## Symbolic Representation

Let's denote the number of Brand A sneakers as $x_1$ and the number of Brand B sneakers as $x_2$. The problem can be represented symbolically as:

```json
{
    'sym_variables': [('x1', 'Brand A sneakers'), ('x2', 'Brand B sneakers')],
    'objective_function': '50*x1 + 75*x2',
    'constraints': [
        '100*x1 + 150*x2 <= 2000', 
        'x1 + x2 <= 15', 
        'x1 >= 0', 
        'x2 >= 0', 
        'x1 <= 15', 
        'x2 <= 15'
    ]
}
```

However, since $x_1$ and $x_2$ represent the number of sneakers, they should be integers. Thus, we should also specify that $x_1$ and $x_2$ are integers.

## Gurobi Code

Here's how you can solve this problem using Gurobi in Python:

```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="Brand_A", vtype=gurobi.GRB.INTEGER, lb=0)
    x2 = model.addVar(name="Brand_B", vtype=gurobi.GRB.INTEGER, lb=0)

    # Objective function: maximize 50*x1 + 75*x2
    model.setObjective(50*x1 + 75*x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(100*x1 + 150*x2 <= 2000, name="budget_constraint")
    model.addConstr(x1 + x2 <= 15, name="total_sneakers_constraint")

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution: x1 = {x1.varValue}, x2 = {x2.varValue}")
        print(f"Maximum profit: {model.objVal}")
    else:
        print("No optimal solution found")

solve_sneaker_problem()
```

This code will solve the optimization problem and print out the optimal number of Brand A and Brand B sneakers to buy and sell, as well as the maximum profit that can be achieved. If no optimal solution is found, it will indicate that the problem is infeasible.