Session 5: thinking in arrays, columns and cohorts#

Zoo models: Nagel-Schreckenberg, age-structured SEIR

Goals#

  • Store the attributes that are updated in bulk as Columns, and update the whole population with numpy.

  • Know when a phase can be vectorised without changing the model (synchronous updates, independent draws), and when it cannot.

  • Replace individuals by cohorts when everybody in the same state faces the same probabilities.

Before the session: Vectorised step phases and Profiling a model.

Exercises#

5.1 Write the Nagel-Schreckenberg step as a loop over cars (for car in cars: ...), with the same rules, and check it gives the same flow as the zoo model for one seed. Which order must the loop follow, and why?

Solution

The update is parallel: every car’s new speed depends on the gaps before anybody moves. Compute all speeds first in one loop, then move all cars in a second loop. With one loop that moves each car straight away, the car behind sees a larger gap: a different model. Draw the dawdle coins in one call, rng.random(n), so both versions consume the same random numbers and give identical flows.

5.2 Profile both versions with modelly profile on a road of 100,000 cells. How much faster is the column version?

Solution 
modelly profile my_traffic/model.py:NagelSchreckenberg --steps 100 --param length=100000

The column version does a handful of numpy operations per step whatever the number of cars; the loop pays a Python iteration per car. Expect one to two orders of magnitude.

5.3 The SEIR model keeps one cohort row per (age group, state). Print model.people.to_dataframe() after a few steps. How many rows are there, and how many people?

Solution 
from modelly.zoo.age_seir.model import AgeSEIR, SEIRScenario

model = AgeSEIR(SEIRScenario(rng=1))
model.run(30)
print(model.people.to_dataframe())

At most 12 rows (3 groups x 4 states) after merge(), for 100,000 people.

5.4 Why would cohorts stop being exact if every person had their own infectiousness?

Solution

A cohort replaces n independent draws with the same probability by one binomial draw. If the probabilities differ within a row, the row is not homogeneous and the binomial has the wrong law. Either key the cohorts on the trait too (one row per infectiousness class) or keep individuals.