Editing Logistic function (section)

=== In medicine: modeling of a pandemic ===
{{main|Compartmental models in epidemiology}}
A novel infectious pathogen to which a population has no immunity will generally spread exponentially in the early stages, while the supply of susceptible individuals is plentiful.  The SARS-CoV-2 virus that causes [[COVID-19]] exhibited exponential growth early in the course of infection in several countries in early 2020.<ref>[https://www.worldometers.info/coronavirus/ Worldometer: COVID-19 CORONAVIRUS PANDEMIC]</ref> Factors including a lack of susceptible hosts (through the continued spread of infection until it passes the threshold for [[herd immunity]]) or reduction in the accessibility of potential hosts through physical distancing measures, may result in exponential-looking epidemic curves first linearizing (replicating the "logarithmic" to "logistic" transition first noted by [[Pierre François Verhulst|Pierre-François Verhulst]], as noted above) and then reaching a maximal limit.<ref>{{Cite arXiv |eprint = 2004.02406|last1 = Villalobos-Arias|first1 = Mario|title = Using generalized logistics regression to forecast population infected by Covid-19|year = 2020|class = q-bio.PE}}</ref>

A logistic function, or related functions (e.g. the [[Gompertz function]]) are usually used in a descriptive or phenomenological manner because they fit well not only to the early exponential rise, but to the eventual levelling off of the pandemic as the population develops a herd immunity. This is in contrast to actual models of pandemics which attempt to formulate a description based on the dynamics of the pandemic (e.g. contact rates, incubation times, social distancing, etc.). Some simple models have been developed, however, which yield a logistic solution.<ref>{{cite journal |last1=Postnikov |first1=Eugene B. |date=June 2020 |title=Estimation of COVID-19 dynamics "on a back-of-envelope": Does the simplest SIR model provide quantitative parameters and predictions? |url= |journal=Chaos, Solitons & Fractals |volume=135 |page=109841 |doi=10.1016/j.chaos.2020.109841 |pmid=32501369 |pmc=7252058 <!--|access-date=July 20, 2020-->|bibcode=2020CSF...13509841P }}</ref><ref>{{Cite medRxiv |last1=Saito |first1=Takesi |date=June 2020 |title=A Logistic Curve in the SIR Model and Its Application to Deaths by COVID-19 in Japan |medrxiv=10.1101/2020.06.25.20139865v2}}</ref><ref name="Reiser2020">{{cite arXiv|eprint=2006.01550 |last1=Reiser |first1=Paul A. |title=Modified SIR Model Yielding a Logistic Solution |year=2020 |class=q-bio.PE }}</ref>

==== Modeling early COVID-19 cases ====
[[File:Combined GLF.jpg|class=skin-invert-image|400px|thumb|[[Generalized logistic function]] (Richards growth curve) in epidemiological modeling]]
A [[generalized logistic function]], also called the Richards growth curve, has been applied to model the early phase of the [[COVID-19]] outbreak.<ref>{{Cite journal |last1=Lee|first1=Se Yoon |first2=Bowen |last2=Lei|first3=Bani|last3=Mallick|  title = Estimation of COVID-19 spread curves integrating global data and borrowing information|journal=PLOS ONE|year=2020|volume=15 |issue=7 |pages=e0236860 |doi=10.1371/journal.pone.0236860|pmid=32726361 |pmc=7390340 |arxiv=2005.00662 |bibcode=2020PLoSO..1536860L |doi-access=free}}</ref> The authors fit the generalized logistic function to the cumulative number of infected cases, here referred to as ''infection trajectory''. There are different parameterizations of the [[generalized logistic function]] in the literature. One frequently used forms is

<math display="block"> f(t ; \theta_1,\theta_2,\theta_3, \xi) = \frac{\theta_1}{{\left[1 + \xi \exp \left(-\theta_2 \cdot (t - \theta_3) \right) \right]}^{1/\xi}}</math>

where <math>\theta_1,\theta_2,\theta_3</math> are real numbers, and <math> \xi </math> is a positive real number. The flexibility of the curve <math>f</math> is due to the parameter <math> \xi </math>: (i) if <math> \xi = 1 </math> then the curve reduces to the logistic function, and (ii) as <math> \xi </math> approaches zero, the curve converges to the [[Gompertz function]]. In epidemiological modeling, <math>\theta_1</math>, <math>\theta_2</math>, and <math>\theta_3</math> represent the final epidemic size, infection rate, and lag phase, respectively. See the right panel for an example infection trajectory when <math>(\theta_1,\theta_2,\theta_3)</math> is set to <math>(10000,0.2,40)</math>.
[[File:COVID_19_Outbreak.jpg|class=skin-invert-image|right|thumb|400x400px|Extrapolated infection trajectories of 40 countries severely affected by COVID-19 and grand (population) average through May 14th]]
One of the benefits of using a growth function such as the [[generalized logistic function]] in epidemiological modeling is its relatively easy application to the [[multilevel model]] framework, where information from different geographic regions can be pooled together.