1. **State the problem:** We want to find the basic reproduction number $R_0$ for the system without using the Next Generation Matrix (NGM) method, given the differential equations: $$\frac{dS}{dt} = B - (a+b+h)S$$ $$\frac{dI_1}{dt} = aS - (c+h)I_1$$ $$\frac{dI_2}{dt} = bS - (d+h)I_2$$ $$\frac{dT}{dt} = cI_1 + dI_2 - (h+l)T$$ 2. **Interpret the model components:** - $S$ is susceptible individuals. - $I_1$ and $I_2$ are two infected classes. - $T$ is the treated/recovered class. - Parameters $a,b,c,d,h,l,B$ are rates. 3. **Find the Disease-Free Equilibrium (DFE):** Set all infected classes and treated class to zero ($I_1=I_2=T=0$) and find $S^*$: $$0 = B - (a+b+h) S^* \implies S^* = \frac{B}{a+b+h}$$ 4. **Calculate individual reproduction terms:** - New infections caused by $I_1$: $$R_1 = \frac{a S^*}{c + h} = \frac{a B}{(a+b+h)(c+h)}$$ - New infections caused by $I_2$: $$R_2 = \frac{b S^*}{d + h} = \frac{b B}{(a+b+h)(d+h)}$$ 5. **Combine to find the basic reproduction number:** Since $I_1$ and $I_2$ contribute independently to new infections, $$R_0 = R_1 + R_2 = \frac{a B}{(a+b+h)(c+h)} + \frac{b B}{(a+b+h)(d+h)}$$ 6. **Interpretation:** $R_0$ represents the average number of secondary infections caused by one infected individual in a fully susceptible population, considering both infection pathways $I_1$ and $I_2$. **Final answer:** $$R_0 = \frac{a B}{(a+b+h)(c+h)} + \frac{b B}{(a+b+h)(d+h)}$$