I have got to prove that a function $$f(S, I)=-\alpha S(t) I(t)$$ is Lipschitz' continuous and then find the Lipschitz' constant. This is all that I have got already: $$\|f(t,S_1,I_1)-f(t,S_2,I_2)\| \leqslant L(|S_1-S_2|+|I_1-I_2|)\\ |-\alpha S_1I_1+\alpha S_2I_2|\leqslant \ldots\leqslant L(|S_1-S_2|+|I_1-I_2|)$$ My biggest problem is how to modify the left side of the equation to get something that would help me prove the Lipschitz continuity and then find the satisfying constant.
Also, I was looking for books/articles where this topic was already described, but couldn't find anything helpful.