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

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<body style="background: #FEFEFE; font-family: "open sans", "Helvetica Neue" , helvetica, arial, sans-serif;">
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<body>
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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">
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<div>
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<div class="bs-component">
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<div class="container"></div>
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<div class="center-block">
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<h1> </h1>
 +
<p> </p>
 +
<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>
 +
</div>
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</div>
 +
</div>
  
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    <div class="navbar" style="top:5px;position:fixed;z-index:5">
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<div>
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        <div class="navbar-nav">
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<h2 id="Assumptions">Assumptions</h2>
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            <ul id="nav-list-ul" class="nav-list">
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<ul>
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<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" />
 +
</div>
  
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                <li>
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<div>
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                    <a href="#">TEAM</a>
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<h2 id="With infection">With infection</h2>
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                    <span class="caret"></span>
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<p>Our model was constructed based on SIR epidemic model (Susceptible, Infectious, Recovered) , following are
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                    <ul class="sub-nav" style="top:-5px">
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some basic properties:</p>
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                        <li><a href="https://2017.igem.org/Team:SZU-China/Team">MEMBERS</a></li>
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<ul>
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                        <li><a href="https://2017.igem.org/Team:SZU-China/Collaborations">COLLABORATIONS</a></li>
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<li>1. Naturally all cockroaches are susceptible individuals, they can infect by M.anisopliae becoming
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                        <li><a href="https://2017.igem.org/Team:SZU-China/Attributions">ATTRIBUTION</a></li>
+
infectious individuals.</li>
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                    </ul>
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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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                </li>
+
simulate by standard incidence .</li>
 +
<li>3. The transition rate between Infectious and dead is ��, its reciprocal (1/��) determines the average
 +
infectious period, which is estimate by experiment data.</li>
  
 +
</ul>
 +
<div class="text-center">
 +
<img width="560px" src="../img/Model_2.png" />
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</div>
  
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                <li>
 
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                    <a href="#">PRACTICE</a>
 
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                    <span class="caret"></span>
 
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                    <ul class="sub-nav" style="top:-5px">
 
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                        <li><a href="https://2017.igem.org/Team:SZU-China/HP/Gold_Integrated">INTEGRATED HP</a></li>
 
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                        <li><a href="https://2017.igem.org/Team:SZU-China/HP/Silver">SILVER HP</a></li>
 
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                        <li><a href="https://2017.igem.org/Team:SZU-China/Engagement">ENGAGEMENT</a></li>
 
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                    </ul>
 
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                </li>
 
  
 +
</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">
 +
<img class="rounded" style="width: 420px;" src="../img/Model_3.png" />
 +
</div>
 +
</div>
 +
<div>
 +
<h2 id="Sensitivity Analysis">Sensitivity Analysis</h2>
 +
<p>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. </p>
  
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                <li>
 
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                    <a href="#">EXPERIMENT</a>
 
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                    <span class="caret"></span>
 
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                    <ul class="sub-nav" style="top:-5px">
 
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                        <li><a href="https://2017.igem.org/Team:SZU-China/Procedure">PROCEDURE</a></li>
 
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                        <li><a href="https://2017.igem.org/Team:SZU-China/Results">RESULTS</a></li>
 
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                        <li><a href="https://2017.igem.org/Team:SZU-China/Demonstrate">DEMONSTRATE</a></li>
 
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                        <li><a href="https://2017.igem.org/Team:SZU-China/Protocol">PROTOCOL</a></li>
 
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                        <li><a href="https://2017.igem.org/Team:SZU-China/Notebook">NOTEBOOK</a></li>
 
  
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                    </ul>
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</div>
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                </li>
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<div class="card-group">
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                <li>
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<div class="card">
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                    <a href="#">PROJECT</a>
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<div class="card-header">
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                    <span class="caret"></span>
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<h3>1. change gamma</h3>
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                    <ul class="sub-nav" style="top:-5px">
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</div>
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                        <li><a href="https://2017.igem.org/Team:SZU-China/Description">DESCRIPTION</a></li>
+
  
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                        <li><a href="https://2017.igem.org/Team:SZU-China/Design">DESIGN</a></li>
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<img class="card-img-top" style="width: 512px;" src="../img/Model_4.png" />
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                        <li><a href="https://2017.igem.org/Team:SZU-China/Model">MODEL</a></li>
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<div class="card card-body">
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                        <li><a href="https://2017.igem.org/Team:SZU-China/Safety">SAFTY</a></li>
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<table>
−
                    </ul>
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<thead class="table table-active">
−
                </li>
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<tr>
 +
<th>change ��</th>
 +
<th>��</th>
 +
<th>��</th>
 +
<th>LT50</th>
 +
<th>��LT50</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 class="card">
 +
<div class="card-header">
 +
<h3>2. change beta</h3>
 +
</div>
 +
<img class="card-img-top" style="width: 512px;" src="../img/Model_5.png" />
 +
<div class="card card-body">
 +
<table>
 +
<thead class="table table-active">
 +
<tr>
 +
<th>change ��</th>
 +
<th>��</th>
 +
<th>��</th>
 +
<th>LT50</th>
 +
<th>��LT50</th>
 +
<th>Ratio</th>
 +
</tr>
 +
</thead>
 +
<tbody>
 +
<tr>
 +
<td>+20%</td>
 +
<td>0.900</td>
 +
<td>0.125</td>
 +
<td>10.300</td>
 +
<td>0.072</td>
 +
<td>0.360</td>
 +
</tr>
 +
<tr>
 +
<td></td>
 +
<td>0.750</td>
 +
<td>0.125</td>
 +
<td>11.100</td>
 +
<td></td>
 +
<td></td>
 +
</tr>
 +
<tr>
 +
<td>-20%</td>
 +
<td>0.600</td>
 +
<td>0.125</td>
 +
<td>12.400</td>
 +
<td>0.117</td>
 +
<td>0.585</td>
 +
</tr>
  
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                <li>
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</tbody>
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                    <a href="#">ACHIEVEMENT</a>
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</table>
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                    <span class="caret"></span>
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</div>
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                    <ul class="sub-nav" style="top:-5px">
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</div>
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                        <li><a href="https://2017.igem.org/Team:SZU-China/Achievement">MEDAL</a></li>
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</div>
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                        <li><a href="https://2017.igem.org/Team:SZU-China/Parts">PARTS</a></li>
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                    </ul>
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                </li>
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            </ul>
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        </div>
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        <div>
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            <a href="https://2017.igem.org/Team:SZU-China"><img src="https://static.igem.org/mediawiki/2017/1/17/T--SZU-China--team-logo.png" style="height: 36px;width: auto;padding-top: 12px;padding-left:10px;"></a>
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        </div>
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    </div>
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    <div style="position: relative;top: 80px;">
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<div>
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        <section id="overview" style="padding:96px 0 40px 0;background-color:white;">
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            <div class="container">
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                <div class="row">
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                    <div class="col-sm-12">
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                        <h3 class="uppercase color-primary mb40 " style="margin-bottom: 40px;font-size:50px"><center>MODEL</center> </h3>
+
  
 +
</div>
 +
<div>
  
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                    </div>
+
</div>
−
                </div>
+
<p>The last term Ratio is the normalized sensitivities-the ratio of the relative change of the output to the
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            </div>
+
relative change of the parameter.</p>
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        </section>
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</div>
 +
</div>
 +
</div>
 +
</div>
  
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<div class="col-2">
  
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        <p></p><br /><br />
 
  
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        <section style="background-color: rgba(245,245,245,0.45); padding: 96px 0; ">
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<ul class=" position-fixed border-left">
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            <div style="position: relative; margin: 30px 15%;">
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<p><a class="text-muted" href="#">Modle</a></p>
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                <center><p class="font1" style="font-weight: 500; text-align: center; font-size: 25px; color: rgb(75, 151, 165); ">Introduction</p></center><br /><br />
+
<p><a class="text-muted" href="#Assumptions">Assumptions</a></p>
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                <p style="font-size:16px">This year, our team creates a mathematical representation of our concrete self-healing system. This representation, or model, constructs a judging scale with which we can utilize to regulate the four main environmental factors affecting our final concrete healing efficiency (reflected by mineralization activity). </p><br /><br />
+
<p><a class="text-muted" href="#Natural condition">Natural condition</a></p>
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                <p style="font-size:16px">
+
−
                    The four factors are: <br /><img src="https://static.igem.org/mediawiki/2017/c/c6/T--SZU-China--equation.jpg " width="25%" /></br>
+
−
                </p>
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                <p style="font-size:16px">Based on this model, we can also design the best ‘package'– the microcapsule shells with adequate nutrition combination.</p>
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                <br />
+
−
               
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−
                <p style="font-size:16px">This modelling process, presented below, can be seen as a feedback from the wet lab (experiment result) and the dry lab(modelling analysis).</p><br /><br />
+
  
−
                <p style="font-size:16px">The following page shows how we conducted modelling approaches to achieve our goals. To begin with, we make some definition.</p><br /><br />
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<p><a class="text-muted" href="#With infection">With infection</a></p>
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                <br /><br />
+
<p><a class="text-muted" href="#Parameters">Parameters</a></p>
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                <center><span class="font1" style="font-weight: 500; text-align: center; font-size: 25px; color: rgb(75, 151, 165); ">Definition</span></center><br /><br />
+
<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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                <br/><br/>
+
</ul>
−
                <ul style="font-size:16px;" class="def">Variables and nomenclature:<li>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;concentration of spores - c[Spore]</li> <li>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;concentration of carbon source(C3H5O3Na) - c[C3H5O3Na]</li><li> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;concentration of nitrogen source(NaNO3) - c[NaNO3]</li><li>&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;pH of the media - pH</li></ul><br />
+
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                <ul style="font-size: 16px;" class="def">Goal of model:<li>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Use preliminary data to guide future experiments.</li></ul>
+
−
                <br /><br />
+
−
                <center><span class="font1" style="font-weight: 500; text-align: center; font-size: 25px; color: rgb(75, 151, 165); ">Procedure</span></center><br /><br />
+
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                <p style="font-size:16px">
+
−
                    1.&nbsp;&nbsp;Standardizing variables: transforming variables into the same scale. Here we utilize the z-score standardizing method, such that
+
−
                </p>
+
−
                <div style="text-align:center"><img src="https://static.igem.org/mediawiki/2017/0/07/T--SZU-China--standardization.jpg" width="30%" ><br /></div><br />
+
−
                <p style="font-size:16px">
+
−
                    where the symbols in the formula are variable, standardized variable, sample mean, andstandard deviation respectively.
+
−
                </p>
+
−
                <p style="font-size:16px">2.&nbsp;&nbsp;Fitting functions of each variable with polynomial function.</p>
+
−
                <p style="font-size:16px">3.&nbsp;&nbsp;Getting the overall relationship using linear least square method.</p><br /><br /><br />
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                <center> <span class="font1" style="font-weight: 500; text-align: center; font-size: 25px; color:rgb(75, 151, 165); ">Results</span></center><br /><br />
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                    <img src="https://static.igem.org/mediawiki/2017/2/24/T--SZU-China--model1.png " width="90%" style="box-shadow: 0px 0px 2px #1E1E1E;">
 
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                <p style="font-size:16px">The graph above depicts the polynomial regression of 4 factors. In general, for each of those four factors, the mineralization activity shows the similar tendency of going up first and down later with the increase of each factor.</p><br /><br />
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                <p style="font-size:16px">The fitted functions are showed as follows:</p>
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                <div style="text-align:center"><img src="https://static.igem.org/mediawiki/2017/1/1d/T--SZU-China--fitting.jpg" width="60%" ><br /></div>
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                <p style="font-size:16px">We then fitted x with y linearly to get the overall regression equation which can describe the weight of four variables respectively.</p><br /><br />
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                <center><span class="font1" style="font-weight:bold;font-size:16px">Overall regression equation:</span></center><br /><br />
 
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                <div style="text-align:center"><img src="https://static.igem.org/mediawiki/2017/a/ad/T--SZU-China--regression.jpg" width="60%" ><br /></div>
 
  
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                <p style="font-size:16px">with</p><br /><br />
 
  
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                <div style="text-align:center"><img src="https://static.igem.org/mediawiki/2017/b/be/T--SZU-China--regressiontransform.jpg" width="30%" ><br /></div>
 
  
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                <p style="font-size:16px">where a<sub>i1</sub>represents the coefficient of the highest power term of curve-fitting equation, such that the coefficient of the highest power term are of the same scale.</p><br /><br />
 
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                <p style="font-size:16px">From the equation above, we can see that nitrogen source has the maximum weight, while pH has the minimum weight, which means nitrogen source is the most essential nutrition for B.subtilis spore. Additionally, the low weight of pH shows that spores are not sensitive to the change of pH in a relatively apt range, although there is a sharp decline in activity when pH reaches 11.</p><br />
 
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                <br /><p style="font-size:16px">In summary, as long as we keep the environment below the boundary high-pH point, it makes no much difference how much we have improved the alkaline resistance of <i>B.subtilis</i> spore. In this way, this model instructs us on how to modify our formulation, such that the ingredients can be made full use of.</p><br />
 
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Revision as of 15:01, 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.