoir (i.e. animals that can harbor the disease),  the disease is over and done with. That’s what happened with smallpox on that glorious day when, on Oct. 26, 1977, in Merka, Somalia the last natural case of Smallpox was diagnosed. As a mathematician would say: \u201cthis is the boundary case for herd immunity, there’s simply no one left to transmit disease to\u201d.<\/p>\n\n\n\n

More generally, I want you to imagine the following situation: you have a bunch of people who are immune and a bunch of people who are infected.  Now also imagine there is a large group of people who are neither infected nor immune, they are the \u201csusceptibles\u201d. OK, an infected person comes into contact with a group of people. Obviously, he or she encounters a certain number of immune people and a certain number of people who are susceptible. Suppose that person infects only one person and subsequently becomes cured. This gives us a steady state (potentially for a very for a long time). Every infected person passes the disease onto one person and gets cured themselves. The disease stays at whatever state it is in – that\u2019s why they call it a \u201csteady state\u201d. Yes, it still infects the susceptible population, but slowly, one person at a time, until there are no more susceptible people. Then and only then does it die out.<\/p>\n\n\n\n

But diseases usually don\u2019t work like that. More likely our one infected person infects more than one person: say two people and then, of course, those two people each infect two more people for a total of four. The next round each of those four infect 8 etc. Then you have exponential growth and disaster ensues quickly. The disease will grow uncontrollably and then you’re down the rabbit hole I talked about this in the parable of the Lily Pond (https:\/\/garycornell.com\/2020\/04\/20\/the-parable-of-the-lily-pond-or-why-you-should-shelter-in-place\/) . <\/p>\n\n\n\n

Since mathematicians and statisticians have no trouble thinking about \u201cfractional people\u201d, I want you to join us and imagine that one person can only infect half a person. In this fantasy world of fractional people, that half a person will only infect a quarter of a person, the quarter person only an eighth of a person and so on. In this case the disease dies out.<\/p>\n\n\n\n

In a nutshell, this is exactly what herd immunity is about:\u00a0 it’s about throwing\u00a0 infected people into a pool of both immune people and susceptible people, having them interact and then thinking about when the disease can no longer spread. More precisely, what the mathematical models try to do is predict when you will be in a situation where the number of immune people is so large that an infected person won’t find one or more susceptible people – even if they succeeded in finding \u201c.99\u201d of a person to infect before they can no longer spread the disease. Once the number of people an infected person infects on average is less than one person, \u201cherd immunity\u201d happens and the disease eventually dies out.\u00a0<\/p>\n\n\n\n

OK now you have seen what it\u2019s all about, let’s try a more realistic situation especially for Covid-19. Imagine a situation where a person will infect at most three people on average. Whether they do or not will obviously depend on the number of susceptible people they encounter.<\/p>\n\n\n\n

Let’s walk through a couple of scenarios:<\/p>\n\n\n\n

Scenario 1: One infected person encounters two susceptible people, both get the disease, and each of these two infect two out of three more people (i.e. they infect four people in the second level). All hell breaks loose if this scenario continues for too long.<\/p>\n\n\n\n

Scenario 2: One infected person\u00a0 always encounters one person who is susceptible and two who are immune. They infect that one person, and that person infects one person and so on. We are in a steady state for )potentially) a long time.<\/p>\n\n\n\n

Scenario 3:  The infected person doesn’t encounter any susceptible people, he\/she gets cured and the disease transmission stops.<\/p>\n\n\n\n

Looking over these scenarios, it is clear that scenario 2 is the \u201ctipping point\u201d. In fact, if you allow me my fractional people then, then if more than two out of three people are immune – even \u201c2.01\/3\u201d people immune\u201d, then the infected person will infect less than 1 person and the disease will (eventually) die out. Similarly, if there were say less than 2 people, say \u201c1.99\/3 people\u201d who were immune (2.01 people who are susceptible) , the infected person would infect more than 1 person and the disease will grow.<\/p>\n\n\n\n

Well, we’ve just proved a special case of the most important basic theorem in disease transmission, it\u2019s called the \u201cThreshold Theorem\u201d i.e. the threshold for herd immunity. While the model that led to this theorem is not necessarily the right one in all situations, nonetheless, we’ve essentially proved that if you have a disease where people can infect 3 people on average (R_{0<\/sub> is 3 as epidemiologists would say), then herd immunity occurs when more than 2 out of 3 (>\u2154, 66.67%) of the people are immune.<\/p>\n\n\n\n}

Next, imagine a scenario where each person can infect 4 people (R_{0<\/sub>=4). Can you see why the tipping point occurs if 3 out of every 4 people (75% = \u00be) are immune?\u00a0 With everyone infecting 5 people (R0<\/sub> =5) the tipping point occurs when 4\/5ths (80%) of the people are immune etc.\u00a0<\/p>\n\n\n\n}

And now I hope you can see that how the general threshold theorem would be proved and here is its statement:<\/p>\n\n\n\n

The threshold theorem: In the basic model of herd immunity, herd immunity occurs when more than <\/em>1-(1\/R_{0<\/sub>) <\/em><\/strong>percent of the people are immune.<\/em><\/p>\n\n\n\n}

Many epidemiologists believe that the threshold theorem with R_{0<\/sub> = 3 – i.e. our first situation actually describes COVID-19 transmission pretty well. So in this case herd immunity to COVID requires roughly \u2154 (66.67%) of the population to be immune.<\/p>\n\n\n\n}

That’s it: you now know more about herd immunity than your average talking head and probably more than your average doctor who doesn\u2019t have an MPh! But I obviously have to end by cautioning you that many epidemiologists want to use different values of R_{0 <\/sub>in the threshold theorem instead of 3 for Covid-19. For example, an R0<\/sub> of 2.5 would give (1- 1\/2.5 = 60%) an R0 <\/sub>of 4 would give 75%. And, more importantly, other epidemiologists want to use much more sophisticated models of disease transmission and throw out the basic threshold theorem completely when it comes to Covid 19. But no epidemiologist has a model that predicts anything like 20 or 25% as the level for herd immunity as far as I know and we have some facts on the ground that make, say, a radiologist who believes that, an inhabitant of the crazy place in the science landscape.\u00a0<\/p>\n","protected":false},"excerpt":{"rendered":"}

Herd immunity is all in the news but in many cases the news media is not really describing the concept clearly. While technically the concept of herd immunity derives from mathematical models of disease transmission, when you drill down, it is just generalizing common-sense thinking about disease transmission. Let\u2019s start by imagining only one person … Continue reading “Herd Immunity 1”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":"","footnotes":""},"categories":[6],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/garycornell.com\/wp-json\/wp\/v2\/posts\/579"}],"collection":[{"href":"https:\/\/garycornell.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/garycornell.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/garycornell.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/garycornell.com\/wp-json\/wp\/v2\/comments?post=579"}],"version-history":[{"count":1,"href":"https:\/\/garycornell.com\/wp-json\/wp\/v2\/posts\/579\/revisions"}],"predecessor-version":[{"id":580,"href":"https:\/\/garycornell.com\/wp-json\/wp\/v2\/posts\/579\/revisions\/580"}],"wp:attachment":[{"href":"https:\/\/garycornell.com\/wp-json\/wp\/v2\/media?parent=579"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/garycornell.com\/wp-json\/wp\/v2\/categories?post=579"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/garycornell.com\/wp-json\/wp\/v2\/tags?post=579"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}