I have a question about the standard epidemic SIR model: $$\frac{dS}{dt}=-\frac{\beta IS}{N},$$ $$\frac{dI}{dt}=\frac{\beta IS}{N}-\gamma I,$$ $$\frac{dR}{dt}=\gamma I.$$ If all initial values ($I_0,R_0$) and parameters ($N,\beta,\gamma$) are the same, expect that we have two different initial $S$, i.e. $S_0>\hat{S}_0$, can we prove that $I(t)\geq\hat{I}(t)$ for any $t>0$?
The simulations show it is true.
Thank you!