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

 
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<h1> </h1>
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<p> </p>
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<h1 id="header">Model</h1>
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<p>We set up a mathematical model to predict the population dynamics of cockroaches before and after using our
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product. By doing so, we can estimate the lethal time of our cockroaches terminator, analyse the relationships
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among each relative factors so as to modify our product.</p>
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<h2 id="Assumptions">Assumptions</h2>
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<div class="col-10 offset-1 shadow">
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<li>1. The number of cockroach has reached the highest value in stable stage</li>
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<div class="text-center">
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<li>2. Ignore natural birth and death rates in our system</li>
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<h1 class="h1" style="color: #469789; text-align: center;">Model</h1>
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<li>3. Infectious individuals can not recover</li>
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</div>
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<li>4. Other factors that may affect the experiment are ignored</li>
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<div class="row">
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<div class="col-8 offset-2">
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<h2 id="Natural condition">Natural condition</h2>
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<p>In natural condition indoors, due to environmental resistance like food, water, space, the population of
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<p style="font-size: 18px; font-weight: bold;">
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cockroaches is more likely to follow a S-shaped growth curve (sigmoid growth curve), which can be formalized
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On this page, two mathematical models were constructed to analysis the efficiency of our producet based on simulation and experiment. <br />Click the frame  to see the details.
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mathematically by logistic function.</p>
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</p>
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<img class="rounded" style="width: 420px;" src="../img/Model_1.png" />
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<div class="col-4 offset-1">
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<h2 id="With infection">With infection</h2>
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<p>Our model was constructed based on SIR epidemic model (Susceptible, Infectious, Recovered) , following are
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<div id="card1" class="card border-3 h-100">
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some basic properties:</p>
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<div class="view text-center">
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<ul>
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<img id="icon" class="card-img-top" style="width: 96px;" src="https://static.igem.org/mediawiki/2018/4/48/T--SZU-China--Model_home1.png"/>
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<li>1. Naturally all cockroaches are susceptible individuals, they can infect by M.anisopliae becoming
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</div>
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infectious individuals.</li>
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<div class="card-body text-center">
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<li>2. The number of individual being infected in a contact between a susceptible and an infectious subject is
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<a style="color: #469789;" href="https://2018.igem.org/Team:SZU-China/Epidemic_Model"><h3 >Epidemic Model </h3></a>
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simulate by standard incidence .</li>
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<p class="card-text" >We developed a epidemic model with ordinary differential equation to predict the population dynamics of cockroaches infected by Metarhizium anisopliae. We then performed numerical simulations on the model and sensitivity analysis on some key parameters to find they impacts.
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<li>3. The transition rate between Infectious and dead is ��, its reciprocal (1/��) determines the average
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</p>
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infectious period, which is estimate by experiment data.</li>
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<img width="560px" src="../img/Model_2.png" />
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<div>
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<h2 id="Parameters">Parameters</h2>
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<p>Model was simulate during 30 days, with total number of 60.</p>
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<table class="table table-bordered">
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<thead>
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<tr class="table-active">
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<th>Parameter</th>
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<th>Value</th>
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<th>Meaning</th>
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</tr>
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</thead>
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<tbody class="text-center">
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<tr>
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<td>S(t)</td>
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<td></td>
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<td>the number of susceptible individuals over time</td>
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</tr>
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<tr>
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<td>I(t)</td>
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<td></td>
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<td>the number of infectious individuals over time</td>
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</tr>
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<tr>
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<td>D(t)</td>
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<td></td>
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<td>the number of dead individuals over time</td>
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</tr>
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<tr>
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<td>��</td>
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<td>0.75</td>
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<td>transmission rate, which is the probability of getting the infection in a contact between susceptible and
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an infectious</td>
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</tr>
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<tr>
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<td>��</td>
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<td>1/8</td>
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<td>mortality, which is the the transition rate between I and D, its reciprocal (1/��) determines the average
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infectious period</td>
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</tr>
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<tr>
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<td>S(0)</td>
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<td>55</td>
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<td>the initial number of susceptible individuals</td>
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</tr>
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<tr>
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<td>I(0)</td>
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<td>5</td>
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<td>the initial number of infectious individuals</td>
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</tr>
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<tr>
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<td>r</td>
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<td>0.3</td>
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<td>growth rate</td>
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</tr>
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<tr>
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<td>N=S+I</td>
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<td></td>
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<td>population size</td>
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</tr>
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<tr>
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<td>K</td>
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<td>70</td>
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<td>carring capacity</td>
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</tr>
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</tbody>
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</table>
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<P>The system without so-called vital dynamics (birth and death) described above can be expressed by the
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following set of ordinary differential equations:</P>
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<p>This system is non-linear, and the analytic solution does not exist, but we can compute the numerical solution
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by MATLAB. (see results)</p>
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</div>
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<div>
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<h2 id="Results">Results</h2>
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<p>The following curves show dynamics number change of each kinds of individuals. We see that the infectious
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individuals grow fast before first 6 day, and then began to drop. The total number of cockroaches continuously
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going down. We specify the median lethal time (LT50), which in this condition is 11.1 days.</p>
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<div class="text-center">
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<img class="rounded" style="width: 420px;" src="../img/Model_3.png" />
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</div>
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</div>
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<div>
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<h2 id="Sensitivity Analysis">Sensitivity Analysis</h2>
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<p>We use sensitivity analysis to analyze the impacts of some important parameter values (��, ��) on our model
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outcomes (LT50). The figures below show the tendency of dead number with respect to each parameter change. </p>
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</div>
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<div class="card-group">
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<div class="card-header">
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<h3>1. change gamma</h3>
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</div>
 
</div>
  
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<img class="card-img-top" style="width: 512px;" src="../img/Model_4.png" />
 
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<div class="card card-body">
 
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<table>
 
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<thead class="table table-active">
 
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<tr>
 
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<th>change ��</th>
 
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<th>��</th>
 
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<th>��</th>
 
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<th>LT50</th>
 
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<th>��LT50</th>
 
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<th>Ratio</th>
 
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</tr>
 
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</thead>
 
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<tbody>
 
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<tr>
 
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<td>+20%</td>
 
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<td>0.750</td>
 
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<td>0.150</td>
 
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<td>10.100</td>
 
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<td>0.090</td>
 
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<td>0.450</td>
 
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</tr>
 
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<tr>
 
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<td></td>
 
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<td>0.075</td>
 
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<td>0.125</td>
 
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<td>11.100</td>
 
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<td></td>
 
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<td></td>
 
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</tr>
 
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<tr>
 
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<td>-20%</td>
 
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<td>0.750</td>
 
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<td>0.100</td>
 
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<td>12.700</td>
 
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<td>0.138</td>
 
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<td>0.690</td>
 
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</tr>
 
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</tbody>
 
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</table>
 
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</div>
 
 
</div>
 
</div>
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<div class="card">
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<div class="card-header">
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<div class="col-4 offset-2 ">
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<h3>2. change beta</h3>
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<div id="card2" class="card border-3 h-100">
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</div>
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<div class="view text-center">
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<img class="card-img-top" style="width: 512px;" src="../img/Model_5.png" />
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<img id="icon" class="card-img-top" style="width: 96px;" src="https://static.igem.org/mediawiki/2018/4/4b/T--SZU-China--Model_home2.png"/>
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<div class="card card-body">
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</div>
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<table>
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<div class="card-body text-center">
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<thead class="table table-active">
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<a style="color: #469789;" href="https://2018.igem.org/Team:SZU-China/Statistic_Model"><h3  >Statistic Model</h3></a>
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<tr>
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<p class="card-text">We constructed a statistical model for our experimental data from simulating room. It can give us a clear and scientific understanding that whether there had significant difference in migration rate, mortality and gnawing rate between each products.</p>
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<th>change ��</th>
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<th>��</th>
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<th>��</th>
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<th>LT50</th>
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<th>��LT50</th>
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<th>Ratio</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>+20%</td>
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<td>0.900</td>
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<td>0.125</td>
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<td>10.300</td>
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<td>0.072</td>
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<td>0.360</td>
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</tr>
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<tr>
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<td></td>
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<td>0.750</td>
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<td>0.125</td>
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<td>11.100</td>
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<td></td>
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<td></td>
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</tr>
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<tr>
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<td>-20%</td>
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<td>0.600</td>
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<td>0.125</td>
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<td>12.400</td>
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<td>0.117</td>
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<td>0.585</td>
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</tr>
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</tbody>
 
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</table>
 
 
</div>
 
</div>
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</div>
  
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<div>
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<div>
 
  
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<p>The last term Ratio is the normalized sensitivities-the ratio of the relative change of the output to the
 
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relative change of the parameter.</p>
 
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<div class="col-2">
 
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<ul class=" position-fixed border-left">
 
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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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Latest revision as of 18:33, 17 October 2018

Model

On this page, two mathematical models were constructed to analysis the efficiency of our producet based on simulation and experiment.
Click the frame to see the details.

Epidemic Model

We developed a epidemic model with ordinary differential equation to predict the population dynamics of cockroaches infected by Metarhizium anisopliae. We then performed numerical simulations on the model and sensitivity analysis on some key parameters to find they impacts.

Statistic Model

We constructed a statistical model for our experimental data from simulating room. It can give us a clear and scientific understanding that whether there had significant difference in migration rate, mortality and gnawing rate between each products.