Wasting dynamic transition model (GBD 2020)¶

Overview ¶

This page contains information pertaining to the joint risk-cause wasting model. We note that this model is built on the GBD 2020 risk exposure model for wasting, and the GBD 2020 protein energy malnutrition (PEM) cause model. GBD stratifies wasting into four categories: TMREL, mild, moderate, and severe wasting. All PEM cases are attributed to moderate and severe wasting, making PEM a PAF-of-1 model. Under the GBD framework, wasting is additionally a risk for measles, diarrheal diseases, and lower respiratory infections. These relationships are detailed under a risk effects page for wasting.

Todo

Link to more info on PAF-of-1 models. Does this exist?

Todo

Link to risk effects page for wasting.

In the sections that follow we describe wasting and PEM: both as understood by the literature, and in the context of GBD 2020. We then include information on how we will model wasting and PEM in vivarium for the CIFF-SAM project.

List of abbreviations
AM	Acute malnutrition
MAM	Moderate acute malnutrtion
SAM	Severe acute malnutrition
TMREL	Theoretical minimum risk exposure level
CGF	Child growth failure composed of wasting stunging and underweight
DD	Diarrheal disease
LRI	Lower respiratory tract infection
MSLS	Measles
PEM	Protein energy malnutrition

Wasting background ¶

Acute Malnutrition ¶

Malnutrition is an imbalance between the body’s needs and its use and intake of nutrients. The imbalance can be caused by poor or lacking diet, poor hygiene, disease states, lack of knowledge, and cultural practices, among others. Underweight, stunting, wasting, obesity, and vitamin and mineral deficiencies are all forms of malnutrition.

Acute malnutrition (AM), also referred to as wasting, is recent rapid weight loss or a failure to gain weight that results from illness, lack of appropriate foods, or other underlying causes. Wasting is mostly typically classified in terms of a weight-for-height z-score. Other forms of under or malnutrition include stunting (low height for age), underweight (low weight for age), and micronutrient deficiencies (a lack of key vitamins or minerals).

For an individual, AM is not a chronic condition: children with AM either recover or die and recovered children can relapse to AM. It is measured in weight-for-height z-scores (WFH) which is a comparison of a child’s WFH from the median value of the global reference population. A z-score between -2 to -3 indicates moderate acute malnutrition (MAM) and a z-score below -3 indicate severe acute malnutrition (SAM). WHZ z-scores range from -7 to +7. Although MAM is less severe, it affects a greater number of children and is associated with more nutrition-related deaths than SAM.

Protein-Energy Malnutrition ¶

Protein-energy malnutrition (PEM) is defined to be moderate or severe acute malnutrition, and takes the form of either marasmus, kwashiorkor, or marasmic kwashiorkor. These conditions are diagnosed by clinical findings, with oedema being the primary marker of kwashiorkor. [UpToDate-malnutrition-Wasting] [WHO-Malnutrition-Wasting] [Dipasquale-et-al-Wasting]

Marasmus is caused by inadequate intake of all nutrients, but particularly total calories. It is characterized by a low weight-for-height and reduced mid-upper arm circumference (MUAC), but the precise deifinition is highly debated. Marasmus is often categorized as (1) WHZ score \(i<3\), or (2) WHZ score \(i<3\) and NO oedema. Signs and symptoms include: [UpToDate-malnutrition-Wasting]

Head that appears large relative to the body, with staring eyes

Emaciated and weak appearance

Irritable and fretful affect

Bradycardia, hypotension, and hypothermia

Thin, dry skin

Shrunken arms, thighs, and buttocks, with redundant skin folds caused by loss of subcutaneous fat

Thin, sparse hair

Kwashiorkor is typically defined to be any form of malnutrition with oedema, regardless of WHZ score. Symmetric peripheral pitting oedema begins in the lower parts of the body and moves upwards, often affecting the presacral area, genitalia, and periorbital area, with or without anasarca (severe generalized oedema). The particular cause or nutrient deficit responsible for oedematous malnutrition was originally attributed to protein deficiency, but is now debated. Kwashiorkor also marked muscle atrophy with normal or even increased body fat. Other signs and symptoms include: [UpToDate-malnutrition-Wasting]

Apathetic, listless affect

Rounded prominence of the cheeks (“moon face”)

Pursed appearance of the mouth

Thin, dry, peeling skin with confluent areas of hyperkeratosis and hyperpigmentation

Dry, dull, hypopigmented hair that falls out or is easily plucked

Hepatomegaly (from fatty liver infiltrates)

Distended abdomen with dilated intestinal loops

Bradycardia, hypotension, and hypothermia

Despite generalized edema, most children have loose inner inguinal skin folds

Implications ¶

Children with acute malnutrition are at greater risk of death from diarrhea and other infectious diseases than well-nourished children. They also face greater risk of morbidity from infectious diseases and delayed physical and cognitive development. AM tends to peak during seasonal hunger, disease outbreaks, or during food security ‘shocks’ (e.g. economic or climatic crises) and stresses including humanitarian crises. However, AM is a problem that not only occurs in emergencies, but also can be endemic in development contexts. When untreated, MAM can deteriorate to SAM and possible death. Furthermore, evidence is emerging that repeated episodes of MAM can have a significant impact on stunting; prevention of wasting could potentially increase height in children.

Wasting Exposure in GBD 2020 ¶

Case definition ¶

Wasting, a sub-component indicator of child growth failure (CGF), is based on a categorical definition using the WHO 2006 growth standards for children 0-59 months. Definitions are based on Z-cores from the growth standards, which were derived from an international reference population. Mild, moderate and severe categorical prevalences were estimated for each of the three indicators. Theoretical minimum risk exposure level (TMREL) for wasting was assigned to be greater than or equal to one standard deviation below the mean (-1 SD) of the WHO 2006 standard weight-for-height curve. This has not changed since GBD 2010.

Wasting category definition (range -7 to +7)
TMREL	>= -1
MILD	< -1 to -2 Z score
MAM	< -2 to -3 Z score
SAM	< -3 Z score

Input data ¶

Two types of input data are used in CGF estimation:

1. Tabulated report data. This data does not report individual anthropometric measurements. It only reports the prevalence of forms of CGF in a sample size. For example, this data would may report a 15% prevalence of moderate stunting out of a nationally representative sample of 5,000 children.

2. Microdata. This data does have individual anthropometric measurements. From these datasources, GBD can see entire distributions of CGF, while also collapsing them down to point prevalences like moderate and severe CGF.

Exposure estimation ¶

In modeling CGF, all data types go into ST-GPR modeling. GBD has STGPR models for moderate, severe, and mean stunting, wasting, and underweight. The output of these STGPR models is an estimate of moderate, severe, and mean stunting, wasting, and underweight for all under 5 age groups, all locations, both sexes, and all years.

They also take the microdata sources and fit ensemble distributions to the shapes of the stunting, wasting, and underweight distributions. They thus find characteristic shapes of stunting, wasting, and underweight curves. Once they have ST-GPR output as well as weights that define characteristic curve shapes, the last step is to combine them. They anchor the curves at the mean output from ST-GPR, use the curve shape from the ensemble distribution modeling, and then use an optimization function to find the standard deviation value that allows them to stretch/shrink the curve to best match the moderate and severe CGF estimates from ST-GPR. The final CGF estimates are the area under the curve for this optimized curve.

Note that the z-score ranges from -7 to +7. If we limit ourselves to Z-scores between -4 and +4, we will be excluding a lot of kids.

Note

In the paper that Ryan (GBD modeller for CGF and LWBSG) is working on right now, he presents the first results ever for “extreme” stunting which we define as kids with stunting Z scores below -4. For Ethiopia, that’s about 7% of kids. So it’s non-trivial!

CGF burden does not start until after neonatal age groups (from 1mo onwards). In the neonatal age groups (0-1mo), burden comes from LBWSG. See risk effects page for details on model structure. The literature on interventions for wasting target age groups 6mo onwards. This coincides with the timing of supplementary food introduction. Prior to 6mo, interventions to reduce DALYs focus on breastfeeding and reduction of LBWSG.

Protein Energy Malnutrition in GBD 2019/2020 ¶

Important

We will use PEM 2019 model (with a 2020 wasting model) because PEM 2020 is not completed. Once PEM 2020 is completed (expected July 30th), we will update to a PEM 2020 model.

PEM is responsible for both fatal and nonfatal outcomes within the GBD framework. GBD maintains a cause of death model called “Nutritional deficiencies” that is split into PEM and Other Nutritional Deficiencies that estimates PEM mortality. Nonfatal PEM cases are modelled independently, using the case definition moderate and severe acute malnutrition, defined in terms of weight-for-height Z-scores (WHZ). All PEM cases are attributed to the GBD Child Growth Failure risk factor, which is not detailed here. We include specifics on the PEM cause models below. [GBD-2019-Capstone-Appendix-Wasting], p789.

PEM Fatal Model ¶

GBD runs a parent CODEm model to estimate deaths attributable to nutritional deficiency, using vital registration and verbal autopsy data as inputs. The applicable ICD codes are as follows: [GBD-2019-Capstone-Appendix-Wasting]

PEM CoD ICD-10 Codes¶
GBD Cause	ICD-10 Code
Protein-energy malnutrition	E40-E46.9 (Kwashiorkor, marasmus, specified and unspecified proteincalorie malnutrition)
Other nutritional deficiencies	D51-D52.0 (vitamin B12 deficiency anaemia and folate deficiency anaemia)
Other nutritional deficiencies	D52.8-D53.9 (other nutritional anaemias)
Other nutritional deficiencies	D64.3 (other sideroblastic anaemias)
Other nutritional deficiencies	E51-E61.9 (thiamine, niacin, other B group vitamins, ascorbic acid, vitamin D, other vitamin, dietary calcium, dietary selenium, dietary zinc, and other nutrient element deficiencies)
Other nutritional deficiencies	E63-E64.0 (other nutritional deficiencies and sequelae of protein-calorie malnutrition)
Other nutritional deficiencies	E64.2-E64.9 (sequelae of vitamin C deficiency, rickets, other nutritional deficiencies, and unspecified nutritional deficiencies)
Other nutritional deficiencies	M12.1-M12.19 (Kashin-Beck disease)
Garbage code	D50, D50.0 and D50.9 (unspecified anaemia)

They then run (1) an under-5 PEM model, (2) a 5-and-over PEM model, and (3) an other nutritional deficiencies model. These models are scaled using CODCorrect to fit the parent nutritional deficiency model. [GBD-2019-Capstone-Appendix-Wasting]

Note that as PEM is defined as “a lack of dietary protein and/or energy”, it includes famines and severe droughts. These result in discontinuities in PEM estimation, which the GBD team accounts for. The appendix specifically mentions using the Tombstone report to estimate deaths due to the famine during the Great Leap Forward in the 1960s in China. [GBD-2019-Capstone-Appendix-Wasting]

PEM Nonfatal Model ¶

GBD’s nonfatal PEM model takes as its case definition “moderate and severe acute malnutrition”, defined in terms of distance from the mean WHZ score given by the WHO 2006 growth standard for children. The relevant ICD 10 codes are E40-E46.9, E64.0, and ICD 9 codes are 260-263.9. PEM is partitioned into the following four sequelae: [GBD-2019-Capstone-Appendix-Wasting]

Nonfatal PEM Sequelae 2019/2020¶
Sequela Name	WHZ range	Clinical description	Disability weights
Moderate wasting without oedema	{WHZ_i \| -3SD < WHZ_i < -2SD}	Asymptomatic	NA
Moderate wasting with oedema	{WHZ_i \| -3SD < WHZ_i < -2SD}	Is very tired and irritable and has diarrhoea	0.051 (0.031–0.079)
Severe wasting without oedema	{WHZ_i \| WHZ_i < -3SD}	Is extremely skinny and has no energy.	0.128 (0.082–0.183)
Severe wasting with oedema	{WHZ_i \| WHZ_i < -3SD}	Is very tired and irritable and has diarrhoea. Is extremely skinny and has no energy.	0.051 (0.031–0.079); 0.128 (0.082–0.183). Applied multiplicatively.

These are mapped onto clinically-defined wasting states as follows:

Clinical definitions 2019/2020¶
Condition	Estimated by GBD sequelae
Kwashiorkor	{Moderate wasting with oedema} + {Severe wasting with oedema}
Marasmus	{Severe wasting without oedema} + {Severe wasting with oedema}

The above table represents GBD definitions. In the literature these definitions are highly debated, often defining marasmus as strictly “severe wasting without oedema”.

The nonfatal estimation pipeline comprises five models:

Nonfatal PEM sub-models 2019/2020¶
Modeled entity	Age	Modeling software
Prevalence of WHZ <-2SD	under-5	STGPR
Prevalence of WHZ <-3SD	under-5	STGPR
Proportion of WHZ <-2SD with oedema	under-5	DisMod
Proportion of WHZ <-3SD with oedema	under-5	DisMod
All WHZ <-2SD (PEM)	All ages	DisMod

For the all-age model, they set the duration of PEM to 9 months after consulting with nutrition experts. The current modelers (as of June 2021 no longer have documentation of these conversations, which took place sometime before 2015). They used a remission rate of 0.25 - 1.25 (remitted cases of PEM per person-year of illness). Note this is a rather wide interval that allowed DisMod to choose a remission rate within the given bounds based on other input data. [GBD-2019-Capstone-Appendix-Wasting]

From the all-age model, they then derived (1) a prevalence:incidence ratio that was applied across all categories of non-fatal PEM, and (2) a moderate:severe wasting ratio for both under and over 5. [GBD-2019-Capstone-Appendix-Wasting]

Todo

What do the modelers do with this mod:sev ratio? How do they get estimates for 5+?

The modelers then assumed that there is zero prevalence of oedema in anyone over 5. [GBD-2019-Capstone-Appendix-Wasting]

Additionally, they calculated the fraction of wasting attributable to severe worm infestation and subtracted this out of all wasting, attributing the remainder to PEM. They assumed no oedema due to worms, and the prevalence:incidence ratio derived from the all-age PEM model. [GBD-2019-Capstone-Appendix-Wasting]

The modelers used child anthropometry data from health surveys, literature, and national reports, from which they estimate the WHZ SDs that correspond with the case definitions. They additionally used SMART datasets to estiamte the proportion under 5 with oedema. In the GBD 2019 Appendix, they note, “Future work in systematically evaluating longitudinal datasets on nutrition and growth failure will allow us to improve the empirical basis for PEM incidence estimates, including improved resolution for the component categories.” [GBD-2019-Capstone-Appendix-Wasting]

Cause Hierarchy ¶

Vivarium Modeling Strategy ¶

We will model wasting in four compartments: TMREL, Mild, Moderate, and Severe. In a given timestep a simulant will either stay put, transition to an adjacent wasting category, or die. In this case of “CAT 1: severe wasting”, simulants can also transition to “CAT 3: Mild wasting” via a treatment arrow, t1.

We will use the GBD 2020 wasting and PEM models to inform this model, in addition to data found in the literature. We will derive the remaining transition rates from a Markov chain model, described in further detail below. Simulants in each wasting category will receive a corresponding relative risk for diarrheal diseases, measles, lower respiratory infections. The vivarium models for these three causes will draw from the corresponding GBD 2019 models, as GBD 2020 is not yet complete at this time (July 2021), and will be subject to updates and reruns. In addition, current scatters indicate that (unlike wasting and PEM), LRI, diarrhea and measles have not undergone significant changes between GBD rounds 2019 and 2020.

Assumptions and Limitations ¶

Todo

Describe the clinical and mathematical assumptions made for this cause model, and the limitations these assumptions impose on the applicability of the model. Flesh out list below.

Markov chain assumption is flawed (remission / incidence isn’t constant over time / memoryless).
Seasonality of data
Unclear if our input data that informs “time to recovery from SAM” ought to be “time to recovery or death from SAM”

Input data ¶

GBD and literature sources ¶

Todo

@Ninicorn will you help fill out this table? i.e. the sources for the remission rates

Wasting model input data sources¶
Variable	Source
TMREL prevalence	GBD wasting model
Mild wasting prevalence	GBD wasting model
MAM prevalence	GBD wasting model
SAM prevalence	GBD wasting model
TMREL mortality rate	Derived from GBD
Mild wasting mortality rate	Derived from GBD
MAM mortality rate	Derived from GBD
SAM mortality rate	Derived from GBD
Incidence of mild wasting from TMREL	Derived using a Markov model
Incidence of MAM from mild wasting	Derived using a Markov model
Incidence of SAM from MAM	Derived using a Markov model
Remission from mild wasting to TMREL
Remission from MAM to mild wasting
Remission from SAM to MAM
Treated remission from SAM to mild wasting
Probability of staying in TMREL	Derived using a Markov model
Probability of staying in Mild wasting	Derived using a Markov model
Probability of staying in MAM	Derived using a Markov model
Probability of staying in SAM	Derived using a Markov model

Deriving wasting transition probabilities ¶

Wasting model ¶

Important

We will model wasting transitions and risk effects only among simulants at least six months of age. Simulants should be initialized into a wasting model state at birth with a birth prevalence equal to the wasting risk exposure among the 1-5 month age group (age_group_id=388, or the postneonatal age_group_id=4 if using GBD 2019 instead of GBD 2020).

All wasting transition rates should equal zero among all ages under 6 months. The relative risks for each wasting risk exposure category and each risk/outcome pair should equal one for all ages under 6 months.

Wasting transition rates should be informed by the data tables below for ages over 6 months. Wasting risk effects for ages over 6 months should be informed by the standard GBD wasting relative risks.

NOTE: When the birthweight and wasting risk exposure at birth correlation is implemented, it will cause simulants with a greater neonatal mortality (due to brithweight exposure) to be initialized into more severe wasting states. This will cause the wasting exposure distribution to shift to less severe wasting states over the neonatal period as simulants with lower birthweights (and more severe wasting states due to the birthweight and wasting exposure correlation) die. The magnitude of the bias introduced by this modeling strategy should be investigated upon implementation to determine if different modeling strategies are necessary. This should be done by comparing the wasting exposure and wasting-affected outcomes in the simulation output to the GBD inputs by age group.

NOTE: The modeling decision not to model wasting transitions among simulants less than six months of age is due to the reliance of the wasting model transition rates on the wasting treatment model and the lack of data to inform treatment-related transition rates among this age group. Note that a sensitivity analysis scenario that includes infants less than six months of age in the treatment model may be performed in the future.

This Markov model comprises 5 compartments: four wasting categories, plus CAT 0. Because we need simulants to die at a higher rate out of CAT 1 than CAT 2, 3, or the TMREL, it is necessary to include death to correctly derive our transition rates. Thus we allow simulants to die into CAT 0. However, because we need to assume equilibrium of our system over time, we allow simulants to “age in” to CATs 1-4, out of CAT 0. We thus set the transition probabilies \(f_i\) equal to the prevalence of the four wasting categories, obtained from GBD.

It is important here to note first that \(f_i\) don’t represent fertility rates: rather, if \(k_i\) sims died in timestep \(k\), we allow \(k_i\) sims to age in in timestep \(k+1\), to replenish those that died. Second, we emphasize that we utilize this method in order to calculate transition probabilities between the different wasting categories. However, the final Vivarium model of wasting will not include a reincarnation pool.

Here we include equations for the transition probabilities, and in the section that follows we will detail how to calculate all the variables used

Wasting transition probability equations¶
Variable	Equation	Description	Source
i1	ap0f2/ap2 + ap0f3/ap2 + ap0f4/ap2 + ap1r2/ap2 + ap1t1/ap2 - d2 - ap3d3/ap2 - ap4*d4/ap2	Daily probability of incidence into cat 1 from cat 2	System of equations
i2	ap0f3/ap3 + ap0f4/ap3 + ap1t1/ap3 + ap2r3/ap3 - d3 - ap4*d4/ap3	Daily probability of incidence into cat 2 from cat 1	System of equations
i3	ap0f4/ap4 + ap3r4/ap4 - d4	Daily probability of incidence into cat 3 from cat 4	System of equations
r2	1 - e^(-(1-sam_tx_coveragesam_tx_efficacy)(1/time_to_sam_ux_recovery))	Daily probability of remission into cat 2 from cat 1 (untreated)	Nicole’s calculations; also referred to as r2ux (get lit source!)
r3	1 - e^(-(mam_tx_coveragemam_tx_efficacy 1/time_to_mam_tx_recovery + (1-mam_tx_coveragemam_tx_efficacy)(1/time_to_mam_ux_recovery)))	Daily probability of remission from cat 2 into cat 3 (treated or untreated)	Nicole’s calculations (get lit source!)
r4	1 - e^{-rate}. 6-12 months: rate = 0.006140 (SD: 0.003015). 1-4 years: rate = 0.005043 (SD: 0.002428). For each rate parameter, use truncated normal distribution of uncertainty with lower bound equal to zero and upper bound equal to 25 standard deviations above the mean (25 standard deviations above the mean was determined to be the upper limit of the python distribution function)	Daily probability of remission from cat 3 into cat 4	From implied transition rate from the KI data. Assume a normal distribution of uncertainty.
t1	1 - e^(-sam_tx_coveragesam_tx_efficacy (1/time_to_sam_tx_recovery))	Daily probability of remission into cat 3 from cat 1 (treated)	Nicole’s calculations (get lit source!)
s1	-r2 - t1 + ap2d2/ap1 + ap3d3/ap1 + ap4*d4/ap1 + (-ap0 + ap1)/ap1	Daily probability of staying in cat 1	System of equations
s2	-ap0f2/ap2 - ap0f3/ap2 - ap0f4/ap2 - ap1r2/ap2 - ap1t1/ap2 - r3 + 1 + ap3d3/ap2 + ap4*d4/ap2	Daily probability of staying in cat 2	System of equations
s3	-ap0f3/ap3 - ap0f4/ap3 - ap1t1/ap3 - ap2r3/ap3 - r4 + 1 + ap4*d4/ap3	Daily probability of staying in cat 3	System of equations
s4	-ap0f4/ap4 - ap3r4/ap4 + 1	Daily probability of staying in cat 4	System of equations

in terms of the following variables:

Variables for transition probabilities¶
Variable	Description	Equation	Notes
\(d_i\)	Death probability out of wasting category \(i\)	\(1 - exp(-1 * (acmr + (\sum_{c\in diar,lri,msl,pem} emr_cprevalence_{ci}) - csmr_c) timestep)\)
\(f_i\)	“Age-in” probability into \(cat_i\)	Prevalence of wasting category i, pulled from GBD	These probabilities were chosen to maintain equilibrium of our system
\(ap_0\)	Adjusted prevalence of \(cat_0\) (the reincarnation pool)	1 - exp(-acmr * 1 / 365)	We set this equal to the number of simulants that die each time step
\(ap_i\) for \(i\in \{1,2,3,4\}\)	Adjusted prevalence of \(cat_i\)	\(f_i/(ap_0 + 1)\)	All category “prevalences” are scaled down, such that the prevalence of cat 0 (the reincarnation pool) and the prevalences of the wasting categories sum to 1
mam_tx_coverage	Proportion of MAM (CAT 2) cases that have treatment coverage	defined here as \(C_{MAM}\)
sam_tx_coverage	Proportion of SAM (CAT 1) cases that have treatment coverage	defined here as \(C_{SAM}\)
sam_tx_efficacy	Proportion of children treated for SAM who successfully respond to treatment	defined here as \(E_{SAM}\)	Baseline scenario value
mam_tx_efficacy	Proportion of children treated for MAM who successfully respond to treatment	defined here as \(E_{MAM}\)	Baseline scenario value
\(time_to_mam_ux_recovery\)	Without treatment or death, average days spent in MAM before recovery	defined here as \(\text{time to recovery}_\text{untreated MAM}\)
time_to_mam_tx_recovery	With treatment and without death, average days spent in MAM before recovery	defined here as \(\text{time to recovery}_\text{treated MAM}\)
time_to_sam_ux_recovery	Without treatment or death, average days spent in SAM before recovery	\(365 / r_{SAM,ux}\)	\(r_{SAM,ux}\) defined in the Annual recovery rate equations table in the wasting treatment intervention document
time_to_sam_tx_recovery	With treatment and without death, average days spent in SAM before recovery	defined here as \(\text{time to recovery}_\text{treated SAM}\)
time_step	Scalar time step conversion to days	1

Calculations for variables in transition equations¶
Variable	Description	Equation
\(prevalence_{ci}\)	The prevalence of cause c among wasting category i	\(incidence_{ci} * duration_c\)
\(duration_c\)	The average duration of cause c, in years	Defined on the respective cause model documents for diarrheal diseases, measles, and lower respiratory infections
\(incidence_{ci}\)	incidence probability of cause c among wasting category i	\(incidence_{c}(1-paf_{c})rr_{ci}\)
\(incidence_c\)	population-level incidence probability of cause c	Pulled from GBD
\(paf_{c}\)	The PAF of cause c attributable to wasting	\(\frac{(\sum_{i} prevalence_{i} * rr_{ci})-1}{\sum_{i} prevalence_{i} * rr_{ci}}\)
\(rr_{ci}\)	The relative risk for incidence of cause c given wasting category i
\(prevalence_{i}\)	the prevalence of wasting category i	Pulled from GBD
\(acmr\)	All-cause mortality probability	Pulled from GBD
\(emr_c\)	Excess mortality probability of cause c	Pulled from GBD
\(csmr_c\)	Cause-specific mortality rate of cause c	Pulled from GBD

We now detail how the above wasting probability transition equations were derived.

Todo

Consider adding all code for calculating above eqns.

We solve our transition probabilities using a Markov Chain transition matrix T.

T =

	cat4	cat3	cat2	cat1	cat0
cat4	s4	i3	0	0	d4
cat3	r4	s3	i2	0	d3
cat2	0	r3	s2	i1	d2
cat1	0	0	r2	s1	d1
cat0	f4	f3	f2	f1	0

\(π_{T}\) =

p4

p3

p2

p1

p0

\(π_{T}\) is the eigenvector at equilibrium

\(π_{T}\times\text{T} = π_{T}\) (the T means transposed, this is a 1 row vector)

\(\sum_{\text{i=p}}\) = \(π_{T}\)

\(π_{i}\) ≥ 0 , these are GBD 2020 age/sex/location/year-specific prevalence for wasting categories 1-4, plus \(p0\), which will equal the number of sims who die in a timestep

Solving a)

\(ap_4s_4 + ap_3r_4 + ap_0f_4 = ap_4\)

\(ap_4i_3 + ap_3s_3 + ap_2r_3 + ap_0f_3 = ap_3\)

\(ap_3i_2 + ap_2s_2 + ap_1r_2 + ap_0f_2 = ap_2\)

\(ap_2i_1 + ap_1s_1 + ap_0f_1 = ap_1\)

\(ap_4d_4 + ap_3d_3 + ap_2d_2 + ap_1d_1=ap_0\)

Rows of the P matrix sums to 1

\(s_4 + i_3 + d-4 = 1\)

\(r_4 + s_3 + i_2 + d_3 = 1\)

\(r_3 + s_2 + i_1 + d_2 = 1\)

\(r_2 + s_1 + d_1 = 1\)

\(f_4+f_3+f_2+f_1=1\)

import numpy as np, pandas as pd
import sympy as sym
from sympy import symbols, Matrix, solve, simplify

# define symbols
s4, i3 = symbols('s4 i3')
r4, s3, i2 = symbols('r4 s3 i2')
r3, s2, i1 = symbols('r3 s2 i1')
r2, s1 = symbols('r2 s1')
d4, d3, d2, d1 = symbols('d4 d3 d2 d1')
f4, f3, f2, f1 = symbols('f4 f3 f2 f1')
ap4, ap3, ap2, ap1, ap0 = symbols('ap4 ap3 ap2 ap1 ap0')
acmr = sym.Symbol('acmr')


# for k linearly independent eqns, sympy will solve the first k unknowns
unknowns = [i2,s1,s2,s3,s4,r3,i1,i3,t1,r4,r2,d1,d2,d3,d4,f1,f2,f3,f4]

def add_eq(terms, y, i, A, v):
  """
  For input equation y = sum([coeff*var for var:coeff in {terms}])
  adds right side of equation to to row i of matrix A

  adds y to row i of vector v
  """
  for x in terms.keys():
      A[x][i] = terms[x]
  v.iloc[i] = y


# # assuming equilibrium:
# p4*s4 + p3*r4 + p0*f4 = p4
eq1 = [{s4:p4, r4:p3, f4:p0}, p4]

# p4*i3 + p3*s3 + p2*r3 + p0*f3 = p3
eq2 = [{i3:p4, s3:p3, r3:p2, f3:p0}, p3]

# p3*i2 + p2*s2 + p1*r2 + p0*f2 = p2
eq3 = [{i2:p3, s2:p2, r2:p1, f2:p0}, p2]

# p2*i1 + p1*s1 + p0*f1 = p1
eq4 = [{i1:p2, s1:p1, f1:p0}, p1]

# p4*d4 + p3*d3 + p2*d2 + p1*d1 + p0*sld = p0
eq5 = [{d4:p4, d3:p3, d2:p2, d1:p1}, p0]


# # rows sum to one:
# s4 + i3 + d4 = 1
eq6 = [{s4:1, i3:1, d4:1}, 1]

# r4 + s3 + i2 + d3 = 1
eq7 = [{r4:1, s3:1, i2:1, d3:1}, 1]

# r3 + s2 + i1 + d2 = 1
eq8 = [{r3:1, s2:1, i1:1, d2:1}, 1]

# r2 + s1 + d1 = 1
eq9 = [{r2:1, s1:1, d1:1}, 1]

# f4 + f3 + f2 + f1 + sld = 1
eq10 = [{f4:1, f3:1, f2:1, f1:1}, 1]


def build_matrix(eqns, unknowns):
  """
  INPUT
  ----
  eqns: a list of sympy equations
  unknowns: a list of sympy unknowns
  ----
  OUTPUT
  ----
  A:  a matrix containing the coefficients of LHS of all eq in eqns.
      nrows = number of equations
      rcols = number of unknowns
  b: an nx1 matrix containing the RHS of all the eqns
  x: a sympy matrix of the unknowns
  """
  n_eqns = len(eqns)
  n_unknowns = len(unknowns)

  # frame for matrix/LHS equations.
  # nrows = n_eqns, ncols = n_unknowns
  A = pd.DataFrame(
      index = range(n_eqns),
      columns = unknowns,
      data = np.zeros([n_eqns,n_unknowns])
  )

  # frame for RHS of equations
  b = pd.DataFrame(index = range(n_eqns), columns = ['val'])

  # populate LHS/RHS
  i = 0
  for eq in eqns:

      add_eq(eq[0], eq[1], i, A, b)
      i += 1

  # convert to sympy matrices
  A = sym.Matrix(A)
  b = sym.Matrix(b)
  x = sym.Matrix(unknowns) #vars to solve for

  return A, x, b

# solve in terms of i3
A0, x0, b0 = build_matrix([eq1,eq2,eq3,eq4,eq5,eq6,eq7,eq8,eq9,eq10,eq11,eq12],
                         unknowns)

result_0 = sym.solve(A0 * x0 - b0, x0)

# solve in terms of duration of cat3 instead of i3:
A1, x1, b1 = build_matrix([eq1,eq2,eq3,eq4,eq5,eq6,eq7,eq8,eq9,eq10],
                       unknowns)
result_1 = sym.solve(A1 * x1 - b1, x1)

Wasting x Disease model ¶

../../../../_images/wasting_state_2x4.svg

Data Description Tables ¶

Wasting State Data¶
State	Measure	Value	Notes
TMREL, MOD, MAM, SAM	birth prevalence	\(prevalence_{240_{cat-1-4}}\)	Use prevalence of age_group_id = 388 (1 to 5 months)

#to pull GBD 2020 category specific prevalence of wasting

 get_draws(gbd_id_type='rei_id',
                 gbd_id=240,
                 source='exposure',
                 year_id=2020,
                 gbd_round_id=7,
                 status='best',
                 location_id = [179],
                 decomp_step = 'iterative')

Wasting State Data¶
State	Measure	Value	Notes
MAM	disability weight	\(\frac{{\sum_{sequelae\in \text{MAM}}} \scriptstyle{\text{disability_weight}_s \times\ \text{prevalence}_s}}{{\sum_{sequelae\in xt{MAM}} \scriptstyle{\text{prevalence}_s}}}\)	disability weight for MAM
SAM	disability weight	\(\frac{{\sum_{sequelae\in \text{SAM}}} \scriptstyle{\text{disability_weight}_s \times\ \text{prevalence}_s}}{{\sum_{sequelae\in \text{SAM}} \scriptstyle{\text{prevalence}_s}}}\)	disability weight for SAM
MAM & SAM	excess mortality	\(\frac{\text{deaths_c387}}{\text{population} \times \text{prevalence_c387}}\)	death counts come from codecorrect
All	cause specific mortality	\(\frac{\text{deaths_c387}}{\text{population}}\)	death counts come from codecorrect

Note

The 2020 Codecorrect model for PEM is not yet completed. Check here on central machinary to see latest codecorrect modeling. https://hub.ihme.washington.edu/pages/viewpage.action?spaceKey=GBD2020&title=GBD+2020+CodCorrect+Tracking

and here for scheduled finishing time (currently scheduled to complete on july 30th- 12July2021) https://hub.ihme.washington.edu/pages/viewpage.action?spaceKey=GBD2020&title=GBD+2020+Release+1+Computation

The following code can be used to access draw-level deaths for PEM

# GBD 2019 (this is the version we will use for PEM for now)

 get_draws(gbd_id_type = 'cause_id',
        gbd_id = [387], #pem
        source = "codcorrect",
        metric_id = 1, #counts
        measure_id = 1, #deaths
        location_id = [179],
        sex_id = [1,2],
        age_group_id = [4,5],
        gbd_round_id = 6,
        year_id  =2019,
        decomp_step = 'step5')


# GBD 2020 (not fully formed)

get_draws(gbd_id_type = 'cause_id',
        gbd_id = [387], #pem
        source = "codcorrect",
        metric_id = 1, #counts
        measure_id = 1, #deaths
        location_id = [179],
        sex_id = [1,2],
        age_group_id = [388,389,238,34],
        gbd_round_id = 7,
        year_id  = 2020,
        decomp_step = 'step3', #this is the latest decomp step,  will get updated
        version_id = 260) #this is the latest version, will get updated

PEM Data Sources and Definitions¶
Variable	Source	Description	Notes
MAM sequelae		{s198, s2033}	Moderate wasting with eodema, moderate wasting without oedema
SAM sequelae		{s2036, s199}	Severe wasting with eodema, severe wasting without oedema

Note

The 2020 Como model for PEM is not yet completed, with only 100 draw. Check here on central machinary to see latest como modeling. https://hub.ihme.washington.edu/display/GBD2020/COMO+tracking

To pull PEM sequelae prevalence, use the following code

#GBD 2019

get_draws(gbd_id_type = 'sequela_id',
         gbd_id = [198,2033,2036,199],
         source = "como",
         location_id = [179],
         sex_id = [1,2],
         age_group_id = [2,3,4,5],
         gbd_round_id = 6,
         decomp_step = 'step5')


#GBD 2020 (currently only 100 draws)

 get_draws(gbd_id_type = 'sequela_id',
         gbd_id = [198,2033,2036,199],
         source = "como",
         location_id = [179],
         sex_id = [1,2],
         age_group_id = [2,3,388,389,238,34],
         gbd_round_id = 7,
         decomp_step = 'iterative')


 #as well as from db_queries

 from db_queries import get_sequela_metadata

 hierarchy_2019 = get_sequela_metadata(sequela_set_id=2, gbd_round_id=6, decomp_step="step4")
 hierarchy_2019.loc[(hierarchy_2019.cause_id==387)] #2019

PEM Restrictions 2019¶
Restriction type	Value	Notes
Male only	False
Female only	False
YLL only	False
YLD only	False
YLL age group start	Post Neonatal	age_group_id = 4
YLL age group end	95 plus	age_group_id = 235
YLD age group start	Early Neonatal	age_group_id = 2
YLD age group end	95 Plus	age_group_id = 235

PEM Restrictions 2020¶
Restriction type	Value	Notes
Male only	False
Female only	False
YLL only	False
YLD only	False
YLL age group start	1-5 months	age_group_id = 388
YLL age group end	95 plus	age_group_id = 235
YLD age group start	Early Neonatal	age_group_id = 2
YLD age group end	95 Plus	age_group_id = 235

Wasting Restrictions 2020¶
Restriction type	Value	Notes
Male only	False
Female only	False
Prevalence age group start	Early Neonatal	age_group_id = 2. This is the earliest age group for which the wasting risk exposure estimates nonzero prevalence.
Burden age group start	28 days - 5 months	age_group_id = 388. This is the earliest age group for which there exist wasting RRs.
Age group end	2 to 4	age_group_id = 34

#age group id differences between 2019 and 2020

#2020 age ids
early nn = 2
late nn = 3
1m-5m = 388   #2019 it was 4 = postneonatal
6m-11m = 389  #2019 it was 4 = postneonatal
12m-23m = 238 #2019 it was 5 = 1-5
2y-4y = 34    #2019 it was 5 = 1-5

As we are building this model before the completion of GBD 2020, we will need to calculate the PAFs ourselves, using the following equation:

\[\frac{(\sum_{wasting\_category_i} prevalence_{i} * rr_{ci})-1}{\sum_{wasting\_category_i} prevalence_{i} * rr_{ci}}\]

PAF equation variable descriptions¶
Variable	Description	Equation
\(rr_{ci}\)	The relative risk for incidence of cause c given wasting category i
\(prevalence_{i}\)	the prevalence of wasting category i	Pulled from GBD

Note the RRs should be pulled as follows:

from get_draws.api import get_draws
get_draws(
  gbd_id_type='rei_id',
  gbd_id=240,
  source='rr',
  location_id=179,
  sex_id=[1,2],
  age_group_id=[2, 3, 388, 389, 34],
  decomp_step='iterative',
  status='best'
)

Transition Data¶
Transition	Source State	Sink State	Value	Notes
ux_rem_rate_sam	CAT 1	CAT 2	\(-log(1 - r2) * 365\)	Untreated remission rate (counts/person-year) from SAM to MAM
tx_rem_rate_sam	CAT 1	CAT 3	\(-log(1 - t1) * 365\)	Treated remission rate (counts/person-year) from SAM to mild wasting
rem_rate_mam	CAT 2	CAT 3	\(-log(1 - r3) * 365\)	Remission rate (counts/person-year) from MAM to mild wasting
rem_rate_mild	CAT 3	CAT 4	\(-log(1 - r4) * 365\)	Remission rate (counts/person-year) from mild wasting to TMREL
inc_rate_sam	CAT 2	CAT 1	\(-log(1 - i1) * 365\)	Incidence rate (counts/person-year) from MAM to SAM
inc_rate_mam	CAT 3	CAT 2	\(-log(1 - i2) * 365\)	Incidence rate (counts/person-year) from mild wasting to MAM
inc_rate_mild	CAT 4	CAT 3	\(-log(1 - i3) * 365\)	Incidence rate (counts/person-year) from TMREL to mild wasting

Validation ¶

Wasting model

prevalence of cat 1-4

the incidences and the recovery rates (with our calibration inputs, can be accessed in interative sim)

death rates per category

relative risks (this would be done in the cause model validation)

SAM and MAM duration (including who recovered from t1 arrow vs. r2 arrow)

fertility (total person-time vs. year)

PEM model

prevalences

csmr

emr

we are not validating against GBD incidence or remission

References ¶

Dipasquale-et-al-Wasting: Dipasquale et al. Acute Malnutrition in Children: Pathophysiology, Clinical Effects and Treatment. Nutrients 2020, 12, 2413; doi:10.3390/nu12082413, https://www.mdpi.com/2072-6643/12/8/2413
GBD-2019-Capstone-Appendix-Wasting(1,2,3,4,5,6,7,8,9,10): Appendix to: GBD 2019 Diseases and Injuries Collaborators. Global burden of 369 diseases and injuries in 204 countries and territories, 1990–2019: a systematic analysis for the Global Burden of Disease Study 2019. The Lancet. 17 Oct 2020;396:1204-1222
UpToDate-malnutrition-Wasting(1,2,3): Retrieved 25 June 2021. https://www-uptodate-com.offcampus.lib.washington.edu/contents/malnutrition-in-children-in-resource-limited-countries-clinical-assessment
WHO-Malnutrition-Wasting: Retrieved 25 June 2021. https://www.who.int/news-room/q-a-detail/malnutri