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

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                        <h3 class="uppercase color-primary mb40 " style="margin-bottom: 40px;font-size:50px"><center>MODEL</center> </h3>
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        <p></p><br /><br />
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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 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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                <center><p class="font1" style="font-weight: 500; text-align: center; font-size: 25px; color: rgb(75, 151, 165); ">Introduction</p></center><br /><br />
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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 />
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                <p style="font-size:16px">
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                    The four factors are: <br /><img src="https://static.igem.org/mediawiki/2017/c/c6/T--SZU-China--equation.jpg " width="25%" /></br>
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<h2 id="Assumptions">Assumptions</h2>
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                </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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<li>1. The number of cockroach has reached the highest value in stable stage</li>
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                <br />
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<li>2. Ignore natural birth and death rates in our system</li>
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<li>3. Infectious individuals can not recover</li>
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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 />
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<li>4. Other factors that may affect the experiment are ignored</li>
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                <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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<h2 id="Natural condition">Natural condition</h2>
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                <br /><br />
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<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>
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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 />
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                <br/><br/>
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<img class="rounded" style="width: 420px;" src="../img/Model_1.png"/>
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                <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>
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                <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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<h2 id="With infection">With infection</h2>
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                <p style="font-size:16px">
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<p>Our model was constructed based on SIR epidemic model (Susceptible, Infectious, Recovered) , following are some basic properties:</p>
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                    1.&nbsp;&nbsp;Standardizing variables: transforming variables into the same scale. Here we utilize the z-score standardizing method, such that
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<li>1. Naturally all cockroaches are susceptible individuals, they can infect by M.anisopliae becoming infectious individuals.</li>
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                <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 />
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<li>2. The number of individual being infected in a contact between a susceptible and an infectious subject is simulate by standard incidence      .</li>
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                <p style="font-size:16px">
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<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>
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                    where the symbols in the formula are variable, standardized variable, sample mean, andstandard deviation respectively.
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                </p>
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                <p style="font-size:16px">2.&nbsp;&nbsp;Fitting functions of each variable with polynomial function.</p>
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                <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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<img width="560px" src="../img/Model_2.png"/>
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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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<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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                <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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<th>Parameter</th>
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<th>Value</th>
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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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<th>Meaning</th>
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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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<td>S(t)</td>
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<td></td>
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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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<td>the number of susceptible individuals over time</td>
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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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                <br/>
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<td>I(t)</td>
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<td></td>
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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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<td>the number of infectious individuals over time</td>
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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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                <span></span><br />
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<td>D(t)</td>
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                <span></span><br />
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                <span></span><br />
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<td>the number of dead individuals over time</td>
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<td>β</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 an infectious</td>
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<td>γ</td>
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            <img src="https://static.igem.org/mediawiki/2017/0/08/2017SZUChina-sponsor1.jpg" class="sponsor-img">
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<td>1/8</td>
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            <img src="https://static.igem.org/mediawiki/2017/d/d4/2017SZUChina-sponsor2.jpg" class="sponsor-img">
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<td>mortality, which is the the transition rate between I and D, its reciprocal (1/γ) determines the average infectious period</td>
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            <img src="https://static.igem.org/mediawiki/2017/d/d5/2017SZUChina-sponsor3.jpg" class="sponsor-img">
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<td>S(0)</td>
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<td>55</td>
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        <div class="footer-info">
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<td>the initial number of susceptible individuals</td>
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            <address style="color:#888">3688 Nanhai Ave, ShenZhen, GuangDong Province, China</address>
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            <span class="copyright">Copyright © 2017.<a href="https://2017.igem.org/Team:SZU-China" class="mainpage">SZU-China</a> iGEM</span>
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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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<td>r</td>
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<td>0.3</td>
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<td>growth rate</td>
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<td>population size</td>
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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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<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>
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<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>
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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 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>
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<img class="rounded" style="width: 420px;" src="../img/Model_3.png"/>
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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 (α, β) 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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<h3>1. change gamma</h3>
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<th>change α</th>
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<th>α</th>
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<th>β</th>
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<th>LT50</th>
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<th>ΔLT50</th>
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<th>Ratio</th>
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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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<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>-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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<div class="card">
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<div class="card-header"><h3>2. change beta</h3></div>
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<img class="card-img-top" style="width: 512px;" src="../img/Model_5.png" />
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<th>change β</th>
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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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<td>0.750</td>
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<td>11.100</td>
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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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<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>
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Revision as of 14:34, 28 September 2018

MODEL



Introduction



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).



The four factors are:

Based on this model, we can also design the best ‘package'– the microcapsule shells with adequate nutrition combination.


This modelling process, presented below, can be seen as a feedback from the wet lab (experiment result) and the dry lab(modelling analysis).



The following page shows how we conducted modelling approaches to achieve our goals. To begin with, we make some definition.





Definition




    Variables and nomenclature:
  •         concentration of spores - c[Spore]
  •         concentration of carbon source(C3H5O3Na) - c[C3H5O3Na]
  •         concentration of nitrogen source(NaNO3) - c[NaNO3]
  •          pH of the media - pH

    Goal of model:
  •         Use preliminary data to guide future experiments.


Procedure


1.  Standardizing variables: transforming variables into the same scale. Here we utilize the z-score standardizing method, such that



where the symbols in the formula are variable, standardized variable, sample mean, andstandard deviation respectively.

2.  Fitting functions of each variable with polynomial function.

3.  Getting the overall relationship using linear least square method.




Results


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.



The fitted functions are showed as follows:


We then fitted x with y linearly to get the overall regression equation which can describe the weight of four variables respectively.



Overall regression equation:



with




where ai1represents 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.




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.



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 B.subtilis spore. In this way, this model instructs us on how to modify our formulation, such that the ingredients can be made full use of.











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