Introduction to the Modeling and Analysis of Complex Systems

112 CHAPTER 7. CONTINUOUS-TIME MODELS II: ANALYSISFind the equilibrium points of the following model of a simple pen-Exercise 7.2dulum:d 2 θdt = − g sin θ (7.6)2 LExercise 7.3 The following model is called a Susceptible-Infected-Recovered(SIR) model, a mathematical model of epidemiological dynamics. S is the numberof susceptible individuals, I is the number of infected ones, and R is the number ofrecovered ones. Find the equilibrium points of this model.dSdtdIdtdRdt= −aSI (7.7)= aSI − bI (7.8)= bI (7.9)7.2 Phase Space VisualizationA phase space of a continuous-time model, once time is discretized, can be visualizedin the exact same way as it was in Chapter 5, using Codes 5.1 or 5.2. This is perfectlyfine. In the meantime, Python’s matplotlib has a specialized function called streamplot,which is precisely designed for drawing phase spaces of continuous-time models. It worksonly for two-dimensional phase space visualizations, but its output is quite neat and sophisticated.Here is how you can use this function:Code 7.1: phasespace-drawing-streamplot.pyfrom pylab import *xvalues, yvalues = meshgrid(arange(0, 3, 0.1), arange(0, 3, 0.1))xdot = xvalues - xvalues * yvaluesydot = - yvalues + xvalues * yvalues