Alzheimer's: Update testing model

docs/source/models/intervention_models/alzheimers/testing_diagnosis.rst

@@ -208,9 +208,20 @@ assign positive/negative diagnosis which will inform treatment in :ref:`Alternat
 Time-specific testing rates
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^
 Testing rates do not vary by location, age or sex. 
-In 2020, 0% of eligible simulants are tested annually. This increases (instantly) to 10% at year 2030, 
-then increases linearly over time in each six-month period to reach 20% in 2035, to 40% in 2040 
-and then maxes out at 60% in 2045. 
+In 2020, 0% of eligible simulants are tested annually. This becomes
+nonzero in 2027, increasing to 10% at year 2030,
+then increases linearly for awhile, then levels off and eventually maxes
+out at 60% after 2045. We will model this as a piecewise linear function
+with knots at the following (year, coverage) values:
+
+* (2020.0, 0%)
+* (2027.0, 0%) -- Note that this is 0% at the *beginning* of 2027, but
+  coverage will become positive on the second time step that year
+* (2030.5, 10%) -- Note that this is 10% at **mid**-year
+* (2045.0, 50%)
+* (2050.0, 57%)
+* (2055.0, 60%)
+* (2100.0, 60%)
 
 .. _bbbm_propensity:
 
@@ -225,42 +236,55 @@ On each timestep, use the following steps to assign BBBM tests:
 
 .. _bbbm_requirements:
 
-1. Assess eligibility based on the following requirements:
-
-  - Simulant is not in MCI or AD dementia state (only susceptible, or pre-clinical)
-  - Simulant age is >=60 and <80
-  - Simulant has not received a BBBM test in the last three years (or
-    six time steps)
-  - Simulant has never received a positive BBBM test
-
-2. If eligible (meets all requirements), check propensity. 
-   If the propensity value is less than the time-specific testing rate, give the test. If not, do not give the test.
-3. Assign a positive diagnosis to 90% of people and a negative diagnosis to 10% of people. This 90% draw should be independent of any previous draws, e.g., people who test negative still have a 90% chance of being positive on a re-test.
-4. Record time of last test, yes/no diagnosis for future testing eligibility.
+#. Assess eligibility based on the following requirements:
+
+   - Simulant is not in MCI or AD dementia state (only susceptible, or pre-clinical)
+   - Simulant age is :math:`\ge 60` and :math:`< 80`
+   - Simulant has not received a BBBM test in the last three years (or
+     six time steps)
+   - Simulant has never received a positive BBBM test
+
+#. If eligible (meets all requirements), check testing propensity. If
+   the propensity value is less than the time-specific testing rate, the
+   simulant has the opportunity to get tested on this time step (but may
+   not be). If not, the simulant won't be tested.
+#. If the simulant has the opportunity to be tested on this time step
+   (their propensity is less than the testing rate), give them a BBBM
+   test with probability 0.5, independently of other random choices
+   (see explanation below).
+#. For those who get tested, assign a positive diagnosis to 90% of people and a negative diagnosis to 10% of people. This 90% draw should be independent of any previous draws, e.g., people who test negative still have a 90% chance of being positive on a re-test.
+#. Record time of last test and yes/no diagnosis for determining future testing eligibility.
+
+The strategy of giving eligible simulants a test with probability 0.5 on
+each time step is to introduce randomness to the time between testing,
+rather than having all simulants be retested at a fixed interval of 3
+years (which caused oscillations in the number of tests over time). The
+probability 0.5 was chosen as a convenient value that will guarantee
+that most people will get retested within 5 years (Lilly requested that
+tests occur every 3-5 years). Specifically, the probability that a
+simulant *doesn't* get retested between 3 and 5 years (i.e., on one of
+the 5 time steps at 3, 3.5, 4, 4.5, 5) is :math:`(1-0.5)^5 = 3.125\%`.
 
 On initialization
 '''''''''''''''''
 
-In order to avoid having all eligible simulants be tested immediately
-upon entering the simulation, we will assign a BBBM testing history to
-each initialized simulant who is eligible for a BBBM test. Since
-simulants are only eligible for testing every three years (more
-precisely, every 6 time steps), we will assign a random test date within
-the last three years before entering the simulation, as follows.
+In order to avoid having a large fraction of eligible simulants be
+tested immediately upon entering the simulation, we will assign a BBBM
+testing history to each initialized simulant who is eligible for a BBBM
+test. Since simulants are only eligible for testing every three years
+(more precisely, every 6 time steps) and are likely to be tested at most
+every five years (10 time steps), we will assign a random test date
+within the last five years before entering the simulation, as follows.
 
 On initialization of each eligible simulant, choose uniformly at random
-from one of the last 6 time steps when they could have been tested,
-omitting any time steps before 2030 when testing is not yet available.
-If there are no such time steps (i.e., all 6 are before 2030), assign
-"not a time" (NaT) for the simulant's previous test date. Otherwise, the
-first time the simulant could be eligible for testing again is 6 time
-steps after the chosen previous test date. We assume for simplicity that
-there were no prior false positive tests among simulants entering the
-simulation, so all previous BBBM tests are negative.
-
-Even with prior BBBM testing history in place, due to test coverage
-jumping from 0% to 10% in 2030, we expect a large group to be
-immediately tested and then a drop-off in testing counts.
+from one of the last 10 time steps when they could have been tested. If
+the chosen time step occurs before the first date in 2027 when testing
+becomes available, assign "not a time" (NaT) for the simulant's previous
+test date. Otherwise, the first time the simulant could be eligible for
+testing again is 6 time steps after the chosen previous test date. We
+assume for simplicity that there were no prior false positive tests
+among simulants entering the simulation, so all previous BBBM tests are
+negative.
 
 .. note::