Mathematical Models in Healthcare

Abstract

In the recent years, healthcare applications of mathematical models have been increasingly developed although the field of healthcare models is not a new area. Current trends could be explained with growing rate of data and computing skills, rising healthcare costs, decreasing resources as well as more complex health problems due to extended life expectancy. In this paper, we survey the mathematical models applied to healthcare problems with a focus on disease applications. Infectious and chronic disease modelling which has been studied for several diseases such as measles, influenza is an important research area. Furthermore, effectiveness and cost-effectiveness of prevention, screening and treatment interventions could be assessed with the help of these models. In this study, we present the definition of mathematical modeling, advantages and disadvantages of modelling and introduce an extensive summary of published literature. We mainly focus on three modeling methodology: Markov models, compartmental models and agent-based simulation.

Keywords

• Agent-based simulation
• compartmental models
• disease modeling
• markov models
• mathematical modeling

Sağlık Araştırmalarında Matematik Model Kullanımı

Abstract

Sağlık araştırmalarında matematik modellerin uygulanması yeni olmamakla beraber son yıllarda oldukça yaygınlaşmıştır. Bu artışın nedeni olarak veriyle hesaplama gücündeki artış kadar sağlık maliyetlerinin artması, kaynakların azalması bununla beraber artan yaşam süresi nedeniyle rastlanan kompleks sağlık sorunları da gösterilebilir. Bu çalışma, matematik modellerin sağlık alanındaki uygulamalarını incelemeyi amaçlamakta olup özellikle klinik uygulamaları ve hastalık modellerine önem vermiştir. Bulaşıcı hastalıklar ve kronik hastalıkların modellenmesi bunlara bağlı olarak tedavi ve korunma yöntemlerinin arasından en etkin ve maliyet etkili olanların belirlenmesi önemli bir alandır. Kızamık, grip, kanser ve HIV gibi birçok hastalık ve halk sağlığı sorunu matematik modeller yardımıyla incelenip var olan kaynakların etkin kullanımını sağlayacak karar destek çalışmaları mevcuttur. Bu çalışmada, bu çalışmaların geniş bir özeti kullanılan matematik modelleme yöntemlerinin sınıflandırılmasıyla verilmiştir. Hastalık model yöntemleri olarak Markov modeller, kompartıman modelleri ve ajan temelli benzetim modelleri metot olarak özetlenmiş ve yapılan önemli çalışmalardan bazıları ve Türkiye’de yapılan uygulamalar incelenmiştir.

Keywords

• Ajan temelli simülasyon
• Hastalık modelleri
• Kompartıman modelleri
• Markov modeller
• Matematik modelleme

References

1. 1. Denton B, Verter V. Health care O.R. OR MS Today. 2010. http://www.lionhrtpub.com/ab/wpgen.shtml. Accessed March 9, 2020.
2. 2. Türkiye Odalar ve Borsalar Birliği. Türkiye Sağlık Sektörüne Genel Bakış.; 2017. https://www.tobb.org.tr/saglik/20171229-tss-genel-bakis-tr.pdf. Accessed March 9, 2020.
3. 3. Fries BE. Bibliography of Operations Research in Health-Care Systems. Oper Res. 1976;24(5):801-814. doi:10.1287/opre.24.5.801
4. 4. Pierskalla WP, Brailer DJ. Applications of operations research in health care delivery. Handbooks Oper Res Manag Sci. 1994;6(C):469-505. doi:10.1016/S0927-0507(05)80094-5
5. 5. Brandeau, Margaret L., Sainfort, Francois, Pierskalla WP. Operations Research and Health Care: A Handbook of Methods and Applications.; 2004. doi:10.1057/jos.2009.8
6. 6. Rais A, Vianaa A. Operations research in healthcare: A survey. Int Trans Oper Res. 2011;18(1):1-31. doi:10.1111/j.1475-3995.2010.00767.x
7. 7. Fakhimi M, Probert J. Operations research within UK healthcare: A review. J Enterp Inf Manag. 2013;26(1):21-49. doi:10.1108/17410391311289532
8. 8. Caro JJ, Briggs AH, Siebert U, Kuntz KM. Modeling good research practices-overview: A report of the ISPOR-SMDM modeling good research practices task force-1. Med Decis Mak. 2012;32(5):667-677. doi:10.1177/0272989X12454577

9. 9. Eddy DM, Hollingworth W, Caro JJ, Tsevat J, McDonald KM, Wong JB. Model transparency and validation: A report of the ISPOR-SMDM modeling good research practices task force-7. Med Decis Mak. 2012;32(5):733-743. doi:10.1177/0272989X12454579
10. 10. Briggs AH, Weinstein MC, Fenwick EAL, Karnon J, Sculpher MJ, Paltiel AD. Model parameter estimation and uncertainty analysis: A report of the ISPOR-SMDM modeling good research practices task force working group-6. Med Decis Mak. 2012;32(5):722-732. doi:10.1177/0272989X12458348
11. 11. Roberts M, Russell LB, Paltiel AD, Chambers M, McEwan P, Krahn M. Conceptualizing a model: A report of the ISPOR-SMDM modeling good research practices task force-2. Med Decis Mak. 2012;32(5):678-689. doi:10.1177/0272989X12454941
12. 12. Karnon J, Stahl J, Brennan A, Caro JJ, Mar J, Möller J. Modeling using discrete event simulation: A report of the ISPOR-SMDM modeling good research practices task force-4. Med Decis Mak. 2012;32(5):701-711. doi:10.1177/0272989X12455462
13. 13. Siebert U, Alagoz O, Bayoumi AM, et al. State-transition modeling: A report of the ISPOR-SMDM modeling good research practices task force-3. Med Decis Mak. 2012;32(5):690-700. doi:10.1177/0272989X12455463
14. 14. Simpson KN, Strassburger A, Jones WJ, Dietz B, Rajagopalan R. Comparison of Markov model and discrete-event simulation techniques for HIV. Pharmacoeconomics. 2009;27(2):159-165. doi:10.2165/00019053-200927020-00006
15. 15. Meltzer MI, Damon I, Leduc JW, Donald Millar J. Modeling Potential Responses to Smallpox as a Bioterrorist Weapon. Vol 7. http://www.cdc.gov/ncidod/eid/vol7no6/. Accessed March 9, 2020.
16. 16. Myers ER, McCrory DC, Nanda K, Bastian L, Matchar DB. Mathematical model for the natural history of human papillomavirus infection and cervical carcinogenesis. Am J Epidemiol. 2000;151(12):1158-1171. doi:10.1093/oxfordjournals.aje.a010166
17. 17. Chhatwal J, Kanwal F, Roberts MS, Dunn MA. Cost-effectiveness and budget impact of hepatitis C virus treatment with sofosbuvir and ledipasvir in the United States. Ann Intern Med. 2015;162(6):397-406.
18. 18. Yaylali E, Ivy JS, Taheri J. Systems engineering methods for enhancing the value stream in public health preparedness: The role of Markov models, simulation, and optimization. Public Health Rep. 2014;129. doi:10.1177/00333549141296S419
19. 19. Sanders GD, Bayoumi AM, Sundaram V, et al. Cost-effectiveness of screening for HIV in the era of highly active antiretroviral therapy. N Engl J Med. 2005;352(6):570-585.
20. 20. Shadick NA, Liang MH, Phillips CB, Fossel K, Kuntz KM. The cost-effectiveness of vaccination against Lyme disease. Arch Intern Med. 2001;161(4):554-561. doi:10.1001/archinte.161.4.554
21. 21. Requena-Méndez A, Bussion S, Aldasoro E, et al. Cost-effectiveness of Chagas disease screening in Latin American migrants at primary health-care centres in Europe: a Markov model analysis. Lancet Glob Heal. 2017;5(4):e439-e447. doi:10.1016/S2214-109X(17)30073-6
22. 22. Yaesoubi R, Cohen T. Generalized Markov models of infectious disease spread: A novel framework for developing dynamic health policies. Eur J Oper Res. 2011;215(3):679-687. doi:10.1016/j.ejor.2011.07.016
23. 23. Maillart LM, Ivy JS, Ransom S, Diehl K. Assessing dynamic breast cancer screening policies. Oper Res. 2008;56(6):1411-1427. doi:10.1287/opre.1080.0614
24. 24. Zhang J, Denton BT, Balasubramanian H, et al. Optimization of PSA-Based Screening Decisions for Prostate Cancer Detection Preventive Follow-up Policies for Cardiovascular Diseases View Project Treatment Planning-Modeling View Project Optimization of PSA-Based Screening Decisions for Prostate Cancer D.; 2009. https://www.researchgate.net/publication/228589623. Accessed March 25, 2020.
25. 25. Denton BT, Kurt M, Shah ND, Bryant SC, Smith SA. Optimizing the start time of statin therapy for patients with diabetes. Med Decis Mak. 2009;29(3):351-367. doi:10.1177/0272989X08329462
26. 26. Kurt M, Denton BT, Schaefer AJ, Shah ND, Smith SA. Type 2 diabetes. IIE Trans Healthc Syst Eng. 2011;1(1):49-65. doi:10.1080/19488300.2010.550180
27. 27. Mason JE, England DA, Denton BT, Smith SA, Kurt M, Shah ND. Optimizing statin treatment decisions for diabetes patients in the presence of uncertain future adherence. Med Decis Mak. 2012;32(1):154-166. doi:10.1177/0272989X11404076
28. 28. Klebanoff MJ, Corey KE, Samur S, et al. Cost-effectiveness Analysis of Bariatric Surgery for Patients With Nonalcoholic Steatohepatitis Cirrhosis. JAMA Netw open. 2019;2(2):e190047. doi:10.1001/jamanetworkopen.2019.0047
29. 29. Steimle LN, Denton BT. Markov decision processes for screening and treatment of chronic diseases. In: International Series in Operations Research and Management Science. Vol 248. Springer New York LLC; 2017:189-222. doi:10.1007/978-3-319-47766-4_6
30. 30. Briggs A, Sculpher M. An introduction to Markov modelling for economic evaluation. Pharmacoeconomics. 1998;13(4):397-409. doi:10.2165/00019053-199813040-00003
31. 31. Faissol DM, Griffin PM, Swann JL. Bias in Markov models of disease. Math Biosci. 2009;220(2):143-156. doi:10.1016/j.mbs.2009.05.005
32. 32. Kirsch F. Economic Evaluations of Multicomponent Disease Management Programs with Markov Models: A Systematic Review. Value Heal. 2016;19(8):1039-1054. doi:10.1016/j.jval.2016.07.004
33. 33. WO Kermack AM. A contribution to the mathematical theory of epidemics. Proc R Soc London Ser A, Contain Pap a Math Phys Character. 1927;115(772):700-721. doi:10.1098/rspa.1927.0118
34. 34. Hethcote HW. Mathematics of infectious diseases. SIAM Rev. 2000;42(4):599-653. doi:10.1137/S0036144500371907
35. 35. Lindsay SW, Hole DG, Hutchinson RA, Richards SA, Willis SG. Assessing the future threat from vivax malaria in the United Kingdom using two markedly different modelling approaches. Malar J. 2010;9(1):70.
36. 36. McLean AR, Anderson RM. Measles in developing countries. Part II. The predicted impact of mass vaccination. Epidemiol Infect. 1988;100(3):419-442. doi:10.1017/S0950268800067170
37. 37. Ferrari MJ, Grais RF, Bharti N, et al. The dynamics of measles in sub-Saharan Africa. Nature. 2008;451(7179):679-684. doi:10.1038/nature06509
38. 38. Zhou L, Wang Y, Xiao Y, Li MY. Global dynamics of a discrete age-structured SIR epidemic model with applications to measles vaccination strategies. Math Biosci. 2019;308:27-37. doi:10.1016/j.mbs.2018.12.003
39. 39. Thakkar N, Gilani SSA, Hasan Q, McCarthy KA. Decreasing measles burden by optimizing campaign timing. Proc Natl Acad Sci U S A. 2019;166(22):11069-11073. doi:10.1073/pnas.1818433116
40. 40. Metcalf CJE, Lessler J, Klepac P, Morice A, Grenfell BT, Bjørnstad ON. Structured models of infectious disease: Inference with discrete data. Theor Popul Biol. 2012;82(4):275-282. doi:10.1016/j.tpb.2011.12.001
41. 41. Pandey A, Atkins KE, Medlock J, et al. Strategies for containing Ebola in West Africa. Science. 2014;346(6212):991-995. doi:10.1126/science.1260612
42. 42. Dye C, Williams BG. The population dynamics and control of tuberculosis. Science. 2010;328(5980):856-861. doi:10.1126/science.1185449
43. 43. Chan CH, McCabe CJ, Fisman DN. Core groups, antimicrobial resistance and rebound in gonorrhoea in North America. Sex Transm Infect. 2012;88(3):200-204. doi:10.1136/sextrans-2011-050049
44. 44. Khurana N, Yaylali E, Farnham PG, et al. Impact of Improved HIV Care and Treatment on PrEP Effectiveness in the United States, 2016–2020. JAIDS J Acquir Immune Defic Syndr. 2018;78(4):399-405.
45. 45. Lipsitch M, Cohen T, Cooper B, et al. Transmission dynamics and control of severe acute respiratory syndrome. Science. 2003;300(5627):1966-1970. doi:10.1126/science.1086616
46. 46. Chowell G, Blumberg S, Simonsen L, Miller MA, Viboud C. Synthesizing data and models for the spread of MERS-CoV, 2013: Key role of index cases and hospital transmission. Epidemics. 2014;9:40-51. doi:10.1016/j.epidem.2014.09.011
47. 47. Coburn BJ, Wagner BG, Blower S. Modeling influenza epidemics and pandemics: insights into the future of swine flu (H1N1). BMC Med. 2009;7(1):30.
48. 48. Bajardi P, Poletto C, Ramasco JJ, Tizzoni M, Colizza V, Vespignani A. Human Mobility Networks, Travel Restrictions, and the Global Spread of 2009 H1N1 Pandemic. Perc M, ed. PLoS One. 2011;6(1):e16591. doi:10.1371/journal.pone.0016591
49. 49. Khazeni N. Effectiveness and Cost-Effectiveness of Vaccination Against Pandemic Influenza (H1N1) 2009. Ann Intern Med. 2009;151(12):829. doi:10.7326/0000605-200912150-00157
50. 50. Fraser C, Donnelly CA, Cauchemez S, et al. Pandemic potential of a strain of influenza A (H1N1): Early findings. Science. 2009;324(5934):1557-1561. doi:10.1126/science.1176062
51. 51. Hethcote HW. An age-structured model for pertussis transmission. Math Biosci. 1997;145(2):89-136. doi:10.1016/S0025-5564(97)00014-X
52. 52. Keeling MJ, Rohani P. Modeling Infectious Diseases in Humans and Animals. Princeton University Press; 2011. doi:10.1016/s1473-3099(08)70147-6
53. 53. Long EF, Brandeau ML, Owens DK. The cost-effectiveness and population outcomes of expanded HIV screening and antiretroviral treatment in the united states. Ann Intern Med. 2010;153(12):778-789. doi:10.7326/0003-4819-153-12-201012210-00004
54. 54. Long EF, Brandeau ML, Owens DK. Potential population health outcomes and expenditures of HIV vaccination strategies in the United States. Vaccine. 2009;27(39):5402-5410. doi:10.1016/j.vaccine.2009.06.063
55. 55. Wu JT, Leung K, Leung GM. Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study. Lancet. 2020;395(10225):689-697. doi:10.1016/S0140-6736(20)30260-9
56. 56. Maier BF, Brockmann D. Effective containment explains subexponential growth in recent confirmed COVID-19 cases in China. Science. 2020;368(6492):742-746. doi:10.1126/science.abb4557
57. 57. Prem K, Liu Y, Russell TW, et al. The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China: a modelling study. Lancet Public Heal. 2020;5(5):e261-e270. doi:10.1016/S2468-2667(20)30073-6
58. 58. Li Y, Lawley MA, Siscovick DS, Zhang D, Pagán JA. Agent-based modeling of chronic diseases: A narrative review and future research directions. Prev Chronic Dis. 2016;13(5). doi:10.5888/pcd13.150561
59. 59. Farnham PG, Gopalappa C, Sansom SL, et al. Updates of lifetime costs of care and quality-of-life estimates for HIV-infected persons in the United States: late versus early diagnosis and entry into care. JAIDS J Acquir Immune Defic Syndr. 2013;64(2):183-189.
60. 60. Gopalappa C, Farnham PG, Chen YH, Sansom SL. Progression and Transmission of HIV/AIDS (PATH 2.0): A New, Agent-ased Model to Estimate HIV Transmissions in the United States. Med Decis Mak. 2016;37(2):224-233. doi:10.1177/0272989X16668509
61. 61. Lee BY, Brown ST, Cooley P, et al. Vaccination deep into a pandemic wave: Potential mechanisms for a “third wave” and the impact of vaccination. Am J Prev Med. 2010;39(5):e21-e29. doi:10.1016/j.amepre.2010.07.014
62. 62. Lee BY, Brown ST, Cooley P, et al. Simulating school closure strategies to titigate an influenza epidemic. J Public Heal Manag Pract. 2010;16(3):252-261. doi:10.1097/PHH.0b013e3181ce594e
63. 63. Grefenstette JJ, Brown ST, Rosenfeld R, et al. FRED (A Framework for Reconstructing Epidemic Dynamics): An open-source software system for modeling infectious diseases and control strategies using census-based populations. BMC Public Health. 2013;13(1):940. doi:10.1186/1471-2458-13-940
64. 64. Merler S, Ajelli M, Fumanelli L, et al. Spatiotemporal spread of the 2014 outbreak of Ebola virus disease in Liberia and the effectiveness of non-pharmaceutical interventions: A computational modelling analysis. Lancet Infect Dis. 2015;15(2):204-211. doi:10.1016/S1473-3099(14)71074-6
65. 65. Olsen J, Jepsen MR. Human papillomavirus transmission and cost-effectiveness of introducing quadrivalent HPV vaccination in Denmark. Int J Technol Assess Health Care. 2010;26(2):183-191. doi:10.1017/S0266462310000085
66. 66. Day TE, Ravi N, Xian H, Brugh A. An Agent-Based Modeling Template for a Cohort of Veterans with Diabetic Retinopathy. Chaum E, ed. PLoS One. 2013;8(6):e66812. doi:10.1371/journal.pone.0066812
67. 67. Li Y, Kong N, Lawley M, Weiss L, Pagán JA. Advancing the use of evidence-based decision-making in local health departments with systems science methodologies. Am J Public Health. 2015;105 Suppl 2(S2):S217-22. doi:10.2105/AJPH.2014.302077
68. 68. Hammond RA, Ornstein JT. A model of social influence on body mass index. Ann N Y Acad Sci. 2014;1331(1):34-42. doi:10.1111/nyas.12344
69. 69. Nianogo RA, Arah OA. Agent-based modeling of noncommunicable diseases: A systematic review. Am J Public Health. 2015;105(3):e20-e31. doi:10.2105/AJPH.2014.302426
70. 70. Sayan M, Hınçal E, Şanlıdağ T, Kaymakamzade B, Sa’ad FT, Baba IA. Dynamics of HIV/AIDS in Turkey from 1985 to 2016. Qual Quant. 2018;52(1):711-723. doi:10.1007/s11135-017-0648-7
71. 71. Kaymakamzade B, Şanlıdağ T, Hınçal E, Sayan M, Sa’ad FT, Baba IA. Role of awareness in controlling HIV/AIDS: a mathematical model. Qual Quant. 2018;52(1):625-637. doi:10.1007/s11135-017-0640-2
72. 72. Örmeci N, Malhan S, Balık İ, Ergör G, Razavi H, Robbins S. Scenarios to manage the hepatitis C disease burden and associated economic impact of treatment in Turkey. Hepatol Int. 2017;11(6):509-516.
73. 73. Yaylali E, Ozdemir B, Lacin N, Ceyil S. Modelling Hepatitis C Infections Among People Who Inject Drugs in Turkey: Is HCV Elimination Possible? In: Calisir F, Korhan O, eds. Industrial Engineering in the Digital Disruption Era: Selected Papers from the Global Joint Conference on Industrial Engineering and Its Application Areas, GJCIE 2019. Springer, Cham; 2020:360-374. doi:10.1007/978-3-030-42416-9_32
74. 74. Koyuncu M, Erol R. Optimal resource allocation model to mitigate the impact of pandemic influenza: A case study for Turkey. J Med Syst. 2010;34(1):61-70. doi:10.1007/s10916-008-9216-y
75. 75. Wolfson LJ, Daniels VJ, Pillsbury M, et al. Cost-effectiveness analysis of universal varicella vaccination in Turkey using a dynamic transmission model. PLoS One. 2019;14(8):e0220921. doi:10.1371/journal.pone.0220921
76. 76. Bakir M, Türel Ö, Topachevskyi O. Cost-effectiveness of new pneumococcal conjugate vaccines in Turkey: A decision analytical model. BMC Health Serv Res. 2012;12(1). doi:10.1186/1472-6963-12-386
77. 77. Marijam A, Olbrecht J, Ozakay A, Eken V, Meszaros K. Cost-Effectiveness Comparison of Pneumococcal Conjugate Vaccines in Turkish Children. Value Heal Reg Issues. 2019;19:34-44. doi:10.1016/j.vhri.2018.11.007
78. 78. Ozmen V, Cakar B, Gokmen E, et al. Cost effectiveness of Gene Expression Profiling in Patients with Early-Stage Breast Cancer in a Middle-Income Country, Turkey: Results of a Prospective Multicenter Study. Eur J Breast Heal. 2019;15(3):183-190. doi:10.5152/ejbh.2019.4761
79. 79. Balçik PY, Şahin B. Cost-effectiveness analysis of pemetrexed and gemcitabine treatment for advanced nonsmall cell lung cancer in turkey. Turkish J Med Sci. 2016;46(1):152-158. doi:10.3906/sag-1408-4
80. 80. Sözmen K, Unal B, Capewell S, Critchley J, O’Flaherty M. Estimating diabetes prevalence in Turkey in 2025 with and without possible interventions to reduce obesity and smoking prevalence, using a modelling approach. Int J Public Health. 2014;60(1):13-21. doi:10.1007/s00038-014-0622-2
81. 81. Islek D, Sozmen K, Unal B, et al. Estimating the potential contribution of stroke treatments and preventative policies to reduce the stroke and ischemic heart disease mortality in Turkey up to 2032: a modelling study. BMC Public Health. 2016;16(1):46. doi:10.1186/s12889-015-2655-8
82. 82. Kretzschmar M, Wallinga J. Mathematical Models in Infectious Disease Epidemiology. In: Krämer A, Kretzschmar M, Krickeberg K, eds. Modern Infectious Disease Epidemiology. New York, NY: Springer; 2009:209-221. doi:10.1007/978-0-387-93835-6_12