Introduction to LAPM

LAPM is a python package for the analysis of linear autonomous pool (compartmental) models. It can be used to obtain a large set of different system-level diagnostics of compartmental models. This can be done either in symbolic or numeric form. In this notebook, we will introduce the basics of the package.

We assume that you have already installed the package following the instructions provided in the download page. After the package is installed, you can import it following the commands:



In [1]:

    
from sympy import *
from LAPM import *
from LAPM.linear_autonomous_pool_model import LinearAutonomousPoolModel

In the second line above, we imported also the linear_autonomous_pool_model module which contains most of the functions required for the examples in this notebook. We will create now a simple two-compartment model with the following syntax



In [2]:

    
lambda_1, lambda_2, alpha, u_1, u_2 = symbols('lambda_1 lambda_2 alpha u_1 u_2', positive=True)
A = Matrix([[     -lambda_1,        0],
            [alpha*lambda_1, -lambda_2]])
u = Matrix(2, 1, [u_1, u_2])

Notice that we created a symblic version of the model with no assignment of values to the parameters. This is useful because we can make computations in symbolic form only, or we can assign parameter values later. With the compartmental matrix and the input vector created above, we can now create a compartmental model as



In [3]:

    
M=LinearAutonomousPoolModel(u, A)

Symbolic computations

We can use now the set of functions available in LAPM with this model. For example, we can compute the mean system age as



In [4]:

    
M.A_expected_value









    Out[4]:





(alpha*u_1/lambda_2 + u_2/lambda_2)/(lambda_2*(alpha*u_1/lambda_2 + u_2/lambda_2 + u_1/lambda_1)) + u_1*(alpha/lambda_2 + 1/lambda_1)/(lambda_1*(alpha*u_1/lambda_2 + u_2/lambda_2 + u_1/lambda_1))

For an output easier to read on the screen



In [5]:

    
pprint(M.A_expected_value)









    



     α⋅u₁   u₂               ⎛α    1 ⎞   
     ──── + ──            u₁⋅⎜── + ──⎟   
      λ₂    λ₂               ⎝λ₂   λ₁⎠   
─────────────────── + ───────────────────
   ⎛α⋅u₁   u₂   u₁⎞      ⎛α⋅u₁   u₂   u₁⎞
λ₂⋅⎜──── + ── + ──⎟   λ₁⋅⎜──── + ── + ──⎟
   ⎝ λ₂    λ₂   λ₁⎠      ⎝ λ₂    λ₂   λ₁⎠

Latex output can be obtained as



In [6]:

    
print(latex(M.A_expected_value))









    



\frac{\frac{\alpha u_{1}}{\lambda_{2}} + \frac{u_{2}}{\lambda_{2}}}{\lambda_{2} \left(\frac{\alpha u_{1}}{\lambda_{2}} + \frac{u_{2}}{\lambda_{2}} + \frac{u_{1}}{\lambda_{1}}\right)} + \frac{u_{1} \left(\frac{\alpha}{\lambda_{2}} + \frac{1}{\lambda_{1}}\right)}{\lambda_{1} \left(\frac{\alpha u_{1}}{\lambda_{2}} + \frac{u_{2}}{\lambda_{2}} + \frac{u_{1}}{\lambda_{1}}\right)}

You can copy and paste this Latex ouput to a markdown or latex document $$ \frac{\frac{\alpha u_{1}}{\lambda_{2}} + \frac{u_{2}}{\lambda_{2}}}{\lambda_{2} \left(\frac{\alpha u_{1}}{\lambda_{2}} + \frac{u_{2}}{\lambda_{2}} + \frac{u_{1}}{\lambda_{1}}\right)} + \frac{u_{1} \left(\frac{\alpha}{\lambda_{2}} + \frac{1}{\lambda_{1}}\right)}{\lambda_{1} \left(\frac{\alpha u_{1}}{\lambda_{2}} + \frac{u_{2}}{\lambda_{2}} + \frac{u_{1}}{\lambda_{1}}\right)} $$

Numerical calculations

In most cases, we actually want to perform numerical computations of system-level diagnostics. In this case, we need first to assign values to the elements of the compartmental system. For instance, we can use the subs function to assign numerical values to existing matrices and vectors:



In [7]:

    
u1=u.subs({u_1: 2, u_2: 4})
A1=A.subs({lambda_1: 0.8, lambda_2: 0.01, alpha: 0.13})

We take now these numerical arguments and create a new compartmental system and compute the mean system age as above



In [8]:

    
M1=LinearAutonomousPoolModel(u1, A1)
M1.A_expected_value









    Out[8]:





99.4997082847141

Other useful system diagnostics are:



In [9]:

    
M1.A_standard_deviation # standard deviation of the system age distribution









    Out[9]:





99.99249461290282



In [10]:

    
M1.A_quantile(0.5) # Median (50% quantile) of the system age distribution









    Out[10]:





68.80680584095865



In [11]:

    
M1.T_expected_value #Mean transit time









    Out[11]:





71.4166666666667



In [12]:

    
M1.T_standard_deviation # standard deviation of the transit time distribution









    Out[12]:





95.45435936730298



In [13]:

    
M1.T_quantile(0.5) # Median (50% quantile) of the transit time distribution









    Out[13]:





35.14291412333978



In [14]:

    
M1.a_expected_value # Mean age vector of individual pools









    Out[14]:





Matrix([
[            1.25],
[100.076291079812]])



In [15]:

    
M1.a_quantile(0.5) # Median age of individual pools









    Out[15]:





array([ 0.86643398, 69.39194502])



In [16]:

    
M1.T_laplace









    Out[16]:





0.232/(s + 0.8) + 0.00666666666666667/(s + 0.01) + 0.000346666666666667/((s + 0.01)*(s + 0.8))



In [17]:

    
M1.A_laplace









    Out[17]:





0.00406067677946324/(s + 0.8) + 0.0099416569428238/(s + 0.01) + 6.06767794632439e-6/((s + 0.01)*(s + 0.8))



In [68]:

    
M1.r_compartments # release flux of individual compartments









    Out[68]:





Matrix([
[1.74],
[4.26]])



In [69]:

    
M1.r_total # Total release flux









    Out[69]:





6.00000000000000



In [74]:

    
M1.entropy_per_jump









    Out[74]:





-0.32626427406199*log(2) + 0.489396411092985*log(3) + 2.21020273307626



In [55]:

    
M1.entropy_per_cycle









    Out[55]:





-0.666666666666667*log(2) + 1.0*log(3) + 2.95715004792764



In [56]:

    
M1.entropy_rate









    Out[56]:





-0.0865384615384615*log(2) + 0.129807692307692*log(3) + 0.383860823529068



In [ ]: