{"id":3910,"date":"2020-12-01T20:51:12","date_gmt":"2020-12-02T01:51:12","guid":{"rendered":"http:\/\/www.goodmath.org\/blog\/?p=3910"},"modified":"2020-12-02T08:52:00","modified_gmt":"2020-12-02T13:52:00","slug":"herd-immunity","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2020\/12\/01\/herd-immunity\/","title":{"rendered":"Herd Immunity"},"content":{"rendered":"\n<div class=\"wp-block-jetpack-markdown\"><p>With COVID running rampant throughout the US, I&#8217;ve seen a bunch of discussions about herd immunity, and questions about what it means. There&#8217;s a simple mathematical concept behind it, so I decided to spend a bit of time explaining.<\/p>\n<p>The basic concept is pretty simple. Let&#8217;s put together a simple model of an infectious disease. This will be an extremely simple model &#8211; we won&#8217;t consider things like variable infectivity, population age distributions, population density &#8211; we&#8217;re just building a simple model to illustrate the point.<\/p>\n<p>To start, we need to model the infectivity of the disease. This is typically done using the name <img src='http:\/\/l.wordpress.com\/latex.php?latex=R_0&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='R_0' style='vertical-align:1%' class='tex' alt='R_0' \/>. <img src='http:\/\/l.wordpress.com\/latex.php?latex=R_0&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='R_0' style='vertical-align:1%' class='tex' alt='R_0' \/> is the average number of susceptible people that will be infected by each person with the disease.<\/p>\n<p><img src='http:\/\/l.wordpress.com\/latex.php?latex=R_0&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='R_0' style='vertical-align:1%' class='tex' alt='R_0' \/> is the purest measure of infectivity &#8211; it&#8217;s the infectivity of the disease in ideal circumstances. In practice, we look for a value <img src='http:\/\/l.wordpress.com\/latex.php?latex=R&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='R' style='vertical-align:1%' class='tex' alt='R' \/>, which is the actual infectivity. <img src='http:\/\/l.wordpress.com\/latex.php?latex=R&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='R' style='vertical-align:1%' class='tex' alt='R' \/> includes the effects of social behaviors, population density, etc.<\/p>\n<p>The <em>state<\/em> of an infectious disease is based on the expected number of new infections that will be produced by each infected individual. We compute that by using a number S, which is the proportion of the population that is susceptible to the disease.<\/p>\n<ul>\n<li>If <em>R S &lt; 1<\/em>, then the disease dies out without spreading throughout the population. More people can get sick, but each wave of infection will be smaller than the last.<\/li>\n<li>If <em>R S = 1<\/em>, then the disease is said to be <em>endemic<\/em>. It continues as a steady state in the population. It never spreads dramatically, but it never dies out, either.<\/li>\n<li>If <em>R S &gt; 1<\/em>, then the disease is <em>pandemic<\/em>. Each wave of infection spreads the disease to a larger subsequent wave. The higher the value of <img src='http:\/\/l.wordpress.com\/latex.php?latex=R&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='R' style='vertical-align:1%' class='tex' alt='R' \/> in a pandemic, the faster the disease will spread, and the more people will end up sick.<\/li>\n<\/ul>\n<p>There are two keys to managing the spread of an infectious disease<\/p>\n<ol>\n<li>Reduce the effective value of <img src='http:\/\/l.wordpress.com\/latex.php?latex=R&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='R' style='vertical-align:1%' class='tex' alt='R' \/>. The value of <img src='http:\/\/l.wordpress.com\/latex.php?latex=R&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='R' style='vertical-align:1%' class='tex' alt='R' \/> can be affected by various attributes of the population, including behavioral ones. In the case of COVID-19, an infected person wearing a mask will spread the disease to fewer others; and if other people are also wearing masks, then it will spread even less.<\/li>\n<li>Reduce the value of <img src='http:\/\/l.wordpress.com\/latex.php?latex=S&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='S' style='vertical-align:1%' class='tex' alt='S' \/>. If there are fewer susceptible people in the population, then even with a high value of <img src='http:\/\/l.wordpress.com\/latex.php?latex=R&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='R' style='vertical-align:1%' class='tex' alt='R' \/>, the disease can&#8217;t spread as quickly.<\/li>\n<\/ol>\n<p>The latter is the key concept behind herd immunity. If you can get the value of <img src='http:\/\/l.wordpress.com\/latex.php?latex=S&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='S' style='vertical-align:1%' class='tex' alt='S' \/> to be small enough, then you can get <img src='http:\/\/l.wordpress.com\/latex.php?latex=R%20%2A%20S&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='R * S' style='vertical-align:1%' class='tex' alt='R * S' \/> to the sub-endemic level &#8211; you can prevent the disease from spreading. You&#8217;re effectively denying the disease access to enough susceptible people to be able to spread.<\/p>\n<p>Let&#8217;s look at a somewhat concrete example. The <img src='http:\/\/l.wordpress.com\/latex.php?latex=R_0&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='R_0' style='vertical-align:1%' class='tex' alt='R_0' \/> for measles is somewhere around 15, which is <em>insanely<\/em> infectious. If 50% of the population is susceptible, and no one is doing anything to avoid the infection, then each person infected with measles will infect 7 or 8 other people &#8211; and they&#8217;ll each infect 7 or 8 others &#8211; and so on, which means you&#8217;ll have epidemic spread.<\/p>\n<p>Now, let&#8217;s say that we get 95% of the population vaccinated, and they&#8217;re immune to measles. Now <img src='http:\/\/l.wordpress.com\/latex.php?latex=R%20%2A%20S%20%3D%2015%20%2A%200.05%20%3D%200.75&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='R * S = 15 * 0.05 = 0.75' style='vertical-align:1%' class='tex' alt='R * S = 15 * 0.05 = 0.75' \/>. The disease isn&#8217;t able to spread. If you had an initial outbreak of 5 infected, then they&#8217;ll infect around 3 people, who&#8217;ll infect around 2 people, who&#8217;ll infect one person, and soon, there&#8217;s no more infections.<\/p>\n<p>In this case, we say that the population has <em>herd immunity<\/em> to the measles. There aren&#8217;t enough susceptible people in the population to sustain the spread of the disease &#8211; so if the disease is introduced to the population, it will rapidly die out. Even if there are individuals who are still susceptible, they probably won&#8217;t get infected, because there aren&#8217;t enough other susceptible people to carry it to them.<\/p>\n<p>There are very few diseases that are as infectious as measles. But even with a disease that is <em>that<\/em> infectious, you can get to herd immunity relatively easily with vaccination.<\/p>\n<p>Without vaccination, it&#8217;s still possible to develop herd immunity. It&#8217;s just extremely painful. If you&#8217;re dealing with a disease that can kill, getting to herd immunity means letting the disease spread until enough people have gotten sick and recovered that the disease can&#8217;t spread any more.  What that means is letting a huge number of people get sick and suffer &#8211; and let some portion of those people die.<\/p>\n<p>Getting back to COVID-19: it&#8217;s got an <img src='http:\/\/l.wordpress.com\/latex.php?latex=R_0&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='R_0' style='vertical-align:1%' class='tex' alt='R_0' \/> that&#8217;s much lower. It&#8217;s somewhere between 1.4 and 2.5.  Of those who get sick, even with good medical care, somewhere between 1 and 2% of the infected end up dying. Based on that <img src='http:\/\/l.wordpress.com\/latex.php?latex=R_0&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='R_0' style='vertical-align:1%' class='tex' alt='R_0' \/>, herd immunity for COVID-19 (the value of S required to make R*S&lt;1) is somewhere around 50% of the population. Without a vaccine, that means that we&#8217;d need to have 150 million people in the US get sick, and of those, around 2 million would die.<\/p>\n<p><em>(UPDATE: Ok, so I blew it here. The papers that I found in a quick search appear to have a really bad estimate. The current CDC estimate of <img src='http:\/\/l.wordpress.com\/latex.php?latex=R_0&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='R_0' style='vertical-align:1%' class='tex' alt='R_0' \/> is around 5.7 &#8211; so the S needed for herd immunity is significantly higher &#8211; upward of 80%, and so the would the number of deaths.)<\/em><\/p>\n<p>A strategy for dealing with an infection disease that accepts the needless death of 2 million people is not exactly a good strategy.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[22],"tags":[],"class_list":["post-3910","post","type-post","status-publish","format-standard","hentry","category-good-math"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-114","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/3910","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=3910"}],"version-history":[{"count":8,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/3910\/revisions"}],"predecessor-version":[{"id":3919,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/3910\/revisions\/3919"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=3910"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=3910"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=3910"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}