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  58       <img src="../logo.svg" class="logo" alt=""><h1>White and Pagano: Utilizing the Serial Distribution</h1>
  59
  60
  61       <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>
  62       <div class="d-none name"><code>wp_serial.Rmd</code></div>
  63     </div>
  64
  65
  66
  67 <p>The serial distribution of an infectious disease is the distribution
  68 of the time from when an infectious individual – the infector – becomes
  69 symptomatic, to when another individual who is infected by the infector
  70 becomes symptomatic. The serial interval refers to a range of likely
  71 values from this distribution, although it is typically reported as the
  72 mean.</p>
  73 <p>In the White and Pagano method, the serial distribution is assumed to
  74 be a discretized, finite version of a gamma distribution. Setting the
  75 parameter <code>serial</code> to <code>TRUE</code> causes this
  76 discretized distribution to be returned in addition to the estimate of
  77 R0. Furthermore, the method can be used whether or not the serial
  78 interval (specified as the parameter <code>mu</code>) is known. When
  79 <code>mu</code> is specified, it is taken to be the mean of a continuous
  80 gamma distribution (i.e., before the discretization). As such, the mean
  81 computed from the returned serial distribution may differ slightly from
  82 <code>mu</code>:</p>
  83 <div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
  84 <code class="sourceCode R"><span><span class="co"># Case counts.</span></span>
  85 <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>
  86 <span></span>
  87 <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>
  88 <span></span>
  89 <span><span class="co"># `supp` is the support of the distribution, and `pmf` is its probability mass</span></span>
  90 <span><span class="co"># function.</span></span>
  91 <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>
  92 <span><span class="co">#&gt; [1] 3.840047</span></span></code></pre></div>
  93 <p>When <code>mu</code> is unspecified (left to its default value of
  94 <code>NA</code>), the method performs a maximum likelihood estimation
  95 over all (discretized) gamma distributions via a grid search, whose
  96 range of parameters are specified via <code>grid_length</code>,
  97 <code>max_shape</code> and <code>max_scale</code> (see <code><a href="../reference/wp.html">?wp</a></code>
  98 for more details). It is useful to return the estimated serial
  99 distribution in this case, as it can provide estimates of the serial
 100 interval when it is unknown:</p>
 101 <div class="sourceCode" id="cb2"><pre class="downlit sourceCode r">
 102 <code class="sourceCode R"><span><span class="co"># The grid search parameters specified below are the default values.</span></span>
 103 <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>
 104 <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>
 105 <span><span class="op">)</span></span>
 106 <span></span>
 107 <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>
 108 <span><span class="va">serial_mean</span></span>
 109 <span><span class="co">#&gt; [1] 3.564191</span></span>
 110 <span></span>
 111 <span><span class="co"># Compute the (discrete) median for an alternative estimate of the serial</span></span>
 112 <span><span class="co"># interval.</span></span>
 113 <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">$</span><span class="va">pmf</span><span class="op">)</span></span>
 114 <span><span class="va">serial_med</span> <span class="op">&lt;-</span> <span class="va">estimate</span><span class="op">$</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>
 115 <span><span class="va">serial_med</span></span>
 116 <span><span class="co">#&gt; [1] 2</span></span></code></pre></div>
 117 <p>Below is a graph of the above results, containing the serial
 118 distribution as well as its mean and median, which could be used as
 119 estimates of the serial interval:</p>
 120 <p><img src="wp_serial_files/figure-html/unnamed-chunk-4-1.png" width="1400"></p>
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