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Welcome to procmodeling!
An easy way to solve and optimize your ODE system.

How to Solve Your ODE

FIRST STEP: Create a .txt file or a docstring variable containing the ODEs using Python equation syntax.

edo_Lotka_Volterra.txt
# Let's use the Lotka-Volterra ODEs where:
# y[0] = prey population
# y[1] = predator population
# dydt[0] = rate of change of prey over time
# dydt[1] = rate of change of predators over time

# Define the EDOs
dydt[0] = a * y[0] - b * y[0] * y[1]
dydt[1] = d * y[0] * y[1] - g * y[1]

# Define the parameters:
a = 1.0    # prey birth rate
b = 0.1    # predation rate
d = 0.075  # predator growth rate
g = 1.5    # predator death rate

Please note the following three rules:

  1. The names y, t, and dy/dt are reserved keywords representing the dependent variables, time, and the ODE system, respectively;
  2. Both y and dy/dt are vectors, where [i] (i being an integer) indicates the index of the dependent variable;
  3. Respect Python’s equation syntax and make sure to define all the variables used in the system's equations. The order of expeditions does not matter.

SECOND STEP: Import and check your equations

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from procmodeling import solver
edo = solver.edo_functions('edo_Lotka_Volterra.txt')
Example Output 

edo_functions: Document was read
code output

THIRD STEP: Call the solve method after setting the time span and initial conditions

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edo.solve( t_span=[0,15], y0=[40, 10] )
Example Output

code output

How to Optimize Your EDO

FIRST STEP: After successfully solving your ODE, organize the data (which will be used for parameter fitting) into a DataFrame following the format below:

Initial Conditions Time Conditions at time = j
\(y_0[1]\) \(y_0[1]\) ... \(y_0[j]\) t \(y[0]\) \(y[1]\) ... \(y[j]\)
y0[0]\(i\) y0[1]\(i\) ... y0[j]\(i\) j y[0]\(i\) y[1]\(i\) ... y[j]\(i\)
... ... ... ... ... ... ... ... ...

Note

This structure allows you to fit your ODE model using multiple initial conditions if needed.

SECOND STEP: Define the search space for the parameters and call the optimizer

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bounds = {
    'param1': (min, max),
    'param2': (min, max),
    ...
    }

initial_conditions_columns = ['y0[0]', 'y0[1]', ... ]

edo.optimize(
    data = generated_data,
    initial_values_columns = initial_conditions_columns,
    target_values_columns = ['y[0]', 'y[1]', ...],
    time_column = 'time',
    bounds = bounds
)

Note

An example of optimizing the Lotka-Volterra ODE is available here