To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints based on the problem statement.

Let's define:
- $x_1$ as the number of Brand A sneakers bought and sold.
- $x_2$ as the number of Brand B sneakers bought and sold.

The objective is to maximize profit. Given that the profit per Brand A sneaker is $50 and per Brand B sneaker is $75, the objective function can be written as:
\[ \text{Maximize: } 50x_1 + 75x_2 \]

The constraints are:
1. The total cost of buying sneakers cannot exceed $2000. Since Brand A sneakers cost $100 each and Brand B sneakers cost $150 each, we have:
\[ 100x_1 + 150x_2 \leq 2000 \]
2. The boy can sell at most 15 sneakers in total, so:
\[ x_1 + x_2 \leq 15 \]
3. Non-negativity constraints since the number of sneakers cannot be negative:
\[ x_1 \geq 0, x_2 \geq 0 \]

In symbolic notation with natural language objects, we have:
```json
{
    'sym_variables': [('x1', 'Number of Brand A sneakers'), ('x2', 'Number of Brand B sneakers')],
    'objective_function': '50*x1 + 75*x2',
    'constraints': ['100*x1 + 150*x2 <= 2000', 'x1 + x2 <= 15', 'x1 >= 0', 'x2 >= 0']
}
```

Now, let's translate this into Gurobi code in Python to solve the optimization problem:

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="Brand_A_Sneakers")
x2 = m.addVar(vtype=GRB.INTEGER, name="Brand_B_Sneakers")

# Set the objective function
m.setObjective(50*x1 + 75*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(100*x1 + 150*x2 <= 2000, "Total_Cost")
m.addConstr(x1 + x2 <= 15, "Total_Sneakers")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of Brand A sneakers: {x1.x}")
    print(f"Number of Brand B sneakers: {x2.x}")
    print(f"Maximum profit: {m.objVal}")
else:
    print("No optimal solution found")
```