In my notes the following SI model is given
$\frac{d}{dt}S(t)=(1-S(t))-R_0I(t)S(t)$
$\frac{d}{dt}I(t)=R_0I(t)S(t)$
$S(t)+I(t)=1$
$t \ge 0, R_0 >0$, $ 0 \le S(t)\le 1$, $ 0 \le I(t)\le 1$ where $S(T)$ is the fraction if Susceptible individuals and $I(t)$ the fraction of Infected Individuals We want to study the the stability euristically using the phase space if $R_0 >1$
Solution:
The equilibrium points are $(\bar u_1, \bar v_1)=(1,0)$ and $(\bar u_2, \bar v_2)=(1/R_0,1-1/R_0)$ if $R_0 >1$
Let $f(S(t),I(t))=(1-S(t))-R_0I(t)S(t)$ and $g(S(t),I(t))= (1-S(t))-R_0I(t)S(t)$
We are interested in the region $(u,v)\in \mathscr S $ with $\mathscr S=[0,1]\times[0,1]$
There is one vertical-tangent nullcline : $I=\frac{1}{R_0S}(1-S)$
and two horizontal-tangent nullclines: $S=\frac{1}{R_0}$ and $I=0$
then we analyse the signs of $f$ and $g$
$f(S,I)>0 $ if $0 <I <\frac{1}{R_0S}(1-S)$
$f(S,I)<0 $ if $\frac{1}{R_0S}(1-S)<I<1$
$g(S,I)<0 $ if $ 0<S<\frac{1}{R_0}$
$g(S,I)>0 $if $\frac{1}{R_0}<S<1$
The phase-space diagram that we obtain with this information is the following:
and then they conclude that $(\bar u_1, \bar v_1)$ is unstable and $(\bar u_2, \bar v_2)$ is asymptotically stable. I agree on the fist one since some arrows are leaving the point, but how can you deduce that for the second one? I don't think this plot is enough to decide.
Besides two example trajectories where provided:
But there is no explanation on how to get them , I might as well think that $(\bar u_2, \bar v_2)$ is a center point, why should it be asymptotically stable?
So how does one determine euristically that $(\bar u_2, \bar v_2)$ is in fact asymptotically stable and not for instance a center point? I already proved it analyzing the eigenvalues of the jacobian so I know this must be like that, but I am interested in the euristic procedure based only on the phase-space diagram. I don't know much about sketching other that the procedure provided in this example and I don't know if there is a way to get the expression for the trajectories in order to understand they are in fact converging to $(\bar u_2, \bar v_2)$.