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<h1>Lyapunov View</h1>
<p>The <em>Lyapunov View</em> allows a user to apply tools developed to analyze
dynamical systems. The RMS comparison view is a useful measure of the average
difference between two scenarios. However, if a model and reference scenario
each describe an epidemic that begins and end in the same state (zero
infectious), the RMS error will eventually fall&nbsp; to zero as a function of
time, even in case of a &#8220;bad&#8221; model. In addition to measuring the average error,
it is useful to look for other measures that might provide a &#8220;fingerprint&#8221; for
the spatiotemporal dynamics of an infectious disease. Like many dynamical
systems, infectious disease is a process of many variables. However, it is often
possible to capture the essential dynamics by looking at just a few system
variables in an appropriate phase space.&nbsp; In its most general formalism, any
dynamical system is defined by a fixed set of equations that govern the time
dependence of the system&#8217;s state variables. The state variables at some instant
in time define a point in phase space. A SEIR model defines a four dimensional
phase space. An SI model defines a two dimensional space. Examining a reduced
set of dimension may be thought of as taking slice through phase space (for
example in the SI plane).
<p class="MsoNormal" style="text-indent:11.5pt">At the state of the system
changes with time, the point (S(t), I(t)) in phase space defines a <i>trajectory</i>
in the SI plane. Consider an epidemic that begins with one infectious person and
virtually the entire population susceptible at t=0,&nbsp; S(0) ~ 1. The trajectory
will begin at time zero along the S axis near 1. As the disease spreads, the
susceptible population (S) will decrease and the infectious population (I) will
increase. The detailed shape of the this trajectory will depend on the time it
takes for the disease to spread to different population centers, as well as the
(susceptible) population density function. The peaks and valleys along the
trajectory in SI phase space proved a signature or fingerprint for an epidemic
the shape of which depends on the disease, the disease vectors, the population
distribution, etc. The mathematics of dynamical systems provide us with a
formalism to compare trajectories in a phase space. Given a single set of rules
(e.g., a disease model), two simulations that begin infinitesimally close
together in phase space may evolve different in time and space. This separation
in phase space can be measure quantitatively. </p>
<p class="MsoNormal" style="text-indent:11.5pt">The Vector</p>
<p class="MsoNormal" style="text-indent:11.5pt" align="center">&nbsp;<span style="font-size:
9.0pt"><img border="0" src="img/lyapEq2.jpg" width="147" height="39"></span> </p>
<p class="MsoNormal" style="text-indent:11.5pt">defines a trajectory in SI space. The initial separation at
time zero be defined as </p>
<p class="MsoNormal" style="text-indent:11.5pt" align="center">
<img border="0" src="img/lyapEq3.jpg" width="66" height="38">.
</p>
<p class="MsoNormal" style="text-indent:11.5pt">The rate of separation of two trajectories in phase space will often obey the
equation</p>
<p class="MsoNormal" align="center" style="text-align:center;text-indent:11.5pt">
<img border="0" src="img/lyapEq1.jpg" width="234" height="48"></p>
<p class="MsoNormal" style="text-indent:0in">&nbsp;where
<span style="font-family:Symbol">l </span>is the Lyapunov Exponent. This
exponent is a characteristic of the dynamical system that defines the rate of
separation of infinitesimally close trajectories in phase space. </p>
<p>To use the Lyapunov view:<ol>
	<li>Enter the <a href="../perspectives/analysis.html">Analysis Perspective</a></li>
	<li>Click on the Lyapunov Tab</li>
	<li>Use the Select Folder buttons to chose the folders containing the data
	you wish to compare. The files should have the following
	<a href="csvloggerview.html">&nbsp;format.</a></li>
	<li>Click &quot;Compute Lyapunov Exponent&quot;</li>
	<li>Two charts will appear, the left hand chart will show the trajectories
	in phase space (I vs. S) for the two data sets. The right hand chart will
	show the Log of the integrated<br>
	difference between the two trajectories as a function of time. The exponent
	is the initial slope of this time varying difference plotted on a semi-log
	plot.</li>
</ol>
</p>
&nbsp;<p><img border="0" src="img/Lyapunov.jpg" width="1000" height="608"> </p>
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