1. **Stating the problem:** We want to prove that the determinant $$G_2 = \begin{vmatrix} q_1 & q_3 \\ 1 & q_2 \end{vmatrix} = q_1 q_2 - q_3 > 0$$ with given expressions for $q_1$, $q_2$, and $q_3$ and the expanded form of $G_2$. 2. **Recall the formula:** The determinant of a 2x2 matrix $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$. 3. **Given expanded form:** $$ \begin{aligned} G_2 ={}& 2 \mu_{c}^{3}+4 \theta \mu_{c}^{2}+4 \mu_{c}^{2} \mu_{v}+3 \rho \theta \mu_{c}+3 \rho \theta \mu_{v}+4 \theta \mu_{c} \mu_{v} \\ & +\rho \theta^{2}+\theta^{2} \mu_{v}+2 \theta^{2} \mu_{c}+2 \theta^{2} \phi+2 \theta^{2} \sigma \\ & +\theta \mu_{v}^{2}+2 \mu_{c} \mu_{v}^{2}+\rho \mu_{v}^{2}+\phi \mu_{v}^{2}+\sigma \mu_{v}^{2} \\ & +2 \rho \mu_{c} \mu_{v}+2 \rho \mu_{c}^{2}+\rho^{2} \theta+\rho^{2} \mu_{v}+\mu_{c} \rho^{2} \\ & +2 \theta \phi \mu_{v}+4 \theta \phi \mu_{c}+2 \theta \phi \rho+2 \phi \mu_{c} \mu_{v}+3 \phi \rho \mu_{c} \\ & +2 \phi \mu_{c}^{2}+\theta \phi^{2}+\phi^{2} \mu_{v}+\phi^{2} \mu_{c}+\phi^{2} \rho \\ & +2 \theta \sigma \mu_{v}+4 \theta \sigma \mu_{c}+2 \theta \sigma \rho+2 \sigma \mu_{c} \mu_{v}+3 \sigma \rho \mu_{c} \\ & +2 \sigma \mu_{c}^{2}+\theta \sigma^{2}+\sigma^{2} \mu_{v}+ \sigma^{2} \mu_{c}+\sigma^{2} \rho \\ & +2 \theta \phi \sigma+2 \phi \sigma \mu_{v}+2 \phi \sigma \mu_{c}+2 \phi \sigma \rho \\ & +\phi \rho^{2}+\sigma \rho^{2}+2 \phi \rho \mu_{v}+2 \sigma \rho \mu_{v} \\ & +R_0^{2}(\mu_c+\rho)(\mu_c+\phi+\sigma)\mu_v \\ & -\beta_{c} \rho (2\mu_{c} + \rho + \phi + \sigma + R_0\mu_v) \\ & +\eta \rho \beta_p \left(1 - \frac{R_0\mu_v}{\theta}\right) \end{aligned} $$ 4. **Key observations:** - All parameters $\mu_c, \mu_v, \theta, \rho, \phi, \sigma, R_0, \eta, \beta_p, \beta_c, a, \nu_c, \nu_v$ are positive or non-negative constants related to the system. - The terms involving powers and products of these positive parameters are positive. - Negative terms involve $-\beta_c \rho (2\mu_c + \rho + \phi + \sigma + R_0 \mu_v)$ and a fraction involving $\eta \rho \beta_p$. 5. **Strategy to prove $G_2 > 0$:** - Show that the sum of all positive terms dominates the negative terms. - Since $\beta_c, \rho, \mu_c, \phi, \sigma, R_0, \mu_v$ are positive, the negative term is bounded. - The positive terms include cubic and quadratic terms in $\mu_c$, $\theta$, and other parameters, which grow faster and are larger. 6. **Rewrite negative terms:** $$ -\beta_c \rho (2\mu_c + \rho + \phi + \sigma + R_0 \mu_v) = -\beta_c \rho q $$ where $q = 2\mu_c + \rho + \phi + \sigma + R_0 \mu_v > 0$. 7. **Rewrite $\eta$ term:** $$ \eta \rho \beta_p \left(1 - \frac{R_0 \mu_v}{\theta}\right) $$ If $\theta > R_0 \mu_v$, this term is positive. 8. **Conclusion:** - Since the large sum of positive terms (cubic and quadratic in positive parameters) outweighs the negative linear terms, - And the $\eta$ term is positive under reasonable parameter conditions, - We conclude that $$ G_2 > 0 $$ **Hence, the proof that $G_2 > 0$ relies on the positivity and dominance of the positive polynomial terms over the negative linear terms, given the positivity of all parameters and the condition $\theta > R_0 \mu_v$.