Introduction
The centralized care approach in China’s nursing homes has shown extensive growth, with the number of institutions increasing from 308 000 to 1.16 million and the number of nursing beds from 1.533 million to 6.272 million from 2006 to 2020 (Xu et al., 2023). Research has shown that habitual caregiving promotes dependency and loss of self-care capacity, and decreases physical functioning in older persons. In addition, behaviors of proactive caregivers are detrimental to older adults’ ability to care for themselves and positively correlate with their dependence due to caregiver overcare. The decline in the daily living skills of the elderly with mild disability aggravates the occupation of public resources and severely affects the general capacity of society. Accordingly, improving the self-care ability of the mildly disabled elderly is critical. It is also an effective way to improve the quality of life (QoL) of the elderly (Dusseljee et al., 2011). The ability of the elderly to take care of themselves is affected by their internal factors and regional and environmental factors. In addition, the elderly of different genders also have commonalities as well as differences in their ability to care for themselves (Harisalo & McInerney, 2008). Systems science provides a comprehensive perspective for understanding health problems. It has been widely used to solve complex problems in medical rehabilitation. They all have their specific advantages in medical rehabilitation. Public health has many complex systems. Complex systems are composed of heterogeneous elements that interact with each other, have emergent properties that cannot be explained by understanding the individual elements of the system, persist over time, and adapt to changing conditions. Public health is beginning to use the results of systems science studies to shape practice and policy (Luke & Stamatakis, 2012). The elderly often suffer from chronic diseases and their QoL is affected by complex physiological, psychological, and environmental factors. Hence, analyzing the influencing factors and the dynamic changes has an essential value in predicting the QoL of elderly patients (Jifeng et al., 2008). The system dynamics model can predict the impact of chronic diseases and disabilities on the QoL of the elderly in different conditions. Accordingly, this study predicts that changes in the level coefficients of physical functioning, motor participation, and external environment are associated with QoL among elderly people with mild disabilities in nursing homes. 

Materials and Methods

Research methodology
System dynamics, founded in 1956 by Professor Forrest at the Massachusetts Institute of Technology, is a cross-cutting discipline that integrates the social and natural sciences (Ciccarello et al., 2022). The most systematic idea of the discipline is that all systems must have a structure, and the system’s structure determines the system’s function, which guides the study of the QoL of mildly disabled elderly in nursing homes. Compared with other research methods, system dynamics has a high accuracy of simulation calculation. In this study, a set of scenarios were generated through system dynamics modeling and tested through simulation (Bastan et al., 2017), which significantly improved the reliability of this study in predicting the QoL of mildly disabled elderly.

Setting and sample
A random sampling method was used to select six nursing homes in the urban areas of Chengdu, China, with 300 or more beds, as the source sites for the survey sample in 2023.
The inclusion criteria were as follows: Stable patients aged ≥65 years, clearly conscious and able to answer questions alone and elderly with mild disability. To determine the disability level of the subjects this study used the activity of daily living scale, a modified version of the Barthel index of the Taiwan Province of China. This index measures the level of dependency and mobility in the activities of daily living, i.e. feeding, bathing, grooming, dressing, bowel control, bladder control, toileting, chair transfer, ambulation, and stair climbing. Each item in this scale is assigned a score of 0, 5, or 10. In this study, the score for mildly disabled elderly is 61–99, moderately disabled elderly 46–60, totally disabled elderly 21–45, and severely disabled elderly 0–20 (Gibson et al., 2022).
Participation in the study was voluntary and the subjects provided their informed consent. The patients on long-term psychotropic medications or co-occurring cognitive impairment and dementia and subjects with severe mental illness and speech or hearing impairments that prevented them from communicating were excluded from the sample. The sample size was determined according to the scope and content of the study (Hennink & Kaiser, 2022) using the Equation 1:



In this study, α=0.05, μα/2=1.96, and π is the incidence of disablement. According to the incidence of disablement among the Chinese elderly and previous studies is 60.5%. Meanwhile, 10% of the allowable error is considered in this study, i.e. δ=0.1π. The calculated sample size is about 77. Also, 10% of invalid questionnaires were considered, and the final sample size was estimated at around 85. 
Construction of a system dynamics model of QoL for mild disabled elderly in nursing homes
Cause-effect diagram, flow diagram, and equations 
To gain insight into the relationship between self-care ability and the QoL of mildly disabled older adults in nursing homes and to explore the related influencing factors, this study uses a system dynamics model for modeling and simulation (Sun et al., 2022). By modeling the interaction between self-care ability, body functioning level, exercise participation level, and external environment level of mild disabled older adults in nursing homes, the effects of different conditional factors are predicted (Nanthamongkolchai et al., 2022). This study will examine how self-care abilities (body function, motor participation, external environment) changes affect the QoL of older adults with mild disabilities in nursing homes.

Causal loop diagram construction
By a causal loop diagram of system dynamics for the mild disabled elderly in nursing homes using the Vensim PLE software (Zhang & Tang, 2022), we can understand the causal relationships and loops between variables and the relationship between the inputs and outputs of each variable through special functions in the program; hence, we can understand the model structure. In system dynamics, causal diagrams are a model-building tool that can help to systematically analyze and describe the causal relationships between different variables. The complex interactions between variables in a system can be better understood using causal diagrams, and the basic patterns of event development can be revealed. In a typical cause-effect diagram, variables are represented as nodes, and the causal relationships are represented as arrows (Arif & MacNeil, 2023). The arrows indicate the effect of one variable on another, allowing for more explicit identification of the main influences and determining how to intervene in the system. The three modules of body functioning level (coordination, autonomy, chronic disease, emotional function, muscular endurance, health condition, urinary function, muscle strength), exercise participation level (education level, bed, and chair movement, free time, body mass index, sports rehabilitation, going outside, joint flexibility), and external environment levels (rehabilitation environment, service attitude, safety facilities, health care, health education, working system, and social welfare) were integrated and optimized to eventually form a general diagram of the causal cycle for the mildly disabled elderly in nursing homes (Figure 1).



After the causal diagram is built, the stock flow diagram needs to be built (Pasaribu, 2021). A stock-flow diagram is a standard model-building tool in system dynamics that provides a way to represent the interactions between the variables in a system. In a stock-flow diagram, each system variable can be considered a stock, and the relationships between them are represented as flows. These flows represent transfers or interactions between variables and enable a better understanding of the system’s behavior and evolutionary processes.
In the system dynamics modeling process, equations between the variables need to be established to calculate the values of each variable (Kim et al., 2014). The main computational model equations are shown below:
• Inventory equation (Equation 2):

2. Stock(t)="∫t0t) [Inflow(s)-Outflow(s)]ds+Stock(t0)
• Flow equation (Equation 3):

3. d(Stock)/dt=Inflow(t)-Outflow(t)
• Auxiliary variable equation (Equation 4):

4. aux(t)=f[lv(t),aux×(t),exo(t),const]

Regarding the variables, Stock(t) represents the quantity of stock at time t, Inflow(s) represents the inflow, Outflow(s) represents the outflow, Stock(t0) shows the quantity of stock at the initial time, aux(t) demonstrates the auxiliary variables at time t, aux(t) are the variables other than the auxiliary variables to be sought, exo(t) is the exogenous variable, and const is the endogenous variable.
The primary role of stock flow diagrams is to help identify and analyze the relationships between different system variables and reveal their complexity (Sun et al., 2022). This graphical representation allows for a better understanding of the behavioral patterns of the system, thus guiding effective policy and decision-making. In addition, stock flow diagrams can be used to predict future system trends and assess the impact of different decisions on the system. The stock-flow of system dynamics of the QoL for mild disabled elderly in nursing homes is provided in Figure 2.




Data analysis
The influences of different variables on the QoL of the mildly disabled elderly were weighted to establish the model equations and the equations of the variables in the model were given values (Table 1). 





3. Results
The model’s validity should be checked before simulating the impact of the model on the QoL of the mildly disabled elderly in nursing homes. At first, the model structure and equations can be checked for accuracy with the “Check Model” function in the Vensim software; if the Model is “OK”, it passes the test (Arif et al., 2023). The validation result is shown in Figure 3.




Simulation analysis of system dynamics for mildly disabled elderly in nursing homes
The trends of changes in QoL by body function were simulated in elderly people with mild disabilities in nursing homes. The initial value of QoL in the model was 0.25, and after simulating the body function in the nursing home, there was a significant increasing change in the body function in the first 30 days of the simulation time and reached 0.82, and then a stable trend was observed (Figure 4).



The level of exercise participation of the mildly disabled elderly in the nursing home was simulated and predicted. The level of mobility and exercise participation significantly increased, starting from the initial value of 0.20, and then there was a brief decline as the body had a recovery period after exercise participation, after the fourth day of the simulation there was a steady increase, which is also consistent with the physiological characteristics of exercise. After 40 days of simulation, a steady trend was observed (Figure 5).



The level of external environmental factors was also simulated and predicted. The trend of the predicted external environmental factors on QoL showed a steady and continuous increase from a starting value of 0.2 to a plateau after 30 days (Figure 6).



By simulating the body function, exercise participation level, and external environment for 50 days, the final proposed evaluation graph of the QoL of the mild disabled elderly in nursing homes was synthesized, and the QoL level showed a linear increase with the increase of simulation time (Figure 7).



The results show that through the 50-day simulation prediction, the dynamic changes of the factors at the level of physical functioning (pain value=0.241, autonomy=0.163, and chronic diseases=0.192), at the level of motor participation (value of education=0.186, rehabilitation partner=0.21) and the level of the external environment (medical care=0.26 and rehabilitation environment=0.184) leads to subsequent changes in the QoL of the mildly disabled elderly in nursing homes, and the values of these indicators are significantly higher than the corresponding indicators, which shows an increase in the indicators. The QoL of the elderly increases after passing the 50-day simulation prediction. 

Discussion
The study showed that the level of physical functioning (pain value, autonomy, and chronic disease), the level of motor participation (education value and rehabilitation partner), and the level of the external environment (medical care value and rehabilitation environment value) are the most critical factors contributing to the level of QoL of the mildly disabled elderly living in the nursing homes, as it was predicted by a 50-day simulation through which the indicators reflected the top factors influencing the QoL of the elderly. 
The problem of pain in the mildly disabled elderly is particularly prominent, as a study by the Pain Society of the Chinese Medical Association showed that there are at least 100 million patients with pain in China and about 80% of elderly patients have experienced pain in one part of the body. Chronic pain causes both physical and mental torture in the mildly disabled elderly, reducing their level of daily activity and causing negative emotions, such as anxiety, depression, and fear. Subsequently, we can focus on thinking about the pain mitigation aspects of the mildly disabled elderly in the future implementation of rehabilitation programs.
According to the results, improving the activity and motor participation levels of mild disabled older adults in nursing homes can effectively improve their quality of life. Chronic diseases reduce the level of self-care ability of the residents, which in turn affects their quality of life. The study showed that a higher number and type of chronic diseases indicate a lower level of self-care ability of the mild disabled elderly, cardiovascular and cerebrovascular diseases (stroke, coronary heart disease, and chronic heart failure) are the leading chronic diseases that affect the QoL of the elderly (Pearlman & Uhlmann, 1988; Chen & Chen, 2017). The poor self-care ability of mildly disabled elderly leads to different degrees of dependence on the care of others, which creates a severe psychological burden and causes anxiety, depression, guilt, and other emotions that weaken the self-care ability.
Nursing staff in nursing homes should consider the development of the self-care ability of the elderly with mild disabilities, actively communicate with them, and provide timely help to stimulate the self-care motivation of these elderly (Song, 2023). 
Improving the rehabilitation environment in the nursing home also has a significant effect on improving the QoL of the elderly with mild disabilities, and this study also precisely showed that improving the level of the external environment can effectively improve the QoL of the elderly with mild disabilities.

Conclusion
The results of this study show that dynamic changes in the level of body functioning (pain, autonomy, and chronic diseases), level of exercise participation (education and joint flexibility), and level of the external environment (medical coverage and rehabilitation environment) lead to positive changes in the QoL of mild disabled older adults in nursing homes. To improve the QoL of the elderly with mild disability, it is suggested that the managers of the nursing homes strengthen the physical functioning of the elderly, increase their participation in exercise, and modify the external environment. Further studies are recommended using nursing homes of multiple types (for-profit, social welfare) with larger samples.
In this research, the variables of the system dynamics model were assigned based on the objective ranking of the weight of the variables of the research subjects, which may have reduced the access to the model due to the wide range of choices of the research subjects, which raises the possibility of differences in the weight of the variables.

Ethical Considerations

Compliance with ethical guidelines
This study did not intervene in the samples and made data predictions through primary data. All subjects gave informed consent to participate in this study.

Funding
This research did not receive any grant from funding agencies in the public, commercial, or non-profit sectors.

Authors' contributions
Conceptualization and methodology: Maoqiang Xu and Asha Hasnimy Mohd Hashim; Questionnaire count: Zhiwei Chen; Writing the original draft: Maoqiang Xu; Review and editing: Asha Hasnimy Mohd Hashim; Final approval: All authors.

Conflict of interest
The authors declared no conflict of interest.

Acknowledgments
The authors thank the Universiti Teknologi Malaysia and the participants of this study.




References
Arif, L., Donoriyanto, D. S. & Aziz, M. H., 2023. Family program policy projection planning and transmigration against population of sukolilo district with a dynamic simulation approach based on vensim software. Central European Management Journal, 31(2), pp. 604-14. [Link]
Arif, S. & MacNeil, M. A., 2023. Applying the structural causal model framework for observational causal inference in ecology. Ecological Monographs, 93(1), pp. e1554. [DOI:10.1002/ecm.1554]
Bastan, M., et al., 2017. Sustainable development of agriculture: A system dynamics model. Kybernetes, 47(1), pp. 142-62. [DOI:10.1108/K-01-2017-0003]
Chen, H. M. & Chen, M. C., 2017. Factors associated with quality of life among older adults with chronic disease in Taiwan. International Journal of Gerontology, 11(1), pp. 12-5. [DOI:10.1016/j.ijge.2016.07.002]
Ciccarello, F., et al., 2022. Quantum collision models: Open system dynamics from repeated interactions. Physics Reports, 954, pp. 1-70. [DOI:10.1016/j.physrep.2022.01.001]
Dusseljee, J. C., et al., 2011. Participation in daytime activities among people with mild or moderate intellectual disability. Journal of Intellectual Disability Research, 55(1), pp. 4-18. [DOI:10.1111/j.1365-2788.2010.01342.x] [PMID]
Harisalo, R. & McInerney, J., 2008. Welfare State: From dream to reality. International Journal of Public Administration, 31(10-11), pp. 1303-26. [DOI:10.1080/01900690801973501]
Hennink, M. & Kaiser, B. N., 2022. Sample sizes for saturation in qualitative research: A systematic review of empirical tests. Social Science & Medicine, 292, pp. 114523. [DOI:10.1016/j.socscimed.2021.114523] [PMID]
Gibson, E., et al., 2022. Occupational therapy for cognitive impairment in stroke patients. The Cochrane Database of Systematic Reviews, 3(3), pp. Cd006430. [DOI:10.1002/14651858.CD006430.pub3] [PMID] 
Jifeng, W., Huapu, L. & Hu, P., 2008. System dynamics model of urban transportation system and its application. Journal of Transportation Systems engineering and Information Technology, 8(3), pp. 83-9. [DOI:10.1016/S1570-6672(08)60027-6]
Kim, Y. J., Norford, L. K. & Kirtley, J. L., 2015. Modeling and analysis of a variable speed heat pump for frequency regulation through direct load control. IEEE Transactions on Power Systems, 30(1), pp. 397-408. [DOI:10.1109/TPWRS.2014.2319310]
Luke, D. A. & Stamatakis, K. A., 2012. Systems science methods in public health: Dynamics, networks, and agents. Annual Review of Public Health, 33, pp. 357–76. [DOI:10.1146/annurev-publhealth-031210-101222] [PMID] 
Nanthamongkolchai, S., et al., 2022. Quality of life of older adults with physical and mobility disabilities during the covid-19 pandemic: A cross-sectional study in Thailand. Sustainability, 14(14), pp. 8525. [DOI:10.3390/su14148525]
Pearlman, R. A. & Uhlmann, R. F., 1988. Quality of life in chronic diseases: Perceptions of elderly patients. Journal of Gerontology, 43(2), pp. M25–30. [DOI:10.1093/geronj/43.2.M25] [PMID]
Pasaribu, J. S., 2021. Development of a web based inventory information system. International Journal of Engineering, Science and Information Technology, 1(2), pp. 24-31. [DOI:10.52088/ijesty.v1i2.51]
Song, S., 2023. Emotion detection of elderly people in nursing homes based on AI robot vision. Soft Computing, pp. 1-14. [Link] 
Sun, T., et al., 2022. Association between self-perceived stigma and quality of life among urban chinese older adults: The moderating role of attitude toward own aging and traditionality. Frontiers in Public Health, 10, pp. 767255. [DOI:10.3389/fpubh.2022.767255] [PMID] 
Sun, Y., Wei, W. & Li, L., 2022. Prediction and analysis of carbon emissions from construction industry in Liaoning Province based on system dynamics. Paper presented at: International Conference on Automation Control, Algorithm, and Intelligent Bionics, Strasbourg, France, 27 June 2022. [DOI:10.1117/12.2639560]
Xu, M., Hashim, A. H. M. & Chen, Z., 2023. Evaluation of rehabilitation service capacity for mildly disabled institutionalized elderly-an analysis based on the hierarchical analysis and entropy method. Middle East Journal of Rehabilitation and Health Studies, 10(3), pp. e133871. [DOI:10.5812/mejrh-133871]
Zhang, X. & Tang, Y., 2022. Study on the impact of prefabricated building cost system based on Vensim PLE. Paper presented at: International Conference on Computer Vision and Pattern Analysis (ICCPA 2021), Guangzhou, China, 2 March 2022.[DOI:10.1117/12.2626941]