Phase portraits (or phase diagrams) provide a powerful tool to visualize the dynamics of ODE systems. You have been given the option of three easy-to-use programs that make drawing phase portraits very simple. In this seciton, use whichever of these you find convenient. Notice that the basic SIR model can be reduced to a two-dimensional system, because the variable R(t) for recovered individuals does not appear in the evolution equations fir the other two variables. The reduced (S, I) system is thus the following: d/dt S = -βSI
d/dt I = βSI – γI dt and can be plotted in a 2D S-I phase diagram without loss of information. For a fiaxed set of parameters, draw a phase portrait with trajectories corresponding to different initial conditions. Observe how the course of the epidemic depends on the initial conditions. Draw trajectories for different values of N. For each simulation, identify the maximum value of I, and draw the point on the phase diagram. It is possible to determine analytically the value of S for which the epidemic reaches its peak. Using Ro, we find A SI is max =γ /β.
Plot these values on your phase diagram, and compare them to the points you obtained previously. Interpret the result. In particular, what happens when So <γ/β?