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<h1>Logistic kinetics</h1>
<small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/R/parent_solutions.R" class="external-link"><code>R/parent_solutions.R</code></a></small>
<div class="hidden name"><code>logistic.solution.Rd</code></div>
</div>
<div class="ref-description">
<p>Function describing exponential decline from a defined starting value, with
an increasing rate constant, supposedly caused by microbial growth</p>
</div>
<div id="ref-usage">
<div class="sourceCode"><pre class="sourceCode r"><code><span><span class="fu">logistic.solution</span><span class="op">(</span><span class="va">t</span>, <span class="va">parent_0</span>, <span class="va">kmax</span>, <span class="va">k0</span>, <span class="va">r</span><span class="op">)</span></span></code></pre></div>
</div>
<div id="arguments">
<h2>Arguments</h2>
<dl><dt>t</dt>
<dd><p>Time.</p></dd>
<dt>parent_0</dt>
<dd><p>Starting value for the response variable at time zero.</p></dd>
<dt>kmax</dt>
<dd><p>Maximum rate constant.</p></dd>
<dt>k0</dt>
<dd><p>Minimum rate constant effective at time zero.</p></dd>
<dt>r</dt>
<dd><p>Growth rate of the increase in the rate constant.</p></dd>
</dl></div>
<div id="value">
<h2>Value</h2>
<p>The value of the response variable at time <code>t</code>.</p>
</div>
<div id="note">
<h2>Note</h2>
<p>The solution of the logistic model reduces to the
<code><a href="SFO.solution.html">SFO.solution</a></code> if <code>k0</code> is equal to <code>kmax</code>.</p>
</div>
<div id="references">
<h2>References</h2>
<p>FOCUS (2006) “Guidance Document on Estimating Persistence
and Degradation Kinetics from Environmental Fate Studies on Pesticides in
EU Registration” Report of the FOCUS Work Group on Degradation Kinetics,
EC Document Reference Sanco/10058/2005 version 2.0, 434 pp,
<a href="http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics" class="external-link">http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics</a>
FOCUS (2014) “Generic guidance for Estimating Persistence
and Degradation Kinetics from Environmental Fate Studies on Pesticides in
EU Registration” Report of the FOCUS Work Group on Degradation Kinetics,
Version 1.1, 18 December 2014
<a href="http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics" class="external-link">http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics</a></p>
</div>
<div id="see-also">
<h2>See also</h2>
<div class="dont-index"><p>Other parent solutions:
<code><a href="DFOP.solution.html">DFOP.solution</a>()</code>,
<code><a href="FOMC.solution.html">FOMC.solution</a>()</code>,
<code><a href="HS.solution.html">HS.solution</a>()</code>,
<code><a href="IORE.solution.html">IORE.solution</a>()</code>,
<code><a href="SFO.solution.html">SFO.solution</a>()</code>,
<code><a href="SFORB.solution.html">SFORB.solution</a>()</code></p></div>
</div>
<div id="ref-examples">
<h2>Examples</h2>
<div class="sourceCode"><pre class="sourceCode r"><code><span class="r-in"><span></span></span>
<span class="r-in"><span> <span class="co"># Reproduce the plot on page 57 of FOCUS (2014)</span></span></span>
<span class="r-in"><span> <span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="kw">function</span><span class="op">(</span><span class="va">x</span><span class="op">)</span> <span class="fu">logistic.solution</span><span class="op">(</span><span class="va">x</span>, <span class="fl">100</span>, <span class="fl">0.08</span>, <span class="fl">0.0001</span>, <span class="fl">0.2</span><span class="op">)</span>,</span></span>
<span class="r-in"><span> from <span class="op">=</span> <span class="fl">0</span>, to <span class="op">=</span> <span class="fl">100</span>, ylim <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0</span>, <span class="fl">100</span><span class="op">)</span>,</span></span>
<span class="r-in"><span> xlab <span class="op">=</span> <span class="st">"Time"</span>, ylab <span class="op">=</span> <span class="st">"Residue"</span><span class="op">)</span></span></span>
<span class="r-in"><span> <span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="kw">function</span><span class="op">(</span><span class="va">x</span><span class="op">)</span> <span class="fu">logistic.solution</span><span class="op">(</span><span class="va">x</span>, <span class="fl">100</span>, <span class="fl">0.08</span>, <span class="fl">0.0001</span>, <span class="fl">0.4</span><span class="op">)</span>,</span></span>
<span class="r-in"><span> from <span class="op">=</span> <span class="fl">0</span>, to <span class="op">=</span> <span class="fl">100</span>, add <span class="op">=</span> <span class="cn">TRUE</span>, lty <span class="op">=</span> <span class="fl">2</span>, col <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></span>
<span class="r-in"><span> <span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="kw">function</span><span class="op">(</span><span class="va">x</span><span class="op">)</span> <span class="fu">logistic.solution</span><span class="op">(</span><span class="va">x</span>, <span class="fl">100</span>, <span class="fl">0.08</span>, <span class="fl">0.0001</span>, <span class="fl">0.8</span><span class="op">)</span>,</span></span>
<span class="r-in"><span> from <span class="op">=</span> <span class="fl">0</span>, to <span class="op">=</span> <span class="fl">100</span>, add <span class="op">=</span> <span class="cn">TRUE</span>, lty <span class="op">=</span> <span class="fl">3</span>, col <span class="op">=</span> <span class="fl">3</span><span class="op">)</span></span></span>
<span class="r-in"><span> <span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="kw">function</span><span class="op">(</span><span class="va">x</span><span class="op">)</span> <span class="fu">logistic.solution</span><span class="op">(</span><span class="va">x</span>, <span class="fl">100</span>, <span class="fl">0.08</span>, <span class="fl">0.001</span>, <span class="fl">0.2</span><span class="op">)</span>,</span></span>
<span class="r-in"><span> from <span class="op">=</span> <span class="fl">0</span>, to <span class="op">=</span> <span class="fl">100</span>, add <span class="op">=</span> <span class="cn">TRUE</span>, lty <span class="op">=</span> <span class="fl">4</span>, col <span class="op">=</span> <span class="fl">4</span><span class="op">)</span></span></span>
<span class="r-in"><span> <span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="kw">function</span><span class="op">(</span><span class="va">x</span><span class="op">)</span> <span class="fu">logistic.solution</span><span class="op">(</span><span class="va">x</span>, <span class="fl">100</span>, <span class="fl">0.08</span>, <span class="fl">0.08</span>, <span class="fl">0.2</span><span class="op">)</span>,</span></span>
<span class="r-in"><span> from <span class="op">=</span> <span class="fl">0</span>, to <span class="op">=</span> <span class="fl">100</span>, add <span class="op">=</span> <span class="cn">TRUE</span>, lty <span class="op">=</span> <span class="fl">5</span>, col <span class="op">=</span> <span class="fl">5</span><span class="op">)</span></span></span>
<span class="r-in"><span> <span class="fu"><a href="https://rdrr.io/r/graphics/legend.html" class="external-link">legend</a></span><span class="op">(</span><span class="st">"topright"</span>, inset <span class="op">=</span> <span class="fl">0.05</span>,</span></span>
<span class="r-in"><span> legend <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/paste.html" class="external-link">paste0</a></span><span class="op">(</span><span class="st">"k0 = "</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0.0001</span>, <span class="fl">0.0001</span>, <span class="fl">0.0001</span>, <span class="fl">0.001</span>, <span class="fl">0.08</span><span class="op">)</span>,</span></span>
<span class="r-in"><span> <span class="st">", r = "</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0.2</span>, <span class="fl">0.4</span>, <span class="fl">0.8</span>, <span class="fl">0.2</span>, <span class="fl">0.2</span><span class="op">)</span><span class="op">)</span>,</span></span>
<span class="r-in"><span> lty <span class="op">=</span> <span class="fl">1</span><span class="op">:</span><span class="fl">5</span>, col <span class="op">=</span> <span class="fl">1</span><span class="op">:</span><span class="fl">5</span><span class="op">)</span></span></span>
<span class="r-plt img"><img src="logistic.solution-1.png" alt="" width="700" height="433"></span>
<span class="r-in"><span></span></span>
<span class="r-in"><span> <span class="co"># Fit with synthetic data</span></span></span>
<span class="r-in"><span> <span class="va">logistic</span> <span class="op"><-</span> <span class="fu"><a href="mkinmod.html">mkinmod</a></span><span class="op">(</span>parent <span class="op">=</span> <span class="fu"><a href="mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"logistic"</span><span class="op">)</span><span class="op">)</span></span></span>
<span class="r-in"><span></span></span>
<span class="r-in"><span> <span class="va">sampling_times</span> <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0</span>, <span class="fl">1</span>, <span class="fl">3</span>, <span class="fl">7</span>, <span class="fl">14</span>, <span class="fl">28</span>, <span class="fl">60</span>, <span class="fl">90</span>, <span class="fl">120</span><span class="op">)</span></span></span>
<span class="r-in"><span> <span class="va">parms_logistic</span> <span class="op"><-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>kmax <span class="op">=</span> <span class="fl">0.08</span>, k0 <span class="op">=</span> <span class="fl">0.0001</span>, r <span class="op">=</span> <span class="fl">0.2</span><span class="op">)</span></span></span>
<span class="r-in"><span> <span class="va">d_logistic</span> <span class="op"><-</span> <span class="fu"><a href="mkinpredict.html">mkinpredict</a></span><span class="op">(</span><span class="va">logistic</span>,</span></span>
<span class="r-in"><span> <span class="va">parms_logistic</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>parent <span class="op">=</span> <span class="fl">100</span><span class="op">)</span>,</span></span>
<span class="r-in"><span> <span class="va">sampling_times</span><span class="op">)</span></span></span>
<span class="r-in"><span> <span class="va">d_2_1</span> <span class="op"><-</span> <span class="fu"><a href="add_err.html">add_err</a></span><span class="op">(</span><span class="va">d_logistic</span>,</span></span>
<span class="r-in"><span> sdfunc <span class="op">=</span> <span class="kw">function</span><span class="op">(</span><span class="va">x</span><span class="op">)</span> <span class="fu"><a href="sigma_twocomp.html">sigma_twocomp</a></span><span class="op">(</span><span class="va">x</span>, <span class="fl">0.5</span>, <span class="fl">0.07</span><span class="op">)</span>,</span></span>
<span class="r-in"><span> n <span class="op">=</span> <span class="fl">1</span>, reps <span class="op">=</span> <span class="fl">2</span>, digits <span class="op">=</span> <span class="fl">5</span>, LOD <span class="op">=</span> <span class="fl">0.1</span>, seed <span class="op">=</span> <span class="fl">123456</span><span class="op">)</span><span class="op">[[</span><span class="fl">1</span><span class="op">]</span><span class="op">]</span></span></span>
<span class="r-in"><span></span></span>
<span class="r-in"><span> <span class="va">m</span> <span class="op"><-</span> <span class="fu"><a href="mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="st">"logistic"</span>, <span class="va">d_2_1</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></span>
<span class="r-in"><span> <span class="fu"><a href="plot.mkinfit.html">plot_sep</a></span><span class="op">(</span><span class="va">m</span><span class="op">)</span></span></span>
<span class="r-plt img"><img src="logistic.solution-2.png" alt="" width="700" height="433"></span>
<span class="r-in"><span> <span class="fu"><a href="https://rdrr.io/r/base/summary.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">m</span><span class="op">)</span><span class="op">$</span><span class="va">bpar</span></span></span>
<span class="r-out co"><span class="r-pr">#></span> Estimate se_notrans t value Pr(>t) Lower</span>
<span class="r-out co"><span class="r-pr">#></span> parent_0 1.057896e+02 1.9023449590 55.610120 3.768360e-16 1.016451e+02</span>
<span class="r-out co"><span class="r-pr">#></span> kmax 6.398190e-02 0.0143201029 4.467978 3.841828e-04 3.929235e-02</span>
<span class="r-out co"><span class="r-pr">#></span> k0 1.612775e-04 0.0005866813 0.274898 3.940351e-01 5.846688e-08</span>
<span class="r-out co"><span class="r-pr">#></span> r 2.263946e-01 0.1718110662 1.317695 1.061043e-01 4.335843e-02</span>
<span class="r-out co"><span class="r-pr">#></span> sigma 5.332935e+00 0.9145907310 5.830952 4.036926e-05 3.340213e+00</span>
<span class="r-out co"><span class="r-pr">#></span> Upper</span>
<span class="r-out co"><span class="r-pr">#></span> parent_0 109.9341588</span>
<span class="r-out co"><span class="r-pr">#></span> kmax 0.1041853</span>
<span class="r-out co"><span class="r-pr">#></span> k0 0.4448749</span>
<span class="r-out co"><span class="r-pr">#></span> r 1.1821120</span>
<span class="r-out co"><span class="r-pr">#></span> sigma 7.3256566</span>
<span class="r-in"><span> <span class="fu"><a href="endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">m</span><span class="op">)</span><span class="op">$</span><span class="va">distimes</span></span></span>
<span class="r-out co"><span class="r-pr">#></span> DT50 DT90 DT50_k0 DT50_kmax</span>
<span class="r-out co"><span class="r-pr">#></span> parent 36.86533 62.41511 4297.853 10.83349</span>
<span class="r-in"><span></span></span>
</code></pre></div>
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