## Step 1: Define the symbolic representation of the variables
The variables are ['milligrams of zinc', 'milligrams of vitamin B12', 'milligrams of vitamin A']. Let's denote them as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $5.59x_1^2 + 5.84x_1x_2 + 1.97x_1x_3 + 8.63x_2^2 + 8.1x_2x_3 + 2.13x_3^2 + 3.17x_1 + 8.16x_2 + 7.15x_3$.

## 3: Define the constraints in symbolic notation
The constraints are:
1. $8x_1 = 8$ (immune support index for $x_1$)
2. $20x_2 = 20$ (immune support index for $x_2$)
3. $4x_3 = 4$ (immune support index for $x_3$)
4. $x_1^2 + x_3^2 \geq 11$
5. $x_1 + x_2 \geq 14$
6. $x_1 + x_2 + x_3 \geq 16$
7. $-5x_2 + 4x_3 \geq 0$
8. $x_1^2 + x_2^2 + x_3^2 \leq 58$

## 4: Create a symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x1', 'milligrams of zinc'), 
        ('x2', 'milligrams of vitamin B12'), 
        ('x3', 'milligrams of vitamin A')
    ], 
    'objective_function': '5.59*x1^2 + 5.84*x1*x2 + 1.97*x1*x3 + 8.63*x2^2 + 8.1*x2*x3 + 2.13*x3^2 + 3.17*x1 + 8.16*x2 + 7.15*x3', 
    'constraints': [
        '8*x1 = 8', 
        '20*x2 = 20', 
        '4*x3 = 4', 
        'x1^2 + x3^2 >= 11', 
        'x1 + x2 >= 14', 
        'x1 + x2 + x3 >= 16', 
        '-5*x2 + 4*x3 >= 0', 
        'x1^2 + x2^2 + x3^2 <= 58'
    ]
}
```

## Step 5: Solve the optimization problem using Gurobi
Given that $x_1 = 1$, $x_2 = 1$, and $x_3 = 1$ from the first three constraints, let's implement the problem in Gurobi.

```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

    # Define variables
    x1 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x1")
    x2 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x2")
    x3 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x3")

    # Objective function
    model.setObjective(5.59*x1**2 + 5.84*x1*x2 + 1.97*x1*x3 + 8.63*x2**2 + 8.1*x2*x3 + 2.13*x3**2 + 3.17*x1 + 8.16*x2 + 7.15*x3)

    # Constraints
    model.addConstr(8*x1 == 8, name="c1")
    model.addConstr(20*x2 == 20, name="c2")
    model.addConstr(4*x3 == 4, name="c3")
    model.addConstr(x1**2 + x3**2 >= 11, name="c4")
    model.addConstr(x1 + x2 >= 14, name="c5")
    model.addConstr(x1 + x2 + x3 >= 16, name="c6")
    model.addConstr(-5*x2 + 4*x3 >= 0, name="c7")
    model.addConstr(x1**2 + x2**2 + x3**2 <= 58, name="c8")

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.Status.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}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```