To analyze the positivity of the system described by equation set (5), we need to ensure that each function or variable remains non-negative over time (specifically $S_g$, $I_g$, $X_g$, and $Y_g$), since these typically represent quantities like population fractions or densities in mathematical models.

Let’s break down the conditions under which the right-hand sides of each equation remain non-negative:

1. Positivity Conditions for $\frac{dS_g}{dt}$

The equation for $\frac{dS_g}{dt}$ is: $\frac{dS_g}{dt} = \mu_g N_g - \rho \left( \beta_1 X_g(t) + \beta_2 Y_g(t) \right) \frac{S_g}{N_g} - \mu_g S_g$ For $S_g$ to remain positive:

• The term $\mu_g N_g$ (replenishment) should balance or exceed the losses due to the infection transmission terms and mortality (the $- \mu_g S_g$ term).

2. Positivity Conditions for $\frac{dI_g}{dt}$

The equation for $\frac{dI_g}{dt}$ is: $\frac{dI_g}{dt} = \rho \left( \beta_1 X_g(t) + \beta_2 Y_g(t) \right) \frac{S_g}{N_g} - \mu_g I_g$ For $I_g$ to remain positive:

• The infection term $\rho \left( \beta_1 X_g(t) + \beta_2 Y_g(t) \right) \frac{S_g}{N_g}$ should be non-negative. This typically holds if $S_g \geq 0$ and $\rho, \beta_1, \beta_2 \geq 0$.
• The natural decay term $-\mu_g I_g$ should not lead $I_g$ to go negative if $I_g$ starts positive.

3. Positivity Conditions for $\frac{dX_g}{dt}$

The equation for $\frac{dX_g}{dt}$ is: $\frac{dX_g}{dt} = \gamma I_g - \frac{\gamma}{2} X_g - \beta_1 X_g$ For $X_g$ to remain positive:

• The recovery term $\gamma I_g$ should be positive if $I_g \geq 0$.
• The decay terms $-\frac{\gamma}{2} X_g - \beta_1 X_g$ should not push $X_g$ to be negative if $X_g$ is initially positive.

4. Positivity Conditions for $\frac{dY_g}{dt}$

The equation for $\frac{dY_g}{dt}$ is: $\frac{dY_g}{dt} = \alpha I_g - \beta_2 Y_g$ For $Y_g$ to remain positive:

• The source term $\alpha I_g$ should be positive if $I_g \geq 0$.
• The decay term $-\beta_2 Y_g$ should not push $Y_g$ to zero if it is initially positive.

Summary of Positivity Conditions

For the system to ensure positivity, the following conditions are needed:

1. $S_g(0), I_g(0), X_g(0), Y_g(0) \geq 0$.
2. Parameters $\mu_g$, $N_g$, $\rho$, $\beta_1$, $\beta_2$, $\gamma$, and $\alpha$ should be chosen so that they maintain non-negative values on the right-hand side of each equation.

Would you like more detailed conditions for specific terms or further analysis?

Here are five questions that might expand on this topic:

1. How can we use initial values to ensure long-term positivity in non-linear systems?
2. What is the significance of each parameter (e.g., $\rho$, $\beta_1$, $\beta_2$) in biological models?
3. How can eigenvalue analysis confirm the stability of positive solutions?
4. What role does mortality ($\mu_g$) play in sustaining positivity?
5. How does coupling between equations impact positivity conditions?

Tip: Always check the parameter ranges for biological interpretability when modeling population dynamics.