I wonder if there are no solutions because of the following:
eq = Expand[{f1[S, e, I1, I2, R] == 0,
f2[S, e, I1, I2, R] == 0,
f3[S, e, I1, I2, R] == 0,
f4[S, e, I1, I2, R] == 0,
f5[S, e, I1, I2, R] == 0}];
Solve[eq[[{1, 2}]], {I1, I2}]
(* {} *)
If there are no solutions for I1
and I2
using just the first two equations, then adding additional equations can't help. Of course, the fact that in the first two equations are both functions of I1+I2
might negate that conclusion.