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<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"><head>
<meta charset="utf-8">
<meta name="generator" content="quarto-1.4.555">
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
<title>Untitled</title>
<style>
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</head>
<body class="fullcontent">
<div id="quarto-content" class="page-columns page-rows-contents page-layout-article">
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<header id="title-block-header" class="quarto-title-block default">
<div class="quarto-title">
<h1 class="title">Untitled</h1>
</div>
<div class="quarto-title-meta">
</div>
</header>
<section id="packages" class="level2">
<h2 class="anchored" data-anchor-id="packages">Packages</h2>
<div class="cell">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(deSolve)</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(purrr)</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(parallel)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</section>
<section id="functions" class="level2">
<h2 class="anchored" data-anchor-id="functions">Functions</h2>
<div class="cell">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>plot2 <span class="ot"><-</span> <span class="cf">function</span>(...) <span class="fu">plot</span>(..., <span class="at">lwd =</span> <span class="dv">2</span>, <span class="at">col =</span> <span class="dv">4</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a>last <span class="ot"><-</span> <span class="cf">function</span>(x) <span class="fu">tail</span>(x, <span class="dv">1</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a>lines2 <span class="ot"><-</span> <span class="cf">function</span>(...) <span class="fu">lines</span>(..., <span class="at">lwd =</span> <span class="dv">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>mclapply2 <span class="ot"><-</span> <span class="cf">function</span>(...) <span class="fu">mclapply</span>(..., <span class="at">mc.cores =</span> <span class="fu">detectCores</span>() <span class="sc">-</span> <span class="dv">1</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</section>
<section id="exponential-growth" class="level2">
<h2 class="anchored" data-anchor-id="exponential-growth">Exponential growth</h2>
<p>A continuous-time model of an exponential growth:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a>continuous_exponential_growth <span class="ot"><-</span> <span class="cf">function</span>(N0, r, times) {</span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a> tibble<span class="sc">::</span><span class="fu">as_tibble</span>(</span>
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">as.data.frame</span>(</span>
<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a> <span class="fu">ode</span>(<span class="fu">c</span>(<span class="at">N =</span> N0),</span>
<span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a> times,</span>
<span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a> <span class="cf">function</span>(time, variables, parameters) {</span>
<span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a> <span class="fu">with</span>(<span class="fu">as.list</span>(<span class="fu">c</span>(variables, parameters)), {</span>
<span id="cb6-8"><a href="#cb6-8" aria-hidden="true" tabindex="-1"></a> dN <span class="ot"><-</span> r <span class="sc">*</span> N</span>
<span id="cb6-9"><a href="#cb6-9" aria-hidden="true" tabindex="-1"></a> <span class="fu">list</span>(dN)</span>
<span id="cb6-10"><a href="#cb6-10" aria-hidden="true" tabindex="-1"></a> })</span>
<span id="cb6-11"><a href="#cb6-11" aria-hidden="true" tabindex="-1"></a> },</span>
<span id="cb6-12"><a href="#cb6-12" aria-hidden="true" tabindex="-1"></a> <span class="fu">c</span>(<span class="at">r =</span> r))</span>
<span id="cb6-13"><a href="#cb6-13" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb6-14"><a href="#cb6-14" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb6-15"><a href="#cb6-15" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>Here is an example use:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="fu">continuous_exponential_growth</span>(<span class="dv">10</span>, .<span class="dv">5</span>, <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">10</span>, .<span class="dv">1</span>)) <span class="sc">|></span> </span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">with</span>(<span class="fu">plot2</span>(time, N, <span class="at">type =</span> <span class="st">"l"</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="explorations_files/figure-html/unnamed-chunk-7-1.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
</div>
<p>A discrete-time version of this model:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a>discrete_exponential_growth <span class="ot"><-</span> <span class="cf">function</span>(N0, r, times) {</span>
<span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a> step <span class="ot"><-</span> <span class="fu">mean</span>(<span class="fu">diff</span>(times))</span>
<span id="cb8-3"><a href="#cb8-3" aria-hidden="true" tabindex="-1"></a> N <span class="ot"><-</span> N0</span>
<span id="cb8-4"><a href="#cb8-4" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> (i <span class="cf">in</span> <span class="fu">remove_first</span>(times)) {</span>
<span id="cb8-5"><a href="#cb8-5" aria-hidden="true" tabindex="-1"></a> N <span class="ot"><-</span> <span class="fu">c</span>(N, <span class="fu">last</span>(N) <span class="sc">*</span> (<span class="dv">2</span> <span class="sc">-</span> <span class="fu">exp</span>(<span class="sc">-</span> r <span class="sc">*</span> step)))</span>
<span id="cb8-6"><a href="#cb8-6" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb8-7"><a href="#cb8-7" aria-hidden="true" tabindex="-1"></a> tibble<span class="sc">::</span><span class="fu">tibble</span>(<span class="at">time =</span> times, <span class="at">N =</span> N)</span>
<span id="cb8-8"><a href="#cb8-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a>discrete_exponential_growth <span class="ot"><-</span> <span class="cf">function</span>(N0, r, times) {</span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a> N <span class="ot"><-</span> N0</span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a> r <span class="ot"><-</span> <span class="fu">exp</span>(r <span class="sc">*</span> <span class="fu">mean</span>(<span class="fu">diff</span>(times)))</span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> (i <span class="cf">in</span> times[<span class="sc">-</span><span class="dv">1</span>]) {</span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a> N <span class="ot"><-</span> <span class="fu">c</span>(N, <span class="fu">last</span>(N) <span class="sc">*</span> r)</span>
<span id="cb9-6"><a href="#cb9-6" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb9-7"><a href="#cb9-7" aria-hidden="true" tabindex="-1"></a> tibble<span class="sc">::</span><span class="fu">tibble</span>(<span class="at">time =</span> times, <span class="at">N =</span> N)</span>
<span id="cb9-8"><a href="#cb9-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a>discrete_exponential_growth <span class="ot"><-</span> <span class="cf">function</span>(N0, r, times) {</span>
<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a> N <span class="ot"><-</span> N0</span>
<span id="cb10-3"><a href="#cb10-3" aria-hidden="true" tabindex="-1"></a> r <span class="ot"><-</span> <span class="fu">exp</span>(r <span class="sc">*</span> <span class="fu">mean</span>(<span class="fu">diff</span>(times)))</span>
<span id="cb10-4"><a href="#cb10-4" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> (i <span class="cf">in</span> times[<span class="sc">-</span><span class="dv">1</span>]) {</span>
<span id="cb10-5"><a href="#cb10-5" aria-hidden="true" tabindex="-1"></a> N <span class="ot"><-</span> <span class="fu">c</span>(N, <span class="fu">last</span>(N) <span class="sc">*</span> r)</span>
<span id="cb10-6"><a href="#cb10-6" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb10-7"><a href="#cb10-7" aria-hidden="true" tabindex="-1"></a> tibble<span class="sc">::</span><span class="fu">tibble</span>(<span class="at">time =</span> times, <span class="at">N =</span> N)</span>
<span id="cb10-8"><a href="#cb10-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>Where the probability per time step is computed as</p>
<p><span class="math display">\[
p = 1 - e^{-r \Delta t}
\]</span></p>
<p>New formula:</p>
<p><span class="math display">\[
p = 1 - e^{(1 - e^r)\Delta t}
\]</span></p>
<p>with <span class="math inline">\(r\)</span> the rate and <span class="math inline">\(\Delta t\)</span> the duration of the time step, both expressed in the same time unit. And here an example use:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="fu">discrete_exponential_growth</span>(<span class="dv">10</span>, .<span class="dv">5</span>, <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">10</span>, <span class="dv">1</span>)) <span class="sc">|></span> </span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">with</span>(<span class="fu">plot2</span>(time, N, <span class="at">type =</span> <span class="st">"o"</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="explorations_files/figure-html/unnamed-chunk-12-1.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
</div>
<p>A function to compare the dynamics of the discrete-time and continuous-time models:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a>compare_dynamics <span class="ot"><-</span> <span class="cf">function</span>(N0, rc, rd, tmin, tmax, length, step) {</span>
<span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a> out <span class="ot"><-</span> <span class="fu">continuous_exponential_growth</span>(N0, rc, <span class="fu">seq</span>(tmin, tmax, <span class="at">le =</span> length))</span>
<span id="cb12-3"><a href="#cb12-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">with</span>(out, <span class="fu">plot2</span>(time, N, <span class="at">type =</span> <span class="st">"n"</span>))</span>
<span id="cb12-4"><a href="#cb12-4" aria-hidden="true" tabindex="-1"></a> step <span class="sc">|></span></span>
<span id="cb12-5"><a href="#cb12-5" aria-hidden="true" tabindex="-1"></a> <span class="fu">walk</span>(<span class="sc">~</span> <span class="fu">discrete_exponential_growth</span>(N0, rd, <span class="fu">seq</span>(tmin, tmax, .x)) <span class="sc">|></span></span>
<span id="cb12-6"><a href="#cb12-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">with</span>(<span class="fu">lines2</span>(time, N, <span class="at">type =</span> <span class="st">"o"</span>, <span class="at">col =</span> <span class="dv">2</span>)))</span>
<span id="cb12-7"><a href="#cb12-7" aria-hidden="true" tabindex="-1"></a> <span class="fu">with</span>(out, <span class="fu">lines2</span>(time, N, <span class="at">col =</span> <span class="dv">4</span>))</span>
<span id="cb12-8"><a href="#cb12-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>And it gives:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="fu">compare_dynamics</span>(<span class="at">N0 =</span> <span class="dv">10</span>, <span class="at">rc =</span> .<span class="dv">5</span>, <span class="at">rd =</span> .<span class="dv">5</span>, <span class="at">tmin =</span> <span class="dv">0</span>, <span class="at">tmax =</span> <span class="dv">10</span>, <span class="at">length =</span> <span class="dv">512</span>,</span>
<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a> <span class="at">step =</span> <span class="fu">c</span>(.<span class="dv">01</span>, <span class="fu">seq</span>(.<span class="dv">1</span>, .<span class="dv">5</span>, .<span class="dv">1</span>)))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="explorations_files/figure-html/unnamed-chunk-14-1.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
</div>
<p>meaning that the larger the time step, the slower the dynamics of the discrete-time version is compared to the continuous-time version. Let’s now explore how the <span class="math inline">\(r\)</span> parameter should be adjusted for the discrete-time model so that it produces the same dynamics as the continuous-time model. Let’s for that consider the following minus log likelihood function:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a>mLL <span class="ot"><-</span> <span class="cf">function</span>(rd, rc, N0, times) {</span>
<span id="cb14-2"><a href="#cb14-2" aria-hidden="true" tabindex="-1"></a> <span class="sc">-</span> <span class="fu">sum</span>(<span class="fu">dpois</span>(<span class="fu">round</span>(<span class="fu">unlist</span>(<span class="fu">discrete_exponential_growth</span>(N0, rd, times)[<span class="sc">-</span><span class="dv">1</span>, <span class="dv">2</span>])),</span>
<span id="cb14-3"><a href="#cb14-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">unlist</span>(<span class="fu">continuous_exponential_growth</span>(N0, rc, times)[<span class="sc">-</span><span class="dv">1</span>, <span class="dv">2</span>]), <span class="cn">TRUE</span>))</span>
<span id="cb14-4"><a href="#cb14-4" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>Let’s try it:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="fu">mLL</span>(<span class="at">rd =</span> .<span class="dv">5</span>, <span class="at">rc =</span> .<span class="dv">5</span>, <span class="at">N0 =</span> <span class="dv">10</span>, <span class="at">times =</span> <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">10</span>, .<span class="dv">1</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 333.5412</code></pre>
</div>
</div>
<p>The following function uses the previous one in order to compute the adjusted <span class="math inline">\(r\)</span> parameter for the discrete-time model:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb17"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a>r_discrete <span class="ot"><-</span> <span class="cf">function</span>(rd, rc, N0, step) {</span>
<span id="cb17-2"><a href="#cb17-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">optimize</span>(mLL, rd, <span class="at">rc =</span> rc, <span class="at">N0 =</span> N0, <span class="at">times =</span> <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">10</span>, step))<span class="sc">$</span>minimum</span>
<span id="cb17-3"><a href="#cb17-3" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>Let’s try it:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb18"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb18-1"><a href="#cb18-1" aria-hidden="true" tabindex="-1"></a><span class="fu">r_discrete</span>(<span class="fu">c</span>(.<span class="dv">0001</span>, <span class="dv">100</span>), .<span class="dv">5</span>, <span class="dv">10</span>, .<span class="dv">1</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 0.4999222</code></pre>
</div>
</div>
<p>Let’s now compute the adjusted value of the <span class="math inline">\(r\)</span> parameter of the discrete-time model for various values of time steps:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb20"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a>step_size <span class="ot"><-</span> <span class="fu">seq</span>(.<span class="dv">01</span>, .<span class="dv">5</span>, .<span class="dv">01</span>)</span>
<span id="cb20-2"><a href="#cb20-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb20-3"><a href="#cb20-3" aria-hidden="true" tabindex="-1"></a>out <span class="ot"><-</span> step_size <span class="sc">|></span></span>
<span id="cb20-4"><a href="#cb20-4" aria-hidden="true" tabindex="-1"></a> <span class="fu">mclapply2</span>(r_discrete, <span class="at">rd =</span> <span class="fu">c</span>(.<span class="dv">0001</span>, <span class="dv">100</span>), <span class="at">rc =</span> .<span class="dv">5</span>, <span class="at">N0 =</span> <span class="dv">10</span>) <span class="sc">|></span> </span>
<span id="cb20-5"><a href="#cb20-5" aria-hidden="true" tabindex="-1"></a> <span class="fu">unlist</span>() <span class="sc">|></span> </span>
<span id="cb20-6"><a href="#cb20-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">matrix</span>(<span class="at">ncol =</span> <span class="dv">1</span>, <span class="at">byrow =</span> <span class="cn">TRUE</span>) <span class="sc">|></span> </span>
<span id="cb20-7"><a href="#cb20-7" aria-hidden="true" tabindex="-1"></a> <span class="fu">as.data.frame</span>() <span class="sc">|></span> </span>
<span id="cb20-8"><a href="#cb20-8" aria-hidden="true" tabindex="-1"></a> <span class="fu">setNames</span>(<span class="st">"r"</span>) <span class="sc">%>%</span></span>
<span id="cb20-9"><a href="#cb20-9" aria-hidden="true" tabindex="-1"></a> <span class="fu">cbind</span>(step_size, .) <span class="sc">|></span> </span>
<span id="cb20-10"><a href="#cb20-10" aria-hidden="true" tabindex="-1"></a> tibble<span class="sc">::</span><span class="fu">as_tibble</span>()</span>
<span id="cb20-11"><a href="#cb20-11" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb20-12"><a href="#cb20-12" aria-hidden="true" tabindex="-1"></a><span class="fu">with</span>(out, <span class="fu">plot</span>(step_size, r, <span class="at">col =</span> <span class="dv">4</span>, <span class="at">ylim =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">1</span>,</span>
<span id="cb20-13"><a href="#cb20-13" aria-hidden="true" tabindex="-1"></a> <span class="at">xlab =</span> <span class="st">"duration of the time step"</span>, <span class="at">ylab =</span> <span class="st">"adjusted value of r"</span>))</span>
<span id="cb20-14"><a href="#cb20-14" aria-hidden="true" tabindex="-1"></a><span class="fu">abline</span>(<span class="at">h =</span> .<span class="dv">5</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="explorations_files/figure-html/unnamed-chunk-19-1.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
</div>
<p>We can verify that these adjusted values of <span class="math inline">\(r\)</span> work OK:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb21"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a>opar <span class="ot"><-</span> <span class="fu">par</span>(<span class="at">mfrow =</span> <span class="fu">c</span>(<span class="dv">10</span>, <span class="dv">5</span>), <span class="at">plt =</span> <span class="fu">c</span>(.<span class="dv">15</span>, .<span class="dv">85</span>, .<span class="dv">15</span>, .<span class="dv">85</span>))</span>
<span id="cb21-2"><a href="#cb21-2" aria-hidden="true" tabindex="-1"></a><span class="fu">walk2</span>(out<span class="sc">$</span>r, out<span class="sc">$</span>step_size,</span>
<span id="cb21-3"><a href="#cb21-3" aria-hidden="true" tabindex="-1"></a> <span class="sc">~</span> <span class="fu">compare_dynamics</span>(<span class="at">N0 =</span> <span class="dv">10</span>, <span class="at">rc =</span> .<span class="dv">5</span>, <span class="at">rd =</span> .x, <span class="at">tmin =</span> <span class="dv">0</span>, <span class="at">tmax =</span> <span class="dv">10</span>, <span class="at">length =</span> <span class="dv">512</span>,</span>
<span id="cb21-4"><a href="#cb21-4" aria-hidden="true" tabindex="-1"></a> <span class="at">step =</span> .y))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="explorations_files/figure-html/unnamed-chunk-20-1.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
<div class="sourceCode cell-code" id="cb22"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a><span class="fu">par</span>(opar)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>Let’s now try to model the adjusted value of the <span class="math inline">\(r\)</span> parameter as a function of the duration of the time step:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb23"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a>mod <span class="ot"><-</span> <span class="fu">lm</span>(<span class="fu">I</span>(r <span class="sc">-</span> .<span class="dv">5</span>) <span class="sc">~</span> step_size <span class="sc">-</span> <span class="dv">1</span>, out)</span>
<span id="cb23-2"><a href="#cb23-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb23-3"><a href="#cb23-3" aria-hidden="true" tabindex="-1"></a><span class="fu">with</span>(out, <span class="fu">plot</span>(step_size, r, <span class="at">col =</span> <span class="dv">4</span>, <span class="at">ylim =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">1</span>,</span>
<span id="cb23-4"><a href="#cb23-4" aria-hidden="true" tabindex="-1"></a> <span class="at">xlab =</span> <span class="st">"duration of the time step"</span>, <span class="at">ylab =</span> <span class="st">"adjusted value of r"</span>))</span>
<span id="cb23-5"><a href="#cb23-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb23-6"><a href="#cb23-6" aria-hidden="true" tabindex="-1"></a>(slope <span class="ot"><-</span> <span class="fu">coef</span>(mod))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> step_size
-0.0003443509 </code></pre>
</div>
<div class="sourceCode cell-code" id="cb25"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb25-1"><a href="#cb25-1" aria-hidden="true" tabindex="-1"></a>xs <span class="ot"><-</span> <span class="fu">seq</span>(<span class="dv">0</span>, .<span class="dv">5</span>, <span class="at">le =</span> <span class="dv">512</span>)</span>
<span id="cb25-2"><a href="#cb25-2" aria-hidden="true" tabindex="-1"></a><span class="fu">abline</span>(.<span class="dv">5</span>, slope, <span class="at">col =</span> <span class="dv">2</span>, <span class="at">lwd =</span> <span class="dv">2</span>)</span>
<span id="cb25-3"><a href="#cb25-3" aria-hidden="true" tabindex="-1"></a><span class="fu">abline</span>(<span class="at">h =</span> .<span class="dv">5</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="explorations_files/figure-html/unnamed-chunk-21-1.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
</div>
</section>
<section id="exponential-decay" class="level2">
<h2 class="anchored" data-anchor-id="exponential-decay">Exponential decay</h2>
<div class="cell">
<div class="sourceCode cell-code" id="cb26"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb26-1"><a href="#cb26-1" aria-hidden="true" tabindex="-1"></a>continuous_exponential_decay <span class="ot"><-</span> <span class="cf">function</span>(N0, r, times) {</span>
<span id="cb26-2"><a href="#cb26-2" aria-hidden="true" tabindex="-1"></a> tibble<span class="sc">::</span><span class="fu">as_tibble</span>(</span>
<span id="cb26-3"><a href="#cb26-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">as.data.frame</span>(</span>
<span id="cb26-4"><a href="#cb26-4" aria-hidden="true" tabindex="-1"></a> <span class="fu">ode</span>(<span class="fu">c</span>(<span class="at">N =</span> N0),</span>
<span id="cb26-5"><a href="#cb26-5" aria-hidden="true" tabindex="-1"></a> times,</span>
<span id="cb26-6"><a href="#cb26-6" aria-hidden="true" tabindex="-1"></a> <span class="cf">function</span>(time, variables, parameters) {</span>
<span id="cb26-7"><a href="#cb26-7" aria-hidden="true" tabindex="-1"></a> <span class="fu">with</span>(<span class="fu">as.list</span>(<span class="fu">c</span>(variables, parameters)), {</span>
<span id="cb26-8"><a href="#cb26-8" aria-hidden="true" tabindex="-1"></a> dN <span class="ot"><-</span> r <span class="sc">*</span> N</span>
<span id="cb26-9"><a href="#cb26-9" aria-hidden="true" tabindex="-1"></a> <span class="fu">list</span>(dN)</span>
<span id="cb26-10"><a href="#cb26-10" aria-hidden="true" tabindex="-1"></a> })</span>
<span id="cb26-11"><a href="#cb26-11" aria-hidden="true" tabindex="-1"></a> },</span>
<span id="cb26-12"><a href="#cb26-12" aria-hidden="true" tabindex="-1"></a> <span class="fu">c</span>(<span class="at">r =</span> r))</span>
<span id="cb26-13"><a href="#cb26-13" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb26-14"><a href="#cb26-14" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb26-15"><a href="#cb26-15" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb27"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb27-1"><a href="#cb27-1" aria-hidden="true" tabindex="-1"></a><span class="fu">continuous_exponential_decay</span>(<span class="dv">1000</span>, .<span class="dv">2</span>, <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">10</span>, .<span class="dv">1</span>)) <span class="sc">|></span> </span>
<span id="cb27-2"><a href="#cb27-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">with</span>(<span class="fu">plot2</span>(time, N, <span class="at">type =</span> <span class="st">"l"</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="explorations_files/figure-html/unnamed-chunk-23-1.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb28"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb28-1"><a href="#cb28-1" aria-hidden="true" tabindex="-1"></a>discrete_exponential_decay <span class="ot"><-</span> <span class="cf">function</span>(N0, r, times) {</span>
<span id="cb28-2"><a href="#cb28-2" aria-hidden="true" tabindex="-1"></a> Delta_t <span class="ot"><-</span> <span class="fu">mean</span>(<span class="fu">diff</span>(times))</span>
<span id="cb28-3"><a href="#cb28-3" aria-hidden="true" tabindex="-1"></a> r <span class="ot"><-</span> <span class="fu">exp</span>(r <span class="sc">*</span> Delta_t)</span>
<span id="cb28-4"><a href="#cb28-4" aria-hidden="true" tabindex="-1"></a> p <span class="ot"><-</span> <span class="dv">1</span> <span class="sc">-</span> <span class="fu">exp</span>(<span class="sc">-</span> r <span class="sc">*</span> Delta_t)</span>
<span id="cb28-5"><a href="#cb28-5" aria-hidden="true" tabindex="-1"></a> <span class="fu">print</span>(p)</span>
<span id="cb28-6"><a href="#cb28-6" aria-hidden="true" tabindex="-1"></a> N <span class="ot"><-</span> N0</span>
<span id="cb28-7"><a href="#cb28-7" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> (i <span class="cf">in</span> times[<span class="sc">-</span><span class="dv">1</span>]) {</span>
<span id="cb28-8"><a href="#cb28-8" aria-hidden="true" tabindex="-1"></a> N <span class="ot"><-</span> <span class="fu">c</span>(N, <span class="fu">last</span>(N) <span class="sc">*</span> (<span class="dv">1</span> <span class="sc">+</span> p))</span>
<span id="cb28-9"><a href="#cb28-9" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb28-10"><a href="#cb28-10" aria-hidden="true" tabindex="-1"></a> tibble<span class="sc">::</span><span class="fu">tibble</span>(<span class="at">time =</span> times, <span class="at">N =</span> N)</span>
<span id="cb28-11"><a href="#cb28-11" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb29"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb29-1"><a href="#cb29-1" aria-hidden="true" tabindex="-1"></a><span class="fu">discrete_exponential_decay</span>(<span class="dv">1000</span>, .<span class="dv">2</span>, <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">10</span>, <span class="dv">1</span>)) <span class="sc">|></span> </span>
<span id="cb29-2"><a href="#cb29-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">with</span>(<span class="fu">plot2</span>(time, N, <span class="at">type =</span> <span class="st">"o"</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 0.7051837</code></pre>
</div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="explorations_files/figure-html/unnamed-chunk-25-1.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb31"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb31-1"><a href="#cb31-1" aria-hidden="true" tabindex="-1"></a>compare_dynamics2 <span class="ot"><-</span> <span class="cf">function</span>(N0, rc, rd, tmin, tmax, length, step) {</span>
<span id="cb31-2"><a href="#cb31-2" aria-hidden="true" tabindex="-1"></a> out <span class="ot"><-</span> <span class="fu">continuous_exponential_decay</span>(N0, rc, <span class="fu">seq</span>(tmin, tmax, <span class="at">le =</span> length))</span>
<span id="cb31-3"><a href="#cb31-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">with</span>(out, <span class="fu">plot2</span>(time, N, <span class="at">type =</span> <span class="st">"n"</span>))</span>
<span id="cb31-4"><a href="#cb31-4" aria-hidden="true" tabindex="-1"></a> step <span class="sc">|></span></span>
<span id="cb31-5"><a href="#cb31-5" aria-hidden="true" tabindex="-1"></a> <span class="fu">walk</span>(<span class="sc">~</span> <span class="fu">discrete_exponential_decay</span>(N0, rd, <span class="fu">seq</span>(tmin, tmax, .x)) <span class="sc">|></span></span>
<span id="cb31-6"><a href="#cb31-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">with</span>(<span class="fu">lines2</span>(time, N, <span class="at">type =</span> <span class="st">"o"</span>, <span class="at">col =</span> <span class="dv">2</span>)))</span>
<span id="cb31-7"><a href="#cb31-7" aria-hidden="true" tabindex="-1"></a> <span class="fu">with</span>(out, <span class="fu">lines2</span>(time, N, <span class="at">col =</span> <span class="dv">4</span>))</span>
<span id="cb31-8"><a href="#cb31-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<div class="cell">
<div class="sourceCode cell-code" id="cb32"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb32-1"><a href="#cb32-1" aria-hidden="true" tabindex="-1"></a><span class="fu">compare_dynamics2</span>(<span class="at">N0 =</span> <span class="dv">1000</span>, <span class="at">rc =</span> .<span class="dv">2</span>, <span class="at">rd =</span> .<span class="dv">2</span>, <span class="at">tmin =</span> <span class="dv">0</span>, <span class="at">tmax =</span> <span class="dv">10</span>, <span class="at">length =</span> <span class="dv">512</span>,</span>
<span id="cb32-2"><a href="#cb32-2" aria-hidden="true" tabindex="-1"></a> <span class="at">step =</span> <span class="fu">c</span>(.<span class="dv">01</span>, <span class="fu">seq</span>(.<span class="dv">1</span>, .<span class="dv">5</span>, .<span class="dv">1</span>)))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="explorations_files/figure-html/unnamed-chunk-27-1.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 0.009969987
[1] 0.09698863
[1] 0.1879247
[1] 0.272798
[1] 0.3516437
[1] 0.4245399</code></pre>
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