To solve for the partial derivatives of $f_2$ with respect to $V, S, E, A, I, H, R,$ and $G$, let's examine $f_2$ and differentiate it with respect to each variable.

The equation for $f_2$ is: $f_2 = \Lambda - \frac{\beta (I + h_1 A + h_2 G) S}{V + S + E + A + I + H + R + G} - (\mu + \eta) S + \omega V$

Let's compute the partial derivatives:

1. Partial Derivative of $f_2$ with respect to $V$:

Differentiating $f_2$ with respect to $V$: $\frac{\partial f_2}{\partial V} = \frac{\partial}{\partial V} \left( \Lambda - \frac{\beta (I + h_1 A + h_2 G) S}{V + S + E + A + I + H + R + G} - (\mu + \eta) S + \omega V \right)$ Only the second and last terms depend on $V$. Applying the quotient rule: $\frac{\partial f_2}{\partial V} = - \frac{\beta (I + h_1 A + h_2 G) S}{(V + S + E + A + I + H + R + G)^2} + \omega$

2. Partial Derivative of $f_2$ with respect to $S$:

Differentiating $f_2$ with respect to $S$: $\frac{\partial f_2}{\partial S} = \frac{\partial}{\partial S} \left( \Lambda - \frac{\beta (I + h_1 A + h_2 G) S}{V + S + E + A + I + H + R + G} - (\mu + \eta) S + \omega V \right)$ Using the product and quotient rules, we get: $\frac{\partial f_2}{\partial S} = - \frac{\beta (I + h_1 A + h_2 G)(V + S + E + A + I + H + R + G) - \beta (I + h_1 A + h_2 G) S}{(V + S + E + A + I + H + R + G)^2} - (\mu + \eta)$

3. Partial Derivative of $f_2$ with respect to $E$:

Only the second term depends on $E$, so: $\frac{\partial f_2}{\partial E} = - \frac{\beta (I + h_1 A + h_2 G) S}{(V + S + E + A + I + H + R + G)^2}$

4. Partial Derivative of $f_2$ with respect to $A$:

$\frac{\partial f_2}{\partial A} = \frac{\partial}{\partial A} \left( - \frac{\beta (I + h_1 A + h_2 G) S}{V + S + E + A + I + H + R + G} \right)$ Applying the product and quotient rules: $\frac{\partial f_2}{\partial A} = - \frac{\beta h_1 S (V + S + E + A + I + H + R + G) - \beta (I + h_1 A + h_2 G) S}{(V + S + E + A + I + H + R + G)^2}$

5. Partial Derivative of $f_2$ with respect to $I$:

$\frac{\partial f_2}{\partial I} = - \frac{\beta S (V + S + E + A + I + H + R + G) - \beta (I + h_1 A + h_2 G) S}{(V + S + E + A + I + H + R + G)^2}$

6. Partial Derivative of $f_2$ with respect to $H$:

Since $H$ only appears in the denominator: $\frac{\partial f_2}{\partial H} = - \frac{\beta (I + h_1 A + h_2 G) S}{(V + S + E + A + I + H + R + G)^2}$

7. Partial Derivative of $f_2$ with respect to $R$:

$\frac{\partial f_2}{\partial R} = - \frac{\beta (I + h_1 A + h_2 G) S}{(V + S + E + A + I + H + R + G)^2}$

8. Partial Derivative of $f_2$ with respect to $G$:

$\frac{\partial f_2}{\partial G} = - \frac{\beta h_2 S (V + S + E + A + I + H + R + G) - \beta (I + h_1 A + h_2 G) S}{(V + S + E + A + I + H + R + G)^2}$

If you'd like more detailed explanations or have any specific questions, feel free to ask!

Here are some related questions to consider:

1. What does each term in $f_2$ represent in a real-world context?
2. How do the parameters $\beta$, $h_1$, and $h_2$ affect the behavior of $f_2$?
3. What role does $S$ (susceptible population) play in the overall dynamics of this system?
4. How would an increase in $\mu$ (possibly death rate) impact $f_2$?
5. What is the significance of each variable in this set of differential equations?

Tip: When differentiating functions with multiple variables, remember to carefully apply the quotient and product rules, especially when the function has complex denominators or nested terms.