The derivation of the endemic equilibrium point for a mathematical model typically involves finding steady-state solutions where the rates of change for all compartments in the system are zero. Based on the document you provided, here's an explanation:

Endemic Point Derivation

1. Model Setup:

   • The model equations are a set of differential equations governing the populations of susceptible and infected red chili plants (vegetative and generative phases) and susceptible and infected insect vectors. These equations incorporate logistic growth, infection rates, and other dynamics.

2. Conditions for the Endemic Point:

   • An endemic equilibrium occurs when the disease persists in the population. Mathematically, this means that at least one infected compartment has a non-zero steady-state value.
   • At equilibrium, $\frac{dS_v}{dt} = \frac{dI_v}{dt} = \frac{dS_g}{dt} = \frac{dI_g}{dt} = \frac{dS_{BT}}{dt} = \frac{dI_{BT}}{dt} = 0$.

3. Basic Reproduction Number ($R_0$):

   • The basic reproduction number ($R_0$) determines the existence of the endemic equilibrium. If $R_0 > 1$, the disease can persist, leading to an endemic state.
   • $R_0$ is derived using the next-generation matrix approach and is expressed in terms of model parameters, such as infection rates, plant and insect mortality, and logistic growth factors.

4. Endemic Point Solution:

   • Solve the system of equations simultaneously, substituting steady-state conditions. This yields expressions for the steady-state populations: $S_v^* = f(\text{parameters}), \quad I_v^* = f(\text{parameters, } S_v^*), \quad \text{and similarly for } S_g^*, I_g^*, S_{BT}^*, I_{BT}^*.$
   • Each expression depends on parameters like infection rates ($\beta_1, \beta_2$), plant growth rate ($\alpha$), mortality rates ($\mu_p, \mu_I$), and the effectiveness of the control agent ($\delta_p$).

5. Stability Analysis:

   • The endemic point's stability is verified by analyzing the Jacobian matrix of the system evaluated at the endemic equilibrium. Eigenvalues with negative real parts indicate local asymptotic stability.

6. Result:

   • The endemic equilibrium exists when $R_0 > 1$. The exact values depend on solving the steady-state equations, often requiring numerical computation due to the complexity of the system.

This process provides the conditions and the parameter-dependent solutions for the populations in the system at equilibrium when the disease is endemic.

Let me know if you would like more detailed derivations or explanations for specific parts. Here are some related questions to explore further:

1. What are the steps for computing $R_0$ using the next-generation matrix method?
2. How does the effectiveness of control measures ($\delta_p$) influence the endemic equilibrium?
3. What numerical methods are used to solve for equilibrium points in complex models?
4. Can you show a simulation of the population dynamics based on the equations?
5. How do parameter changes shift the stability of the endemic point?

Tip: Always validate model assumptions and parameters