Alzheimer’s disease with preclinical and MCI stages (GBD 2023)

Abbreviations
Abbreviation	Definition
ACMR	All-Cause Mortality Rate
AD	Alzheimer’s Disease
BBBM	Blood-Based Biomarker
CSMR	Cause-Specific Mortality Rate
CSU	Client Services Unit
DW	Disability Weight
FHS	Future Health Scenarios
GBD	Global Burden of Disease
MCI	Mild Cognitive Impairment
YLD	Years Lived with Disability
YLL	Years of Life Lost

The IHME dementia modelers use DisMod to estimate the incidence, prevalence and excess mortality of a “dementia envelope” (modelable entity ID 24351) comprising all types of dementia combined, and then they estimate what proportion of the envelope corresponds to each subtype of dementia. They attribute the proportions of dementia due to stroke, Parkinson’s disease, Down’s syndrome, and traumatic brain injury to those GBD causes, and then they use the remaining dementia in the envelope for the GBD cause “Alzheimer’s disease and other dementias” (cause ID 543).

For this simulation, we use the dementia envelope incidence, prevalence, and excess mortality from GBD 2023 (release ID 16) and multiply incidence and prevalence by the proportion due at least partially to Alzheimer’s disease to obtain AD-specific estimates. See the Data Values and Sources section for details.

Restrictions 

The following table describes restrictions in GBD 2023 on the effects of the cause “Alzheimer’s disease and other dementias” (such as being only fatal or only nonfatal), as well as restrictions on the ages and sexes to which the cause applies.

GBD 2023 Cause Restrictions
Restriction Type	Value	Notes
Male only	False
Female only	False
YLL only	False
YLD only	False
YLL age group start	40 to 44	age_group_id = 13
YLL age group end	95 plus	age_group_id = 235
YLD age group start	40 to 44 for Dementia-AD cause state No a priori age restriction for MCI-AD cause state	Restriction to age_group_id = 13 (40 to 44) for Dementia-AD cause state is from GBD. However, due to simulation dynamics, it is possible for simulants to enter this state before age 40.
YLD age group end	95 plus	age_group_id = 235

Vivarium Modeling Strategy 

This cause model extends GBD’s Alzheimer’s disease dementia with two predementia states: BBBM-AD (preclinical AD detectable via blood-based biomarkers) and MCI-AD (mild cognitive impairment due to AD). GBD does not model these predementia states, so we calibrate transition rates and prevalences using GBD data combined with additional evidence on state durations. See below for details.

To obtain Alzheimer’s-specific estimates, we multiply the GBD 2023 dementia envelope by the proportion due at least in part to Alzheimer’s disease. See the Data Values and Sources section for details.

Note

The dementia envelope includes “mixed” dementias, which involve two or more causes, and we assume that 94% of these mixed dementias include Alzheimer’s Disease combined with another etiology. While it seems logical that the modeled treatment will not help for these mixed dementias, it is impractical to exclude them from a trial population, and therefore a treatment efficacy measured in a trial will as include a similar proportion of patients with mixed dementias.

Cause Model Diagram 

State Definitions
State	State Name	Definition
S	Susceptible	Simulant does not have Alzheimer’s disease or any of its precursors
BBBM-AD	Blood-Based-Biomarker-preclinical Alzheimer’s Disease	Simulant has preclinical Alzheimer’s disease that is detectable using blood-based biomarkers but causes no cognitive impairment
MCI-AD	Mild Cognitive Impairment due to Alzheimer’s Disease	Simulant has mild cognitive impairment due to Alzheimer’s disease
Dementia-AD	Alzheimer’s Disease dementia	Simulant has mild, moderate, or severe dementia due to Alzheimer’s disease
Death (not pictured)	Death	Simulant has died

Transition Definitions
Transition	Definition	Notes
\(h_{S \to \text{BBBM}}\)	Hazard rate of transitioning from susceptible to BBBM-AD	Obtained from rate calibration
\(h_{\text{BBBM} \to \text{MCI}}\)	Hazard rate of transitioning from BBBM-AD to MCI-AD	Time-dependent hazard based on dwell time in BBBM-AD state; see \(h_\text{MCI}(t)\) in Data Values and Sources; approximated as a constant hazard during rate calibration
\(h_{\text{MCI} \to D}\)	Hazard rate of transitioning from MCI-AD to Dementia-AD	Constant hazard, defined below as the inverse of the average duration of MCI-AD, \(1 / \Delta_\text{MCI}\)
\(m\)	Background mortality rate (non-AD mortality)	Applies to all states; computed by mortality component in the simulation
\(f\)	Excess mortality rate due to AD dementia	Obtained from rate calibration; applies only to Dementia-AD state

State and Transition Data 

The tables in this section describe the data needed for the cause model drawn in the Cause Model Diagram section above. The variables in the tables are defined in the the Data Values and Sources section below.

The following tables describe the data for each state and transition if modeling only simulants with AD dementia or predementia AD as described in the Alzheimer’s population model:

State data when modeling only simulants with AD dementia or predementia AD
State	Initial prevalence	Entrance prevalence	Excess mortality rate	Disability weight
S	0	0	0	0
BBBM-AD	\(\delta_\text{BBBM}\)	1	0	0
MCI-AD	\(\delta_\text{MCI}\)	0	0	\(\text{DW}_\text{MCI}\)
Dementia-AD	\(1 - \delta_\text{BBBM} - \delta_\text{MCI}\)	0	\(f\)	\(\text{DW}_\text{c543}\)

Note: The conditional prevalences \(\delta_\text{BBBM}\) and \(\delta_\text{MCI}\), and the excess mortality rate \(f\), are obtained from the rate calibration. The disability weights \(\text{DW}_\text{MCI}\) and \(\text{DW}_\text{c543}\) are defined in the data values and sources table below.

Transition Data
Transition	Source State	Sink State	Value
\(h_{S \to \text{BBBM}}\)	S	BBBM-AD	From rate calibration. When modeling only AD simulants, this is not used in the cause model itself, but is essential for the Alzheimer’s population model to ensure the correct number of appropriately aged simulants are added in the BBBM-AD state on each time step.
\(h_{\text{BBBM} \to \text{MCI}}\)	BBBM-AD	MCI-AD	\(h_\text{MCI}(t - T_\text{BBBM})\), where \(t\) is the current simulation time and \(T_\text{BBBM}\) is when the simulant entered BBBM-AD. Adjusted in Hypothetical Alzheimer’s Treatment scenario.
\(h_{\text{MCI} \to D}\)	MCI-AD	Dementia-AD	\(1 / \Delta_\text{MCI}\)
\(m\)	Any state	Death	\(\text{ACMR} - \text{CSMR}\)
\(f\)	Dementia-AD	Death	From rate calibration; total mortality in Dementia-AD state is \(m + f\)

Note: \(h_\text{MCI}(t)\) is the time-dependent hazard function for transitioning from BBBM-AD to MCI-AD, and \(\Delta_\text{MCI}\) is the average duration of MCI-AD in the absence of mortality; see the data values and sources table below.

Simulants initialized into the BBBM-AD state need an assigned value for \(T_\text{BBBM}\) to determine their dwell time. For simulants in BBBM-AD at time \(t=0\), assign \(T_\text{BBBM}\) uniformly in the interval \([-\Delta_\text{BBBM}, 0]\), where \(\Delta_\text{BBBM}\) is the average duration of BBBM-AD in the absence of mortality, defined in the data values and sources table below.

Attention

If we model the entire population including susceptible simulants, the state data should be modified as follows.

The rate calibration provides \(p\), the total prevalence of any AD state (BBBM + MCI + dementia) in the total population, along with conditional prevalences \(\delta_\text{BBBM}\) and \(\delta_\text{MCI}\). The prevalence of each AD state in the total population is then \(p\) multiplied by the corresponding conditional prevalence. The following table shows the resulting initial prevalences when modeling the total population, as well as the birth prevalences, which replace the entrance prevalences. The excess mortality rate and disability weight of each state remain the same.

State data when modeling entire population including susceptible simulants
State	Initial prevalence	Birth prevalence
S	\(1 - p\)	1
BBBM-AD	\(p \cdot \delta_\text{BBBM}\)	0
MCI-AD	\(p \cdot \delta_\text{MCI}\)	0
Dementia-AD	\(p \cdot (1 - \delta_\text{BBBM} - \delta_\text{MCI})\)	0

Note

The calibrated value of \(p\) (total prevalence of any AD state) is needed to compute the model scale and initialize the correct number of simulants in each demographic subgroup. In the notation on the Alzheimer’s population model page, \(p\) for a specific demographic subgroup \(g\) and year \(t\) corresponds to \(p_{g,t}\) on that page.

Data Values and Sources 

Unless otherwise noted, all data values depend on year, location, age group, and sex, as defined by GBD.

The following paths on the cluster contain the data files listed in the table below:

population_agg.nc and mortality_all.nc from FHS team
squeezed_proportions_to_sim_sci.csv from dementia modelers
all.hdf disability weight file saved by Simulation Science team

# Data folder for Alzheimer's sim, including data from FHS team and
# dementia modelers (see README.txt for data provenance)
/mnt/team/simulation_science/pub/models/vivarium_csu_alzheimers/data

# Disability weights saved by Simscience team:
/mnt/team/simulation_science/costeffectiveness/auxiliary_data/GBD_2021/02_processed_data/disability_weight/sequela/all/all.hdf

Data values and sources
Variable	Definition	Source or value	Notes
proportion_AD	The proportion of the dementia envelope that is Alzheimer’s disease dementia	`squeezed_proportions_to_sim_sci.csv`	Point estimate stratified by age group and sex for ages 40+. Includes proportions for all subtypes of dementia — filter to type_label == “Alzheimer’s disease”. Note: These estimates were provided by the dementia modelers and are not yet published, so they should not be stored directly in the Artifact or any other public location.
proportion_mixed	The proportion of the dementia envelope that is due to mixed dementia (i.e. more than one type of dementia simultaneously)	`squeezed_proportions_to_sim_sci.csv`	Point estimate stratified by age group and sex for ages 40+. Includes proportions for all subtypes of dementia — filter to type_label == “Mixed dementia”. Note: These estimates were provided by the dementia modelers and are not yet published, so they should not be stored directly in the Artifact or any other public location.
prevalence_m24351	Prevalence of GBD 2023 dementia envelope in year 2023	get_draws( source=”epi”, gbd_id_type = “modelable_entity_id”, gbd_id=24351, release_id=16, year_id=2023, measure_id=5 )	The dementia envelope represents the combined prevalence all types of dementia. By contrast, the GBD cause “Alzheimer’s disease and other dementias” (c543) does not include certain dementias that result from other modeled GBD causes.
prevalence_AD	Prevalence of AD dementia in total population	prevalence_m24351 \(\cdot\) (proportion_AD + 0.94 \(\cdot\) proportion_mixed)	Used as input data in rate calibration. We assume that 94% of mixed dementias include Alzheimer’s disease, based on this mixed dementias presentation from the dementia modelers.
\(p\)	Total prevalence of any AD state (BBBM + MCI + dementia) in total population	From rate calibration, artifact key `cause.alzheimers_consistent.prevalence_any`
\(\delta_\text{BBBM}\), \(\delta_\text{MCI}\)	Conditional prevalences of BBBM and MCI states among all AD cases	From rate calibration, artifact keys `cause.alzheimers_consistent.bbbm_conditional_prevalence` and `cause.alzheimers_consistent.mci_conditional_prevalence`	Prevalence of each state in the total population is \(p\) times the conditional prevalence
incidence_m24351	Total-population incidence rate for GBD 2023 dementia envelope in year 2023	get_draws( source=”epi”, gbd_id_type = “modelable_entity_id”, gbd_id=24351, release_id=16, year_id=2023, measure_id=6 )	Raw value from get_draws, different from susceptible-population incidence rate automatically calculated by Vivarium Inputs
incidence_AD	Total-population incidence rate of AD dementia	incidence_m24351 \(\cdot\) (proportion_AD + 0.94 \(\cdot\) proportion_mixed)	Used as input data in rate calibration; formerly used in a previous version of the AD population model to calculate BBBM-AD incidence. We are assuming the prevalence proportions can be applied to incidence. We are assuming the AD dementia incidence rate is constant over time in each demographic group. We assume 94% of mixed dementias include Alzheimer’s, based on this mixed dementias presentation from the dementia modelers.
population_forecast	Forecasted average population during specified year	`population_agg.nc`	Draw-level, age-specific forecasts from GBD 2021 Forecasting Capstone. Numerically equal to person-years. Used in AD population model to calculate BBBM-AD incidence counts. See Abie’s population and mortality forecasts notebook for a demonstration of how to load and transform the `.nc` file.
\(\text{ACMR}\)	All-cause mortality rate	`mortality_all.nc`	Draw-level, age-specific forecasts from GBD 2021 Forecasting Capstone. See Abie’s population and mortality forecasts notebook for a demonstration of how to load and transform the `.nc` file. Used as input data in rate calibration, as well as in mortality component.
emr_m24351	Excess mortality rate of dementia from GBD 2023 dementia envelope in year 2023	get_draws( source=”epi”, gbd_id_type = “modelable_entity_id”, gbd_id=24351, release_id=16, year_id=2023, measure_id=9 )	Used as input data in rate calibration. This EMR from DisMod is a true excess mortality rate, including all deaths associated with dementia, as opposed to only those deaths caused by dementia, which is what we usually use in our simulations
\(\text{CSMR}\)	Cause-specific mortality rate of AD	From rate calibration, artifact key `cause.alzheimers_consistent.cause_specific_mortality_rate`	Defined to be \(f \cdot p_\text{dementia}\), where \(f\) and \(p_\text{dementia}\) are the calibrated excess mortality rate of AD and prevalence of AD dementia, respectively. Since \(f\) is derived from emr_m24351 from DisMod, this “CSMR” is not necessarily a true “cause-specific” mortality rate, as it will include deaths associated with AD dementia, rather than just those caused by AD dementia.
\(m\)	Background mortality rate (non-AD mortality)	\(\text{ACMR} - \text{CSMR}\)	Applies to all states. Calculated in the simulation by mortality component.
\(f\)	Excess mortality rate due to AD dementia	From rate calibration, artifact key `cause.alzheimers_consistent.excess_mortality_rate`	Applies only to Dementia-AD state; total mortality in that state is \(m + f\). Derived from emr_m24351 from DisMod.
sequelae_c543	Sequelae of Alzheimer’s disease and other dementias	Set of 3 sequelae: s452, s453, s454	Obtained from gbd_mapping. Sequela names are “Mild,” “Moderate,” or “Severe Alzheimer’s disease and other dementias,” respectively. Same for all years, locations, age groups, and sexes.
\(\text{prevalence}_s\)	Prevalence of sequela \(s\)	como
\(\text{DW}_s\)	Disability weight of sequela \(s\)	`all.hdf` disability weight file in our team’s auxiliary data	Disability weights are stored as draws and do not vary by year, location, age group, or sex. For reference, the values are: s452: 0.069 (0.046-0.099) s453: 0.377 (0.252-0.508) s454: 0.449 (0.304-0.595)
\(\text{DW}_\text{c543}\)	Average disability weight of AD dementia	\(\sum\limits_{s\in \text{sequelae\_c543}} \text{DW}_s \cdot \text{prevalence}_s\)	Prevalence-weighted average disability weight over sequelae, computed automatically by Vivarium Inputs. Used to calculate YLDs.
\(\text{DW}_\text{motor}\)	Disability weight for health state “motor impairment, mild”	`all.hdf` disability weight file in our team’s auxiliary data	Disability weights are stored as draws and do not vary by year, location, age group, or sex. See Abie’s disability weight notebook for details on pulling the correct value.
\(\text{DW}_\text{motor+cog}\)	Disability weight for health state “motor plus cognitive impairments, mild”	`all.hdf` disability weight file in our team’s auxiliary data	Disability weights are stored as draws and do not vary by year, location, age group, or sex. See Abie’s disability weight notebook for details on pulling the correct value.
\(\text{DW}_\text{MCI}\)	Disability weight of mild cognitive impairment	\(\displaystyle \frac{\text{DW}_\text{motor+cog} - \text{DW}_\text{motor}} {1 - \text{DW}_\text{motor}}\)	Disability weights are stored as draws and do not vary by location, age group, or sex. For reference, the value is 0.021 (0.013, 0.032) Obtained by removing DW of “motor impairment, mild” from DW of “motor plus cognitive impairments, mild,” at the draw level. See Abie’s disability weight notebook for details, and see the derivation below for further explanation.
\(T_X\)	The time at which a simulant enters the cause state \(X\)	Determined within the simulation	Random variable for each simulant. \(T_\text{BBBM}\) is used to determine how long a simulant has been in the BBBM-AD state, in order to compute the hazard rate of transitioning to MCI-AD at a given simulation time \(t\).
\(D_\text{BBBM}\)	Dwell time in cause state BBBM-AD	\(T_\text{MCI} - T_\text{BBBM}\)	Random variable for each simulant, constructed implicitly through simulation dynamics to have approximately a Weibull distribution with shape parameter \(k\) and scale parameter \(\lambda\)
\(k\), \(\lambda\)	Shape and scale parameters, respectively, of Weibull distribution for \(D_\text{BBBM}\)	\(k = 1.22\) \(\lambda = 6.76\)	Chosen to match client’s specification for \(D_\text{BBBM}\): The probability of progression from BBBM-AD to MCI-AD is about 50% at 5 years and 80% at 10 years, corresponding to an average annual rate of progression of approximately 15% . Use the same parameters for all years, locations, age groups, and sexes.
bbbm_dist	Python object representing the Weibull distribution for \(D_\text{BBBM}\)	scipy.stats.weibull_min(k, scale=λ)	An instance of SciPy’s Weibull distribution class.
\(h_\text{MCI}(t)\)	Hazard function for transitioning into the MCI-AD state from BBBM-AD	bbbm_dist.pdf(t) / bbbm_dist.sf(t), or exp( bbbm_dist.logpdf(t) — bbbm_dist.logsf(t) ), an equivalent expression that may help avoid underflow	Equal to \(\frac{k}{\lambda} \left(\frac{t}{\lambda}\right)^{k-1}\), but can also be computed as the ratio of the probability density function to the survival function, using the methods defined in SciPy’s Weibull distribution class
\(\Delta_\text{BBBM}\)	Average duration of BBBM-preclinical AD in the absence of mortality	bbbm_dist.mean()	Equal to \(\lambda \Gamma(1 + 1/k)\), where \(\Gamma\) is the gamma function. Can be computed using scipy.special.gamma, but using bbbm_dist.mean() is more general if we update the underlying distribution. Does not vary by year, location, age group, or sex.
\(\Delta_\text{MCI}\)	Average duration of MCI due to AD in the absence of mortality	3.85 years	Obtained from Table 3 in Potashman et al., assuming a constant hazard rate of transitioning to AD dementia. Corresponds to an annual conditional probability of 0.771 of staying in MCI-AD given that you don’t die within one year, since \(\exp(-1 / 3.85) \approx 0.771\). Does not vary by year, location, age group, or sex. Note: The paper reports a 68.2% chance of staying in MCI and a 5.3% chance of returning to asymptomatic—these probabilities have been combined to get an annual probability of 73.5% of staying in MCI since our model assumes that a backwards transition is not possible. The conditional probability above is computed as \(0.771 = 0.735 / (1 - 0.047)\) since the paper reports a 4.7% chance of dying within a year when starting in the MCI state.

Calibrating Consistent Rates 

GBD provides dementia prevalence, incidence, and mortality, but our model includes predementia states (BBBM-AD and MCI-AD) not directly measured by GBD. To derive internally consistent rates for these states, we use Bayesian inference with NumPyro/JAX to fit disease progression rates to GBD data while enforcing ODE-based consistency constraints.

To elaborate, we use MCMC optimization to sample from the likely values of the seven parameters listed in the “model parameters” section below, subject to the constraints imposed by the DisMod ODEs. Many of these parameters vary as a function of age, and the system of ODEs describes how the age patterns are related (such as prevalent cases at age a+1 are the prevalent cases at age a, minus deaths at age a, plus incident cases. The MCMC algorithm draws samples from an objective that can be interpreted as a Bayesian posterior distribution (joint across all parameters), which ends up being a relatively high dimensional distribution, since many of the model parameters have different values for different ages. Unlike most Bayesian computation, there is not new data to encode in a likelihood function, and the evidence synthesis is focused on finding parameters that satisfy the DisMod equations as well as the “priors” from GBD and other sources.

The calibration is implemented in consistent_rates.py in the vivarium_csu_alzheimers repository. We fit separate models for males and females.

Model Parameters 

The NumPyro model defines 7 age-varying parameters, each with truncated normal priors on \([0, 1]\):

\(p\): Total prevalence of any AD state
\(\delta_\text{BBBM}\), \(\delta_\text{MCI}\): Conditional prevalences of BBBM and MCI states among AD cases
\(h_{S \to \text{BBBM}}\): Transition rate from susceptible to BBBM
\(i\): Total-population incidence rate of dementia
\(f\): Excess mortality rate of dementia
\(m\): Background (non-AD) mortality rate

The model solves for the parameters for ages 30 to 100 in 5-year intervals.

ODE Consistency Constraints 

The calibration produces consistent parameters by solving a 5-compartment ODE system starting with initial conditions at age \(a\) to find the implied values at age \(a + 5\); we include the squared difference between these implied values and the parameter values in the MCMC objective. This is basically an implementation of the method of multiple shootings for solving boundary value problems.

The state variables are S (susceptible), BBBM, MCI, D (dementia), and \(D_\text{new}\) (cumulative incident dementia, which is used to calibrate the total-population incidence rate of AD dementia). The DisMod ODE system is:

\[\begin{split}\frac{dS}{dt} &= -m \cdot S - h_{S \to \text{BBBM}} \cdot S \\ \frac{d(\text{BBBM})}{dt} &= h_{S \to \text{BBBM}} \cdot S - m \cdot \text{BBBM} - h_{\text{BBBM} \to \text{MCI}} \cdot \text{BBBM} \\ \frac{d(\text{MCI})}{dt} &= h_{\text{BBBM} \to \text{MCI}} \cdot \text{BBBM} - m \cdot \text{MCI} - h_{\text{MCI} \to D} \cdot \text{MCI} \\ \frac{dD}{dt} &= h_{\text{MCI} \to D} \cdot \text{MCI} - (m + f) \cdot D \\ \frac{dD_\text{new}}{dt} &= h_{\text{MCI} \to D} \cdot \text{MCI}.\end{split}\]

The transition rates \(h_{\text{BBBM} \to \text{MCI}} = 1 / \Delta_\text{BBBM}\) and \(h_{\text{MCI} \to D} = 1 / \Delta_\text{MCI}\) are fixed based on literature values in the data values table above and the assumption that the Weibull distribution is approximated acceptably by a constant hazard for calibration purposes.

Note that the model parameters \(p\), \(\delta_\text{BBBM}\), and \(\delta_\text{MCI}\) can be represented in terms of compartment sizes as follows:

\[p = \frac{\text{BBBM + MCI} + D}{S + \text{BBBM + MCI} + D}\]

\[\delta_\text{BBBM} = \frac{\text{BBBM}}{\text{BBBM + MCI} + D}\]

\[\delta_\text{MCI} = \frac{\text{MCI}}{\text{BBBM + MCI} + D}.\]

Also note that \(D_\text{new}\) is not required for solving this system of differential equations; it is included for convenience to allow us to easily calibrate the total-population incidence rate of AD dementia, which has the same numerator as \(h_{\text{MCI} \to D}\), but a different population for the denominator:

\[\begin{split}\begin{aligned} i &= h_{\text{MCI} \to D} \cdot \frac{\text{MCI}}{S + BBBM + MCI + D} \\ &= \frac{d D_\text{new}}{dt} \cdot \frac{1}{S + BBBM + MCI + D}. \end{aligned}\end{split}\]

Non-ODE Consistency Constraints 

In addition to the logical constraints that the DisMod ODEs impose on rates at different ages, we have constraints imposed by the data we get from GBD and FHS. The following auxiliary quantities are computed to match against this input data.

Dementia prevalence derives from total AD prevalence and the conditional prevalences of the predementia states:

\[\begin{split}\begin{aligned} p_\text{dementia}(a) &= p(a) \cdot \left(1 - \delta_\text{BBBM}(a) - \delta_\text{MCI}(a)\right)\\ &= p(a) \cdot \delta_D(a), \end{aligned}\end{split}\]

where \(\delta_D = 1 - \delta_\text{BBBM} - \delta_\text{MCI}\) is the conditional prevalence of AD dementia among AD cases.

All-cause mortality combines background mortality with excess mortality weighted by dementia prevalence:

\[\begin{split}\begin{aligned} m_\text{all}(a) &= m(a) + f(a) \cdot p_\text{dementia}(a) \\ &= m(a) + \text{CSMR}(a), \end{aligned}\end{split}\]

where \(\text{CSMR} = f \cdot p_\text{dementia}\) is the “cause-specific” mortality rate of AD dementia. Note that this CSMR is not necessarily causal (i.e., it includes deaths associated with AD, not just caused by AD) since \(f\) is calibrated to the excess mortality rate we get from DisMod, which is not necessarily causal. Despite this limitation, we use the CSMR derived from this rate calibration model as the causal CSMR of AD dementia for the Alzheimer’s disease cause model.

The Inputs and Outputs section below lists the input data used to calibrate the different model variables.

Loss Function 

After solving the ODE with initial values for age \(a\) to find the ODE-implied values at \(a + 5\), the calibration computes the root-sum-squared log-difference between these ODE-impled values and the parameter values \(\delta_\text{BBBM}\), \(\delta_\text{MCI}\), \(p_\text{dementia}\), and \(i\) in the MCMC objective:

\[\epsilon(a) = \sqrt{ \left(\log \hat\delta_\text{BBBM} - \log \delta_\text{BBBM}(a+5)\right)^2 + \left(\log \hat\delta_\text{MCI} - \log \delta_\text{MCI}(a+5)\right)^2 + \left(\log \hat p_\text{dementia} - \log p_\text{dementia}(a+5)\right)^2 + \left(\log \hat\imath - \log i(a+2.5)\right)^2 }\]

where \(\hat\delta_\text{BBBM}\), \(\hat\delta_\text{MCI}\), \(\hat p_\text{dementia}\), and \(\hat\imath\) are the values at age \(a+5\) implied by the ODE solution with initial values from the parameters for age \(a\). The calibration applies a penalty for large errors by assuming a priori that the ODE error \(\epsilon(a) \sim \mathcal{N}(0, \sigma)\) with \(\sigma = 0.005\).

Numerical Methods 

We solve the ODE using diffrax (Dopri5) and sample using NUTS with 500 warmup and 500 sample iterations.

Inputs and Outputs 

The calibration uses the following input data to constrain certain model parameters. The variable names in the first table column are from the Data values and sources table above, which contains additional details about the input data. The last column shows which model parameter is constrained by the data; the model parameters are defined in the Model Parameters and Non-ODE Consistency Constraints sections above.

Calibration Input Data
Input Data	Definition and source	Artifact key	Model parameter
prevalence_AD	Prevalence of AD dementia in year 2023 (derived from GBD 2023 dementia envelope)	`cause.alzheimers.prevalence`	\(p_\text{dementia}\)
incidence_AD	Total-population incidence rate of AD dementia in year 2023 (derived from GBD 2023 dementia envelope)	`cause.alzheimers.population_incidence_rate`	\(i\)
emr_m24351	Excess mortality rate of dementia in year 2023 (from GBD 2023 dementia envelope)	`cause.alzheimers.excess_mortality_rate`	\(f\)
\(\text{ACMR}_{2025}\)	All-cause mortality rate in year 2025 (from FHS forecasts)	`cause.all_causes.cause_specific_mortality_rate`	\(m_\text{all}\)

Note that we include only a single year of input data for each of the calibration targets, and the calibration assumes that these rates remain constant over time.

The calibration generates the following outputs for use in the Vivarium simulation and for verification and validation. The output variables are defined in the Model Parameters and Non-ODE Consistency Constraints sections above.

Calibrated Outputs (written to artifact for year 2025)
Artifact key	Description
`cause.alzheimers_consistent.population_incidence_any`	\(h_{S \to \text{BBBM}}\): S to BBBM transition rate
`cause.alzheimers_consistent.prevalence_any`	\(p\): Total prevalence of any AD state
`cause.alzheimers_consistent.susceptible_to_bbbm_transition_count`	\(h_{S \to \text{BBBM}} \cdot (1 - p) \cdot\) population_forecast: Annual count of incident cases entering BBBM
`cause.alzheimers_consistent.population_incidence_dementia`	\(i\): Population incidence rate of dementia
`cause.alzheimers_consistent.excess_mortality_rate`	\(f\): Excess mortality rate of AD
`cause.alzheimers_consistent.cause_specific_mortality_rate`	\(\text{CSMR}\): Cause-specific mortality rate, defined as \(f \cdot p_\text{dementia}\)
`cause.alzheimers_consistent.bbbm_conditional_prevalence`	\(\delta_\text{BBBM}\): Proportion of AD cases in BBBM state
`cause.alzheimers_consistent.mci_conditional_prevalence`	\(\delta_\text{MCI}\): Proportion of AD cases in MCI state
`cause.alzheimers_consistent.dementia_conditional_prevalence`	\(\delta_D\): Proportion of AD cases in \(D\) state
`cause.alzheimers_consistent.ode_errors`	ODE consistency residuals for validation (should be < 0.01)

Running the Calibration 

from vivarium.framework.artifact import Artifact
from vivarium_csu_alzheimers.data.consistent_rates import generate_consistent_rates

art = Artifact("path/to/artifact.hdf")
generate_consistent_rates(art)  # Takes several minutes

Deriving a disability weight for MCI 

Todo

Derive the formula for the disability weight of MCI, and include Abie’s plot comparing DWs of various relevant health states.