Simulation Approach to Forecasting Population Ageing on Regional Level
The paper discusses the simulation model that uses system dynamics method to study the demographic changes forecasted for the population, based on the aging chain approach. The goal of the study was to elaborate the method to overcome the drainage problem that manifests itself in the smaller number of individuals belonging to the simulated cohorts, as compared to the real population data. The solution for the drainage problem is presented. We propose the original modelling methodology that assumes the coexistence of main and elementary population cohorts. The simulation model was verified based on the historical data for Wrocław Region population and the results of the experiments prove the high degree of compatibility of the simulated age and gender related characteristics with the empirical data, which entitles us to formulate the perspectives of using this approach in the next stages of our research.
KeywordsSimulation System dynamics Ageing chain Population
A wide range of economic studies requires the credible demographic forecasts to be performed before the essential stage of research is started. When the quantitative evaluations of micro or macroeconomic performances depend on the effects of demographic trends then the proper examination of long-term implications of population dynamics can heavily influence the quality of final conclusions formulated for economic and managerial decision problems.
Financial sustainability of social security systems and, more generally, the burden on the state’s budget due to the pensions paid to senior citizens, depends on the size of the older population and the percentage of insured employees . The credible forecasting of the future demographic trends enables to indicate the optimal retirement age that would stop the budget deficit to grow.
It is proved , that there is a strong correlation between demography and savings, and between the population ageing and the economic growth. The authors investigated the life cycle hypothesis and empirically confirmed that population ageing has the negative influence on the society’s savings. The working period favors the savings to grow and the retirement gradually weakens and stops this trend.
Population projections are necessary when studying the healthcare management problems. Lagergren  analyzed the relations between the increase of the older population and the growth of the number of seriously ill patients. Based on the demographic data it was possible to estimate the total resource requirements for Swedish healthcare system in the future. Senese et al.  used the population projections to forecast the growing needs for human (personnel) resources. Ansah et al.  stressed the importance of demographic analysis when estimating the requirements for long-term care services. The knowledge about the structure and the number of the population with certain diagnose (i.e. dementia) enables to carefully plan the financial and human resources needed to cover the future demand for treatment. Jagger et al.  emphasized the importance of planning not only the number but also the certain types of medical specialties to best suit the population needs.
In this paper, we present the aging chain model that replicates population evolution according to continuous simulation paradigm. In our previous research [13, 14], we discussed the implementation of the cohort modeling approach using the system dynamics (SD) method. The projections of long-term population evolutions were performed on the aggregated data and the analysis was focused on pre-specified age-gender cohorts. The demographic groups were described using demographic parameters such as birth and death rates, life expectancy, and migration descriptors.
The challenge to be solved when using the SD approach to model the chronological aging of the population is the wastage of individuals that belong to particular cohorts. Using a series of stocks to represent population, the level of individuals in every cohort and the total population are smaller than expected if there are intermediate outflows that drain the cohorts located in the middle of the chain . This phenomenon is particularly evident when the aging chain is defined for a long-run horizon. In our model, the cohort drainage is a troubling issue because every cohort is drained by at least two intermediate outflows (i.e. deaths and emigration).
To overcome the drainage problem we propose the method that eliminates the differences between the historical and the simulated values of age-gender cohorts in the aging chain population model. The goal of the paper is to introduce the solution and verify its credibility by comparing the empirical data, taken from  and the results of the simulation.
2 Aging Chain Modeling
Aging chains population models are usually built using SD method, the continuous simulation approach developed by Forrester . The SD demographic model consists of the series of stocks to represent the sub-groups of population being at the similar age and the same gender. The stocks accumulate individuals that move through the system. An individual is born and immediately becomes a member of the youngest cohort. After the pre-specified length of time, a person grows up and is moved to the older cohort. The maturation is continued along the aging chain and at the end, after the lifetime, an individual reaches the last age-gender subgroup. In steady state the average age of population is equal the average lifetime which is also the average age of people dying. The flows in demographic SD model are used to represent the movement of individuals over a specified period of time, which is interpreted as the average residence time needed for example for a child to leave the youngest cohort and enter the subsequent one (i.e. from cohort 0–4 to cohort 5–9).
Each person in each moment of simulation belongs to only one age-gender cohort and in the next point of time a person can only be moved forward (the age of an individual can only increase). An individual becomes a member of the particular age-gender cohort when one of the following events takes place: birth (concerns only the youngest cohort, i.e. the cohort starting with the age zero), maturation (a person leaves the previous cohort and enters the next one when the maximum age, as defined for the younger cohort, is reached), immigration (the sources for immigration are irrelevant here). A person ceases to be a member of the cohort when she/he dies, is old enough to become a member of the older cohort, emigrates (the target for the emigration is irrelevant here).
The maturation time is equal to the difference between the oldest age that entitles to stay in the cohort and the age the person entered the cohort plus one. In other words, an individual stays in the cohort for the time that equals the age range of this cohort plus one, unless the person dies or emigrates.
3 Modelling Technology
3.1 Basic Assumptions
The simulation model was built according to SD approach , using ExtendSim software, based on the demographic data from Wrocław Region. The model is a part of the master hybrid model, in which the continuous changes of the population dynamics influence the discrete events of the health care model .
Cohorts are described by the number of people of the same gender and the similar age. In particular, the initial values for each of the 36 main cohorts and for 14 historical years (from the year 2002 to 2015) were collected.
Birth rates describe number of children born during each year. These parameters depend on the number of women from particular female cohorts.
Death rates describe number of people belonging to particular cohort who died during each year. For the last cohort the average life expectancy is used instead.
The volume of immigrating and emigrating people depend on the size of each cohort.
the time ranges for each cohort were extended,
the birth and death rates were modified,
the corrections within the computation algorithm were suggested,
the maturation times were extended based on the genetic optimization algorithm.
The results were satisfactory from the quantitative (numerical) point of view, however each of the above mentioned solutions was difficult to accept because of the non-transparent explanations that lie behind them. For example, when applying such a solution for another region, the tedious calculations would have to be made each time the new data were added to the input data base.
Eberlein and Thompson  used the continuous cohorting approach and they assumed that every input flow entering the cohort is delayed by the time that has to be spent in the previous cohort. Extendsim delivers the block that acts just as the conveyor type solution applied by Eberlein and Thompson , however this block can not be used for discrete simulation and therefore its use is excluded in hybrid modeling as well. For this reason such a solution was rejected from our study. Sato et al.  suggested to consider the year of birth instead of the age of the person when defining the consecutive cohorts, however such a solution forces the entirely new approach to be used when defining the input statistical parameters. For example, the migration rates for people who were born in the same year change over time and they would have to be modified accordingly.
In our model we decided to divide the WR population into 105 elementary cohorts for both genders. One cohort simulates one year of aging of males and (separately) females. To properly run the simulation, the specially designed blocks were developed, tested and implemented in the simulation model.
3.3 Hierarchical Blocks - General Background
The population model was developed in Extendsim 9.2 using the blocks from Value library. The set of hierarchical blocks was designed to represent the elementary and the main cohorts. These new blocks are ready to be used in the future implementations of Extendsim environment. The blocks can be compiled, incorporated into the users’ libraries and used repeatedly. Their equivalent in object-oriented programming languages is object plus private variables, methods for processing information and the mechanism such as encapsulation/inheritance and instantiation. Every modification made in the structure or the internal code of the hierarchical block is automatically replicated in its instances. This approach enabled us to create and run the model composed of over one hundred hierarchical blocks for both genders. Each block simulates one year aging of the population cohort. The concept of hierarchical modeling and the example of the hierarchical block designed by the user is discussed in .
The model works inside the master model and has to cooperate with the discrete submodel and therefore the compatibility with the library Item has to be ensured. One of the requirements was the adoption of named connection approach. When the different model regions work simultaneously, then the correction of the names of the connections is required in order to avoid names’ duplication.
3.4 Key Elementary Blocks in Extendsim Environment
Modeling in Extendsim environment is done through selecting the block from the library, inserting the block’s instance into the model, parametrization using the dialog boxes and input connectors, and creating the relations between the instances through the connectors. The connectors are divided into input, output, informative and object, (c.f. [8, 9]). The blocks usually used in the continuous simulation modeling are: HOLDING TANK, CONSTANT, SELECT VALUE IN, and EQUATION.
One of the most useful block is HOLDING TANK. This block is equivalent to stock in SD approach. It accumulates the dynamic objects (for example the individuals in aging chain simulation) coming through input connector and it could be withdrawn through the output connector want. The initial value of holding tank can be defined through init connector. Another connector (RS) is used to reset the tank’s value and get connector enables to read the tank’s level recorded in the previous simulation step.
Another key block, i.e. EQUATION enables the complex computation of the variables and sends the results of simulation to submodel’s output or next block input.
4 Modelling Approach
4.1 Main and Elementary Population Cohorts
For simplicity it was assumed that demographic factors are constant for each elementary cohort being a part of the main cohort. For example, the death rates are the same for every elementary cohort that forms the 5–9 main cohort, however the initial state of the main cohort is evenly distributed between each of the elementary cohort. For the oldest cohorts, when an individual reaches a maximum age, he leaves the population (he/she dies).
4.2 Feedback Loops
Positive feedback loops between the birth input flow of the main youngest cohort (i.e. 0–4 for both genders separately) and the number of women from all main female cohorts. The intensities of these feedback loops are described by the products of birth rates and the number of women in main female cohorts.
Negative feedback loops between the death output flows and the sizes of the main cohorts. The intensity is described by the product of death rate and the size of the main cohort or by the quotient of the length of the average life expectancy (this applies to the oldest cohort) and the size of the oldest main cohort.
Optional positive or negative feedback loops between migration flows and the sizes of main cohorts. The type of the feedback depends on the migration direction.
There are 18 main cohorts for each gender (36 in total) and each main cohort decomposes into a chain of elementary cohorts. Elementary cohort is the key construct of the model and it represents individuals being at the same age (described as the integer number) and the same gender. There are 105 elementary cohorts for each gender designed and programmed in the form of the hierarchical blocks.
Positive feedback loops between the birth input flow of the elementary youngest cohort and the number of women from all main female cohorts. The intensities of these feedback loops are described by the products of birth rates and the number of women in main female cohorts.
Negative feedback loops between the death output flows and the sizes of the elementary cohorts. The intensity is described by the product of death rate (death rates inside the main cohort are constant) and the size of the elementary cohort or by the by the quotient of the length of the average life expectancy (this applies to the oldest elementary cohort) and the size of the oldest elementary cohort.
Negative feedback loops between the maturation flows and the sizes of elementary cohorts. This feedback loop is absent on the main cohort level. The intensity of the maturation feedback loop depends on the age range of the elementary cohort and the variant discussed in Subsect. 4.1 (it is equal to one in the discussed model and is constant inside the main cohort).
Optional positive or negative feedback loops between migration flows and the sizes of the cohort. The type of the feedback depends on the migration direction and it is constant inside the main cohort.
4.3 Input Data
the oldest age of the individuals belonging to the main cohort: end age,
the youngest age of the individuals belonging to the main cohort: begin age,
the maturation flow between two consecutive elementary cohorts: younger ones,
the size of immigration/emigration flows: immigration/emigration,
the annual death rates: death rate,
the initial state of the cohort: initial state,
the variant selected to divide the main cohort into elementary cohorts: mode.
the size of elementary cohort: size,
the maturation flow between two consecutive elementary cohorts: elderly,
the information whether the cohort is active or inactive. The cohort is activated based on the user’s decision that defines the size of the main cohort. The wider age range of the main cohort, the higher number of elementary cohorts: active.
4.4 Internal Relations
x is total number of main cohorts of one gender,
y is time range that describes the age of individuals in the main cohort + 1,
i is the number of cohort,
j is the number of cohort from which the mother of the new-born child comes from,
elemental is the number of elementary cohort,
5 or 20 is the number of elementary cohorts in the main cohort, however the last main cohort is different because it encompasses 20 elementary cohorts (85–105).
4.5 Input and Output Data Bases
the sizes of female/male cohorts for the years 2002–2014,
the birth rates for females/males in relation to mothers’ age for the years 2002–2014,
the death rates for female/male main cohorts for the years 2002–2014,
the migration rates for female/male main cohorts for the years 2002–2014.
the table with the sizes of main cohorts calculated every 0.2 year step,
the table with the sizes of main cohorts calculated at the beginning of each year (starting from 2006).
4.6 Hierarchical Blocks
the beginning age of the main cohort: begin age,
the ending age of the main cohort: end age,
the intensity of the flow of individuals leaving the younger main cohort and entering the older main cohort, because of the maturation process: output/input,
the intensity of the migrations and the mode of including migration parameters: migration/migration mode,
the death rate for the main cohort: death rate,
the initial size of the main cohort: initial,
the variant of the division of main cohort into elementary cohorts (as mentioned earlier in Subsect 4.1): mode.
the size of main cohort: state,
the intensity of maturation flow between two consecutive min cohorts. In the case of the last main cohort the individuals older than 105 years are not considered: output/input.
5 Simulation Results
The drawback of the new solution is the longer simulation time caused by the modular structure of the computations, however this cost is not very annoying. SD simulation requires only one replication and the more important issue here is the high accuracy and the credibility of the computations. Moreover, the model became independent from the particular age-gender real population structure, the main cohorts are more precisely described, the birth rates are connected to all female cohorts and the final structure of the model became transparent and intuitively understandable.
6 Future Works
Population models are applied to many fields : biology, epidemiology, health economics, management and many others. This research builds upon previous studies on using simulation approach to properly address the demographic evolution. We managed to significantly improve our previous model by introducing the hierarchical structure of age-gender cohorts. The main age-gender cohorts define the basic aging chain of the population, however each main cohort decomposes into a range of one-year elementary cohorts. This solution enabled us to overcome the drainage problem and at the same time to keep the transparency of the model. The results demonstrate the usefulness of the approach in capturing the population evolution.
The accuracy of the output data and the technology applied offer a well-defined starting point for future research. The several research topics seem to be warranted. First, we would like to concentrate on the more credible integration of SD approach with discrete modeling. We aim at the development of a hybrid simulation model that would allow alignment of continuous demographic forecasts with the discrete model to simulate the demand for healthcare services on the regional level. This would require the better integration of the two modules driven by different simulation paradigms and the definition of the relations between the size of the particular cohort and the intensity of the demand generated by the population of this cohort. We expect the frequency distributions of the demand to be defined individually for each classified disease unit with regard to demographic, temporal, and geographical factors. The verification of such model will be based on the comparison of the historical data describing number of patients arriving to healthcare system during the predefined time unit with the results of the simulation.
Next, we would like to enhance the time range of the simulation horizon and to forecast the expected population evolution for the consecutive years. This would enable us to define the influence of the population factor on the morbidity level of the particular disease unit on the regional level.
The final research direction leads to the economic analysis that would enable to forecast the future material/financial/human resources needed to cover the healthcare demand being the result of the demographic changes observed within the population.
This project was financed by the grant Simulation modeling of the demand for healthcare services from the National Science Centre, Poland, and was awarded based on the decision 2015/17/B/HS4/00306.
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