Editing Logistic function (section)

==== 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.