Is There Another Way To Code The Idea of a MAX Constraint Without The Use of Binary Variables?

I have a constraint of the following form that describes the growth of trees, where the population of trees in period $$t$$ is the previous period's population minus some trees infected with a virus:

$$S_{t}^{ij} = S_{t-1}^{ij} - I_{t-1}^{ij}$$ $$S_{t}\ge0,I_{t} \ge 0$$

From this constraint, you can see that it is possible for $$I_{t-1}^{ij}$$ to be larger than $$S_{t-1}^{ij}$$ , but I need to be able to incorporate the fact that in any period $$S_{t}^{ij}$$ can never be less than 0 because you cannot have negative trees.

I am wondering if there is any way to incorporate this besides including a max constraint of the form :$$S_{t}^{ij} = \max\{S_{t-1}^{ij} - I_{t-1}^{ij}, 0 \}$$

When I include the max constraint and then linearize it, my problem becomes very challenging to solve, so I am wondering if there's maybe some other way I can try to reformulate the constraint so to avoid the binary variables.

Why not just impose $$S_t^{ij} \ge 0$$?
As @RobPratt states why not impose $$S^{ij}_t\geq0$$? Especially it seems rather reasonable to inforce this through $$S^{ij}_t\geq I^{ij}_t$$ given that there probably cannot be more infected trees than there are trees.