Editing Herd immunity (section)

==Theoretical basis==
{{See also|Basic reproduction number}}
{{Further|Mathematical modelling of infectious diseases}}
{{herd_immunity_threshold_vs_r0.svg}}

The critical value, or threshold, in a given population, is the point where the disease reaches an [[Mathematical modelling of infectious disease#Endemic steady state|endemic steady state]], which means that the infection level is neither growing nor declining [[Exponential growth|exponential]]ly. This threshold can be calculated from the [[effective reproduction number]] ''R''<sub>e</sub>, which is obtained by taking the product of the [[basic reproduction number]] ''R''<sub>0</sub>, the average number of new infections caused by each case in an entirely susceptible population that is homogeneous, or well-mixed, meaning each individual is equally likely to come into contact with any other susceptible individual in the population,<ref name="pmid15627236"/><ref name="pmid24175217"/><ref name="pmid19197342"/> and ''S'', the proportion of the population who are susceptible to infection, and setting this product to be equal to 1:

: <math> R_0 \cdot S=1. </math>

''S'' can be rewritten as (1 − ''p''), where ''p'' is the proportion of the population that is immune so that ''p'' + ''S'' equals one. Then, the equation can be rearranged to place ''p'' by itself as follows:{{cn|date=May 2023}}

: <math> R_0 \cdot (1-p)=1, </math>
: <math> 1-p=\frac {1} {R_0}, </math>
: <math> p_c=1 - \frac {1} {R_0}. </math>

With ''p'' being by itself on the left side of the equation, it can be renamed as ''p''<sub>c</sub>, representing the critical proportion of the population needed to be immune to stop the transmission of disease, which is the same as the "herd immunity threshold" HIT.<ref name=pmid15627236/> ''R''<sub>0</sub> functions as a measure of contagiousness, so low ''R''<sub>0</sub> values are associated with lower HITs, whereas higher ''R''<sub>0</sub>s result in higher HITs.<ref name=pmid24175217/><ref name=pmid19197342/> For example, the HIT for a disease with an ''R''<sub>0</sub> of 2 is theoretically only 50%, whereas a disease with an ''R''<sub>0</sub> of 10 the theoretical HIT is 90%.<ref name=pmid24175217/>

When the effective reproduction number ''R''<sub>e</sub> of a contagious disease is reduced to and sustained below 1 new individual per infection, the number of cases occurring in the population gradually decreases until the disease has been eliminated.<ref name=pmid15627236/><ref name=pmid24175217/><ref name=dabmago>{{cite book|vauthors=Dabbaghian V, Mago VK|date=2013|title=Theories and Simulations of Complex Social Systems|url=https://books.google.com/books?id=AdLBBAAAQBAJ&pg=PA134|publisher=Springer|pages=134–35|isbn=978-3642391491|access-date=29 March 2015|archive-date=1 May 2021|archive-url=https://web.archive.org/web/20210501053021/https://books.google.com/books?id=AdLBBAAAQBAJ&pg=PA134|url-status=live}}</ref> If a population is immune to a disease in excess of that disease's HIT, the number of cases reduces at a faster rate, outbreaks are even less likely to happen, and outbreaks that occur are smaller than they would be otherwise.<ref name=pmid21427399/><ref name=pmid15627236/> If the population immunity falls below the herd immunity threshold, where the effective reproduction number increases to above 1, the population is said to have an "immunity gap",<ref>{{cite journal |author1=Joseph L. Melnick |author2=Larry H. Taber |title=Developing Gap in Immunity to Poliomyelitis in an Urban Area |journal=Journal of the American Medical Association |date=1969 |volume=209 |issue=8 |pages=1181–1185 |access-date=20 September 2024 |publisher=American Medical Association |doi=10.1001/jama.1969.03160210013003 |pmid=5819667 |url=https://jamanetwork.com/journals/jama/article-abstract/347965}}</ref> and then the disease is neither in a steady state nor decreasing in [[Incidence (epidemiology)|incidence]], but is actively spreading through the population and infecting a larger number of people than usual.<ref name=pmid20667876/><ref name=dabmago/> 

An assumption in these calculations is that populations are homogeneous, or well-mixed, meaning that every individual is equally likely to come into contact with any other individual, when in reality populations are better described as social networks as individuals tend to cluster together, remaining in relatively close contact with a limited number of other individuals. In these networks, transmission only occurs between those who are geographically or physically close to one another.<ref name=pmid21427399/><ref name=pmid19197342/><ref name=pmid20667876/> The shape and size of a network is likely to alter a disease's HIT, making incidence either more or less common.<ref name=pmid24175217/><ref name=pmid19197342/> Mathematical models can use contact matrices to estimate the likelihood of encounters and thus transmission.<ref>{{Citation |last=von Csefalvay |first=Chris |title=Host factors |date=2023 |work=Computational Modeling of Infectious Disease |pages=93–119 |publisher=Elsevier |language=en |doi=10.1016/b978-0-32-395389-4.00012-8 |isbn=978-0-323-95389-4 |doi-access=free }}</ref>

In heterogeneous populations, ''R''<sub>0</sub> is considered to be a measure of the number of cases generated by a "typical" contagious person, which depends on how individuals within a network interact with each other.<ref name=pmid21427399/> Interactions within networks are more common than between networks, in which case the most highly connected networks transmit disease more easily, resulting in a higher ''R''<sub>0</sub> and a higher HIT than would be required in a less connected network.<ref name=pmid21427399/><ref name=pmid20667876/> In networks that either opt not to become immune or are not immunized sufficiently, diseases may persist despite not existing in better-immunized networks.<ref name=pmid20667876/>
{{#section:Basic reproduction number|r0hittable}}

===Overshoot===
The cumulative proportion of individuals who get infected during the course of a disease outbreak can exceed the HIT. This is because the HIT does not represent the point at which the disease stops spreading, but rather the point at which each infected person infects fewer than one additional person on average. When the HIT is reached, the number of additional infections does not immediately drop to zero. The excess of the cumulative proportion of infected individuals over the theoretical HIT is known as the '''overshoot'''.<ref>{{cite journal|vauthors=Handel A, Longini IM, Antia R|title=What is the best control strategy for multiple infectious disease outbreaks?|journal=Proceedings. Biological Sciences|volume=274|issue=1611|pages=833–7|date=March 2007|pmid=17251095|pmc=2093965|doi=10.1098/rspb.2006.0015|quote=In general, the number of infecteds grows until the number of susceptibles has fallen to S<sub>th</sub>. At this point, the average number of secondary infections created by an infected person drops below 1 and therefore the number of infecteds starts to decrease. However, right at this inflection point, the maximum number of infecteds is present. These infecteds will create on average less than 1, but still more than zero further infections, leading to additional depletion of susceptibles and therefore causing an overshoot.}}</ref><ref>{{cite journal|vauthors=Fung IC, Antia R, Handel A|title=How to minimize the attack rate during multiple influenza outbreaks in a heterogeneous population|journal=PLOS ONE|volume=7|issue=6|pages=e36573|date=11 June 2012|pmid=22701558|pmc=3372524|doi=10.1371/journal.pone.0036573|bibcode=2012PLoSO...736573F|doi-access=free}}</ref><ref>{{Cite news|vauthors=Bergstrom CT, Dean N|date=1 May 2020|title=Opinion: What the Proponents of 'Natural' Herd Immunity Don't Say|language=en-US|work=The New York Times|url=https://www.nytimes.com/2020/05/01/opinion/sunday/coronavirus-herd-immunity.html|access-date=30 May 2020|archive-date=3 June 2020|archive-url=https://web.archive.org/web/20200603150038/https://www.nytimes.com/2020/05/01/opinion/sunday/coronavirus-herd-immunity.html|url-status=live}}</ref>