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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 to zero as a function of |
| time, even in case of a “bad” model. In addition to measuring the average error, |
| it is useful to look for other measures that might provide a “fingerprint” 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. In its most general formalism, any |
| dynamical system is defined by a fixed set of equations that govern the time |
| dependence of the system’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, 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"> <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"> 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"> format.</a></li> |
| <li>Click "Compute Lyapunov Exponent"</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> |
| <p><img border="0" src="img/Lyapunov.jpg" width="1000" height="608"> </p> |
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