Hi

Between t and t+dt

A goes from A(t) to A(t+dt) = A(t) - a A(t) dt

B goes from B(t) to B(t+dt) = B(t) - b B(t) dt + a A(t) dt

C goes from C(t) to C(t+dt) = C(t) + b B(t) dt

It is clear that for every t, A(t) + B(t) + C(t) is constant

A(t+dt) = A(t) - a A(t) dt

B(t+dt) = B(t) - b B(t) dt + a A(t) dt

General solution of equation is

Particular solution of is

if a and b are different

To be checked !!

But I think that the general idea is not so far !