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  64       <img src="../logo.svg" class="logo" alt=""><h1>White and Pagano: Utilizing the Serial Distribution</h1>
  65
  66
  67       <small class="dont-index">Source: <a href="https://github.com/MI2YorkU/Rnaught/blob/master/vignettes/wp_serial.Rmd" class="external-link"><code>vignettes/wp_serial.Rmd</code></a></small>
  68       <div class="d-none name"><code>wp_serial.Rmd</code></div>
  69     </div>
  70
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  73 <p>The serial distribution of an infectious disease is the distribution
  74 of the time from when an infectious individual – the infector – becomes
  75 symptomatic, to when another individual who is infected by the infector
  76 becomes symptomatic. The serial interval refers to a range of likely
  77 values from this distribution, although it is typically reported as the
  78 mean.</p>
  79 <p>In the White and Pagano method, <code><a href="../reference/wp.html">wp()</a></code>, the serial
  80 distribution is assumed to be a discretized, finite version of a gamma
  81 distribution. Setting the parameter <code>serial</code> to
  82 <code>TRUE</code> causes this discretized distribution to be returned in
  83 addition to the estimate of R0. Furthermore, the method can be used
  84 whether or not the serial interval (specified as the parameter
  85 <code>mu</code>) is known. When <code>mu</code> is specified, it is
  86 taken to be the mean of a continuous gamma distribution (i.e., before
  87 the discretization). As such, the mean computed from the returned serial
  88 distribution may differ slightly from <code>mu</code>:</p>
  89 <div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
  90 <code class="sourceCode R"><span><span class="co"># Case counts.</span></span>
  91 <span><span class="va">cases</span> <span class="op">&lt;-</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">1</span>, <span class="fl">4</span>, <span class="fl">10</span>, <span class="fl">5</span>, <span class="fl">3</span>, <span class="fl">4</span>, <span class="fl">19</span>, <span class="fl">3</span>, <span class="fl">3</span>, <span class="fl">14</span>, <span class="fl">4</span><span class="op">)</span></span>
  92 <span></span>
  93 <span><span class="va">estimate</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/wp.html">wp</a></span><span class="op">(</span><span class="va">cases</span>, mu <span class="op">=</span> <span class="fl">3.333</span>, serial <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
  94 <span></span>
  95 <span><span class="co"># `supp` is the support of the distribution, and `pmf` is its probability mass</span></span>
  96 <span><span class="co"># function.</span></span>
  97 <span><span class="fu"><a href="https://rdrr.io/r/base/sum.html" class="external-link">sum</a></span><span class="op">(</span><span class="va">estimate</span><span class="op">$</span><span class="va">supp</span> <span class="op">*</span> <span class="va">estimate</span><span class="op">$</span><span class="va">pmf</span><span class="op">)</span></span>
  98 <span><span class="co">#&gt; [1] 3.840047</span></span></code></pre></div>
  99 <p>When <code>mu</code> is unspecified (left to its default value of
 100 <code>NA</code>), the method performs a maximum likelihood estimation
 101 over all (discretized) gamma distributions via a grid search, whose
 102 range of parameters are specified via <code>grid_length</code>,
 103 <code>max_shape</code> and <code>max_scale</code> (see <code><a href="../reference/wp.html">?wp</a></code>
 104 for more details). It is useful to return the estimated serial
 105 distribution in this case, as it can provide estimates of the serial
 106 interval when it is unknown:</p>
 107 <div class="sourceCode" id="cb2"><pre class="downlit sourceCode r">
 108 <code class="sourceCode R"><span><span class="co"># The grid search parameters specified below are the default values.</span></span>
 109 <span><span class="va">estimate</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/wp.html">wp</a></span><span class="op">(</span><span class="va">cases</span>, serial <span class="op">=</span> <span class="cn">TRUE</span>,</span>
 110 <span>  grid_length <span class="op">=</span> <span class="fl">100</span>, max_shape <span class="op">=</span> <span class="fl">10</span>, max_scale <span class="op">=</span> <span class="fl">10</span></span>
 111 <span><span class="op">)</span></span>
 112 <span></span>
 113 <span><span class="va">serial_mean</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/sum.html" class="external-link">sum</a></span><span class="op">(</span><span class="va">estimate</span><span class="op">$</span><span class="va">supp</span> <span class="op">*</span> <span class="va">estimate</span><span class="op">$</span><span class="va">pmf</span><span class="op">)</span></span>
 114 <span><span class="va">serial_mean</span></span>
 115 <span><span class="co">#&gt; [1] 3.564191</span></span>
 116 <span></span>
 117 <span><span class="co"># Compute the (discrete) median for an alternative estimate of the serial</span></span>
 118 <span><span class="co"># interval.</span></span>
 119 <span><span class="va">cdf</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/cumsum.html" class="external-link">cumsum</a></span><span class="op">(</span><span class="va">estimate</span><span class="op">$</span><span class="va">pmf</span><span class="op">)</span></span>
 120 <span><span class="va">serial_med</span> <span class="op">&lt;-</span> <span class="va">estimate</span><span class="op">$</span><span class="va">supp</span><span class="op">[</span><span class="fu"><a href="https://rdrr.io/r/base/which.html" class="external-link">which</a></span><span class="op">(</span><span class="va">cdf</span> <span class="op">&gt;=</span> <span class="fl">0.5</span> <span class="op">&amp;</span> <span class="va">estimate</span><span class="op">$</span><span class="va">pmf</span> <span class="op">-</span> <span class="va">cdf</span> <span class="op">+</span> <span class="fl">1</span> <span class="op">&gt;=</span> <span class="fl">0.5</span><span class="op">)</span><span class="op">]</span></span>
 121 <span><span class="va">serial_med</span></span>
 122 <span><span class="co">#&gt; [1] 2</span></span></code></pre></div>
 123 <p>Below is a graph of the above results, containing the serial
 124 distribution as well as its mean and median, which could be used as
 125 estimates of the serial interval:</p>
 126 <p><img src="wp_serial_files/figure-html/unnamed-chunk-4-1.png" width="1400"></p>
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