To solve the given optimization problem using Gurobi, we need to translate the natural language description into a mathematical formulation that can be implemented in Python. The objective is to minimize the function $2x_0^2 + 5x_0$, where $x_0$ represents the milligrams of vitamin B4, subject to several constraints involving both $x_0$ (milligrams of vitamin B4) and $x_1$ (milligrams of vitamin D).

The constraints are as follows:
1. The immune support index for milligrams of vitamin B4 is 1.
2. The immune support index for milligrams of vitamin D is 8.
3. The total combined immune support index from the squares of milligrams of vitamin B4 and vitamin D must be at least 16: $x_0^2 + 64x_1^2 \geq 16$.
4. The total combined immune support index from milligrams of vitamin B4 and vitamin D must be greater than or equal to 16: $x_0 + 8x_1 \geq 16$.
5. Ten times the number of milligrams of vitamin B4 minus ten times the number of milligrams of vitamin D should be no less than zero: $10x_0 - 10x_1 \geq 0$.
6. The total combined immune support index from milligrams of vitamin B4 and vitamin D should not exceed 19: $x_0 + 8x_1 \leq 19$.

Given these conditions, the problem can be formulated as a quadratic programming problem. However, to ensure clarity and adherence to the specified format, let's directly proceed to formulating this in Gurobi Python code:

```python
from gurobipy import *

# Create a new model
m = Model("Vitamin_Optimization")

# Define variables
x0 = m.addVar(lb=-GRB.INFINITY, ub=GRB.INFINITY, name="milligrams_of_vitamin_B4")
x1 = m.addVar(lb=-GRB.INFINITY, ub=GRB.INFINITY, name="milligrams_of_vitamin_D")

# Objective function: Minimize 2*x0^2 + 5*x0
m.setObjective(2*x0**2 + 5*x0, GRB.MINIMIZE)

# Constraints
m.addConstr(x0**2 + 64*x1**2 >= 16, name="immune_support_index_squares")
m.addConstr(x0 + 8*x1 >= 16, name="total_immune_support_index_min")
m.addConstr(10*x0 - 10*x1 >= 0, name="vitamin_b4_vs_vitamin_d")
m.addConstr(x0 + 8*x1 <= 19, name="total_immune_support_index_max")

# Solve the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Milligrams of Vitamin B4: {x0.x}")
    print(f"Milligrams of Vitamin D: {x1.x}")
else:
    print("No optimal solution found")
```