Editing Basic reproduction number (section)

==Limitations of <math>R_0</math>==
Use of <math>R_0</math> in the popular press has led to misunderstandings and distortions of its meaning. <math>R_0</math> can be calculated from many different [[mathematical models]]. Each of these can give a different estimate of <math>R_0</math>, which needs to be interpreted in the context of that model.<ref name="Vegvari">{{cite journal | vauthors = Vegvari C | title = Commentary on the use of the reproduction number R during the COVID-19 pandemic | journal = Stat Methods Med Res | date = 2022 | volume = 31 | issue = 9 | pages = 1675–1685 | doi = 10.1177/09622802211037079 | pmid = 34569883| pmc = 9277711 }}</ref> Therefore, the contagiousness of different infectious agents cannot be compared without recalculating <math>R_0</math> with invariant assumptions. <math>R_0</math> values for past outbreaks might not be valid for current outbreaks of the same disease. Generally speaking, <math>R_0</math> can be used as a threshold, even if calculated with different methods: if <math>R_0 < 1</math>, the outbreak will die out, and if <math>R_0 > 1</math>, the outbreak will expand. In some cases, for some models, values of <math>R_0 < 1</math> can still lead to self-perpetuating outbreaks. This is particularly problematic if there are intermediate vectors between hosts (as is the case for [[Zoonosis|zoonoses]]), such as [[malaria]].<ref name="Li & Blakely">{{cite journal | vauthors = Li J, Blakeley D, Smith RJ | title = The failure of R0 | journal = Computational and Mathematical Methods in Medicine | volume = 2011 | issue = 527610 | pages = 527610 | year = 2011 | pmid = 21860658 | pmc = 3157160 | doi = 10.1155/2011/527610 | doi-access = free }}</ref> Therefore, comparisons between values from the "Values of <math>R_0</math> of well-known contagious diseases" table should be conducted with caution.

Although <math>R_0</math> cannot be modified through vaccination or other changes in population susceptibility, it can vary based on a number of biological, sociobehavioral, and environmental factors.<ref name="Delamater"/> It can also be modified by physical distancing and other public policy or social interventions,<ref name="Byrne" /><ref name="Delamater"/> although some historical definitions exclude any deliberate intervention in reducing disease transmission, including nonpharmacological interventions.<ref name=":0" /> And indeed, whether nonpharmacological interventions are included in <math>R_0</math> often depends on the paper, disease, and what if any intervention is being studied.<ref name="Delamater" /> This creates some confusion, because <math>R_0</math> is not a constant; whereas most mathematical parameters with "nought" subscripts are constants.

<math>R</math> depends on many factors, many of which need to be estimated. Each of these factors adds to uncertainty in estimates of <math>R</math>. Many of these factors are not important for informing public policy. Therefore, public policy may be better served by metrics similar to <math>R</math>, but which are more straightforward to estimate, such as [[doubling time]] or  [[half-life]] (<math>t_{1/2}</math>).<ref name = "Balkew">{{cite thesis | vauthors = Balkew TM |date=December 2010 |title=The SIR Model When S(t) is a Multi-Exponential Function |publisher=East Tennessee State University |url=https://dc.etsu.edu/etd/1747 }}</ref><ref name = "Ireland">{{cite book | veditors = Ireland MW |date=1928 |title=The Medical Department of the United States Army in the World War, vol. IX: Communicable and Other Diseases |location=Washington: U.S. |publisher=U.S. Government Printing Office |pages=116–7}}</ref>

Methods used to calculate <math>R_0</math> include the [[survival function]], rearranging the largest [[eigenvalue]] of the [[Jacobian matrix]], the [[Next-generation matrix|next-generation method]],<ref>{{cite book |vauthors=Diekmann O, Heesterbeek JA |chapter=The Basic Reproduction Ratio |pages=73–98 |title=Mathematical Epidemiology of Infectious Diseases : Model Building, Analysis and Interpretation| publisher=New York: Wiley|year=2000 |isbn=0-471-49241-8 |chapter-url=https://books.google.com/books?id=5VjSaAf35pMC&pg=PA73 }}</ref> calculations from the intrinsic growth rate,<ref>{{cite journal | vauthors = Chowell G, Hengartner NW, Castillo-Chavez C, Fenimore PW, Hyman JM | title = The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda | journal = Journal of Theoretical Biology | volume = 229 | issue = 1 | pages = 119–26 | date = July 2004 | pmid = 15178190 | doi = 10.1016/j.jtbi.2004.03.006 | arxiv = q-bio/0503006 | bibcode = 2004JThBi.229..119C | s2cid = 7298792 }}</ref> existence of the endemic equilibrium, the number of susceptibles at the endemic equilibrium, the average age of infection<ref>{{cite journal | vauthors = Ajelli M, Iannelli M, Manfredi P, Ciofi degli Atti ML | title = Basic mathematical models for the temporal dynamics of HAV in medium-endemicity Italian areas | journal = Vaccine | volume = 26 | issue = 13 | pages = 1697–707 | date = March 2008 | pmid = 18314231 | doi = 10.1016/j.vaccine.2007.12.058 | name-list-style = amp }}</ref> and the final size equation.<ref>{{Citation |last=von Csefalvay |first=Chris |title=2 - Simple compartmental models: The bedrock of mathematical epidemiology |date=2023-01-01 |url=https://www.sciencedirect.com/science/article/pii/B9780323953894000116 |work=Computational Modeling of Infectious Disease |pages=19–91 |editor-last=von Csefalvay |editor-first=Chris |publisher=Academic Press |language=en |doi=10.1016/b978-0-32-395389-4.00011-6 |isbn=978-0-323-95389-4 |access-date=2023-03-02|url-access=subscription }}</ref> Few of these methods agree with one another, even when starting with the same system of [[differential equations]].<ref name = "Li & Blakely" /> Even fewer actually calculate the average number of secondary infections. Since <math>R_0</math> is rarely observed in the field and is usually calculated via a mathematical model, this severely limits its usefulness.<ref>{{cite journal | vauthors = Heffernan JM, Smith RJ, Wahl LM | title = Perspectives on the basic reproductive ratio | journal = Journal of the Royal Society, Interface | volume = 2 | issue = 4 | pages = 281–93 | date = September 2005 | pmid = 16849186 | pmc = 1578275 | doi = 10.1098/rsif.2005.0042 }}</ref>