Dietary Iron Deficiency (Iron Deficiency Anemia)¶

Disease Description¶

Generally, anemia is a condition defined by a deficiency of red blood cells or a deficiency of hemoglobin in the blood. Anemia is typically classified by hemoglobin concentrations below a defined threshold that varies by age and sex. Severity of anemia is similarly classified according to ranges of hemoglobin concentrations. Anemia is associated with increased morbidity and mortality and symptoms of anemia often include weakness, fatigue, and difficulty concentrating [Kassebaum-et-al-2016].

Notably, anemia may be caused by many diverse factors. Examples of factors that may cause anemia include genetic mutations in hemoglobin genes, acute or chronic blood loss, altered red blood cell morphology, inadequate nutritional intake, and others [Kassebaum-et-al-2016].

Iron deficiency anemia is a type of anemia that is due to insufficient iron levels, which lead to a deficiency of hemoglobin in the blood. Notably, iron deficiency anemia can occur when dietary intake of iron is insufficient, although it may occur in other situations as well, such as when iron is lost through bleeding (ex: menstrual disorders, hookworm disease, etc.). Iron deficiency anemia is the most common cause of anemia globally in most populations.

Dietary iron deficiency anemia is a specific type of iron deficiency anemia that is due to inadequate dietary intake of iron, leading to inadequate iron levels in the body and a subsequent deficiency of hemoglobin in the blood.

Modeling Iron Deficiency in GBD 2017¶

In GBD 2017, there is an anemia impairment that represents all forms of anemia that are attributable to several causes, including causes such as hemoglobinopathies and hemolytic anemias that are not considered iron deficiency anemias.

The dietary iron deficiency cause is a population attributable fraction (PAF) of 1 cause with the iron deficiency risk factor. This means that 100% of the dietary iron deficiency cases are attributable to the iron deficiency risk factor. Notably, the iron deficiency risk factor affects maternal disorder causes, although these relationships are outside of the scope of this document.

Anemia Impairment¶

The anemia impairment in GBD 2017 represents the total prevalence of anemia due to all causes modeled in GBD (ex: dietary iron deficiency anemia, anemia due to maternal hemorrhage, sickle cell anemia, etc.). Estimating the total prevalence of the anemia impairment for a given population is the first step in modeling anemia in GBD 2017. This is done by fitting a distribution of hemoglobin levels for that population from primary input data based on the population’s hemoclobin concentration mean and standard deviation. For GBD 2017, an ensemble distribution was used, which was 40% gamma and 60% mirror gumbel. Envelope source code.

Once a distribution is fit to hemoglobin levels for a particular age-, sex-, and location-specific demographic group, the prevalence of anemia (by severity level) in each group is determined by the WHO hemoglobin thresholds defined in the following table.

WHO Hemoglobin Thresholds (g/L) [Kassebaum-et-al-2016]¶
Group	Mild Anemia	Moderate Anemia	Severe Anemia
Males and Females <1 month	150	130	90
Males and Females 1 month - 4 years	110	100	70
Males and Females 5-14 years	115	110	80
Males 15+ years	130	110	80
Females 15+ years, non-pregnant	120	110	80
Females 15+ years, pregnant	110	80	70

Note

The threshold values in this table are not inclusive. For instance, if a male less than one month old has a hemoglobin level of exactly 130 g/L, he is not considered mildly anemic. If he has a hemoglobin level of 129 g/ L, he is considered moderately anemic.

The prevalence of anemia as calculated in the process described above serves as the overall anemia envelope for a age-, sex-, and location-specific demographic groups, and prevalent cases of anemia in the anemia envelope are then causally attributed to various causes in GBD 2017 that have anemia as seqeulae. This is done through a process described in the [GBD-2017-YLD-Appendix-IDA]. Notably, the causes iron deficiency anemia, other infectious diseases, other neglected tropical diseases, other hemoglobinopathies andhemolytic anemias, and other endorcine, nutrition, blood, and immune disorders are not directly modeled via the causal attribution process. Rather, these causes were allocated to the residual anemia envelope following the causal attribution process for all other anemic causes [GBD-2017-YLD-Appendix-IDA]. A minimum of 10% of all anemia was assigned to residual categories (a figure selected based on data from the United States) [GBD-2017-YLD-Appendix-IDA].

Notably, early neonatal and late neonatal age groups (age group IDs 2 and 3) are excluded from this process; instead, these age groups are assigned the anemia prevalence from the postneonatal age group (age group ID 4).

Additionally, a pregnancy correction is performed for women of reproductive age. Therefore, additional considerations beyond the scope of the current documentation will need to be made if planning to model hemoglobin among women of reproductive age.

Dietary Iron Deficiency Cause¶

The dietary iron deficiency cause in GBD 2017 is 100% attributable to the iron deficiency risk factor. The dietary iron deficiency cause in GBD is a YLD-only cause, meaning that it contributes to morbidity, but not mortality.

Modeling Strategy for the Dietary Iron Deficiency Cause¶

As noted above, the dietary iron deficiency cause in GBD 2017 is not modeled directly. Rather, it is assigned to a portion of the residual anemia envelope following causal attribution of other anemic causes.

Cause Hierarchy¶

Health States and Sequela¶

The sequela associated with the dietary iron deficiency cause in GBD 2017 include mild iron deficiency anemia, moderate iron deficiency anemia, and severe iron deficiency anemia. The severity of iron deficiency anemia is determined by the WHO age- and sex- specific hemoglobin concentrations, as described in the WHO hemoglobin tresholds table.

Iron Deficiency Risk Factor¶

In GBD 2017, the iron deficiency risk factor is used for two applications. The first is the PAF of 1 relationship with the dietary iron deficiency anemia cause and the second is a risk outcome relationship with maternal disorder causes.

Notably, the iron deficiency risk factor in GBD 2017 represents the age-, sex-, and location-specific mean hemoglobin concentration among the total population. The mean value for the iron deficiency risk factor is stored under modelable entity ID 10487 (also REI ID 95) and the standard deviation is stored under modelable entity ID 10488. The iron deficiency risk factor (population hemoglobin concentration) follows a 40% gamma and 60% mirror Gumbel ensemble distribution.

NOTE:

The values stored in the iron deficiency risk factor (i.e. population hemoglobin concentration parameters) are used in the GBD modeling process to calculate risk-deleted population hemoglobin concentration where the risk is all iron deficiency (i.e. iron responsive anemias), which serves as the population TMREL, in order to calculate the population attributable fraction between the iron deficiency risk factor and maternal disorder causes. See the GBD 2017 Risk Factor Methods Appendix for more information. However, this process is not relevant for the simulation science team use of the iron deficiency risk factor as it relates to the dietary iron deficiency cause in GBD 2017.

Todo

Add citation for the GBD risk factor methods appendix.

Risk Factor Hierarchy¶

../../../_images/iron_risk_hierarchy2.svg

Iron Responsive Anemias in GBD¶

Notably, not all causes of anemia in the GBD anemia impairment are considered iron responsive (i.e. will respond to iron supplementation). A list of causes with iron responsive anemia health states along with their cause and anemia-afflicated sequelae IDs are included in the table below.

Iron Responsive Anemia Causes¶
Cause	Cause ID	Anemia-Afflicated Sequela ID
Dietary Iron Deficiency	390	206, 207, 208
Endocrine, Metabolic, Blood, and Immune Disorders	619	537, 538, 539
Uterine Fibroids	604	1106, 1107, 1108
Other Gynecological Diseases	612	525, 526, 527
Hookworm disease	363	172, 173, 174
Schistosomiasis	351	144, 145, 146
Other Neglected Tropical Diseases	365	177, 178, 179
Other Unspecified Infectious Diseases	408	240, 241, 242
Maternal Hemorrhage	367	182, 183, 184
Vitamin A Deficiency	389	5393, 5396, 5399
Peptic Ulcer Disease	527	4952, 4955, 4958, 4961, 4964, 4967, 4976, 4979, 4982, 5627, 5630, 5633, 7202, 7205, 7208
Gastritis and Duodenitis	528	4985, 4988, 4991, 4994, 4997, 5000, 5009, 5012, 5015, 5678, 5681, 5684, 7214, 7217, 7220
Chronic Kidney Disease	589 (591, 592, 593, 997, 998)	1004, 1005, 1006, 1008, 1009, 1010, 1012, 1013, 1014, 1016, 1017, 1018, 1020, 1021, 1022, 1024, 1025, 1026, 1028, 1029, 1030, 1032, 1033, 1034, 1361, 1364, 1367, 1373, 1376, 1379, 1385, 1388, 1391, 1397, 1400, 1403, 1409, 1412, 1415, 1421, 1424, 1427, 1433, 1436, 1439, 1445, 1448, 1451, 5213, 5216, 5219, 5222, 5225, 5228, 5237, 5240, 5243, 5246, 5249, 5252, 5261, 5264, 5267, 5270, 5273, 5276

Note

According to the GBD modelers, ESRD - Dialysis, Crohn’s disease, and ulcerative colitis were also included in this list, although there do not appear to be results for these causes in GBD 2017. Additionally, according to the GBD modelers, cirrhosis should be included in this list, although there do not appear to be any anemia-afflicted sequelae with results in GBD 2017 within any of the cirrhosis causes.

Vivarium Modeling Strategy¶

Model Scope¶

The scope of the Vivarium modeling strategy detailed in this document is to sample the hemoglobin concentration for an individual simulant (who is not a woman of reproductive age) and evaluate if that simulant’s hemoglobin concentration will respond to iron supplementation (iron responsive).

The modeling strategy detailed in this document aims to evaluate all iron responsive anemias (collection of causes) rather than the singular cause of dietary iron deficiency anemia. However, the modeling strategy described in this docuemnt can be modified to include only dietary iron deficiency anemia ( PAF of 1 cause), if desired, by assuming that dietary iron deficiency anemia is the only iron responsive cause of anemia, rather than all of the causes listed in the iron responsive anemia causes table and that the dietary iron deficiency anemia sequelae are the only iron responsive anemia sequelae in the anemia sequelae IDs table.

Note

The Vivarium modeling strategy described here is a strategy to model the PAF-of-one GBD cause dietary iron deficiency (attributable to the iron deficiency risk factor). The modeling strategy described here does not consider the realtionship between the GBD iron deficiency risk factor and other causes (i.e. maternal disorders).

Initialization¶

At the start of a Vivarium simulation, each simulant must be initalized with two parameters, including 1) a hemoglobin concentration, and 2) an indicator of whether the simulant will respond to iron supplementation. Details on how to intialize these parameters are included in the following sections.

Notably, the initialization of a simulant’s hemoglobin concentration should occur before the initialization of iron responsiveness.

Todo

Confirm order in which initialization occurs with research team, and explain reasoning. (More simple to assign a hemoglobin concentration based full hemoglobin distribution; bounds are universally between zero and 1. Also fits more easily with propensity score approach.)

Hemoglobin Concentration¶

In order to initialize an individual’s hemoglobin concentration, each simulant should be assigned a random number between 0 and 1 (random_number_i). This number will represent the percentile of hemoglobin concentration for that individual simulant relative to the baseline population distribution of hemoglobin concentrations (from the GBD iron deficiency risk factor rei_92) for the remainder of the simulation. The corresponding hemoglobin concentration for that percentile should then be assigned to the simulant using the methodology described in the reaminder of this section.

Any shifts in hemoglobin concentration (due to baseline coverage or intervention effects) should be applied after an individual’s hemoglobin concentration is sampled from the population distribution as described above. The post-shift hemoglobin concentration will then act as the simulant’s assigned hemoglobin concentration.

Notably, because the mean and standard deviation for the population hemoglobin concentration varies by age group, an individual’s assigned hemoglobin concentration will vary as they transition between age groups, although their assigned percentile within that population hemoglobin concentration distribution will not vary as the simulant ages.

The ensemble distribution of population hemoglobin concentrations can be recreated with the following equations and code:

Population Hemoglobin Parameters¶
Parameter	Value	Note
hemoglobin_mean	rei_92_exposure	meid_10487
hemoglobin_sd	rei_92_sd	meid_10488
w_gamma	0.4	Ensemble weight for gamma distribution
w_mirror_gumbel	0.6	Ensemble weight for mirror gumbel distribution
eulers_constant	0.57721566
xmax	220	Defined by GBD anemia modelers
pi	3.14…..	Use math.pi for all significant figures
gamma_shape	(hemoglobin_mean)^2 / (hemoglobin_sd)^2
gamma_rate	(hemoglobin_mean) / (hemoglobin_sd)^2
mirror_gumbel_alpha	xmax - (hemoglobin_mean) - eulers_constant * (hemoglobin_sd) * sqrt(6) / pi
mirror_gumbel_scale	(hemoglobin_sd) * sqrt(6) / pi
random_number_i	random number between 0 and 1	Assigned to an individual simulant

import scipy.stats


# TO-DO: WRITE SOME CODE THAT ACCURATELY SAMPLES FROM THE ENSEMBLE DIST.
# BASED ON ASSIGNED RANDOM PERCENTILE

Iron Responsiveness¶

As previously discussed, not all anemias in the anemia impairment are iron deficiency anemias, meaning that not all anemias will respond to iron supplementation. Therefore, the probability that a simulant with mild, moderate, or severe anemia (based on their sampled hemoglobin concentration and WHO anemia threshold values) will respond to iron supplementation/ fortification can be measured by:

\[\frac{\text{prevalence}_\text{iron responsive anemia}}{\text{prevalence}_\text{total anemia}}\]

Where prevalence_iron_responsive_anemia and prevalence_total_anemia are equal to the severity-, age-, sex-, and location-specific prevalence (from COMO) summed across all iron responsive anemia and all total anemia sequela IDs, respectively. Sequela IDs for each category are listed in the table below.

Sequela IDs¶
Anemia Severity	All Anemia Sequela	Iron Responsive Anemia Sequela
Mild	144, 172, 177, 182, 206, 240, 525, 531, 537, 645, 648, 651, 654, 1016, 1024, 1032, 1057, 1061, 1065, 1069, 1079, 1089, 1099, 1106, 1120, 1373, 1385, 1397, 1421, 1433, 1445, 4952, 4955, 4976, 4985, 4988, 5009, 5018, 5027, 5036, 5051, 5063, 5075, 5087, 5099, 5111, 5123, 5225, 5228, 5249, 5252, 5273, 5276, 5393, 5567, 5579, 5606, 5627, 5678, 7202, 7214	144, 172, 177, 182, 206, 240, 525, 537, 1016, 1024, 1032, 1106, 1373, 1385, 1397, 1421, 1433, 1445, 4952, 4955, 4976, 4985, 4988, 5009, 5225, 5228, 5249, 5252, 5273, 5276, 5393, 5567, 5579, 5627, 5678, 7202, 7214
Moderate	145, 173, 178, 183, 207, 241, 526, 532, 538, 646, 649, 652, 655, 1017, 1025, 1033, 1058, 1062, 1066, 1070, 1080, 1090, 1100, 1107, 1121, 1376, 1388, 1400, 1424, 1436, 1448, 4958, 4961, 4979, 4991, 4994, 5012, 5021, 5030, 5039, 5054, 5066, 5078, 5090, 5102, 5114, 5126, 5219, 5222, 5243, 5246, 5267, 5270, 5396, 5570, 5582, 5609, 5630, 5681, 7205, 7217	145, 173, 178, 183, 207, 241, 526, 538, 1017, 1025, 1033, 1107, 1376, 1388, 1400, 1424, 1436, 1448, 4958, 4961, 4979, 4991, 4994, 5012, 5219, 5222, 5243, 5246, 5267, 5270, 5396, 5570, 5582, 5630, 5681, 7205, 7217
Severe	146, 174, 179, 184, 208, 242, 527, 533, 539, 647, 650, 653, 656, 1018, 1026, 1034, 1059, 1060, 1063, 1064, 1067, 1068, 1071, 1074, 1075, 1077, 1081, 1083, 1085, 1087, 1091, 1093, 1095, 1097, 1101, 1108, 1122, 1379, 1391, 1403, 1427, 1439, 1451, 4964, 4967, 4982, 4997, 5000, 5015, 5024, 5033, 5042, 5057, 5069, 5081, 5093, 5105, 5117, 5129, 5213, 5216, 5237, 5240, 5261, 5264, 5399, 5573, 5585, 5612, 5633, 5684, 7208, 7220	146, 174, 179, 184, 208, 242, 527, 539, 1018, 1026, 1034, 1108, 1379, 1391, 1403, 1427, 1439, 1451, 4964, 4967, 4982, 4997, 5000, 5015, 5213, 5216, 5237, 5240, 5261, 5264, 5399, 5573, 5585, 5633, 5684, 7208, 7220

Therefore, each simulant should be initialized as either iron responsive (1) or non-iron responsive (0) according to the following rules:

if hb_i < severe_threshold:
        if random_number_i =< prevalence_severe_ira / prevalence_total_severe_anemia:
                iron_responsive_i = 1
        else:
                iron_responsive_i = 0
elif hb_i < moderate_threshold:
        if random_number_i =< prevalence_moderate_ira / prevalence_total_moderate_anemia:
                iron_responsive_i = 1
        else:
                iron_responsive_i = 0
elif hb_i < mild_threshold:
        if random_number_i =< prevalence_mild_ira / prevalence_total_mild_anemia:
                iron_responsive_i = 1
        else:
                iron_responsive_i = 0
else:
        iron_responsive_i = 1

# NOTE: use <, not =< for anemia thresholds

Note

Final else statement indicates that we assume all individuals without anemia will be iron responsive. See calculations that investigate this assumption here.

Where:

Parameters¶
Parameter	Description	Note
hb_i	An individual simulant’s hemoglobin distribution	Sampled from population hemoglobin distribution
random_number_i	An independent random number between 0 and 1 assigned to an individual simulant	Generated in Vivarium
iron_responsive_i	An individual simulant’s value for the iron responsive indicator variable	1=iron responsive, 0=not iron responsive
{severity}_threshold	Age-, sex-, and severity-specific hemoglobin anemia threshold	Defined in WHO treshold table
prevalence_{severity}_ira	Severity-specific prevalence of iron responsive anemia	Sum of severity-specific iron responsive anemia sequelae
prevalence_total_{severity}_anemia	Severity-specific prevalence of all anemia	Sum of all severity-specific anemia sequelae
prevalence_c_316	Prevalence of hemoglobinopathies and hemolytic anemias	COMO (NOT a most detailed cause)
prevalence_c_298	Prevalence of HIV/AIDs	COMO (NOT a most detailed cause)
prevalence_c_345	Prevalence of malaria	COMO (most detailed cause)

Then, effect sizes for iron supplementation or fortification interventions as shifts in mean hemoglobin concentrations should be applied only to those who are initialized in the model as iron responsive (iron_responsive_i = 1) based on the methodology described here.

Todo

Describe how to handle the changing prev_ira/prev_total_anemia across age groups, which generally decreases by about 2% between the early NN and 1-4 year age groups for locations of interest (India, Nigeria, Ethiopia). (Also, cite this 2% statistic.)

Tentative approach: initialize only once using the minimum probability of iron responsiveness across all age groups in simulation.

Confirm with research team.

Other Model Notes/Strategies¶

Neonatal Age Groups¶

Neonatal age groups should be excluded from the process described in this document. Simulants should be initialized with a hemoglobin value and an iron responsiveness indicator at the start of the simulation and/or when they age into the postneonatal age group (age group ID #4). The reasoning behind this is as follows:

GBD did not directly collect data for neonates but instead just copied estimates from the post-neonatal age group
The GBD modelers warned us that there might be problems with this approach (anemia impairment prevalence will likely not agree with anemia prevalence calculated from hemoglobin distribution). Based on an initial investigation, the anemia prevalence for the neonatal age groups do not validate to GBD anemia impairment prevalence as well as for the other under five age groups (see here)
We are currently using this model of iron responsive anemia to estimate the effect of large-scale food fortification with iron, and we so far do not have data to support that neonates’ hemoglobin concentrations are affected by their mothers consumption of iron-fortified foods
The neonatal period lasts only 28 days and therefore has a relatively
small impact on YLDs and anemia in GBD does not affect mortality among neonates
Since there may be problems with the data, and we don’t know whether our intervention should affect neonates anyway, the simplest thing is to not bother modeling hemoglobin levels in neonates

Note

If upon further investigation the hemoglobin distribution among neonates can be validated to the GBD anemia impairment prevalence and it is applicable for a given simulation/intervention, the model may be adjusted accordingly.

Tracking Years Lived with Disability due to Anemia¶

Person time in mild, moderate, and severe anemic states (based on assigned simulant hemoglobin concentration and age- and sex- specific hemoglobin thresholds for anemia) should be tracked in the Vivarium simulation and multiplied by the severity-spefic anemia disability weight to obtain a measure of YLDs due to anemia in the model. This can be done for all anemia or iron-responsive anemias (among simulants with iron_responsive_i = 1 only), depending on model needs (use corresponding anemia prevalences for validation).

Model Assumptions and Limitations¶

If any causes with anemia health state sequelae are included in the Vivarium simulation, any disability associated with anemic sequela of that cause will be counted both through the process described in this document as well as through the disability weight associated with that cause. The impact of this double counting should be considered when this is the case before implementation in a model and recorded as a model limitation if applicable.

Our approach of assigning individual simulants propensity scores (percentiles within the population hemoglobin concentration distribution) is a limitation of our modeling strategy in that it assumes that this remains constant over time and age groups.

We assume that all individuals who are not anemic are iron responsive. This assumption is based on two additional assumptions, including a) asymptomatic individuals with non-iron-responsive causes will remain asymptomatic, and b) the prevalence of individuals with non-iron-responsive causes without anemia is neglible (see a notebook investigating the prevalence of these groups here).

These assumptions may cause our model to improve the hemoglobin status via an iron supplementaion/foritifaction intervention among simulants who may not actually respond to iron supplmentation and otherwise would be borderline non-anemic based on their hemoglobin threshold (and may go on to develop mild anemia as they age and the hemoglobin distribution and anemia threshold values change). Therefore, this assumption may slightly overestimate the impact of iron supplementation/fortification interventions.

Because we are not modeling individual causes of anemia (and their associated mortality), we assume that all simulants have the same mortality rate regardless of their hemoglobin value, when in reality, those with lower hemoglobin values will have higher mortality rates than those with higher hemoglobin values. Notably, deaths due to causes with iron responsive anemia sequelae account for approximately 1.1% of deaths in the first five years of life globally (see the calculations here).

Because hemoglobin concentrations are not directly modeled among the early and late neonatal age groups in GBD, the prevalence of mild, moderate, and severe anemia are assumed to be equal to the prevalence in the post-neonatal age group. Therefore, this model is limited when applied to neonatal age groups.

The modeling strategy currently described in this document does not consider the effect of pregnancy on hemoglobin concentration and therefore is limited in that is should not be used to model women of reproductive age.

The modeling strategy both as conducted by the GBD modelers and as described in this document assume a constant shape and standard deviation in the hemoglobin distribution throughout the modeling process. This is a limitation of our modeling strategy in that we assume the distribution before a shift is applied maintains the same shape after a shift due to the intervention is applied.

Essentially, both the GBD modeling process and our Vivarium implementation assume that hemoglobin shifts are constant regardless of an individual’s starting hemoglobin concentration. The implication is that the shift applies uniformly to all individuals regardless of their position in the hemoglobin distribution (imagine just pushing a normal distribution to the right, but its shape stays the same). However, we might actually expect the left end of the distribution to get pushed farther than the right end of the distribution in reality, so the shape would change from a normal distribution to a more lop-sided distribution if this were to happen (we assume it doesn’t). Note, however, that the the shape is only maintained in our simulation when stratified by iron responsiveness (because non-iron-responsive individuals will not get pushed at all).

Further, the model is limited due to GBD not directly modeling the prevalence of dietary iron deficiency, which may cause error in the estimation of the prevalence of this cause.

Validation Criteria¶

The overall prevalence and YLDs of anemia should be equal between:

The anemia impairment (rei_192 for all anemia, rei_205 for mild anemia, rei_206 for moderate anemia, and rei_207 for severe anemia)
The sum across all anemia sequlae (overall and severity-specific)
The result of anemia prevalence calculated from the population hemoglobin distribution as described in the modeling strategy for prevalence, and prevalence multiplied by the disability weight(s) for YLDs (overall and severity-specific)

The prevalence of anemia using the population hemoglobin distribution can be calculated using the code below using the parameters defined earlier in this document and assuming age- and sex- specific anemia_threshold values as defined in the WHO hemoglobin tresholds table and using the parameters defined in the population hemoglobin parameters table:

Note

The following code has been validated for the age groups and locations relevant to the Large Scale Food fortification project (1 month - 5 years ( age group IDs 4 and 5) in India, Ethiopia, and Nigeria). See the relevant notebook here. However, it requires further validation before application to additional demographic groups, as this method did not validate for all age groups or locations.

import scipy.stats


# overall anemia prevalence
gamma_prev = scipy.stats.gamma(gamma_shape, loc=0,
                        scale=1/gamma_rate).cdf(mild_anemia_threshold)
mirror_gumbel_prev = 1 - scipy.stats.gumbel_r(mirror_gumbel_alpha,
                        mirror_gumbel_scale).cdf(xmax - mild_anemia_threshold)
ensemble_prev = w_gamma * gamma_prev + w_mirror_gumbel * mirror_gumbel_prev


# severe anemia prevalence
gamma_severe_prev = scipy.stats.gamma(gamma_shape, loc=0,
                        scale=1/gamma_rate).cdf(severe_anemia_threshold)
mirror_gumbel_severe_prev = 1 - scipy.stats.gumbel_r(mirror_gumbel_alpha,
                        mirror_gumbel_scale).cdf(xmax - severe_anemia_threshold)
ensemble_severe_prev = w_gamma * gamma_severe_prev + w_mirror_gumbel * mirror_gumbel_severe_prev


# moderate anemia prevalence
gamma_moderate_prev = scipy.stats.gamma(gamma_shape, loc=0,
                        scale=1/gamma_rate).cdf(moderate_anemia_threshold) - gamma_severe_prev
mirror_moderate_severe_prev = 1 - scipy.stats.gumbel_r(mirror_gumbel_alpha,
                        mirror_gumbel_scale).cdf(xmax - moderate_anemia_threshold) - gamma_severe_prev
ensemble_moderate_prev = w_gamma * gamma_moderate_prev + w_mirror_gumbel * mirror_gumbel_moderate_prev


# mild anemia prevalence
gamma_mild_prev = scipy.stats.gamma(gamma_shape, loc=0,
                        scale=1/gamma_rate).cdf(mild_anemia_threshold) - gamma_moderate_prev
mirror_mild_severe_prev = 1 - scipy.stats.gumbel_r(mirror_gumbel_alpha,
                        mirror_gumbel_scale).cdf(xmax - mild_anemia_threshold) - gamma_moderate_prev
ensemble_mild_prev = w_gamma * gamma_mild_prev + w_mirror_gumbel * mirror_mild_moderate_prev

References¶

Kassebaum-et-al-2016(1,2,3): View Kassebaum et al. 2016

Kassebaum NJ, GBD 2013 Anemia Collaborators. The Global Burden of Anemia. Hematol Oncol Clin North Am. 2016 Apr;30(2):247-308. doi: https://doi.org/10.1016/j.hoc.2015.11.002

GBD-2017-YLD-Appendix-IDA(1,2,3): Pages 763-774 in Supplementary appendix 1 to the GBD 2017 YLD Capstone:

(GBD 2017 YLD Capstone) GBD 2017 Disease and Injury Incidence and Prevalence Collaborators. Global, regional, and national incidence, prevalence, and years lived with disability for 354 diseases and injuries for 195 countries and territories, 1990–2017: a systematic analysis for the Global Burden of Disease Study 2017. Lancet 2018; 392: 1789–858. DOI: https://doi.org/10.1016/S0140-6736(18)32279-7