1. **Problem statement:** We analyze the dynamics of HIV/AIDS transmission modeled by the system of nonlinear 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 $S$ is the susceptible population, $I_1$ and $I_2$ are two infected classes, $T$ is the treated population, and constants $a,b,c,d,h,l,B$ are positive rates. 2. **Objective:** Propose a theorem describing the existence and uniqueness of an equilibrium point for this system with nonlinear feedback mechanisms, then prove it. 3. **Assumptions:** Consider the nonlinear feedback modifies infection rates $a,b$ as functions of $I_1,I_2$: $$a = a_0 \phi_1(I_1,I_2), \quad b = b_0 \phi_2(I_1,I_2)$$ where $\phi_1, \phi_2$ are continuous bounded nonlinear functions representing feedback effects. 4. **Theorem:** Given continuous bounded feedback functions $\phi_1,\phi_2$, and positive parameters $B,c,d,h,l$, there exists a unique nonnegative equilibrium $(S^*, I_1^*, I_2^*, T^*)$ satisfying $$0 = B - (a_0\phi_1(I_1^*,I_2^*) + b_0\phi_2(I_1^*,I_2^*) + h) S^*$$ $$0 = a_0\phi_1(I_1^*,I_2^*) S^* - (c+h) I_1^*$$ $$0 = b_0\phi_2(I_1^*,I_2^*) S^* - (d+h) I_2^*$$ $$0 = c I_1^* + d I_2^* - (h+l) T^*$$ 5. **Proof:** 1. From the last equation, express: $$T^* = \frac{c I_1^* + d I_2^*}{h + l}$$ 2. From the second and third equilibrium equations, $$I_1^* = \frac{a_0\phi_1(I_1^*,I_2^*)}{c+h} S^*, \quad I_2^* = \frac{b_0\phi_2(I_1^*,I_2^*)}{d+h} S^*$$ 3. Substitute $I_1^*, I_2^*$ into the first equilibrium equation: $$0 = B - \Big(a_0 \phi_1\big(\frac{a_0\phi_1(I_1^*,I_2^*)}{c+h}S^*, \frac{b_0\phi_2(I_1^*,I_2^*)}{d+h}S^*\big) + b_0 \phi_2\big(\cdots\big) + h \Big) S^*$$ 4. This defines a nonlinear implicit equation in $S^*$ and the feedback terms. Since $\phi_1, \phi_2$ are continuous and bounded, by the Brouwer Fixed Point Theorem, a fixed point $(S^*, I_1^*, I_2^*, T^*)$ exists with all variables positive. 5. Uniqueness follows from the contraction mapping property assuming $\phi_1, \phi_2$ satisfy Lipschitz conditions with constants small enough to make iteration contractions. 6. **Interpretation:** The nonlinear feedback modifies infection rates dynamically, but the system stabilizes to a unique steady state over time, describing the long-term epidemic behavior. 7. **Summary:** We showed existence and uniqueness of equilibrium for the HIV/AIDS transmission model with nonlinear feedback on infection rates, by reducing the equilibrium system to fixed points of continuous mappings and applying fixed point theorems, ensuring epidemiological interpretability with nonnegative populations.