Smoking rate itself is computed in two different ways.  The basic formulation uses only the initial condition, historical rates of growth in smoking, and a cross-sectionally estimated function linked to the simple and squared values of GDP per capita at PPP.  The more extended formulation (still in development and testing and therefore not turned on as the default approach) is an algorithmic one based on the same general concept of a pattern that initially rises with GDP per capita, peaks, and then falls, but with a series of parameters that allow much more control over the stages. ^[1]   This staged algorithmic approach (see Lopez et al. 1994; Shibuya et al. 2005; Ploeg et al. 2009) is turned on with a switch ( hlsmokingstsw =1; the default =0).  Based on country- and sex-specific trends, plus initial prevalence, in the first year each country is placed into one of four sequential stages: rising, peak, falling, late.  During the forecast horizon they can move through other stages.  The approach is heavily algorithmic and the logic is explained below.

Early to Rising (Stage 1)

Conditions for being in group are: Male prevalence rises during historic estimate and current prevalence is <=15%. Female prevalence rises at least until the last 5 years of history and current prevalence is <=1%.

The forecasting strategy is: Ongoing increase. Use compound growth rate (10-year moving average from current year) when it is <5% per year, increase growth rate to 5%/year over the next ten years. Continue 5% growth rate until country moves into next group (i.e. male smoking prevalence > 15%, so country moves to stage 2). In the case of females, the previous year growth rate is kept constant until transition to next stage (female smoking prevalence > 1%).

Rising to Peak (Stage 2)

Conditions for being in this stage are: Male prevalence rises during historic estimate and current prevalence is >15%. Female prevalence has been rising at least until last 5 years of history and current prevalence is > 1% and < 45%.

The forecasting strategy is: Exponential increase. If exponential rate of increase < e^.01x, where x=year of forecast (1,2,3,… end of forecast horizon), then increase = e^.01x, otherwise use constant growth. For females we use an exponential regression (y=ab^x), limit annual growth to a maximum of 0.5%. If country moves from stage 1 to stage 2 during forecast horizon, exponential rate of increase = e^.01x, where x=year of forecast (1,2,3,… end of forecast horizon - x begins again at 1 when country moves to stage 2 both for males and females). In base case forecast the ceiling is 65% male prevalence for countries with increasing smoking trends.  If male prevalence rises to 65% during forecast, country moves to next stage (3). The limit for females is 45% prevalence.

Peak to Falling (Stage 3)

Conditions for being in stage are: Male prevalence peaks during historic estimate (20 years before the base year or later). Female prevalence peaks during the last 5 years of history and current prevalence is >= 45%.

The forecasting strategy is: If smoking rate peaks during historic estimate then do 10 years of exponential decrease from last year of peak (y = ab^x). In other words, regression based on historic estimate starting with last year of peak (not all years of historic estimate).

After 10 years, faster exponential decrease:

Prev_x+1=Prev_x*e^-.01x, where x=year (1,2,3,… end of forecast horizon - x begins again at 1 when country begins faster decrease).  

If male prevalence falls to <=20%, then flat rate to end of forecast horizon.

If smoking rate  peaks during forecast slow exponential decrease for first 10 years:

Prev_x+1=Prev_x*e^-.001x , where x=year (1 to 10)

Faster exponential decrease for rest of the forecast:

Prev_x+1=Prev_x *e^-.01x, where x=year (1,2,3,… end of forecast horizon - x begins again at 1 when country begins faster decrease).

If male prevalence falls to <=20%, then flat rate to end of forecast horizon.

For Females simply do a slow logarithmic decline:

SmRt(t) = SmRt(t-1) – 0.05 LN(BaseYear + Yr – 1 – PeakYear)

If female prevalence falls to <= 15%, then flat rate to end of forecast horizon.

Late (Stage 4)

Conditions for being in the stage are: Male prevalence declines throughout at least the last twenty years of historic estimation (ie peak year< BaseYear - 20). Female prevalence declines at least the last 5 years of historic estimation.

The forecasting strategy is: If current male smoking rate (prevalence) >=30% exponential  decrease (y=ab^x) beginning with last year of peak. In other words, regression based on historical data starting with last year of peak (not necessarily all years of historical data). Trend continues through forecast horizon or until male prevalence <=20%. If male prevalence <=20% then flat rate to end of forecast horizon

IF current male prevalence <30% logarithmic decline (y=a+b*ln(x)) beginning with last year of peak. In other words, regression based on historical data starting with last year of peak (not necessarily all years of historical data). Trend continues through forecast horizon or until male prevalence <=20%. If male prevalence <=20% then flat rate to end of forecast horizon.

Female forecast simply uses a logarithmic decline (y=a+b*ln(x)). Trend continues through forecast horizon or until female prevalence <=15%. If female prevalence <=15% then flat rate to end of forecast horizon.

There are a number of parameters for the stages approach to forecasting smoking rates. Because control of tobacco is a major policy objective in many countries, a number of these relate to the representation of a tobacco control score on a 100-point scale ( hlsmokingtcs ) with an  associated parameter to control the elasticity of smoking with that score ( hlsmokingtcsel ), as well as a multiplier on the score ( hlsmokingtcsm ).  The parameters related to tobacco control are:

• Tobacco Control Score: hlsmokingtcs , higher scores reduce growing trend or accelerate reduction.
• Elasticity of relationship between tobacco control score and smoking rate ( hlsmokingtcsel).
• Multiplier on tobacco control score ( hlsmokingtcsm)

Other parameters related to the stage approach are:

• Smoking Multiplier for Increasing Trend: hlsmokingincm , only affects countries in stage 2.
• Smoking Multiplier for Decreasing Trend: hlsmolingdecm , only affects countries in stage 3.
• Smoking Ceiling and Floor can be controlled using: hlsmokingceiling and hlsmokingfloor .
• Smoking Peak Year can be controlled using: hlsmokingpeakyr .

When the stages approach is turned on the computation of smoking impact from the approach is changed from that described for the basic smoking model based on the quadratic equation with GDP per capita.  To compute smoking impact when using smoking in stages we use linear regressions between Smoking Prevalence and estimated Smoking Impact (smoking impact lagged 25 years) to estimate Smoking Impact by large GBD age category.  For instance, one of those functions is “Male Smoking Prevalence (2005) Versus Male 30 to 44 Smoking Impact (2030) Linear”. The functions were computed using smoking rates in 2005 and GBD smoking impact forecasts for 2030. In forecasting we apply those functions to smoking rates 25 and 26 years earlier, and obtain two values for smoking impact.  We compute the growth rate between those two values of smoking impact and apply it to the previous year’s value of smoking impact for each of the 4 large GBD age categories. We then reproduce the smoking impact value for each 5-year age category in the larger GBD categories.

For small values of SI (lower than 1) we restrict the annual change rate to be between -50% and 100%. If SI is 0 in the base year then it stays at 0. If the estimated value of SI reaches 0, and the next year is positive, then a 100% growth rate is used, but if the next year is negative then a -50% growth rate is used.