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<!DOCTYPE html>
<html>
<head><meta charset="utf-8" />
<title>SIRmodel</title>
<script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
<style type="text/css">
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<h1 id="Simple-model-of-epidemic-dynamics:-SIR">Simple model of epidemic dynamics: SIR<a class="anchor-link" href="#Simple-model-of-epidemic-dynamics:-SIR">¶</a></h1><h3 id="Prof.-Marco-Arieli-Herrera-Valdez,">Prof. Marco Arieli Herrera-Valdez,<a class="anchor-link" href="#Prof.-Marco-Arieli-Herrera-Valdez,">¶</a></h3><h4 id="Facultad-de-Ciencias,-Universidad-Nacional-Autónoma-de-México">Facultad de Ciencias, Universidad Nacional Autónoma de México<a class="anchor-link" href="#Facultad-de-Ciencias,-Universidad-Nacional-Autónoma-de-México">¶</a></h4><p>Created March 7, 2016</p>
<p>Let $x$, $y$, and $z$ represent the fraction of susceptibles, infected, and recovered individuals within a population. Assume homogeneous mixing with a probability of infection given a contact with an infected individual given by $\beta$ and an average removal time $\beta^{-1}$ from the infected group, by recovery or death due to infection. The population dynamics are given by
\begin{eqnarray}
\partial_t x &=& -\alpha xy
\
\partial_t y &=& \left( \alpha x - \beta \right) y
\
\partial_t x &=& \beta y
\end{eqnarray}</p>
<p>Notice that the population size does not matter because it is kept constant.</p>
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<div class=" highlight hl-ipython3"><pre><span class="c">#Import the necessary modules and perform the necessary tests</span>
<span class="kn">import</span> <span class="nn">scipy</span> <span class="k">as</span> <span class="nn">sc</span>
<span class="kn">import</span> <span class="nn">pylab</span> <span class="k">as</span> <span class="nn">gr</span>
<span class="n">sc</span><span class="o">.</span><span class="n">test</span><span class="p">(</span><span class="s">"all"</span><span class="p">,</span><span class="n">verbose</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
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<pre>Running unit tests for scipy
NumPy version 1.10.4
NumPy relaxed strides checking option: False
NumPy is installed in /Library/Frameworks/Python.framework/Versions/3.5/lib/python3.5/site-packages/numpy
SciPy version 0.17.0
SciPy is installed in /Library/Frameworks/Python.framework/Versions/3.5/lib/python3.5/site-packages/scipy
Python version 3.5.1 (v3.5.1:37a07cee5969, Dec 5 2015, 21:12:44) [GCC 4.2.1 (Apple Inc. build 5666) (dot 3)]
nose version 1.3.7
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<pre>----------------------------------------------------------------------
Ran 1 test in 2.492s
OK
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<p>Setup a python function that specifies the dynamics</p>
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<div class=" highlight hl-ipython3"><pre><span class="k">def</span> <span class="nf">SIR</span><span class="p">(</span><span class="n">U</span><span class="p">,</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">):</span>
<span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="o">=</span><span class="n">U</span>
<span class="n">yNew</span><span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="s">"alpha"</span><span class="p">]</span> <span class="o">*</span> <span class="n">y</span> <span class="o">*</span> <span class="n">x</span>
<span class="n">zNew</span><span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="s">"beta"</span><span class="p">]</span> <span class="o">*</span> <span class="n">y</span>
<span class="n">dx</span> <span class="o">=</span> <span class="o">-</span><span class="n">yNew</span>
<span class="n">dy</span> <span class="o">=</span> <span class="n">yNew</span> <span class="o">-</span> <span class="n">zNew</span>
<span class="n">dz</span> <span class="o">=</span> <span class="n">zNew</span>
<span class="k">return</span> <span class="n">dx</span><span class="p">,</span> <span class="n">dy</span><span class="p">,</span> <span class="n">dz</span>
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<p>The function SIR above takes three arguments, $U$, $t$, and $p$ that represent the states of the system, the time and the parameters, respectively.</p>
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<h3 id="Outbreak-condition">Outbreak condition<a class="anchor-link" href="#Outbreak-condition">¶</a></h3><p>The condition
\begin{equation}
\frac{\alpha}{\beta}x(t)>1 , \quad y>0
\end{equation}
defines a threshold for a full epidemic outbreak. An equivalent condition is
\begin{equation}
x>\frac{\beta}{\alpha }, \quad y>0
\end{equation}</p>
<p>Therefore, with the parameters $(\alpha,\beta)$=(0.5,0.1), there will be an outbreak if the initial condition for $x(t)>1/5$ with $y>0$.
Notice that the initial value for $z$ can be interpreted as the initial proportion of immune individuals within the population.</p>
<p>The dynamics related to the oubreak condition can be studied by defining a variable $B(t) = x(t) \alpha/\beta$, called by some authors "<em>effective reproductive number</em>". If $x(t)\approx 1$, the corresponding $B(t)$ is called "<em>basic reproductive number</em>", or $R_o$.</p>
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<p>Let's define a python dictionary containing parameters and initial conditions to perform simulations.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">p</span><span class="o">=</span><span class="p">{</span><span class="s">"alpha"</span><span class="p">:</span> <span class="mf">0.15</span><span class="p">,</span> <span class="s">"beta"</span><span class="p">:</span><span class="mf">0.1</span><span class="p">,</span> <span class="s">"timeStop"</span><span class="p">:</span><span class="mf">300.0</span><span class="p">,</span> <span class="s">"timeStep"</span><span class="p">:</span><span class="mf">0.01</span> <span class="p">}</span>
<span class="n">p</span><span class="p">[</span><span class="s">"Ro"</span><span class="p">]</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s">"alpha"</span><span class="p">]</span><span class="o">/</span><span class="n">p</span><span class="p">[</span><span class="s">"beta"</span><span class="p">]</span>
<span class="n">p</span><span class="p">[</span><span class="s">"sampTimes"</span><span class="p">]</span><span class="o">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">p</span><span class="p">[</span><span class="s">"timeStop"</span><span class="p">],</span><span class="n">p</span><span class="p">[</span><span class="s">"timeStep"</span><span class="p">])</span>
<span class="n">N</span><span class="o">=</span> <span class="mi">1</span><span class="n">e4</span><span class="p">;</span> <span class="n">i0</span><span class="o">=</span> <span class="mi">1</span><span class="n">e1</span><span class="p">;</span> <span class="n">r0</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span> <span class="n">s0</span><span class="o">=</span><span class="n">N</span><span class="o">-</span><span class="n">i0</span><span class="o">-</span><span class="n">r0</span>
<span class="n">x0</span><span class="o">=</span><span class="n">s0</span><span class="o">/</span><span class="n">N</span><span class="p">;</span> <span class="n">y0</span><span class="o">=</span><span class="n">i0</span><span class="o">/</span><span class="n">N</span><span class="p">;</span> <span class="n">z0</span><span class="o">=</span><span class="n">r0</span><span class="o">/</span><span class="n">N</span><span class="p">;</span>
<span class="n">p</span><span class="p">[</span><span class="s">"ic"</span><span class="p">]</span><span class="o">=</span><span class="p">[</span><span class="n">x0</span><span class="p">,</span><span class="n">y0</span><span class="p">,</span><span class="n">z0</span><span class="p">]</span>
<span class="nb">print</span><span class="p">(</span><span class="s">"N=%g with initial conditions (S,I,R)=(%g,%g,%g)"</span><span class="o">%</span><span class="p">(</span><span class="n">N</span><span class="p">,</span><span class="n">s0</span><span class="p">,</span><span class="n">i0</span><span class="p">,</span><span class="n">r0</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s">"Initial conditions: "</span><span class="p">,</span> <span class="n">p</span><span class="p">[</span><span class="s">"ic"</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s">"B(0)=%g"</span><span class="o">%</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s">"ic"</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="s">"Ro"</span><span class="p">]))</span>
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<pre>N=10000 with initial conditions (S,I,R)=(9990,10,0)
Initial conditions: [0.999, 0.001, 0.0]
B(0)=1.4985
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<p>Integrate numerically and plot the results</p>
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<div class=" highlight hl-ipython3"><pre><span class="c"># Numerical integration</span>
<span class="n">xyz</span><span class="o">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">integrate</span><span class="o">.</span><span class="n">odeint</span><span class="p">(</span><span class="n">SIR</span><span class="p">,</span> <span class="n">p</span><span class="p">[</span><span class="s">"ic"</span><span class="p">],</span> <span class="n">p</span><span class="p">[</span><span class="s">"sampTimes"</span><span class="p">],</span> <span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="n">p</span><span class="p">,))</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
<span class="c"># Calculate the outbreak indicator</span>
<span class="n">B</span><span class="o">=</span> <span class="n">xyz</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="s">"alpha"</span><span class="p">]</span><span class="o">/</span><span class="n">p</span><span class="p">[</span><span class="s">"beta"</span><span class="p">]</span>
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<div class=" highlight hl-ipython3"><pre><span class="c"># Figure</span>
<span class="n">fig</span><span class="o">=</span><span class="n">gr</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span><span class="mi">5</span><span class="p">))</span>
<span class="n">gr</span><span class="o">.</span><span class="n">ioff</span><span class="p">()</span>
<span class="n">rows</span><span class="o">=</span><span class="mi">1</span><span class="p">;</span> <span class="n">cols</span><span class="o">=</span><span class="mi">2</span>
<span class="n">ax</span><span class="o">=</span><span class="nb">list</span><span class="p">()</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">sc</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">rows</span><span class="o">*</span><span class="n">cols</span><span class="p">):</span>
<span class="n">ax</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="n">rows</span><span class="p">,</span><span class="n">cols</span><span class="p">,</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s">"sampTimes"</span><span class="p">],</span> <span class="n">xyz</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="s">'k'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">r"$(t,x(t))$"</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s">"sampTimes"</span><span class="p">],</span> <span class="n">xyz</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="s">'g'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">r"$(t,y(t))$"</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s">"sampTimes"</span><span class="p">],</span> <span class="n">xyz</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="s">'b'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">r"$(t,z(t))$"</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s">"sampTimes"</span><span class="p">],</span> <span class="n">B</span><span class="p">,</span> <span class="s">'r'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">r"$(t,B(t))$"</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="n">p</span><span class="p">[</span><span class="s">"timeStop"</span><span class="p">]],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'k--'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.4</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xyz</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">xyz</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="s">'g'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">r"$(x(t),y(t))$"</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xyz</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">xyz</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="s">'b'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">r"$(x(t),z(t))$"</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xyz</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">B</span><span class="p">,</span> <span class="s">'r'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">r"$(x(t),B(t))$"</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'k--'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.4</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">();</span> <span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s">"upper left"</span><span class="p">)</span>
<span class="n">gr</span><span class="o">.</span><span class="n">ion</span><span class="p">();</span> <span class="n">gr</span><span class="o">.</span><span class="n">draw</span><span class="p">()</span>
</pre></div>
</div>
</div>
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<p>Notice that $y$ reaches its maximum when $B(t)$ crosses 1. That is, the epidemic starts to wine down when the $B(t)<1$.</p>
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<h3 id="Exercises:">Exercises:<a class="anchor-link" href="#Exercises:">¶</a></h3><p>Setup two simulations for which there is no outbreak, such that:</p>
<p>(a) The initial density of "immune" individuals is large enough to prevent an epidemic.</p>
<p>(b) The initial density of "immune" individuals is really small but there is no epidemic outbreak.</p>
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