To find the disease-free equilibrium (DFE) of the system represented in equation (5), we need to set all infection-related terms to zero, which corresponds to a scenario where there is no infected population (i.e., all $I_g$ and $Y_g$ terms are zero).

Steps for Disease-Free Equilibrium (DFE)

1. Set Infection Variables to Zero: In a disease-free state, $I_g = 0$ and $Y_g = 0$. Thus, we focus on the terms that have $I_g$ and $Y_g$ in the equations.

2. Equilibrium Condition: Set the derivatives of each compartment equal to zero to represent the steady state. We then solve for $S_g$ and $X_g$ under the conditions $I_g = 0$ and $Y_g = 0$.

Apply to the Given System

Given the system: $\frac{dS_g}{dt} = \mu_g N_g - \rho \left( \beta_1 X_g(t) + \beta_2 Y_g(t) \right) \frac{S_g}{N_g} - \mu_g S_g,$ $\frac{dI_g}{dt} = \rho \left( \beta_1 X_g(t) + \beta_2 Y_g(t) \right) \frac{S_g}{N_g} - \mu_g I_g,$ $\frac{dX_g}{dt} = \gamma I_g - \frac{\gamma}{2} X_g - \beta_1 X_g,$ $\frac{dY_g}{dt} = \alpha I_g - \beta_2 Y_g.$

Disease-Free Equilibrium Conditions:

1. Set $I_g = 0$ and $Y_g = 0$.
2. Set each of the differential equations to zero.

By substituting $I_g = 0$ and $Y_g = 0$ into the equations and solving for $S_g$ and $X_g$, you will obtain the values of $S_g$ and $X_g$ that satisfy the disease-free equilibrium.

Would you like a detailed, step-by-step solution for each equation in the system?

Related Questions:

1. How does the basic reproduction number $R_0$ relate to the stability of the disease-free equilibrium?
2. What is the biological interpretation of the parameters $\mu_g$, $\beta_1$, $\beta_2$, $\rho$, and $\gamma$?
3. How would this system change if we introduced vaccination into the model?
4. Can this system exhibit oscillatory behavior near the endemic equilibrium?
5. What methods are commonly used to analyze the stability of disease-free and endemic equilibria?

Tip: Setting up a Jacobian matrix at the DFE and examining its eigenvalues is a standard approach to study stability.