## Step 1: Define the symbolic representation of the problem
The problem involves variables: ['milligrams of vitamin B6', 'milligrams of vitamin B2', 'milligrams of calcium']. Let's denote them as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Translate the objective function into symbolic notation
The objective function to minimize is $3.09x_1^2 + 8.41x_1x_3 + 2.01x_3^2 + 8.39x_3$.

## 3: Define the constraints in symbolic notation
The constraints are:
- $16x_1 = 16$ (immune support index of $x_1$)
- $20x_2 = 20$ (immune support index of $x_2$)
- $22x_3 = 22$ (immune support index of $x_3$)
- $20x_2 + 22x_3 \geq 52$
- $16x_1 + 20x_2 + 22x_3 \geq 52$
- $16x_1 + 22x_3 \leq 101$
- $x_1$ can be non-whole, $x_2$ must be whole, $x_3$ can be non-whole.

## 4: Correct and simplify constraints based on given attributes
Given $x_0 = 16$, $x_1 = 20$, $x_2 = 22$, and upper bounds, we realize the constraints can be directly written as:
- $x_1 = 1$ (since $16x_1 = 16$ implies $x_1 = 1$)
- $x_2 = 1$ (since $20x_2 = 20$ implies $x_2 = 1$)
- $x_3 = 1$ (since $22x_3 = 22$ implies $x_3 = 1$)
- $20x_2 + 22x_3 \geq 52$ becomes $20*1 + 22*1 \geq 52$ or $42 \geq 52$ which is incorrect, reevaluating: it seems there was a misunderstanding in interpreting $x_0, x_1, x_2$ as they relate to vitamins B6, B2, and calcium respectively.
- Correct interpretation:
  - The immune support index for vitamin B6 ($x_1$) is 16.
  - For vitamin B2 ($x_2$) is 20.
  - For calcium ($x_3$) is 22.

## 5: Reinterpret constraints and objective function correctly
Let's correctly denote:
- $x_1$ as milligrams of vitamin B6
- $x_2$ as milligrams of vitamin B2
- $x_3$ as milligrams of calcium

The objective function remains $3.09x_1^2 + 8.41x_1x_3 + 2.01x_3^2 + 8.39x_3$.

## 6: Correctly formulate constraints
- $16x_1 \leq 201$ and similar for others but directly using given values:
  - $16x_1 = 16$ implies $x_1 = 1$,
  - $20x_2 = 20$ implies $x_2 = 1$,
  - $22x_3 = 22$ implies $x_3 = 1$.
- However, it seems there was confusion. Direct constraints from the problem:
  - $20x_2 + 22x_3 \geq 52$
  - $16x_1 + 20x_2 + 22x_3 \geq 52$
  - $16x_1 + 22x_3 \leq 101$

## 7: Symbolic representation
```json
{
'sym_variables': [('x1', 'milligrams of vitamin B6'), ('x2', 'milligrams of vitamin B2'), ('x3', 'milligrams of calcium')],
'objective_function': '3.09*x1^2 + 8.41*x1*x3 + 2.01*x3^2 + 8.39*x3',
'constraints': ['20*x2 + 22*x3 >= 52', '16*x1 + 20*x2 + 22*x3 >= 52', '16*x1 + 22*x3 <= 101', 'x2 % 1 == 0']
}
```

## 8: Gurobi Code
```python
import gurobi

def optimize_problem():
    model = gurobi.Model()
    
    # Define variables
    x1 = model.addVar(name="x1", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, vtype=gurobi.GRB.CONTINUOUS)
    x2 = model.addVar(name="x2", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, vtype=gurobi.GRB.INTEGER)
    x3 = model.addVar(name="x3", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, vtype=gurobi.GRB.CONTINUOUS)
    
    # Objective function
    model.setObjective(3.09*x1**2 + 8.41*x1*x3 + 2.01*x3**2 + 8.39*x3, gurobi.GRB.MINIMIZE)
    
    # Constraints
    model.addConstr(20*x2 + 22*x3 >= 52, name="constraint_1")
    model.addConstr(16*x1 + 20*x2 + 22*x3 >= 52, name="constraint_2")
    model.addConstr(16*x1 + 22*x3 <= 101, name="constraint_3")
    
    # Solve the problem
    model.optimize()
    
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"x1: {x1.varValue}")
        print(f"x2: {x2.varValue}")
        print(f"x3: {x3.varValue}")
        print(f"Objective: {model.objVal}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("The problem is infeasible.")
    else:
        print("The problem has no optimal solution.")

optimize_problem()
```