Diarrheal diseases risk effects

Risk Overview

The diarrheal diseases cause model will impact the dynamic child wasting exposure model incidence rates for the acute malnutrition simulation as part of the “vicious cycle”/positive feedback model between child wasting and diarrheal diseases (note that diarrheal diseases is an affected outcome of child wasting).

Todo

Include brief literature background

Vivarium Modeling Strategy

Note

This section will describe the Vivarium modeling strategy for risk effects. For a description of Vivarium modeling strategy for risk exposure, see the diarrheal diseases cause model document.

Wasting transition rates

Estimation of risk effects

The relative risk for wasting transition rates between wasting exposure states by diarrheal disease status was estimated under the assumption of a steady state disease model and resulting system of equations that is described in this word document.

The estimation of the prevalence ratios of wasting states by diarrheal disease status relied on evidence from [Troeger-et-al-2018] and details on the calculation can be found here. Notably, due to data availability of the WHZ scores, the prevalence ratios are estimated among the 2-5 year age group and it is assumed that they do not vary by age group (prevalence ratios are defined in this document in a table in the next section).

Todo

Perform additional runs of this simulation to get uncertainty about the prevalence ratios

The system of equations relating to the steady state model were solved symbolically for the state prevalence values here and for the incidence rates here.

Using the resulting equations from the processes described above, the incidence rates specific to diarrheal disease status for each age/sex/year/location demographic group in the acute malnutrition simulation model were estimated in the notebook found here using GBD data. These incidence rates were then used to estimate age and sex specific relative risks for wasting state incidence rates, which are described below.

Calculation of risk effects

Calculation of risk effects should be performed for simulants aged six to 59 months.

The prevalence ratios used in the calculation of the risk effects are listed in the table below. The excess risk in these ratios (PR - 1) represents the portion of the population afflicted with diarrheal diseases that occupies that wasting state as a result of the diarrheal disease episode. In other words, the PR represents the multiplicative likelihood that an individual will occupy a specific child wasting category if they experienced a diarrheal diasease epidsode relative to if they had not experienced a diarrheal disease episode.

Prevalence ratio values

Parameter

Population

Value

Note

PR_1

Males and females 0.5-5 years

1.060416

Wasting cat1 (SAM)

PR_2

Males and females 0.5-5 years

1.061946

Wasting cat2 (MAM)

PR_3

Males and females 0.5-5 years

1.044849

Wasting cat3 (mild child wasting)

PR_4

Males and females 0.5-5 years

0.990530

Wasting cat4 (susceptible to child wasting)

Todo

Incorporate uncertainty into prevalence ratio values

The following parameters should be age/sex/location-specific:

Additional input parameter definitions

Parameter

Description

Value

Note

exposure_wasting_cat{1, 2, 3, 4}

category-specific child wasting exposure

defined on the child wasting exposure page

prevalence_diarrheal_diseases

prevalence of diarrheal diseases

Defined on the diarrheal diseases cause model document

prevalence of state I

remission_diarrheal_diseases

diarrheal diseases remission rate (per person-year in the population infected with diarrheal diseases)

Defined on the diarrheal diseases cause model document

Transition rate from state I to state S

incidence_diarrheal_diseases

susceptible diarrheal diseases incidence rate (per person-year in the population susceptible to diarrheal diseases)

Defined on the diarrheal diseases cause model document

Transition rate from state S to state I

csmr_{diarrheal_diseases, pem, lri, measles}

cause-specific mortality rate

Defined for respective causes on the diarrheal diseases, protein energy malnutrition, lower respiratory infections, and measles documents

emr_{diarrheal_diseases, pem}

cause-specific excess mortality rate

Defined for respective causes on the diarrheal diseases and protein energy malnutrition documents

paf_wasting_{diarrheal_diseases, lri ,measles}

PAF of child wasting on affected causes

As described on the child wasting exposure page

Currently custom calculated, but may update to GBD PAFs following finalization of GBD 2020

RR_wasting_{diarrheal_diseases, lri, measles}_{cat1, cat2, cat3}

Wasting category-specific relative risks for cause-specific affected outcomes

As described on the child wasting exposure page

ACMR

All-cause mortality rate

All-cause mortality rate for a given age/sex/location/year group from GBD

i1, i2, i3, r4, r3, r2, t1

Wasting transition rates

Defined on the child wasting exposure page

(defined in terms of transitions per person-year in the source state)

The following code block provides equations to solve for the relative risks attributable to diarrheal disease infection for each of the wasting state incidence rates according to the prevalence ratio values defined above and artifact data. For reference, the tables below outline the notation of the intermediate variables included in the equations.

Intermediate variable notation: states

Parmeter

Notation

Note

Susceptible to diarrheal diseases

S{wasting state}

Infected with diarrheal diseases

D{wasting state}

Wasting TMREL (cat4)

{diarrheal status}4

Mild wasting (cat3)

{diarrheal status}3

Moderate wasting/MAM (cat2)

{diarrheal status}2

Severe wasting/SAM (cat1)

{diarrheal status}1

Note

All of the transition rates in the table below are defined in terms of the count of transitions per person-time unit in the entire model system (not specific to person-time in the source state).

Intermediate variable notation: transitions

Parameter

Definition

Notation

Note

Mortality rate

Deaths from source state per total population person time

m_{source state}

Birth rate

Rate of aging into source state per total population person time

b_{sink state}

“Reincarnation” or “aging into” states to keep population size stable

Diarrheal disease incidence rate

Incident diarrheal disease cases from a given wasting category per total population person time

di_{wasting state}

Note that wasting does not affect diarrheal disease incidence rates (it affects excess mortality rates instead)

Diarrheal disease remission rate

Remitted diarrheal disease cases from a given wasting category per total population person time

dr_{wasting state}

Wasting incidence

Cases that transition to a more severe wasting state per total population person time

i_{sink state}

Note that diarrheal disease status does not change upon this transition

Wasting remission

Cases that transition to a more severe wasting state per total population person time

r_{source state}

Transitions out of wasting cat1 are dependent of wasting treatment coverage (treated: r_S1tx and r_D1tx, untreated: r_S1ux and r_D1ux). Note that diarrheal disease status does not change upon this transition

 
p_D1 = (PR_1 * exposure_wasting_cat1 * prevalence_diarrheal_diseases) / (PR_1 * prevalence_diarrheal_diseases - prevalence_diarrheal_diseases + 1)
p_D2 = (PR_2 * exposure_wasting_cat2 * prevalence_diarrheal_diseases) / (PR_2 * prevalence_diarrheal_diseases - prevalence_diarrheal_diseases + 1)
p_D3 = (PR_3 * exposure_wasting_cat3 * prevalence_diarrheal_diseases) / (PR_3 * prevalence_diarrheal_diseases - prevalence_diarrheal_diseases + 1)
p_S1 = (-exposure_wasting_cat1 * prevalence_diarrheal_diseases + exposure_wasting_cat1) / (PR_1 * prevalence_diarrheal_diseases - prevalence_diarrheal_diseases + 1)
p_S2 = (-exposure_wasting_cat2 * prevalence_diarrheal_diseases + exposure_wasting_cat2) / (PR_2 * prevalence_diarrheal_diseases - prevalence_diarrheal_diseases + 1)
p_S3 = (-exposure_wasting_cat3 * prevalence_diarrheal_diseases + exposure_wasting_cat3) / (PR_3 * prevalence_diarrheal_diseases - prevalence_diarrheal_diseases + 1)
p_D4 = prevalence_diarrheal_diseases - p_D1 - p_D2 - p_D3
p_S4 = (1 - prevalence_diarrheal_diseases) - p_S1 - p_S2 - p_S3
m_D1 = (ACMR - csmr_diarrheal_diseases + emr_diarrheal_diseases * (1 - paf_wasting_diarrheal_diseases) * RR_wasting_diarrheal_diseases_cat1
        - csmr_pem + emr_pem
        - csmr_lri + csmr_lri * (1 - paf_wasting_lri) * RR_wasting_lri_cat1
        - csmr_measles + csmr_measles * (1 - paf_wasting_measles) * RR_wasting_measles_cat1) * p_D1
m_D2 = (ACMR - csmr_diarrheal_diseases + emr_diarrheal_diseases * (1 - paf_wasting_diarrheal_diseases) * RR_wasting_diarrheal_diseases_cat2
        - csmr_pem + emr_pem
        - csmr_lri + csmr_lri * (1 - paf_wasting_lri) * RR_wasting_lri_cat2
        - csmr_measles + csmr_measles * (1 - paf_wasting_measles) * RR_wasting_measles_cat2) * p_D2
m_D3 = (ACMR - csmr_diarrheal_diseases + emr_diarrheal_diseases * (1 - paf_wasting_diarrheal_diseases) * RR_wasting_diarrheal_diseases_cat3
        - csmr_pem
        - csmr_lri + csmr_lri * (1 - paf_wasting_lri) * RR_wasting_lri_cat3
        - csmr_measles + csmr_measles * (1 - paf_wasting_measles) * RR_wasting_measles_cat3) * p_D3
di_1 = incidence_diarrheal_diseases * p_S1
di_2 = incidence_diarrheal_diseases * p_S2
di_3 = incidence_diarrheal_diseases * p_S3
di_4 = incidence_diarrheal_diseases * p_S4
dr_1 = remission_diarrheal_diseases * p_D1
dr_2 = remission_diarrheal_diseases * p_D2
dr_3 = remission_diarrheal_diseases * p_D3
dr_4 = remission_diarrheal_diseases * p_D4
b_D1 = ACMR * p_D1
b_D2 = ACMR * p_D2
b_D3 = ACMR * p_D3
r_D1tx = t1 * p_D1
r_D1ux = r2 * p_D1
r_D2 = r3 * p_D2
r_D3 = r4 * p_D3
i_S1 = b_D1 + di_1 - dr_1 + i1*exposure_wasting_cat2 - m_D1 - r_D1tx - r_D1ux
i_S2 = b_D1 + b_D2 + 2.0*di_1 - dr_1 - dr_2 + i2*exposure_wasting_cat3 - m_D1 - m_D2 - r_D1tx - r_D2
i_S3 = b_D1 + b_D2 + b_D3 + 2.0*di_1 + di_3 - dr_1 - dr_2 - dr_3 + i3*exposure_wasting_cat4 - m_D1 - m_D2 - m_D3 - r_D3
i_D1 = -b_D1 - di_1 + dr_1 + m_D1 + r_D1tx + r_D1ux
i_D2 = -b_D1 - b_D2 - 2.0*di_1 + dr_1 + dr_2 + m_D1 + m_D2 + r_D1tx + r_D2
i_D3 = -b_D1 - b_D2 - b_D3 - 2.0*di_1 - di_3 + dr_1 + dr_2 + dr_3 + m_D1 + m_D2 + m_D3 + r_D3

RR_i3 = (i_D3 * p_S4) / (i_S3 * p_D4)
RR_i2 = (i_D2 * p_S3) / (i_S2 * p_D3)
RR_i1 = (i_D1 * p_S2) / (i_S1 * p_D2)

Application of risk effects

The age- and sex-specific values for RR_{i3, i2, i1} calculated as described above should be applied in the following manner:

For \(x\) in \([1,2,3]:\)

\[\text{p_diarrhea}_\text{{x+1}} = \frac{p_\text{D{x+1}}}{p_\text{D{x+1}}+p_\text{S{x+1}}}\]
\[PAF_\text{i{x}} = \frac{RR_\text{i{x}} * \text{p_diarrhea}_\text{{x+1}} + (1 - \text{p_diarrhea}_\text{{x+1}}) - 1}{RR_\text{i{x}} * \text{p_diarrhea}_\text{{x+1}} + (1 - \text{p_diarrhea}_\text{{x+1}})}\]
\[\text{i{x}}_i = \text{i{x}} * (1 - PAF_\text{i{x}}) * RR_\text{i{x}_i}\]

Note

This proposed strategy uses wasting state-specific diarrheal prevalence in the source state for each wasting transition in calculation of the PAF for that wasting transition in order to avoid bias in the PAF estimation by using the exposure in the “at-risk” population for the affected transition.

However, the estimated impact of this strategy is small (as shown in this notebook), so it was not implemented. The implemented version of diarrheal diseases risk effects in the acute malnutrition simulation instead used the standard GBD PAFs for wasting on diarrheal diseases.

Validation and Verification Criteria

  1. Verification and validation criteria from the diarrheal diseases cause model should remain true.

  2. Verification and validation criteria from the dynamic child wasting exposure model should remain true.

Todo

List additional V&V criteria

Assumptions and Limitations

  1. We assume that the GBD 2019 relative risks of child wasting on mortality due to diarrheal diseases applies entirely to the excess mortality rate rather than the incidence rate. There is evidence of increased diarrheal disease severity by child nutritional status that supports this assumption [TODO: include citations]. However, there is also evidence that child nutritional status impacts the incidence of diarrheal diseases [TODO: include citations].

  2. We assume that the evidence from [Troeger-et-al-2018] represents a causal impact of diarrheal diseases on child wasting. If this is not the case, we are overestimating the prevalence ratios.

  3. We scale the effect size from [Troeger-et-al-2018] to an average duration of diarrheal diseases episode of 6 days as implied from the GBD remission rate. However, given that child wasting is associated with increased diarrheal disease severity, this may be an overlysimplistic assumption (with the effect size conditional on baseline wasting status).

  4. We assume that the prevalence ratios of wasting states by diarrheal disease status do not vary by age group.

  5. We assume that diarrheal disease status does not affect the remission rate of child wasting. However, there is evidence that this may be the case [TODO: include citation]

Todo

Add complexity to this model so that child wasting remission rates are included as an additional risk effect of diarrheal diseases

Todo

List additional assumptions and limitations

References

Troeger-et-al-2018(1,2,3)

Troeger C, Colombara DV, Rao PC, Khalil IA, Brown A, Brewer TG, Guerrant RL, Houpt ER, Kotloff KL, Misra K, Petri WA Jr, Platts-Mills J, Riddle MS, Swartz SJ, Forouzanfar MH, Reiner RC Jr, Hay SI, Mokdad AH. Global disability-adjusted life-year estimates of long-term health burden and undernutrition attributable to diarrhoeal diseases in children younger than 5 years. Lancet Glob Health. 2018 Mar;6(3):e255-e269. doi: 10.1016/S2214-109X(18)30045-7. PMID: 29433665; PMCID: PMC5861379. Troeger et al 2018 available here