The model consists of three components. In the initialization component the parameter values and the initial distribution of the population over all model states are calculated from the input data. In the simulation component the 1-year changes of the model state prevalence numbers are calculated for each cohort separately. Each cohort is defined by its initial age value, i.e. a(0). These changes in the number of persons for each state are the result of transitions between the risk factor classes and between the disease states. The numbers of transitions are computed as the 1-year transition probabilities times the state prevalence numbers at the start of the 1-year time-interval. Finally, in the post-processing component the model output variables are calculated from the results of the simulation component. All model parameters and variables are specified by gender and age, but we omit the index for gender below for reasons of readability.
The parameters calculated here are the 1-year baseline disease incidence rates, i.e. the rates of getting the disease for never smokers, and the mortality rates for other causes of death. The disease incidence rates for never smokers are calculated by dividing the overall incidence rates by the weighted sum of relative risks.
c index over all smoking classes, c = never, current, former
id,0(a) data disease d incidence rate at age a
id,base(a) disease d incidence rate for never smokers at age a
nc,0(a) initial smoking class probability values at age a
RRd(c,a) relative risk of incidence of disease d for smoking class c
The relative risks for never and current smokers are input data (i.e. 'given'). The relative risks of former smokers are calculated (see Appendix 3). The other causes mortality rates describe the mortality rates for causes of death other than the smoking-related chronic diseases included in the model.
d index over diseases
μtot(a) all cause mortality rates at age a
μother(a) mortality rates for other causes of death at age a
pd,0(a) initial disease d prevalence rate values at age a
amd(a) disease d related attributable mortality rates at age a
The parameter amd(a) describes the mortality rates uniquely attributable to disease d. This parameter is defined as the additional mortality rate of persons with disease d compared to persons without disease d, with gender, age, and risk factor classes and states for other diseases being equal. The initial numbers of never and current smokers and the initial disease prevalence rates are calculated from input data:
a(0) age of cohort at initial time point t = 0
Nc(t) number of persons in smoking class c at time t,
for c = never, current
Npop,0(a) initial total population numbers
pd(t) disease d prevalence rates at time t
nc,0(a) input proportions of population in smoking classes c
pd,0(a) input disease d prevalence rates at age a
The initial values of the numbers of former smokers, stratified by time since cessation, were generated in a pre-processing step by running the model for a birth cohort without any disease included. In this way we calculated the distribution of former smokers nformer(s,a) over all cessation classes (s), specified by gender and age a. Doing so, we implicitly assumed that all smoking class transition probabilities are constant over time. Thus:
Nformer(0,s) = nformer,0(a(0))nformer(s,a(0)))Npop,0(a(0))
Nformer(t,s) number of former smokers at time t in former smoking class s
s index over classes for time since smoking cessation, s = 1,...,S
E.g., s = 2 means former smoker stopped 1–2 years ago.
nformer,0(a) data proportion of former smokers in the population at age a
The simulation component describes the changes of the prevalence numbers in all smoking classes distinguished, as well as the changes of the prevalence rates for all chronic diseases included in the model. These changes are formulated as differential equations with 1-year time steps.
1-year changes of smoking class prevalence numbers
The mortality rates for all smoking classes at time t (μc(t)) depend on the disease prevalence rates. The disease prevalence rates for each smoking class are found by distributing the prevalent disease cases at time t over all smoking classes using relative risks and the smoking class distribution at time t. To do so, first the mean relative risk values are computed for time t.
t time parameter, with 1-year steps
RRd(s,a) disease d relative risk of former smoking class s
at age a, see Appendix 3
E(RRd(former),t) mean disease d relative risk value at time t for former
smokers, at time t
E(RRd,t) mean disease d relative risk value in entire population at time t
Note that the relative risks of never and current smokers (RRd(c,a), c = never,current) are constant values, whilte those of former smokers (E(RRd(former,t))) depend on the distribution over all time since cessation classes and thus are re-calculated each year. We assumed for each disease that the distributions of the prevalent and incident disease cases over the smoking classes are equal. Adding mortality from other causes results in the total mortality rates for each smoking class at time t.
with μc(t) all cause mortality rates for smoking class c at time t
These mortality rates are transformed to 1-year mortality probabilities assuming constant rate values over the year. Using these mortality probabilities, still denoted as μc(t), the following equations describe the 1-year change of the prevalence numbers of never and current smokers (Nc(t)), and of former smokers specified by time since smoking cessation (Nfomer(t,s)).
λstart(a), λstop(a) 1-year start and stop smoking probabilities respectively,
at age a
λrelapse(s) smoking relapse probabilities that depend on time since cessation class s, see appendix 2
The number of former smokers in the last class S at the end of the year are the sum of the numbers in the last and second last class at the start of the year that do not relapse or die.
1-year changes in disease prevalence rates
We describe the 1-year change in prevalence rates instead of numbers for each disease included. Since the mortality rates for other causes of death are assumed equal for persons with and without the disease, the change in the rate values depends only on the disease incidence and disease related excess mortality rates. The current disease incidence rates are the baseline disease incidence rates times the current mean relative risk value.
id(t) = E(RRd,t)id,base(a(t))
id(t) disease d incidence rate at time t
id,base(a) baseline disease d incidence rate at age a
1-year event incidence probabilities were calculated from the rate values assuming constant values over the year. These probabilities are still denoted as id
(t). The prevalence rates change as a result of incidence and mortality. This equation is known as the DisMod-equation [28
pd(t + 1) = pd(t) + id(t)(1 - pd(t)) - emd(a(t))pd(t)(1 - pd(t))
with: emd(a) disease d related excess mortality probability at age a.
The parameter emd
(a) is the excess mortality related to disease d. It describes the additional mortality in the population with disease d as compared to the population without disease d (see the online appendix of [22
]). The parameter amd
(a), that was defined before, describes the additional mortality rate, conditional on all risk factors and other disease states. The parameter amd
(a) can be interpreted as the mortality that uniquely can be attributed to disease d. It adjusts the excess mortality for mortality due to co-morbid diseases. The part (1 - pd
(t)) in the last term of the equation comes from describing changes of prevalence rates instead of numbers.
Model post-processing component
The model output variables are computed from the results in the simulation component. These are the following:
Disease incidence numbers
QALYs generated (accumulated)
Id(t) disease d incidence numbers during 1-year period [t,t+1)
QALY quality-adjusted life years
wd(a) disease d weight coefficients at age a
The weight coefficients wd(a) describe the relative loss of quality of life value due to the disease. A value 0 means there is no loss of quality of life due to the disease; a value 1 indicates there is complete loss of quality of life, and that having the disease is no better than being dead.