Abstract
Chronic diseases impose a tremendous global health problem of the 21st century. Epidemiological and public health models help to gain insight into the distribution and burden of chronic diseases. Moreover, the models may help to plan appropriate interventions against risk factors. To provide accurate results, models often need to take into account three different time-scales: calendar time, age, and duration since the onset of the disease. Incidence and mortality often change with age and calendar time. In many diseases such as, for example, diabetes and dementia, the mortality of the diseased persons additionally depends on the duration of the disease. The aim of this work is to describe an algorithm and a flexible software framework for the simulation of populations moving in an illness-death model that describes the epidemiology of a chronic disease in the face of the different times-scales. We set up a discrete event simulation in continuous time involving competing risks using the freely available statistical software R. Relevant events are birth, the onset (or diagnosis) of the disease and death with or without the disease. The Lexis diagram keeps track of the different time-scales. Input data are birth rates, incidence and mortality rates, which can be given as numerical values on a grid. The algorithm manages the complex interplay between the rates and the different time-scales. As a result, for each subject in the simulated population, the algorithm provides the calendar time of birth, the age of onset of the disease (if the subject contracts the disease) and the age at death. By this means, the impact of interventions may be estimated and compared.
Introduction
Chronic diseases impose a tremendous global health problem of the 21st century. The World Health Organization estimates that 63% of all deaths in 2008 were caused by chronic diseases [1]. Besides taking measures in politics and society, research efforts are needed to oppose this threat. In studying the characteristics of chronic diseases from a public health perspective, it is often important to consider different time-scales [2]. The age of the subjects in a population is a risk factor for many diseases. Changes in life-style and medical care influence the risk of contracting and dying from the disease on a secular scale. Thus, the incidence rate of the chronic disease as well as the mortality rates (with and without the disease) depend on the calendar time. Moreover, the mortality of people with the disease often depends on the duration since its onset. Examples are diabetes [3], [4], dementia [5], depression [6], and systemic lupus erythematosus [7]. For decision-makers all time-scales may be important.
As a hypothetical example, consider the question which health programme to choose from two possibilities A and B if the outcome of interest is the gain of life-years. Possibility A is known to decrease the incidence of the disease by 15%, and possibility B lowers the mortality of those having the disease for more than ten years by 50%. The decision depends on several factors. If, on the one hand, the incidence rate is non-zero for children only and the birth rate in the population is low, possibility A may have little effect with respect to the gain of life-years. If, on the other hand, the chronic disease has very few people reaching ten-year survival after onset, programme B can be nearly useless. In problems similar to the example, the decision-maker may face a complex interplay of epidemiological and demographical considerations.
The aim of this work is to describe an algorithm and a flexible software framework for simulation of populations moving in a multi-state model (illness-death model) that describes a chronic disease. The simulation takes into account the different time-scales: calendar time, age, and duration of the disease. Although simulations using multi-state models are subject to recent textbooks [8], to our knowledge no algorithm has been described that incorporates the effects of all the different time-scales.
Methods
A popular framework for studying irreversible diseases is the illness-death model (IDM) consisting of the three states Normal, Disease and Death, [9]–[11]. The associated transition rates, synonymously densities (in units “per person-time”, not to be confused with risks or probabilities [12]), are the incidence 
, and the mortality rates 
and 
(Figure 1). In general, these rates depend on calendar time 
, age 
and in case of 
on the duration of the disease 
.
Figure 1. Three states model of normal (healthy), diseased and dead subjects.
The transition rates may depend on calender time 
age 
and in case of 
also on the duration 
of the disease.
This article presents a method for simulating populations moving in the IDM. The motivation for the algorithms comes from analytical epidemiology where relations between common epidemiological measures are studied. Examples for those measures are the prevalence, the duration of a disease, the age of onset (or diagnosis), and lost life-years (due to the disease). A typical question may be: what is the mean age of diagnosis of subjects born in a certain time period? What is their mean age at death? Another interesting aim is the estimation of the incidence rate 
from cross-sectional information. At a specific point in time 
, each of the subjects 
has a unique “status”. Neglecting those who are unborn or dead at 
, the status in the IDM is either normal (non-diseased) or diseased. Thus, the status can be seen as a binary random variable, and data of this kind are typically called current status data
[13]. The current status is closely linked with the incidence 
and the mortalities 
and 
before 
. Estimating the incidence from current status data, for example, has been a topic in research for decades [14]. The framework presented here may be useful in this field.
Overview of the simulation algorithm
The simulation is a microsimulation, i.e., it treats each person in the population as an autonomous unit. For each person, indexed 
the relevant events diagnosis and death are simulated. This is accomplished in two steps:
- Contracting the disease or dying without the disease is modelled as competing risk [11]. Given the time
of birth of person 
, the cumulative distribution function 
of the first failure time
is
The term first failure time
(1) 
refers to the time of diagnosis or death without disease and is measured in time units after birth of person 
. Thus, 
is the age at which the first transition from the state Normal occurs. Given that a transition occurs at 
for person 
, then the odds of moving into state Disease versus moving into state Death is
- If the event at
is the death (without the disease), the simulation for person 
is finished. If, however, the event is the diagnosis of the disease, the “second failure time” 
to death (with disease) has the distribution function 
(2)
The next section describes in detail how the integrals in Equations (1) and (2) may be calculated in the simulation. After calculating the integrals, the question arises how the times 
and 
can be obtained from 
and 
. This is done by the inverse transform sampling method: Let 
be a cumulative distribution function and 
. For 
it holds: If 
is a uniform random variable on 
, then 
follows the distribution 
. Thus, the simulation of 
and 
is easy, if a random number generator for 
such as runif in R is available.
For each of the 
persons in the population we store four pieces of data:
a unique identifier
the date
of birth (dob) of person 
the age at diagnosis (adi) of person
andthe age at death (ade) of person
If the person 
does not contract the disease, the age at diagnosis adi is set to NA (missing). In summary, we get the Algorithm 1.
Algorithm 1 Simulation of populations moving in the IDM
1: for
to
do
2: dob 
3: calculate event time 
according to Equation (1)
4: simulate type of event that has happened at 
by Equation (1)
5: if event is diagnosis then
6: adi 
7: calculate time 
of death using Equation (2)
8: ade 
9: else
10: adi 
NA
11: ade 
12: end if
13: write j, dob, adi, ade to file
14: end for
Calculating line integrals
In this section the calculation of the integrals in Equations (1) and (2) is described. The situation in which analytical expressions for these integrals exist, is straightforward. The first simulation in the next section is an example. However, in real world applications, analytical expressions for the integrals are rarely given. For convenience, mathematical functions (e.g., splines) may be fitted to the data and integration is accomplished with the fitted functions. Since the aim of this work is a flexible way of treating the incidence and mortality rates, we assume that the rates are given as numerical values on a regular grid. Here we focus on the most general case, which is characterized by:
None of the time-scales
and 
is negligible, andthe values of
are given as data points only.
For many chronic diseases, we think this is the most relevant case: The mortality rates 
and 
depend on 
and 
. Since age is a risk factor for many diseases, the dependency on age is obvious. Healthier life-style and medical progress in many countries lead to secular trends in 
and 
. In addition, disease duration 
is likely to have an impact on 
in many chronic diseases. Thus, none of the time-scales is negligible, which is covered in the second and third example.
In the most general case, the integrands 
and 
are given by data points only. We assume that the numerical values are located on a regular grid. The grid is two-dimensional in case of 
and 
which depend on two time-scales 
and 
and the grid is three-dimensional in case of 
which depends on 
and 
In event history analysis [2], a useful concept is the Lexis diagram, which is a co-ordinate system with axes calendar time t (abscissa) and age 
(ordinate). The t-dimension sometimes is referred to as period. Each subject is represented by a line segment from time and age at entry to time and age at exit. Entry and exit may be birth and death, respectively, or entry and exit in a epidemiological study or clinical trial. There are excellent and extensive introductions about the theory of Lexis diagrams (see for example [9], [15], [16] and references therein), which allows us to be short here. In irreversible diseases, the common two-dimensional Lexis diagram with axes in t- and a-direction may be generalized to a three-dimensional co-ordinate system with disease duration 
represented by the applicate (i.e., the z-axis). If a subject does not contract the disease during lifetime, the life line remains in the t-a-plane parallel to the line bisecting abscissa and ordinate. In other words, the life line for the time without disease is parallel to 
(where the triple 
denotes the co-ordinates in time, age and duration direction, respectively). However if at a certain point in time 
the disease is diagnosed, the life line changes its direction, henceforth runs parallel to 
.
The situation is illustrated in Figure 2. The life lines of two subjects are shown in the three-dimensional Lexis space. At birth (denoted 
) both subjects are disease-free; both life lines are parallel to 
The first subject contracts the disease at 
Henceforth, the life line is parallel to 
until death at 
. The second subject remains disease-free until death at 
.
Figure 2. Three-dimensional Lexis diagram with two life lines.
Abscissa, ordinate and applicate (z-axis) represent calendar time 
, age 
and duration 
, respectively. The life lines start at birth 
and end at death 
The first subject (blue line segments) contracts the disease at 
. Then, the life line changes its direction. The second subject (red line segment) does not contract the disease, the life line remains in the t-a-plane.
Having the concept of the Lexis diagram at hand, we observe that 
and 
in Equations (1) and (2) are line integrals in the Lexis space. We start with calculating the first failure times 
For subject 
the associated life line starts at 
. We chose an age 
when it is certain that a transition to one of the states Disease or Death has occurred, say 
(years). For calculating 
we trace the hypothetical life line from 
to 
Thus, the hypothetical life line has a representation
 |
Following the life line is related to the method of ray-tracing in the field of computer graphics, where efficient algorithms for this purpose exist. In Siddon's algorithm [17], the key idea is to follow 
by calculating intersections with volume elements (voxels), which form a regular partition of the Lexis space. Let
 |
with 
be a parametrization of the points where 
intersects the voxel faces plus the start and end points 
and 
. Details for the calculation of 
are described in the supporting information to this article. The parametrization 
is ideally suited for approximating the integral in Equation (1) by the trapezoidal rule [18]. The reason lies in the fact that in calculating 
the values 
are a by-product. Algorithm 2 shows the necessary steps.
Algorithm 2 Calculating 
1: for
to
do
2: calculate 
3: 
4: 
5: 
6: 
7: for
to
do
8: 
9: 
10: 
11: 
12: end for
13: end for
Since the values of 
and 
are given on the voxel grid only, the calculation of 
needs bilinear interpolation of the values of the adjacent voxels [19].
After preparing 
the values of the times 
can be calculated by the inverse transform sampling method. Since we have 
calculated at points 
the inverse transform sampling would yield only those 
. A better accuracy can be obtained by (linear) inverse interpolation [18]. For 
let 
Then, it holds
 |
Thus, for 
drawn from a uniform distribution, we find the unique 
such that 
and set
 |
as the inverse 
For those subjects 
who contract the disease, the associated 
can be derived in a similar way as in Algorithm 2. The associated line segment starts at 
. Again, a hypothetical maximal disease duration 
is assumed, say 
(years), such that the line segment ends at 
. Thus, the line segment is parallel to 
The Siddon algorithm computes the corresponding set of intersections with the voxel grid accordingly. The ages 
at death with disease are obtained from Algorithm 2 mutatis mutandis. The interpolation of 
needs to be trilinear.
Examples
This section shows the results of different simulation settings. The associated R [20] code is provided with this article. The first simulation is about a hypothetical chronic disease with rates 
only depending on 
The corresponding age-specific prevalence can be calculated analytically, which allows cross-checking the results of the simulation. In the second example, another hypothetical disease is treated with mortality rates depending on 
and 
Here we use the ray-tracing approach in the Lexis diagram. Again, the outcomes of the simulation are compared with analytical results. The third simulation is about a relatively rare rheumatic disease. A hypothetical birth cohort of 100000 women is followed from birth to death and examined if the disease is diagnosed. Finally, the last simulation demonstrates applicability in the context of medical decision-making. Two interventions are compared with respect to the outcome life-years gained.
Simulation 1: Analytical example
In the first simulation, only one time-scale is involved. Assuming 
and
 |
one can show that the age-specific prevalence 
is given by 
[21]. The notation 
means the positive part of 
The integrals of 
and 
can easily be expressed analytically. Figure 3 shows the age-specific prevalence in a simulated cohort with 
subjects. For comparison, the true (analytical) prevalence is depicted as a solid line. The result of the simulation agrees very well with the analytical age profile of the prevalence, which indicates the correctness of the implemented R code.
Figure 3. Theoretical and simulated age-specific prevalence.
Simulation 1 comprises 
persons. The resulting age-specific prevalence (black crosses) is compared to the analytically calculated prevalence (blue solid line). The example shows the very good agreement between the simulation and the theoretical results.
Simulation 2: Mortality depends on duration
The second simulation is about a hypothetical disease with all time-scales 
and 
playing a role. We aim at calculating epidemiological measures that describe the population of the diseased. For example, age of onset, mean duration of the disease and age at death may be important to plan resource allocation (e.g., inpatient facilities).
In each of sixty consecutive years 
2000 persons are born and followed from birth to death. The incidence of a hypothetical chronic disease is assumed to be 
, the age-specific mortality rate of the non-diseased is chosen to be 
and the mortality of the diseased is 
In total, 42299 of the 120000 simulated persons contract the disease. The simulated data allows the derivation of important epidemiological measures. For example, the histograms of the age of onset and age at death are shown in Figure 4.
Figure 4. Histograms of the age of onset and age at death in a hypothetical chronic disease.
In Simulation 2 the empirical distributions of the age of onset (left) and the age at death of the diseased persons (right) are estimated.
The median age at death of those who contracted the disease is 77.20 (years) whereas the median age at death of those without the disease is 79.67 (years). The median duration of the disease in the 42299 ill subjects is 
, the interquartile range is 7.46–17.60 (years).
Finally, we can cross-check the results of the simulation by comparison with an analytical calculation. In year 
exactly 76548 persons are alive, 8802 of those having the hypothetical disease. Figure 5 shows the age-specific prevalence at 
The black lines indicate the prevalence of several age groups together with 95% confidence bounds as given by the simulation. The blue line represents the prevalence calculated analytically by the exact formula in ([9], Section 7.2). The results agree quite well within the confidence bounds.
Figure 5. Calculated and simulated prevalence in Simulation 2.
If we mimic a cross-sectional study at year 
we obtain the age-specific prevalence as indicated by the black bars (with 95% confidence bounds). For comparison the analytically calculated age-specific prevalence is shown as blue line.
Simulation 3: Systemic lupus erythematosus in the UK
The third simulation is about a hypothetical cohort of 100000 women born in the United Kingdom in 1945. The disease under consideration is systemic lupus erythematosus (SLE), which is an autoimmune disorder with significant morbidity and increased mortality. Women are more often affected than men; in terms of age-standardized incidence, the ratio is 5∶1 [22]. The peak of the age-specific incidence in British women is located at the age of about 50. Since Somers et al. could not find a significant secular trend in the age-standardized incidence in 1990–1999, we assume that the incidence does not depend on 
, [22]. The aim of this example is to study the interplay between age of onset and the mortality of the diseased women in a realistic setting.
Mortality 
has been taken from the official life tables of the British Office for National Statistics [23]. The calendar time trend in 
has been simplified by assuming that the yearly decrease in mortality is 
for all age groups. This is the geometric mean of the decrease in female mortality taken over all reported age groups in the past 60 years. Mortality 
of the women with SLE has been modelled according to [7], which takes into account several covariables: sex, age, duration of SLE, calendar time of diagnosis and geographical region. Unfortunately, an interaction analysis of these factors has not been reported, which forces us to make assumptions. We assume a multiplicative model of the impact of sex, age, region and SLE duration:
 |
The constant 
reflects the impact of sex and the region, 
and 
are hazard ratios. The hazard ratio 
of SLE duration has been interpolated affine-linearly on a logarithmic scale from the published values and backtransformed. The age dependency of 
has been treated similarly. Due to the weak (and possibly insignificant) effect, the impact of the calendar time of diagnosis has been neglected.
From the 100000 women in the simulation, 513 contract SLE. This corresponds to a lifetime risk of about 0.5%. For comparison, a recent publication about women in the US estimated 0.9% [24]. The study [24] included women with African ancestors, who are known to have a higher risk. The median age of onset in the simulation is 46.07 years, which is in nearly perfect agreement with the result of 46.1 years based on another simulation using the same data but a different method [25]. Table 1 shows the interplay between age of diagnosis and duration of SLE in the simulation.
Table 1. Age at diagnosis and disease duration of 513 British women with SLE in a simulated cohort.
| Age at | SLE duration (years) | ||||||
| diagnosis | <1 | 1–2.4 | 2.5–4 | 5–9 | 10–24 | ≥25 | |
| 5–34 | (n = 134) | 12.7% | 9.0% | 11.9% | 14.9% | 19.4% | 32.1% |
| 35–44 | (n = 108) | 15.7% | 10.2% | 14.8% | 13.9% | 19.4% | 25.9% |
| 45–59 | (n = 149) | 16.1% | 14.1% | 14.8% | 13.4% | 26.8% | 14.8% |
| ≥60 | (n = 122) | 29.5% | 16.4% | 21.3% | 23.0% | 9.0% | 0.8% |
Example how to read the table: 108 of the women are diagnosed in the age group 35–44, 13.9% of those die 5–9 years after diagnosis.
From Table 1 it is apparent that ten-year survival is not easy to reach. Substantial loss of life time is also indicated by the age at death: Median age at death for the diseased and the non-diseased women is 61.4 and 77.1 (years), respectively. This indicates a considerable loss of lifetime in the population of the diseased compared to the non-diseased. Fortunately, this situation has changed in the last years with better medical care for SLE patients. Especially the introduction of optimized treatment regimes in the last decade lead to an enormous reduction of mortality [26]. These effects have not been included in the simulation.
Simulation 4: Effectiveness of two interventions
We take Simulation 2 as the basecase and compare the effectiveness of two hypothetical intervention programmes A and B with respect to the total gain of life-years. Assume that the primary prevention programme A gradually lowers the incidence rate for all 
in the calendar years 
by 15% and remains at the 85% level for 
. Programme B is assumed to reduce the mortality 
of the diseased people gradually by 50% starting after six years with the disease. B could be achieved, for example, by listing a new drug on the formulary. After running the simulation, we find that in total the 120000 simulated persons in the basecase, in intervention A, and intervention B, have 9275008, 9348941, and 9351338 life-years, respectively. Hence, intervention A and B yield about 73900 and 76300 life-years more than the basecase. With respect to the chosen outcome parameter, gain of life-years, programme B turns out to be superior to programme A. Thus, primary prevention by programme A, in this example, is less effective than lowering the mortality of the diseased. This result is hardly predictable without a simulation and demonstrates the usefulness of our algorithm in decision making.
To obtain the empirical distributions of the total gain of life-years in both interventions, 5000 random samples of size 120000 have been drawn (with replacement from the simulated 120000 persons) [27]. Figure 6 shows the resulting densities and the superiority of programme B.
Figure 6. Esimtated densities of the gain in life-years in two intervention programmes.
Simulation 4 compares two intervention programmes in a population of 120000 persons. The estimated densities of the total gain in life-years (compared to a basecase) are shown as red and blue curve for programmes A and B, respectively.
Conclusions
This article is about simulating populations in an illness-death model consisting of the three states Normal, Disease, and Death. The relevant events, diagnosis and death, in the most general case depend on three time-scales: calendar time, age and disease duration.
After birth of a subject in the population, two cases may occur:
the subject dies without the disease, or
the subject contracts the disease and dies with the disease.
Both situations can be represented in the Lexis space, a common tool in event history analysis. In the first case, the life line is solely located in the time-age-plane. In the second case, the life line changes its direction after onset of the disease, which allows to model the duration of the disease. In many diseases, the duration plays an important role for the mortality. Beside systemic lupus erythematosus as treated above, other diseases such as diabetes [3], [4], depression [6] and dementia [5] may serve as examples.
In the most general case, the simulation requires to numerically solve line integrals. Therefore, a synthesis between a ray-tracing technique and numerical integration is exploited. The method provides a fast way to follow the individual life lines in the Lexis diagram. Computation time is an issue, because the number of simulated subjects in the population may be large (several thousands). For example, Simulation 4 takes almost nine minutes (525 seconds) on an Intel i3 personal computer with 3.3 GHz and 8 GB RAM. Simulations 1, 2, and 3 take 40, 170 and 85 seconds, respectively.
Beside the areas mentioned above, we think the method may be applicable in following the fields:
In epidemiology, the simulation may be used to study the interplay between characteristics in chronic diseases: prevalence, mean disease duration, age of onset, age at death of the non-diseased and diseased population.
The algorithms may serve as a test bench for estimation methods. For example, in [21] the age-specific incidence is derived from prevalence data. The simulation may be used to study the performance of this and related methods.
In health economics, the result of our simulation allow the application of cost weights to each subject of the population. For example, in diabetes it is well-known that disease related costs depend on the duration since onset [28]. For each individual the disease related costs may be calculated at a specific point in time. By summing over all subjects, the total costs may be estimated easily.
Similarly, for many chronic diseases, the health related quality of life depends on the duration of the disease [29]. By assigning utility weights to each individual and summing them up, the total quality-adjusted life-years (QALY) may be calculated.
The last two points are related to health economic modelling. We think that our algorithm may yield a contribution in that domain, because often health economic models are Markov models. Due to memoryless property of Markov models, the dependency of the relevant outcomes on the duration cannot be included directly ([30], Sec. 4.2.3). In the field of diabetes, for example, at least six out of ten important economic models are not capable to accurately account for diabetes duration [31]. This may not be necessary in every research question, but if highly accurate results are needed, modelling the disease duration should be considered.
Supporting Information
(PDF)
(ZIP)
Acknowledgments
We dedicate this work to the statistician and demographer Wilhelm Lexis on the 100th anniversary of his death.
Data Availability
The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper and its Supporting Information files.
Funding Statement
Financial support for this study was provided by a grant from the Hiller Foundation, Erkrath, Germany. The funder had no role in study design, data collection, analysis and interpretation, decision to publish, or preparation of the manuscript.
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