Logistic curve fitting in epidemiology: Difference between revisions

Logistic curve fitting in epidemiology (view source)

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elided the 1st task, corrected 2 misspellings, used a bigger font to show the super- and subscripts, (fine print), used a consistent font for variables in/from expressions.corrected 2 uses of n sub oh-->n sub zero, moved 2 definitions closer to their use.

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rosettacode>Gerard Schildberger

Revision as of 23:20, 8 September 2020 (view source) Robbie (talk \| contribs) (→‎{{header\|Wren}}) ← Older edit		Revision as of 00:20, 9 September 2020 (view source) rosettacode>Gerard Schildberger m (elided the 1st task, corrected 2 misspellings, used a bigger font to show the super- and subscripts, (fine print), used a consistent font for variables in/from expressions.corrected 2 uses of n sub oh-->n sub zero, moved 2 definitions closer to their use.) Newer edit →
Line 12: including the growth of epidemics, are curves akin to the logistic curve: ~~'''f~~ <big><big><big><b> ƒ(x) = L / (1 + e<sup>-k(x-x<sub>0</sub>)</sup>)~~'''~~ </b></big></big></big> Though predictions based on fitting to such curves may ~~err~~error, especially if used to extrapolate from incomplete data, curves similar to the logistic curve have had good fits in population studies, including modeling the growth of past epidemics. ~~The task:~~ ~~;Task~~ * Given the following daily world totals since December 31, 2019 for persons ▼ ~~who have become infected with the novel coronavirus Covid-19:~~ ▲* Given the following daily world totals since <tt> December 31, 2019 </tt> for persons who have become infected with the novel coronavirus Covid-19: <pre> Daily totals: Line 38 ⟶ 34: </pre> * Use the following variant of the logistic curve as a formula:▼ ▲* Use the following variant of the logistic curve as a formula: ~~'''f~~ <big><big><big><b> ƒ(t) = n<sub>0</sub> e<sup>(r t)</sup> / ((1 + n<sub>0</sub> (e<sup>(r t)</sup> - 1) / K)~~'''~~ </b></big></big></big> Where: ~~Where~~::*   <big><b> r </b></big>   is the rate of growth of the infection in the population. ::*   <big><b> K </b></big>   is the world population, about 7.8 billion.▼ ::* n0  <big><b> n<sub>0</sub> </b></big>   is 27, the number of cases found in China at the start of the pandemic. ▼ The R0 of an infection (different from r above) is a measure of how many ▼ ▲The   <big><b> R0 </b></big>   of an infection (different from   <big><b>r</b></big>   above) is a measure of how many new individuals will become infected for every individual currently infected. It is an important measure of how quickly an infectious disease may spread. <big><b>R0</b></big>   is related to the logistic curve's   <big><b>r</b></big>   parameter by the formula: ~~'''~~ <big><big><big><b> r ≈ ln(R0) / G~~'''~~ </b></big></big></big> where   <big><b> G, </b></big>   the generation time, is roughly the sum of the incubation time, perhaps 5 days, and the mean contagion period, perhaps 7 days, so, for covid-19, roughly we have: <big><big><big><b> R0 ≈ e<sup>12r</sup> </b></big></big></big> ~~'''R0 ≈ e<sup>12r</sup>'''~~ ~~Assume the following constants hold in the formula above:~~ ▲* K is the world population, about 7.8 billion ▲* n0 is 27, the number of cases found in China at the start of the pandemic. ;Task: * Demonstrate code that finds a least-squares fits of the curve to the data. * Show the calculated   <big><b> r </b></big>   for the logistic curve. * Show the final   <big><b> R0 </b></big>   parameter you calculate from the logistic curve   <big><b> r </b></big>   value parameter. ;See also▼ ▲;See also :;[[https://en.wikipedia.org/wiki/Basic_reproduction_number Basic reproduction number]] :;[[https://en.wikipedia.org/wiki/Logistic_function Logistic functions]] Line 73 ⟶ 72: :;[[https://ourworldindata.org/coronavirus#all-charts-preview World covid-19 case tallies]] :;[[https://www.zoology.ubc.ca/~bio310/Blank%20Page%204_files/DL%20R%20and%20r.htm Calculating r and R0]] <br><br> =={{header\|C}}==