If I have the system:
$S’=-bSI$
$I’=bSI-aI$
Where all parameters $a,b,c>0$ and the initial conditions are $S(0)=s_0 >0 , I(0)=i_0>0$ with $s_0+i_0=c$.
In case that I know that the solutions exist for all positive times, then I know that $S$ and $I$ are bounded and the monotonic behaviour of $S$. Again, if I know the existence.
But how can I determine that the solutions do really exist on $[0,\infty)$ in the first place?
I know that $S(t)=0$ and $I(t)=0$ are solutions of that system for $s_0=i_0=0$ and if $s_0,i_0>0$, then by drawing a phase portrait in a typical triangle where $S \ge 0$,$I \ge 0$ and $S+I=c$, then $S$ and $I$ are bounded and therefore should exist.
But is there another more practical way to know this without trying to do phase portrait and also without trying to express the solutions themselves? Because in general we can have much harder systems where we can’t even dream in trying to do so for example. (Expressing the solutions I mean).
Maybe there are some theorems that I can use in systems to determine such existence? I thought maybe Picard-Lindelöf theorem could help but it sorts of require me knowing that $S$ and $I$ are bounded so it kind of doesn’t really help me.