1. **State the problem:** We analyze the dynamics of HIV/AIDS transmission with a linear feedback mechanism represented by the system of differential equations: $$\frac{dS}{dt} = B - (a+b+h)S$$ $$\frac{dI_1}{dt} = aS - (c+h)I_1$$ $$\frac{dI_2}{dt} = bS - (d+h)I_2$$ $$\frac{dT}{dt} = cI_1 + dI_2 - (h+l)T$$ where $S$, $I_1$, $I_2$, $T$ represent different population states and parameters $a,b,c,d,h,l,B$ are positive constants associated with transmission, treatment, and death rates. 2. **Theorem (Existence of equilibrium):** There exists a unique biologically feasible equilibrium point $(S^*, I_1^*, I_2^*, T^*)$ such that: $$S^* = \frac{B}{a+b+h}$$ $$I_1^* = \frac{aS^*}{c+h} = \frac{aB}{(c+h)(a+b+h)}$$ $$I_2^* = \frac{bS^*}{d+h} = \frac{bB}{(d+h)(a+b+h)}$$ $$T^* = \frac{cI_1^* + dI_2^*}{h+l} = \frac{c\frac{aB}{(c+h)(a+b+h)} + d\frac{bB}{(d+h)(a+b+h)}}{h+l}$$ 3. **Proof:** 1. At equilibrium, derivatives are zero: $$0 = B - (a+b+h)S^* \implies S^* = \frac{B}{a+b+h}$$ 2. Substitute $S^*$ into the equation for $I_1$: $$0 = aS^* - (c+h)I_1^* \implies I_1^* = \frac{aS^*}{c+h} = \frac{aB}{(c+h)(a+b+h)}$$ 3. Substitute $S^*$ into the equation for $I_2$: $$0 = bS^* - (d+h)I_2^* \implies I_2^* = \frac{bS^*}{d+h} = \frac{bB}{(d+h)(a+b+h)}$$ 4. Substitute $I_1^*$ and $I_2^*$ into the equation for $T$: $$0 = cI_1^* + dI_2^* - (h+l)T^* \implies$$ $$T^* = \frac{cI_1^* + dI_2^*}{h+l} = \frac{c\frac{aB}{(c+h)(a+b+h)} + d\frac{bB}{(d+h)(a+b+h)}}{h+l}$$ 4. **Interpretation:** The steady-state values represent balances of population inflows and outflows for each compartment. All are positive if parameters and $B$ are positive, ensuring biological feasibility. This theorem establishes the existence and explicit formulas for the equilibrium states in the HIV/AIDS dynamics model with linear feedback.