To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables and formulating the objective function and constraints in terms of these variables.

Let's define:
- \(x_1\) as the number of blue gummies purchased,
- \(x_2\) as the number of red gummies purchased.

The objective is to minimize the total cost, which can be represented by the objective function:
\[ \text{Minimize} \quad 2x_1 + 3x_2 \]

The constraints based on the nutritional requirements are:
1. Calcium: \(3x_1 + 4x_2 \geq 40\)
2. Vitamin A: \(2x_1 + 3x_2 \geq 45\)
3. Iron: \(5x_1 + 2x_2 \geq 20\)

Additionally, since the patient cannot purchase a negative number of gummies, we have:
- \(x_1 \geq 0\)
- \(x_2 \geq 0\)

The symbolic representation in JSON format is:

```json
{
    'sym_variables': [('x1', 'number of blue gummies'), ('x2', 'number of red gummies')],
    'objective_function': '2*x1 + 3*x2',
    'constraints': ['3*x1 + 4*x2 >= 40', '2*x1 + 3*x2 >= 45', '5*x1 + 2*x2 >= 20', 'x1 >= 0', 'x2 >= 0']
}
```

Now, let's implement this optimization problem using Gurobi in Python:

```python
from gurobipy import *

# Create a model
m = Model("Gummy_Vitamins")

# Define variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="blue_gummies", lb=0)
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="red_gummies", lb=0)

# Set the objective function
m.setObjective(2*x1 + 3*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(3*x1 + 4*x2 >= 40, "calcium")
m.addConstr(2*x1 + 3*x2 >= 45, "vitamin_A")
m.addConstr(5*x1 + 2*x2 >= 20, "iron")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Blue Gummies: {x1.x}")
    print(f"Red Gummies: {x2.x}")
    print(f"Total Cost: ${m.objVal:.2f}")
else:
    print("No optimal solution found")
```