Prevalence-based years lived with disability k years after incidence of a fixed-duration condition changes

Signature

YLD_prev = (k * I_new + (L - k) * I_old) * DW

Function

Years lived with disability counted from incident cases or from prevalent cases of a sequela

Computational function

• Computational function: yearly incidence-based and prevalence-based YLDs from a history of incident cases

  Turns a history of yearly incident cases into the incidence-based and prevalence-based YLD counts for every year that has at least L years of history behind it, for a condition of fixed duration L. The inputs differ from the formulas': a whole series of yearly incidence, in place of one or two incidence levels, so any pattern of rises and falls can be followed, with an optional continuous discount rate for the incidence-based count.

  Inputs and outputs: inc: Incident cases in each year, oldest first; required, at least L years long. Unit: cases.; L: Fixed duration of the condition, a whole number of years. Unit: years.; DW: Disability weight. Unit: none.; r: Continuous discount rate for the incidence-based count; default 0. Unit: proportion per year.; year: Position of the reference year in the series. Unit: year.; yld_inc: Incidence-based YLDs (HE-FM-YLD-001, or HE-FM-YLD-002 when r is above 0). Unit: years.; yld_prev: Prevalence-based YLDs from the last L cohorts (HE-FM-YLD-003). Unit: years.

  Assumption: Every case lasts exactly L years with no deaths and contributes one full person-year in each of them; the weight is the same for every case and year.

  Worked example (New cases halved after ten years of 100 a year): With 100 cases a year for years 1 to 10 and 50 a year for years 11 to 20, a duration of 10 years and a weight of 0.20, both counts are 200 in year 10; in year 15, the fifth year of the programme, the incidence-based count is 100 and the prevalence-based count 150, as in the article; the prevalence-based count falls by 10 a year and meets the incidence-based count at 100 in year 20 (later years computed here for illustration). inc = 100 for years 1 to 10, 50 for years 11 to 20; L = 10; DW = 0.2; yld_inc_15 = 100; yld_prev_15 = 150; yld_prev_20 = 100

  Worked example (Same history with discounting at 3 per cent): The incidence-based count becomes 172.8 in year 10 and 86.4 from year 11, while the prevalence-based counts are unchanged, since they have no future stream to discount. r = 0.03; yld_inc_10 = 172.8; yld_inc_15 = 86.4; yld_prev_15 = 150

  Excel: With incident cases by year in a column, the incidence-based count in the row of each year is that row's cases times DisWeight times DurationYrs, and the prevalence-based count is DisWeight times the SUM of the cases in that row and the nine rows above it for a 10-year duration, copied down from the tenth year; the two results fill columns named YLDIncVec and YLDPrevVec.

  R: yld_series <- function(inc, L, DW, r = 0) { yrs <- L:length(inc); f <- if (r > 0) (1-exp(-r*L))/r else L; data.frame(year = yrs, incident = inc[yrs], yld_inc = inc[yrs]*DW*f, yld_prev = sapply(yrs, function(y) DW*sum(inc[(y-L+1):y]))) } Base R only; yld_series(c(rep(100, 10), rep(50, 10)), 10, 0.20) returns the first example.

  Python: def yld_series(inc, L, DW, r=0): f = (1-math.exp(-r*L))/r if r > 0 else L; return [{"year": y+1, "incident": inc[y], "yld_inc": inc[y]*DW*f, "yld_prev": DW*sum(inc[y-L+1:y+1])} for y in range(L-1, len(inc))] Needs import math; returns the same values as the R function, with years numbered from 1.

  Test (Counts agree after L years of constant incidence): In the first reported year, preceded by 10 years of 100 cases, YLDIncVec and YLDPrevVec are equal. Expected result: TRUE. FALSE shows a prevalence window of 9 rows in place of 10, which gives 180. Excel check: =ABS(INDEX(YLDIncVec,1)-INDEX(YLDPrevVec,1))<1E-9

  Common error (Starting the prevalence count before L years of history): In the first years of a series fewer than L cohorts are available, so a prevalence-based count summed over them understates the people living with the condition; the function reports only years with a full window.

  Source: World Health Organization. WHO methods and data sources for global burden of disease estimates 2000-2019. Global Health Estimates Technical Paper WHO/DDI/DNA/GHE/2020.3. Geneva: WHO; 2020 (full text read). Sections 2 and 2.4; Vos T, Flaxman AD, Naghavi M, Lozano R, Michaud C, Ezzati M, et al. Years lived with disability (YLDs) for 1160 sequelae of 289 diseases and injuries 1990-2010: a systematic analysis for the Global Burden of Disease Study 2010. The Lancet. 2012;380(9859):2163-2196. doi:10.1016/S0140-6736(12)61729-2 (full text read). Methods; Larson BA. Calculating disability-adjusted-life-years lost (DALYs) in discrete-time. Cost Effectiveness and Resource Allocation. 2013;11:18. doi:10.1186/1478-7547-11-18 (full text read). Equation 2.

  yld_inc_y = inc_y * DW * L, or inc_y * DW * (1 - exp(-r * L)) / r when r > 0; yld_prev_y = DW * sum_(t=y-L+1)^y inc_t

Try this function

Implementations

• Excel

  Prevalence-based years lived with disability after a change in incidence from named cells

  With YearsSince, IncNew, IncOld, DurationYrs and DisWeight named, the formula returns the prevalence-based YLDs for the reference year, held in YLDPrev.

  =(YearsSince*IncNew+(DurationYrs-YearsSince)*IncOld)*DisWeight

Assumptions

• Fixed duration with one full person-year per cohort per year

  Every case lasts exactly L years, causes no deaths and then resolves completely, and each annual cohort contributes one full person-year in each of its L years, as in the article's illustrative condition.

• One step change in incidence

  Incidence was I_old for at least L years before the change and I_new in every year since, so the people living with the condition this year come from these two cohort sizes only.

Worked examples

• Five years after new cases halved from 100 to 50 a year

  Five cohorts of 50 and five of 100 give 750 person-years and (5 x 50 + 5 x 100) x 0.20 = 150 YLDs, a fall of only a quarter while the incidence-based count has halved to 100, as in the article.

  k = 5; I_new = 50; I_old = 100; L = 10; DW = 0.2; YLD_prev = 150

• Year before the change

  With k = 0 every cohort is of the old size, 1,000 person-years, and the count is 200, equal to the incidence-based count of the steady state.

  k = 0; I_new = 50; I_old = 100; L = 10; DW = 0.2; YLD_prev = 200

• Ten years after the change

  When the last of the larger cohorts has recovered, all 10 cohorts are of 50 and the count is 100, matching the incidence-based count, five years after the article's reference year.

  k = 10; I_new = 50; I_old = 100; L = 10; DW = 0.2; YLD_prev = 100

Common errors

• Reading the change in annual burden as the effect of a programme

  Neither annual count is the programme's effect: a model credits each year of the programme with 50 cases prevented, or 100 YLDs averted over the 10 years each case would have lasted (86.4 discounted at 3 per cent), whatever the burden figure shows in that year.

• Comparing YLD trends across GBD rounds

  Each round re-estimates the whole series: GBD 2010 estimated 777 million YLDs worldwide in 2010, while GBD 2021 puts the 2010 figure at 738 million, so trends should be read within one round.

• Reading YLDs per person as the share of people with a disability

  GBD 2010's YLDs per person of 28.6 per cent at age 80 are average health loss per person, not the proportion of people aged 80 who have a disability.

Sources

• Vos and colleagues on prevalence-based YLDs in GBD 2010

  Vos T, Flaxman AD, Naghavi M, Lozano R, Michaud C, Ezzati M, et al. Years lived with disability (YLDs) for 1160 sequelae of 289 diseases and injuries 1990-2010: a systematic analysis for the Global Burden of Disease Study 2010. The Lancet. 2012;380(9859):2163-2196. doi:10.1016/S0140-6736(12)61729-2 (full text read). Methods: YLDs are computed as the prevalence of a sequela multiplied by the disability weight for that sequela without age weighting or discounting. Findings: 777 million YLDs from all causes in 2010, up from 583 million in 1990; YLDs per person rose from 5.4% at age 5 to 28.6% at age 80.

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• GBD 2021 on YLDs from sequela prevalence and the re-estimated 2010 total

  GBD 2021 Diseases and Injuries Collaborators. Global incidence, prevalence, years lived with disability (YLDs), disability-adjusted life-years (DALYs), and healthy life expectancy (HALE) for 371 diseases and injuries in 204 countries and territories and 811 subnational locations, 1990-2021. The Lancet. 2024;403(10440):2133-2161. doi:10.1016/S0140-6736(24)00757-8 (full text read). Methods: crude YLD rates were estimated by multiplying sequela-specific prevalence by their respective disability weights. Results: global YLDs increased from 738 million in 2010.

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• WHO on why prevalence-based counts reflect current burden

  World Health Organization. WHO methods and data sources for global burden of disease estimates 2000-2019. Global Health Estimates Technical Paper WHO/DDI/DNA/GHE/2020.3. Geneva: WHO; 2020 (full text read). Section 2.4: the incidence-based approach will not reflect the current prevalent burden of disabling sequelae for a condition for which incidence has been substantially reduced; prevalence-based YLDs were already used for period healthy life expectancy.

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Canonical Identity