I understand how these models can work for most traditional diseases, where infected individuals are no longer contagious after some period, and are generally immune, themselves. However, for a condition such as AIDS, where infected individuals are permanently infected (and without intervention, permanently contagious), what happens to the "gI" term in the equation for dI/dt (R being nonexistent).

If one were modelling HIV/AIDS, would they want to do something different with g?

Furthermore, in the equation for dS/dt, if beta is multiplied by S and I, shouldn't B be multiplied by the total population, and dS be multiplied by S?

Any clarification would be much appreciated. Thanks.