Difference between revisions of "Team:SZU-China/Model"

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<div class="center-block">
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<h1> </h1>
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<p> </p>
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<h1 id="header">Model</h1>
 +
<p>We set up a mathematical model to predict the population dynamics of cockroaches before and after using our
 +
product. By doing so, we can estimate the lethal time of our cockroaches terminator, analyse the relationships
 +
among each relative factors so as to modify our product.</p>
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</div>
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</div>
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</div>
  
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</head>
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<div>
 +
<h2 id="Assumptions">Assumptions</h2>
 +
<ul>
 +
<li>1. The number of cockroach has reached the highest value in stable stage</li>
 +
<li>2. Ignore natural birth and death rates in our system</li>
 +
<li>3. Infectious individuals can not recover</li>
 +
<li>4. Other factors that may affect the experiment are ignored</li>
 +
</ul>
 +
<h2 id="Natural condition">Natural condition</h2>
 +
<p>In natural condition indoors, due to environmental resistance like food, water, space, the population of
 +
cockroaches is more likely to follow a S-shaped growth curve (sigmoid growth curve), which can be formalized
 +
mathematically by logistic function.</p>
 +
</div>
 +
<div class="text-center">
 +
<img class="rounded" style="width: 420px;" src="../img/Model_1.png" />
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</div>
  
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<div class="row">
 
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<div class="col-9 offset-1">
 
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<div class="bs-docs-section">
 
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  <div data-spy="scroll" data-target="#list-example" data-offset="0" class="scrollspy-example">
 
 
<div>
 
<div>
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<div class="bs-component">
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<h2 id="With infection">With infection</h2>
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<div class="jumbotron">
+
<p>Our model was constructed based on SIR epidemic model (Susceptible, Infectious, Recovered) , following are
−
<div class="container"></div>
+
some basic properties:</p>
−
<div class="center-block">
+
<ul>
−
<h1> </h1>
+
<li>1. Naturally all cockroaches are susceptible individuals, they can infect by M.anisopliae becoming
−
<p> </p>
+
infectious individuals.</li>
−
<h1 id="header" >Model</h1>
+
<li>2. The number of individual being infected in a contact between a susceptible and an infectious subject is
−
<p>We set up a mathematical model to predict the population dynamics of cockroaches before and after using our product. By doing so, we can estimate the lethal time of our cockroaches terminator, analyse the relationships among each relative factors so as to modify our product.</p>
+
simulate by standard incidence .</li>
−
</div>
+
<li>3. The transition rate between Infectious and dead is ��, its reciprocal (1/��) determines the average
−
</div>
+
infectious period, which is estimate by experiment data.</li>
−
</div>
+
 
−
+
</ul>
−
<div>
+
<div class="text-center">
−
<h2 id="Assumptions">Assumptions</h2>
+
<img width="560px" src="../img/Model_2.png" />
−
<ul>
+
−
<li>1. The number of cockroach has reached the highest value in stable stage</li>
+
−
<li>2. Ignore natural birth and death rates in our system</li>
+
−
<li>3. Infectious individuals can not recover</li>
+
−
<li>4. Other factors that may affect the experiment are ignored</li>
+
−
</ul>
+
−
<h2 id="Natural condition">Natural condition</h2>
+
−
<p>In natural condition indoors, due to environmental resistance like food, water, space, the population of cockroaches is more likely to follow a S-shaped growth curve (sigmoid growth curve), which can be formalized mathematically by logistic function.</p>
+
 
</div>
 
</div>
 +
 +
 +
</div>
 +
<div>
 +
<h2 id="Parameters">Parameters</h2>
 +
<p>Model was simulate during 30 days, with total number of 60.</p>
 +
<table class="table table-bordered">
 +
<thead>
 +
<tr class="table-active">
 +
<th>Parameter</th>
 +
<th>Value</th>
 +
<th>Meaning</th>
 +
</tr>
 +
</thead>
 +
<tbody class="text-center">
 +
<tr>
 +
<td>S(t)</td>
 +
<td></td>
 +
<td>the number of susceptible individuals over time</td>
 +
</tr>
 +
<tr>
 +
<td>I(t)</td>
 +
<td></td>
 +
<td>the number of infectious individuals over time</td>
 +
</tr>
 +
<tr>
 +
<td>D(t)</td>
 +
<td></td>
 +
<td>the number of dead individuals over time</td>
 +
</tr>
 +
<tr>
 +
<td>��</td>
 +
<td>0.75</td>
 +
<td>transmission rate, which is the probability of getting the infection in a contact between susceptible and
 +
an infectious</td>
 +
</tr>
 +
<tr>
 +
<td>��</td>
 +
<td>1/8</td>
 +
<td>mortality, which is the the transition rate between I and D, its reciprocal (1/��) determines the average
 +
infectious period</td>
 +
</tr>
 +
<tr>
 +
<td>S(0)</td>
 +
<td>55</td>
 +
<td>the initial number of susceptible individuals</td>
 +
</tr>
 +
<tr>
 +
<td>I(0)</td>
 +
<td>5</td>
 +
<td>the initial number of infectious individuals</td>
 +
</tr>
 +
<tr>
 +
<td>r</td>
 +
<td>0.3</td>
 +
<td>growth rate</td>
 +
</tr>
 +
<tr>
 +
<td>N=S+I</td>
 +
<td></td>
 +
<td>population size</td>
 +
</tr>
 +
<tr>
 +
<td>K</td>
 +
<td>70</td>
 +
<td>carring capacity</td>
 +
</tr>
 +
</tbody>
 +
</table>
 +
<P>The system without so-called vital dynamics (birth and death) described above can be expressed by the
 +
following set of ordinary differential equations:</P>
 +
<p>This system is non-linear, and the analytic solution does not exist, but we can compute the numerical solution
 +
by MATLAB. (see results)</p>
 +
</div>
 +
<div>
 +
<h2 id="Results">Results</h2>
 +
<p>The following curves show dynamics number change of each kinds of individuals. We see that the infectious
 +
individuals grow fast before first 6 day, and then began to drop. The total number of cockroaches continuously
 +
going down. We specify the median lethal time (LT50), which in this condition is 11.1 days.</p>
 
<div class="text-center">
 
<div class="text-center">
−
<img class="rounded" style="width: 420px;" src="../img/Model_1.png"/>
+
<img class="rounded" style="width: 420px;" src="../img/Model_3.png" />
 
</div>
 
</div>
−
+
</div>
−
<div>
+
<div>
−
<h2 id="With infection">With infection</h2>
+
<h2 id="Sensitivity Analysis">Sensitivity Analysis</h2>
−
<p>Our model was constructed based on SIR epidemic model (Susceptible, Infectious, Recovered) , following are some basic properties:</p>
+
<p>We use sensitivity analysis to analyze the impacts of some important parameter values (��, ��) on our model
−
<ul>
+
outcomes (LT50). The figures below show the tendency of dead number with respect to each parameter change. </p>
−
<li>1. Naturally all cockroaches are susceptible individuals, they can infect by M.anisopliae becoming infectious individuals.</li>
+
 
−
<li>2. The number of individual being infected in a contact between a susceptible and an infectious subject is simulate by standard incidence      .</li>
+
 
−
<li>3. The transition rate between Infectious and dead is γ, its reciprocal (1/γ) determines the average infectious period, which is estimate by experiment data.</li>
+
</div>
−
+
<div class="card-group">
−
</ul>
+
<div class="card">
−
<div class="text-center">
+
<div class="card-header">
−
<img width="560px" src="../img/Model_2.png"/>
+
<h3>1. change gamma</h3>
 
</div>
 
</div>
−
+
 
−
+
<img class="card-img-top" style="width: 512px;" src="../img/Model_4.png" />
−
</div>
+
<div class="card card-body">
−
<div>
+
<table>
−
<h2 id="Parameters">Parameters</h2>
+
<thead class="table table-active">
−
<p>Model was simulate during 30 days, with total number of 60.</p>
+
<tr>
−
<table class="table table-bordered">
+
<th>change ��</th>
−
<thead>
+
<th>��</th>
−
<tr class="table-active">
+
<th>��</th>
−
<th>Parameter</th>
+
<th>LT50</th>
−
<th>Value</th>
+
<th>��LT50</th>
−
<th>Meaning</th>
+
<th>Ratio</th>
−
</tr>
+
</tr>
−
</thead>
+
</thead>
−
<tbody class="text-center">
+
<tbody>
−
<tr>
+
<tr>
−
<td>S(t)</td>
+
<td>+20%</td>
−
<td></td>
+
<td>0.750</td>
−
<td>the number of susceptible individuals over time</td>
+
<td>0.150</td>
−
</tr>
+
<td>10.100</td>
−
<tr>
+
<td>0.090</td>
−
<td>I(t)</td>
+
<td>0.450</td>
−
<td></td>
+
</tr>
−
<td>the number of infectious individuals over time</td>
+
<tr>
−
</tr>
+
<td></td>
−
<tr>
+
<td>0.075</td>
−
<td>D(t)</td>
+
<td>0.125</td>
−
<td></td>
+
<td>11.100</td>
−
<td>the number of dead individuals over time</td>
+
<td></td>
−
</tr>
+
<td></td>
−
<tr>
+
</tr>
−
<td>β</td>
+
<tr>
−
<td>0.75</td>
+
<td>-20%</td>
−
<td>transmission rate, which is the probability of getting the infection in a contact between susceptible and an infectious</td>
+
<td>0.750</td>
−
</tr>
+
<td>0.100</td>
−
<tr>
+
<td>12.700</td>
−
<td>γ</td>
+
<td>0.138</td>
−
<td>1/8</td>
+
<td>0.690</td>
−
<td>mortality, which is the the transition rate between I and D, its reciprocal (1/γ) determines the average infectious period</td>
+
</tr>
−
</tr>
+
 
−
<tr>
+
</tbody>
−
<td>S(0)</td>
+
</table>
−
<td>55</td>
+
−
<td>the initial number of susceptible individuals</td>
+
−
</tr>
+
−
<tr>
+
−
<td>I(0)</td>
+
−
<td>5</td>
+
−
<td>the initial number of infectious individuals</td>
+
−
</tr>
+
−
<tr>
+
−
<td>r</td>
+
−
<td>0.3</td>
+
−
<td>growth rate</td>
+
−
</tr>
+
−
<tr>
+
−
<td>N=S+I</td>
+
−
<td></td>
+
−
<td>population size</td>
+
−
</tr>
+
−
<tr>
+
−
<td>K</td>
+
−
<td>70</td>
+
−
<td>carring capacity</td>
+
−
</tr>
+
−
</tbody>
+
−
</table>
+
−
<P>The system without so-called vital dynamics (birth and death) described above can be expressed by the following set of ordinary differential equations:</P>
+
−
<p>This system is non-linear, and the analytic solution does not exist, but we can compute the numerical solution by MATLAB. (see results)</p>
+
−
</div>
+
−
<div>
+
−
<h2 id="Results">Results</h2>
+
−
<p>The following curves show dynamics number change of each kinds of individuals. We see that the infectious individuals grow fast before first 6 day, and then began to drop. The total number of cockroaches continuously going down. We specify the median lethal time (LT50), which in this condition is 11.1 days.</p>
+
−
<div class="text-center">
+
−
<img class="rounded" style="width: 420px;" src="../img/Model_3.png"/>
+
 
</div>
 
</div>
 
</div>
 
</div>
−
<div>
+
<div class="card">
−
<h2 id="Sensitivity Analysis">Sensitivity Analysis</h2>
+
<div class="card-header">
−
<p>We use sensitivity analysis to analyze the impacts of some important parameter values (α, β) on our model outcomes (LT50). The figures below show the tendency of dead number with respect to each parameter change. </p>
+
<h3>2. change beta</h3>
−
+
−
+
−
</div>
+
−
<div class="card-group">
+
−
<div class="card">
+
−
<div class="card-header">
+
−
<h3>1. change gamma</h3>
+
−
</div>
+
−
+
−
<img class="card-img-top" style="width: 512px;"  src="../img/Model_4.png" />
+
−
<div class="card card-body">
+
−
<table >
+
−
<thead class="table table-active">
+
−
<tr>
+
−
<th>change α</th>
+
−
<th>α</th>
+
−
<th>β</th>
+
−
<th>LT50</th>
+
−
<th>ΔLT50</th>
+
−
<th>Ratio</th>
+
−
</tr>
+
−
</thead>
+
−
<tbody>
+
−
<tr>
+
−
<td>+20%</td>
+
−
<td>0.750</td>
+
−
<td>0.150</td>
+
−
<td>10.100</td>
+
−
<td>0.090</td>
+
−
<td>0.450</td>
+
−
</tr>
+
−
<tr>
+
−
<td></td>
+
−
<td>0.075</td>
+
−
<td>0.125</td>
+
−
<td>11.100</td>
+
−
<td></td>
+
−
<td></td>
+
−
</tr>
+
−
<tr>
+
−
<td>-20%</td>
+
−
<td>0.750</td>
+
−
<td>0.100</td>
+
−
<td>12.700</td>
+
−
<td>0.138</td>
+
−
<td>0.690</td>
+
−
</tr>
+
−
+
−
</tbody>
+
−
</table>
+
−
</div>
+
 
</div>
 
</div>
−
<div class="card">
+
<img class="card-img-top" style="width: 512px;" src="../img/Model_5.png" />
−
<div class="card-header"><h3>2. change beta</h3></div>
+
<div class="card card-body">
−
<img class="card-img-top" style="width: 512px;" src="../img/Model_5.png" />
+
<table>
−
<div class="card card-body">
+
<thead class="table table-active">
−
<table >
+
<tr>
−
<thead class="table table-active">
+
<th>change ��</th>
−
<tr>
+
<th>��</th>
−
<th>change β</th>
+
<th>��</th>
−
<th>α</th>
+
<th>LT50</th>
−
<th>β</th>
+
<th>��LT50</th>
−
<th>LT50</th>
+
<th>Ratio</th>
−
<th>ΔLT50</th>
+
</tr>
−
<th>Ratio</th>
+
</thead>
−
</tr>
+
<tbody>
−
</thead>
+
<tr>
−
<tbody>
+
<td>+20%</td>
−
<tr>
+
<td>0.900</td>
−
<td>+20%</td>
+
<td>0.125</td>
−
<td>0.900</td>
+
<td>10.300</td>
−
<td>0.125</td>
+
<td>0.072</td>
−
<td>10.300</td>
+
<td>0.360</td>
−
<td>0.072</td>
+
</tr>
−
<td>0.360</td>
+
<tr>
−
</tr>
+
<td></td>
−
<tr>
+
<td>0.750</td>
−
<td></td>
+
<td>0.125</td>
−
<td>0.750</td>
+
<td>11.100</td>
−
<td>0.125</td>
+
<td></td>
−
<td>11.100</td>
+
<td></td>
−
<td></td>
+
</tr>
−
<td></td>
+
<tr>
−
</tr>
+
<td>-20%</td>
−
<tr>
+
<td>0.600</td>
−
<td>-20%</td>
+
<td>0.125</td>
−
<td>0.600</td>
+
<td>12.400</td>
−
<td>0.125</td>
+
<td>0.117</td>
−
<td>12.400</td>
+
<td>0.585</td>
−
<td>0.117</td>
+
</tr>
−
<td>0.585</td>
+
 
−
</tr>
+
</tbody>
−
+
</table>
−
</tbody>
+
−
</table>
+
−
</div>
+
 
</div>
 
</div>
 
</div>
 
</div>
−
 
−
<div>
 
−
 
−
</div>
 
−
<div>
 
−
 
−
</div>
 
−
<p>The last term Ratio is the normalized sensitivities-the ratio of the relative change of the output to the relative change of the parameter.</p>
 
 
</div>
 
</div>
−
</div>
+
 
−
</div>
+
<div>
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</div>
+
 
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−
<div class="col-2">
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<div class="position-fixed text-muted">
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<a class="text-muted" href="#">Modle</a>
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    <a class="text-muted" href="#Assumptions">Assumptions</a>
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    <a class="" href="#Natural condition">Natural condition</a>
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    <a class="" href="#With infection">With infection</a>
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−
    <a class="" href="#Parameters">Parameters</a>
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    <a class="" href="#Results">Results</a>
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    <a class="" href="#Sensitivity Analysis">Sensitivity Analysis</a>
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<p><a class="text-muted" href="#">Modle</a></p>
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<p><a class="text-muted" href="#Assumptions">Assumptions</a></p>
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<p><a class="text-muted" href="#Natural condition">Natural condition</a></p>
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<p><a class="text-muted" href="#With infection">With infection</a></p>
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<p><a class="text-muted" href="#Parameters">Parameters</a></p>
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<p><a class="text-muted" href="#Results">Results</a></p>
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<p><a class="text-muted" href="#Sensitivity Analysis">Sensitivity Analysis</a></p>
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<nav id="navbar" class="sidebar flex-column position-fixed border-left ">
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  <a class="nav-link text-muted" href="#">Modle</a>
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    <a class="nav-link text-muted" href="#Assumptions">Assumptions</a>
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    <a class="nav-link" href="#Natural condition">Natural condition</a>
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    <a class="nav-link" href="#With infection">With infection</a>
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    <a class="nav-link" href="#Parameters">Parameters</a>
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    <a class="nav-link" href="#Results">Results</a>
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    <a class="nav-link" href="#Sensitivity Analysis">Sensitivity Analysis</a>
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<div id="list-example" class="list-group list-group-flush position-fixed">
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<a class="list-group-item list-group-item-action active" href="#">Model</a>
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<a class="list-group-item list-group-item-action" href="#Assumptions">Assumptions</a>
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            <a class="list-group-item list-group-item-action" href="#Natural condition">Natural condition</a>
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            <a class="list-group-item list-group-item-action" href="#With infection">With infection</a>
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          <a class="list-group-item list-group-item-action" href="#Parameters">Parameters</a>
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          <a class="list-group-item list-group-item-action" href="#Results">Results</a>
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            <a class="list-group-item list-group-item-action" href="#Sensitivity Analysis">Sensitivity Analysis</a>
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−
         
+
 
</div>
 
</div>
 +
<div>
 +
 +
</div>
 +
<p>The last term Ratio is the normalized sensitivities-the ratio of the relative change of the output to the
 +
relative change of the parameter.</p>
 
</div>
 
</div>
−
-->
+
</div>
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</div>
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<p><a class="text-muted" href="#">Modle</a></p>
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<p><a class="text-muted" href="#Assumptions">Assumptions</a></p>
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<p><a class="text-muted" href="#Natural condition">Natural condition</a></p>
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<p><a class="text-muted" href="#With infection">With infection</a></p>
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<p><a class="text-muted" href="#Parameters">Parameters</a></p>
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<p><a class="text-muted" href="#Results">Results</a></p>
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<p><a class="text-muted" href="#Sensitivity Analysis">Sensitivity Analysis</a></p>
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Revision as of 14:49, 28 September 2018

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Model

We set up a mathematical model to predict the population dynamics of cockroaches before and after using our product. By doing so, we can estimate the lethal time of our cockroaches terminator, analyse the relationships among each relative factors so as to modify our product.

Assumptions

  • 1. The number of cockroach has reached the highest value in stable stage
  • 2. Ignore natural birth and death rates in our system
  • 3. Infectious individuals can not recover
  • 4. Other factors that may affect the experiment are ignored

Natural condition

In natural condition indoors, due to environmental resistance like food, water, space, the population of cockroaches is more likely to follow a S-shaped growth curve (sigmoid growth curve), which can be formalized mathematically by logistic function.

With infection

Our model was constructed based on SIR epidemic model (Susceptible, Infectious, Recovered) , following are some basic properties:

  • 1. Naturally all cockroaches are susceptible individuals, they can infect by M.anisopliae becoming infectious individuals.
  • 2. The number of individual being infected in a contact between a susceptible and an infectious subject is simulate by standard incidence .
  • 3. The transition rate between Infectious and dead is ��, its reciprocal (1/��) determines the average infectious period, which is estimate by experiment data.

Parameters

Model was simulate during 30 days, with total number of 60.

Parameter Value Meaning
S(t) the number of susceptible individuals over time
I(t) the number of infectious individuals over time
D(t) the number of dead individuals over time
�� 0.75 transmission rate, which is the probability of getting the infection in a contact between susceptible and an infectious
�� 1/8 mortality, which is the the transition rate between I and D, its reciprocal (1/��) determines the average infectious period
S(0) 55 the initial number of susceptible individuals
I(0) 5 the initial number of infectious individuals
r 0.3 growth rate
N=S+I population size
K 70 carring capacity

The system without so-called vital dynamics (birth and death) described above can be expressed by the following set of ordinary differential equations:

This system is non-linear, and the analytic solution does not exist, but we can compute the numerical solution by MATLAB. (see results)

Results

The following curves show dynamics number change of each kinds of individuals. We see that the infectious individuals grow fast before first 6 day, and then began to drop. The total number of cockroaches continuously going down. We specify the median lethal time (LT50), which in this condition is 11.1 days.

Sensitivity Analysis

We use sensitivity analysis to analyze the impacts of some important parameter values (��, ��) on our model outcomes (LT50). The figures below show the tendency of dead number with respect to each parameter change.

1. change gamma

change �� �� �� LT50 ��LT50 Ratio
+20% 0.750 0.150 10.100 0.090 0.450
0.075 0.125 11.100
-20% 0.750 0.100 12.700 0.138 0.690

2. change beta

change �� �� �� LT50 ��LT50 Ratio
+20% 0.900 0.125 10.300 0.072 0.360
0.750 0.125 11.100
-20% 0.600 0.125 12.400 0.117 0.585

The last term Ratio is the normalized sensitivities-the ratio of the relative change of the output to the relative change of the parameter.