1. **State the problem:** We analyze the stability of the disease-free equilibrium (DFE) of the system given by: $$\frac{dS}{dt}=B-(a+b+h)S$$ $$\frac{dI_1}{dt}=aS-(c+h)I_1$$ $$\frac{dI_2}{dt}=bS-(d+h)I_2$$ $$\frac{dT}{dt}=cI_1+dI_2-(h+l)T$$ where $S, I_1, I_2, T$ are population compartments and $B,a,b,c,d,h,l$ are parameters. 2. **Find the Disease-Free Equilibrium (DFE):** At DFE, $I_1=0, I_2=0, T=0$, and $S=S^*$ is constant. From first equation: $$0 = B - (a+b+h)S^* \implies S^* = \frac{B}{a+b+h}$$ 3. **Linearize around the DFE:** Jacobian matrix $J$ at $(S^*,0,0,0)$ is: $$J = \begin{bmatrix} -(a+b+h) & 0 & 0 & 0 \\ a & -(c+h) & 0 & 0 \\ b & 0 & -(d+h) & 0 \\ 0 & c & d & -(h+l) \end{bmatrix}$$ 4. **Analyze eigenvalues:** The DFE is stable if all eigenvalues of $J$ have negative real parts. - The first eigenvalue is $\lambda_1 = -(a+b+h)<0$. - The submatrix for $(I_1,I_2,T)$ is: $$M = \begin{bmatrix} -(c+h) & 0 & 0 \\ 0 & -(d+h) & 0 \\ c & d & -(h+l) \end{bmatrix}$$ 5. **Eigenvalues of the submatrix:** - The matrix $M$ is lower block-triangular. - The eigenvalues are $\lambda_2=-(c+h)<0$, $\lambda_3=-(d+h)<0$. - For the $T$ row, substitute eigenvalues and check stability: The eigenvalue associated with $T$ satisfies: $$\lambda_4 = -(h+l) < 0$$ 6. **Conclusion:** All eigenvalues have negative real parts. **Therefore, the disease-free equilibrium is locally asymptotically stable.**