Advanced Search

Review: Mathematical Models Supporting Control of COVID-19

View author affiliations
  • Abstract

    Mathematical models have played an important role in the management of the coronavirus disease 2019 (COVID-19) pandemic. The aim of this review is to describe uses of COVID-19 mathematical models, their classification, and the advantages and disadvantages of different types of models. We conducted subject heading searches of PubMed and China National Knowledge Infrastructure with the terms “COVID-19,” “Mathematical Statistical Model,” “Model,” “Modeling,” “Agent-based Model,” and “Ordinary Differential Equation Model” and classified and analyzed the scientific literature retrieved in the search. We categorized the models as data-driven or mechanism-driven. Data-driven models are mainly used for predicting epidemics, and have the advantage of rapid assessment of disease instances. However, their ability to determine transmission mechanisms is limited. Mechanism-driven models include ordinary differential equation (ODE) and agent-based models. ODE models are used to estimate transmissibility and evaluate impact of interventions. Although ODE models are good at determining pathogen transmission characteristics, they are less suitable for simulation of early epidemic stages and rely heavily on availability of first-hand field data. Agent-based models consider influences of individual differences, but they require large amounts of data and can take a long time to develop fully. Many COVID-19 mathematical modeling studies have been conducted, and these have been used for predicting trends, evaluating interventions, and calculating pathogen transmissibility. Successful infectious disease modeling requires comprehensive considerations of data, applications, and purposes.

  • loading...
  • Funding: Partly supported by the National Key Research and Development Program of China (2021YFC2301604, 2021ZD0113903)
  • [1] Alsharif W, Qurashi A. Effectiveness of COVID-19 diagnosis and management tools: a review. Radiography (Lond) 2021;27(2):682 − 7. http://dx.doi.org/10.1016/j.radi.2020.09.010.
    [2] Surkova E, Nikolayevskyy V, Drobniewski F. False-positive COVID-19 results: hidden problems and costs. Lancet Respir Med 2020;8(12):1167 − 8. http://dx.doi.org/10.1016/S2213-2600(20)30453-7.
    [3] Tang B, Wang X, Li Q, Bragazzi NL, Tang SY, Xiao YN, et al. Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions. J Clin Med 2020;9(2):462. http://dx.doi.org/10.3390/jcm9020462.
    [4] 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 − 97. http://dx.doi.org/10.1016/S0140-6736(20)30260-9.
    [5] Yang ZF, Zeng ZQ, Wang K, Wong SS, Liang WH, Zanin M, et al. Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions. J Thorac Dis 2020;12(3):165 − 74. http://dx.doi.org/10.21037/jtd.2020.02.64.
    [6] Scott N, Palmer A, Delport D, Abeysuriya R, Stuart RM, Kerr CC, et al. Modelling the impact of relaxing COVID-19 control measures during a period of low viral transmission. Med J Aust 2021;214(2):79 − 83. http://dx.doi.org/10.5694/mja2.50845.
    [7] Roosa K, Lee Y, Luo R, Kirpich A, Rothenberg R, Hyman JM, et al. Real-time forecasts of the COVID-19 epidemic in China from February 5th to February 24th, 2020. Infect Dis Model 2020;5:256 − 63. http://dx.doi.org/10.1016/j.idm.2020.02.002.
    [8] Attanayake AMCH, Perera SSN, Jayasinghe S. Phenomenological modelling of COVID-19 epidemics in Sri Lanka, Italy, the United States, and Hebei Province of China. Comput Math Methods Med 2020;2020:6397063. http://dx.doi.org/10.1155/2020/6397063.
    [9] Zou Y, Pan S, Zhao P, Han L, Wang XX, Hemerik L, et al. Outbreak analysis with a logistic growth model shows COVID-19 suppression dynamics in China. PLoS One 2020;15(6):e0235247. http://dx.doi.org/10.1371/journal.pone.0235247.
    [10] Hoertel N, Blachier M, Blanco C, Olfson M, Massetti M, Rico MS, et al. A stochastic agent-based model of the SARS-CoV-2 epidemic in France. Nat Med 2020;26(9):1417 − 21. http://dx.doi.org/10.1038/s41591-020-1001-6.
    [11] Koo JR, Cook AR, Park M, Sun YXH, Sun HY, Lim JT, et al. Interventions to mitigate early spread of SARS-CoV-2 in Singapore: a modelling study. Lancet Infect Dis 2020;20(6):678 − 88. http://dx.doi.org/10.1016/S1473-3099(20)30162-6.
    [12] Lavine JS, Bjornstad ON, Antia R. Immunological characteristics govern the transition of COVID-19 to endemicity. Science 2021;371(6530):741 − 5. http://dx.doi.org/10.1126/science.abe6522.
    [13] Chowell G, Tariq A, Hyman JM. A novel sub-epidemic modeling framework for short-term forecasting epidemic waves. BMC Med 2019;17(1):164. http://dx.doi.org/10.1186/s12916-019-1406-6.
    [14] Chowell G, Hincapie-Palacio D, Ospina J, Pell B, Tariq A, Dahal S, et al. Using phenomenological models to characterize transmissibility and forecast patterns and final burden of Zika epidemics. PLoS Curr 20168. http://dx.doi.org/10.1371/currents.outbreaks.f14b2217c902f453d9320a43a35b9583.
    [15] Liu Q, Li ZQ, Ji Y, Martinez L, Zia UH, Javaid A, et al. Forecasting the seasonality and trend of pulmonary tuberculosis in Jiangsu Province of China using advanced statistical time-series analyses. Infect Drug Resist 2019;12:2311 − 22. http://dx.doi.org/10.2147/IDR.S207809.
    [16] Dansana D, Kumar R, Das Adhikari J, Mohapatra M, Sharma R, Priyadarshini I, et al. Global forecasting confirmed and fatal cases of COVID-19 outbreak using autoregressive integrated moving average model. Front Public Health 2020;8:580327. http://dx.doi.org/10.3389/fpubh.2020.580327.
    [17] Duan XD, Zhang XL. ARIMA modelling and forecasting of irregularly patterned COVID-19 outbreaks using Japanese and South Korean data. Data Brief 2020;31:105779. http://dx.doi.org/10.1016/j.dib.2020.105779.
    [18] Benvenuto D, Giovanetti M, Vassallo L, Angeletti S, Ciccozzi M. Application of the ARIMA model on the COVID-2019 epidemic dataset. Data Brief 2020;29:105340. http://dx.doi.org/10.1016/j.dib.2020.105340.
    [19] Dhamodharavadhani S, Rathipriya R, Chatterjee JM. COVID-19 mortality rate prediction for India using statistical neural network models. Front Public Health 2020;8:441. http://dx.doi.org/10.3389/fpubh.2020.00441.
    [20] Wang YD, Yan ZH, Wang D, Yang MT, Li ZQ, Gong XR, et al. Prediction and analysis of COVID-19 daily new cases and cumulative cases: times series forecasting and machine learning models. BMC Infect Dis 2022;22(1):495. http://dx.doi.org/10.1186/s12879-022-07472-6.
    [21] Namasudra S, Dhamodharavadhani S, Rathipriya R. Nonlinear neural network based forecasting model for predicting COVID-19 cases. Neural Process Lett 2021. https://doi.org/10.1007/S11063-021-10495-W.
    [22] Naeem M, Yu J, Aamir M, Khan SA, Adeleye O, Khan Z. Comparative analysis of machine learning approaches to analyze and predict the COVID-19 outbreak. PeerJ Comput Sci 2021;7:e746. http://dx.doi.org/10.7717/peerj-cs.746.
    [23] de Melo GC, de Araújo Neto RA, de Araújo KCGM. Forecasting the rate of cumulative cases of COVID-19 infection in Northeast Brazil: a Boltzmann function-based modeling study. Cad Saude Publica 2020;36(6):e00105720. http://dx.doi.org/10.1590/0102-311x00105720.
    [24] Fu XM, Ying Q, Zeng TY, Long T, Wang Y. Simulating and forecasting the cumulative confirmed cases of SARS-CoV-2 in China by Boltzmann function-based regression analyses. J Infect 2020;80(5):578 − 606. http://dx.doi.org/10.1016/j.jinf.2020.02.019.
    [25] Li Q, Feng W, Quan YH. Trend and forecasting of the COVID-19 outbreak in China. J Infect 2020;80(4):469 − 96. http://dx.doi.org/10.1016/j.jinf.2020.02.014.
    [26] Anastassopoulou C, Russo L, Tsakris A, Siettos C. Data-based analysis, modelling and forecasting of the COVID-19 outbreak. PLoS One 2020;15(3):e0230405. http://dx.doi.org/10.1371/journal.pone.0230405.
    [27] Xiao LS, Zhang WF, Gong MC, Zhang YP, Chen LY, Zhu HB, et al. Development and validation of the HNC-LL score for predicting the severity of coronavirus disease 2019. eBioMedicine 2020;57:102880. http://dx.doi.org/10.1016/j.ebiom.2020.102880.
    [28] Liang WH, Liang HR, Ou LM, Chen BF, Chen AL, Li CC, et al. Development and validation of a clinical risk score to predict the occurrence of critical illness in hospitalized patients with COVID-19. JAMA Intern Med 2020;180(8):1081 − 9. http://dx.doi.org/10.1001/jamainternmed.2020.2033.
    [29] Shahid F, Zameer A, Muneeb M. Predictions for COVID-19 with deep learning models of LSTM, GRU and Bi-LSTM. Chaos Solitons Fractals 2020;140:110212. http://dx.doi.org/10.1016/j.chaos.2020.110212.
    [30] Shapiro MB, Karim F, Muscioni G, Augustine AS. Adaptive susceptible-infectious-removed model for continuous estimation of the COVID-19 infection rate and reproduction number in the United States: modeling study. J Med Internet Res 2021;23(4):e24389. http://dx.doi.org/10.2196/24389.
    [31] Kermack WO, McKendrick AG. Contributions to the mathematical theory of epidemics-I. Bull Math Biol 1991;53(1–2):33 − 55. http://dx.doi.org/10.1016/S0092-8240(05)80040-0.
    [32] Liu WK, Ye WJ, Zhao ZY, Liu C, Deng B, Luo L, et al. Modelling the emerging COVID-19 epidemic and estimating intervention effectiveness — Taiwan, China, 2021. China CDC Wkly 2021;3(34):716 − 9. http://dx.doi.org/10.46234/ccdcw2021.177.
    [33] Li KG, Zhao ZY, Wei HJ, Rui J, Huang JF, Guo XH, et al. Feasibility of booster vaccination in high-risk populations for controlling coronavirus variants — China, 2021. China CDC Wkly 2021;3(50):1071 − 4. http://dx.doi.org/10.46234/ccdcw2021.259.
    [34] Zhao ZY, Niu Y, Luo L, Hu QQ, Yang TL, Chu MJ, et al. The optimal vaccination strategy to control COVID-19: a modeling study in Wuhan City, China. Infect Dis Poverty 2021;10(1):140. http://dx.doi.org/10.1186/s40249-021-00922-4.
    [35] Hao XJ, Cheng SS, Wu DG, Wu TC, Lin XH, Wang CL. Reconstruction of the full transmission dynamics of COVID-19 in Wuhan. Nature 2020;584(7821):420 − 4. http://dx.doi.org/10.1038/s41586-020-2554-8.
    [36] Li ZY, Lin SN, Rui J, Bai Y, Deng B, Chen QP, et al. An easy-to-use public health-driven method (the generalized logistic differential equation model) accurately simulated COVID-19 epidemic in Wuhan and correctly determined the early warning time. Front Public Health 2022;10:813860. http://dx.doi.org/10.3389/fpubh.2022.813860.
    [37] Nguemdjo U, Meno F, Dongfack A, Ventelou B. Simulating the progression of the COVID-19 disease in Cameroon using SIR models. PLoS One 2020;15(8):e0237832. http://dx.doi.org/10.1371/journal.pone.0237832.
    [38] McMahon A, Robb NC. Reinfection with SARS-CoV-2: discrete SIR (susceptible, infected, recovered) modeling using empirical infection data. JMIR Public Health Surveill 2020;6(4):e21168. http://dx.doi.org/10.2196/21168.
    [39] Neipel J, Bauermann J, Bo S, Harmon T, Jülicher F. Power-law population heterogeneity governs epidemic waves. PLoS One 2020;15(10):e0239678. http://dx.doi.org/10.1371/journal.pone.0239678.
    [40] Sharov KS. Trends in adaptation of fifteen European countries population to SARS-CoV-2 in March-May 2020: can Taiwanese experience be adopted? J Formos Med Assoc 2021;120(1):679-87. http://dx.doi.org/10.1016/j.jfma.2020.07.038.
    [41] Borowiak M, Ning F, Pei J, Zhao S, Tung HR, Durrett R. Controlling the spread of COVID-19 on college campuses. Math Biosci Eng 2020;18(1):551 − 63. http://dx.doi.org/10.3934/mbe.2021030.
    [42] Anderez DO, Kanjo E, Pogrebna G, Kaiwartya O, Johnson SD, Hunt JA. A COVID-19-based modified epidemiological model and technological approaches to help vulnerable individuals emerge from the lockdown in the UK. Sensors (Basel) 2020;20(17):4967. http://dx.doi.org/10.3390/s20174967.
    [43] Fan RG, Wang YB, Luo M, Zhang YQ, Zhu CP. SEIR-based COVID-19 transmission model and inflection point prediction analysis. J Univ Electron Sci Technol China 2020;49(3):369-74. https://kns.cnki.net/kcms/detail/detail.aspx?dbcode=CJFD&dbname=CJFDLAST2020&filename=DKDX202003009&uniplatform=NZKPT&v=nYvDwcq-NBNBL28ubxcQm3MRbXiaKSmVHMuxHcs-vFefAzL6ngQ4nUXAFtl060Z7. (In Chinese).
    [44] Cai J, Jia HY, Wang K. Prediction of development trend of COVID-19 in Wuhan based on SEIR model. Shandong Med J 2020,60(6):1-4. https://kns.cnki.net/kcms/detail/detail.aspx?dbcode=CJFD&dbname=CJFDLAST2020&filename=SDYY202006001&uniplatform=NZKPT&v=8xaANp8M0BwIV90t8tMfOLHObOlIyVZ__8mnxBW2Gk1_DvrEr8BYh7ZyuG5gH_06. (In Chinese).
    [45] Zhang Y, Tian WL, Wu ZG, Chen ZW, Wang J. Transmission mechanism of COVID-19 epidemic along traffic routes based on improved SEIR model. J Traffic Transp Eng 2020;20(3):150 − 8. http://dx.doi.org/10.19818/j.cnki.1671-1637.2020.03.014. (In Chinese).
    [46] Ahmad Z, Arif M, Ali F, Khan I, Nisar KS. A report on COVID-19 epidemic in Pakistan using SEIR fractional model. Sci Rep 2020;10(1):22268. http://dx.doi.org/10.1038/s41598-020-79405-9.
    [47] Niu Y, Rui J, Wang QP, Zhang W, Chen ZW, Xie F, et al. Containing the transmission of COVID-19: a modeling study in 160 countries. Front Med (Lausanne) 2021;8:701836. http://dx.doi.org/10.3389/fmed.2021.701836.
    [48] Zhao ZY, Zhu YZ, Xu JW, Hu SX, Hu QQ, Lei Z, et al. A five-compartment model of age-specific transmissibility of SARS-CoV-2. Infect Dis Poverty 2020;9(1):117. http://dx.doi.org/10.1186/s40249-020-00735-x.
    [49] Chen TM, Zhao ZY, Rui J, Yu SS, Zhu YZ, Xu JW, et al. Estimating the transmissibility of coronavirus disease 2019 and assessing the effectiveness of the countermeasures to control the disease in Xiamen City. J Xiamen Univ (Nat Sci) 2020;59(3):298-303. https://kns.cnki.net/kcms/detail/detail.aspx?dbcode=CJFD&dbname=CJFDLAST2020&filename=XDZK202003002&uniplatform=NZKPT&v=zviJ1kZOe2jhn9QwSusdNIs_Q5A03drMNkbR5XqbESvnBaRDkvIBuTPfU2Iyq6rf. (In Chinese).
    [50] Leung CC, Lam TH, Cheng KK. Mass masking in the COVID-19 epidemic: people need guidance. Lancet 2020;395(10228):945. http://dx.doi.org/10.1016/S0140-6736(20)30520-1.
    [51] Hens N, Vranck P, Molenberghs G. The COVID-19 epidemic, its mortality, and the role of non-pharmaceutical interventions. Eur Heart J Acute Cardiovasc Care 2020;9(3):204 − 8. http://dx.doi.org/10.1177/2048872620924922.
    [52] Flaxman S, Mishra S, Gandy A, Unwin HJT, Mellan TA, Coupland H, et al. Estimating the effects of non-pharmaceutical interventions on COVID-19 in Europe. Nature 2020;584(7820):257 − 61. http://dx.doi.org/10.1038/s41586-020-2405-7.
    [53] Liu PJ, Rahman MU, Din A. Fractal fractional based transmission dynamics of COVID-19 epidemic model. Comput Methods Biomech Biomed Engin 20221 − 18. http://dx.doi.org/10.1080/10255842.2022.2040489.
    [54] Lin SN, Rui J, Chen QP, Zhao B, Yu SS, Li ZY, et al. Effectiveness of potential antiviral treatments in COVID-19 transmission control: a modelling study. Infect Dis Poverty 2021;10(1):53. http://dx.doi.org/10.1186/s40249-021-00835-2.
    [55] Ngonghala CN, Iboi E, Eikenberry S, Scotch M, MacIntyre CR, Bonds MH, et al. Mathematical assessment of the impact of non-pharmaceutical interventions on curtailing the 2019 novel Coronavirus. Math Biosci 2020;325:108364. http://dx.doi.org/10.1016/j.mbs.2020.108364.
    [56] Kucharski AJ, Klepac P, Conlan AJK, Kissler SM, Tang ML, Fry H, et al. Effectiveness of isolation, testing, contact tracing, and physical distancing on reducing transmission of SARS-CoV-2 in different settings: a mathematical modelling study. Lancet Infect Dis 2020;20(10):1151 − 60. http://dx.doi.org/10.1016/S1473-3099(20)30457-6.
    [57] Hachtel GD, Stack JD, Hachtel JA. Forecasting and modeling of the COVID-19 pandemic in the USA with a timed intervention model. Sci Rep 2022;12(1):4339. http://dx.doi.org/10.1038/s41598-022-07487-8.
    [58] Eikenberry SE, Mancuso M, Iboi E, Phan T, Eikenberry K, Kuang Y, et al. To mask or not to mask: modeling the potential for face mask use by the general public to curtail the COVID-19 pandemic. Infect Dis Model 2020;5:293 − 308. http://dx.doi.org/10.1016/j.idm.2020.04.001.
    [59] Iboi EA, Sharomi O, Ngonghala CN, Gumel AB. Mathematical modeling and analysis of COVID-19 pandemic in Nigeria. Math Biosci Eng 2020;17(6):7192 − 220. http://dx.doi.org/10.3934/mbe.2020369.
    [60] Shen MW, Zu J, Fairley CK, Pagán JA, Ferket B, Liu B, et al. Effects of New York's executive order on face mask use on COVID-19 infections and mortality: a modeling study. J Urban Health 2021;98(2):197 − 204. http://dx.doi.org/10.1007/s11524-021-00517-2.
    [61] Keeling MJ, Hill EM, Gorsich EE, Penman B, Guyver-Fletcher G, Holmes A, et al. Predictions of COVID-19 dynamics in the UK: short-term forecasting and analysis of potential exit strategies. PLoS Comput Biol 2021;17(1):e1008619. http://dx.doi.org/10.1371/journal.pcbi.1008619.
    [62] Suthar DL, Habenom H, Aychluh M. Effect of vaccination on the transmission dynamics of COVID-19 in Ethiopia. Results Phys 2022;32:105022. http://dx.doi.org/10.1016/j.rinp.2021.105022.
    [63] Shen MW, Zu J, Fairley CK, Pagán JA, An L, Du ZW, et al. Projected COVID-19 epidemic in the United States in the context of the effectiveness of a potential vaccine and implications for social distancing and face mask use. Vaccine 2021;39(16):2295 − 302. http://dx.doi.org/10.1016/j.vaccine.2021.02.056.
    [64] Li YX, Ge LQ, Zhou Y, Cao X, Zheng JY. Toward the impact of non-pharmaceutical interventions and vaccination on the COVID-19 pandemic with time-dependent SEIR model. Front Artif Intell 2021;4:648579. http://dx.doi.org/10.3389/frai.2021.648579.
    [65] Laubenbacher R, Hinkelmann F, Oremland M. Chapter 5 - Agent-based models and optimal control in biology: a discrete approach. In: Robeva R, Hodge TL, editors. Mathematical concepts and methods in modern biology. Amsterdam: Academic Press. 2013143 − 78. http://dx.doi.org/10.1016/B978-0-12-415780-4.00005-3.
    [66] Hoertel N, Blachier M, Blanco C, Olfson M, Massetti M, Rico MS, et al. Lockdown exit strategies and risk of a second epidemic peak: a stochastic agent-based model of SARS-CoV-2 epidemic in France. medRxiv 2020. http://dx.doi.org/10.1101/2020.04.30.20086264.
    [67] Mahdizadeh Gharakhanlou N, Hooshangi N. Spatio-temporal simulation of the novel coronavirus (COVID-19) outbreak using the agent-based modeling approach (case study: Urmia, Iran). Inform Med Unlocked 2020;20:100403. http://dx.doi.org/10.1016/j.imu.2020.100403.
    [68] Nguyen LKN, Howick S, McLafferty D, Anderson GH, Pravinkumar SJ, Van Der Meer R, et al. Evaluating intervention strategies in controlling coronavirus disease 2019 (COVID-19) spread in care homes: an agent-based model. Infect Control Hosp Epidemiol 2021;42(9):1060 − 70. http://dx.doi.org/10.1017/ice.2020.1369.
    [69] Cuevas E. An agent-based model to evaluate the COVID-19 transmission risks in facilities. Comput Biol Med 2020;121:103827. http://dx.doi.org/10.1016/j.compbiomed.2020.103827.
    [70] Rockett RJ, Arnott A, Lam C, Sadsad R, Timms V, Gray KA, et al. Revealing COVID-19 transmission in Australia by SARS-CoV-2 genome sequencing and agent-based modeling. Nat Med 2020;26(9):1398 − 404. http://dx.doi.org/10.1038/s41591-020-1000-7.
    [71] Ying FB, O'Clery N. Modelling COVID-19 transmission in supermarkets using an agent-based model. PLoS One 2021;16(4):e0249821. http://dx.doi.org/10.1371/JOURNAL.PONE.0249821.
    [72] Araya F. Modeling the spread of COVID-19 on construction workers: an agent-based approach. Saf Sci 2021;133:105022. http://dx.doi.org/10.1016/j.ssci.2020.105022.
    [73] Lombardo G, Pellegrino M, Tomaiuolo M, Cagnoni S, Mordonini M, Giacobini M, et al. Fine-grained agent-based modeling to predict COVID-19 spreading and effect of policies in large-scale scenarios. IEEE J Biomed Health Inform 2022;26(5):2052 − 62. http://dx.doi.org/10.1109/JBHI.2022.3160243.
    [74] Thompson J, McClure R, Blakely T, Wilson N, Baker MG, Wijnands JS, et al. Modelling SARS-CoV-2 disease progression in Australia and New Zealand: an account of an agent-based approach to support public health decision-making. Aust N Z J Public Health 2022;46(3):292 − 303. http://dx.doi.org/10.1111/1753-6405.13221.
  • FIGURE 1.  Classification of mathematical models in COVID-19.

    Abbreviation: COVID-19=coronavirus disease 2019; SIR=susceptible-infectious-removed; SEIR=susceptible-exposed-infectious-removed; SEIAR=susceptible-exposed-infectious-asymptomatic-removed.

Citation:

通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Turn off MathJax
Article Contents

Article Metrics

Article views(11691) PDF downloads(73) Cited by()

Share

Related

Mathematical Models Supporting Control of COVID-19

View author affiliations

Abstract

Mathematical models have played an important role in the management of the coronavirus disease 2019 (COVID-19) pandemic. The aim of this review is to describe uses of COVID-19 mathematical models, their classification, and the advantages and disadvantages of different types of models. We conducted subject heading searches of PubMed and China National Knowledge Infrastructure with the terms “COVID-19,” “Mathematical Statistical Model,” “Model,” “Modeling,” “Agent-based Model,” and “Ordinary Differential Equation Model” and classified and analyzed the scientific literature retrieved in the search. We categorized the models as data-driven or mechanism-driven. Data-driven models are mainly used for predicting epidemics, and have the advantage of rapid assessment of disease instances. However, their ability to determine transmission mechanisms is limited. Mechanism-driven models include ordinary differential equation (ODE) and agent-based models. ODE models are used to estimate transmissibility and evaluate impact of interventions. Although ODE models are good at determining pathogen transmission characteristics, they are less suitable for simulation of early epidemic stages and rely heavily on availability of first-hand field data. Agent-based models consider influences of individual differences, but they require large amounts of data and can take a long time to develop fully. Many COVID-19 mathematical modeling studies have been conducted, and these have been used for predicting trends, evaluating interventions, and calculating pathogen transmissibility. Successful infectious disease modeling requires comprehensive considerations of data, applications, and purposes.

  • 1. State Key Laboratory of Molecular Vaccinology and Molecular Diagnostics, School of Public Health, Xiamen University, Xiamen City, Fujian Province, China
  • 2. Chinese Center for Disease Control and Prevention, Beijing, China
  • Corresponding authors:

    Tianmu Chen, 13698665@qq.com

    Qun Li, liqun@chinacdc.cn

  • Funding: Partly supported by the National Key Research and Development Program of China (2021YFC2301604, 2021ZD0113903)
  • Online Date: October 07 2022
    Issue Date: October 07 2022
    doi: 10.46234/ccdcw2022.186
    • Coronavirus disease 2019 (COVID-19) has had a significant impact on public health and economies worldwide (1-2). Understanding early epidemic trends, providing theoretical support for prevention and control, and evaluating the effect of interventions have been essential foci of COVID-19 research (3-5). Mathematical models play an important role in the study of transmission mechanisms of infectious diseases; dynamic analyses and model-based predictions help guide selection of interventions (6-9). However, classification and applications of existing COVID-19 models are not fully described. For example, some researchers use time-series models to predict epidemic trends, while other researchers use stochastic models for the same purpose (10-12). This situation causes ambiguity in the applicability of models. Classification and applications should be better delineated and standardized, and conditions for appropriate use of specific model types need to be clarified.

      For this review we conducted subject heading searches on PubMed and China National Knowledge Infrastructure with search terms “COVID-19,” “Mathematical Statistical Model,” “Model,” “Modeling,” “Agent-based Model,” and “Ordinary Differential Equation Model.” The searches returned 36,021 papers related to mathematical models in the control of COVID-19; 89 are met criteria and are included in this review. We reclassified the models described in each paper, summarized their applications to COVID-19, and analyzed advantages and disadvantages of the models to provide a new classification scheme and a selection basis for future modeling research to further promote use of models in COVID-19 prevention and control.

    • We classified mathematical models as either data-driven or mechanism-driven. Data-driven models include time series models, generalized additive models, and artificial neural network (ANN) models, among others. Mechanism-driven models include ordinary differential equation (ODE) and agent-based models. Figure 1 shows our model classification scheme. In this review, we focus on data-driven, ODE, and agent-based models.

      Figure 1. 

      Classification of mathematical models in COVID-19.

      Abbreviation: COVID-19=coronavirus disease 2019; SIR=susceptible-infectious-removed; SEIR=susceptible-exposed-infectious-removed; SEIAR=susceptible-exposed-infectious-asymptomatic-removed.
    • “Data-driven model” is a general term for a collection of models that explore the relationship between instances of diseases and time. These models focus on the relationship between number of infections and time and other factors to predict disease trends. Applications in COVID-19 include growth curve models, generalized logistic growth models (GLM), and other real-time prediction models (13-14). These models can be used to describe processes of disease development, evaluate uncertainty of disease incidence, and improve the accuracy of trend predictions (7). Accuracy can be assessed with indicators such as the Akaike information criterion (AIC), Bayesian information criterion (BIC), and r-squared (R2).

      Autoregressive integrated moving average (ARIMA) models are autoregressive sliding average models based on time-series data. They have been widely used for COVID-19 predictions (15-18). The basic idea of an ARIMA model is to systematically smooth time-series sampled data to estimate and infer the state of a phenomenon at some future time by revealing the underlying pattern between a target variable and time (19). ARIMA models are adaptable and accurate at predicting trends (20). We found that for COVID-19 research, ARIMA models focused on comparison of predicted time-series data in different regions.

      Artificial neural network models have played an important role in COVID-19 prevention and control research (19). ANN models capture uncertainty in time-series data and are a powerful technique for handling nonlinear data, thus complementing traditional linear models and yielding results superior to some other models (21-22). As a stochastic nonlinear recurrent neural network model, the Boltzmann machine (BM) model is a good fit for analyzing cumulative COVID-19 infection data (23-25).

      Data-driven models have also been used in other scenarios, including prediction and assessment of cumulative deaths prevented in the pandemic (26-27), optimizing allocation of healthcare resources, and prediction of clinical risk during treatment of COVID-19 (28-30).

    • ODE models are also known as compartment models (31). They reflect the dynamics of infectious diseases based on occurrence and routes of transmission. ODE models reveal epidemiological routes, predict development trends, analyze key factors of disease epidemics, and seek optimal strategies for prevention and control. The most common ODE model is a system of differential equations based on susceptible-infectious-removed (SIR) and susceptible-exposed-infectious-removed (SEIR) compartments (32). ODE models can be developed by combining population dynamics, vaccination dynamics, isolation, and population exposures (33-34).

      The basic reproduction number (R0) is considered a key variable in ODE models that is related to the spread of infectious diseases (35-36). SIR models have been widely used worldwide, and many studies have calculated R0 to help predict COVID-19 trends (37-40). SIR models are commonly used to evaluate interventions and calculate transmissibility (41-42). The SEIR model has been used to predict and analyze outbreaks (43-44). It has also been used to analyze transmission mechanisms in different epidemic scenarios and to predict development of an epidemic (45-46). Compared with SIR and SEIR models, the susceptible-exposed-infectious-asymptomatic-removed (SEIAR) model is more applicable for COVID-19 because of its asymptomatic compartment. This model is more suitable for early outbreak predictions (47-48). Developing models for different age groups, genders, and occupations helps scholars understand important differences in the spread of COVID-19 under several scenarios (49).

      Developing effective infectious disease prevention strategies requires consideration of disease characteristics, intervention mechanisms, and resource availability. COVID-19 prevention and control measures include non-pharmaceutical interventions (NPIs) and pharmaceutical interventions (PIs) (50-54). NPIs are many, and include isolation, social distancing, wearing masks, and enclosing places of activity (55). In our review, we found that nearly 80% of ODE-based research assessed the effects of isolation, doing so by introducing an isolated population into the base-case model and comparing differences in transmission between isolated and non-isolated populations (11,3556). Evaluating the effects of other NPIs is achieved by decomposing transmission rate into the probability of transmission of a single contact and the degree of exposure (57). For example, wearing a mask reduces the probability of single-contact transmission, whereas increasing social distance reduces the degree of exposure.

      An ODE model that incorporated big data on mobile populations was used to simulate changes in epidemic trends in closed and open cities and provide recommendations for policymakers (58-60). The most prominent application of ODE models of pharmaceutical interventions is to predict a vaccination campaign’s duration, coverage, and impact that will be sufficient to reduce need for NPIs and facilitate recovery of a damaged economy and nation (61). Pandemics tend to decrease only gradually after vaccination (61-62). Key indicators for vaccination campaigns are vaccine effectiveness and coverage attained, with an inverse relationship between the two indicators, so that higher coverage can partially compensate for lower effectiveness (61). Selection of target populations for a vaccination campaign is important. For example, prioritizing vaccination of older age groups is more beneficial for reducing morbidity and mortality (63-64).

    • Data-driven and ODE models use population characteristics and do not consider heterogeneity of individuals. Agent-based modeling uses a spatial database of the geographical environment of the study area (65) and is usually applied in evaluations of prevention and control measures (10,5666). These models can refine studies of interventions, demonstrating, for example, that a combination of testing and close follow-up is more effective than mass testing or self-isolation alone (67). Agent-based models makes it possible to evaluate the impact of individual healthcare workers’ hand hygiene practices on COVID-19 transmission (68). To predict trends in individuals’ risk of infection and propose prevention and control strategies, it is necessary to combine individual profiles and demographic data in the model and identify changes in COVID-19 transmission routes in different scenarios (69-74). However, data limitations make it difficult to generalize agent-based model results for prediction of COVID-19 trajectory or evaluation of interventions in a city or country.

    • Data-driven models have unique advantages in modeling prevention and control of infectious diseases. A wide selection of data-driven models is available, with many featuring ease of use and fast performance for rapid diagnosis in early stages of an outbreak. These models are driven by historical data and statistics and are less affected by parameter changes. However, most data-driven models do not consider natural transmission characteristics and clinical features of the disease. In addition, some key disease parameters lack rigorous interpretation in terms of transmission mechanism or disease etiology, and the models require high-quality historical data, which can challenge assessing effectiveness of some interventions.

    • Compared to agent-based models, ODE models are more flexible, provide more reliable predictions, and are less computationally demanding. After setting parameters and initial values, ODE models run quickly and are adaptable to the needs of an early and rapid response to a new infectious disease. Additions and deletions can be made according to actual situations, for example, considering different disease states, time-lags, age structures, and birth and death rates. Such adaptations can make the model fit better with reality. The effects of interventions can also be assessed visually by graphing model compartments while changing parameter values. ODE models have some drawbacks. They ignore individual-level heterogeneity and randomness in the disease characteristics; they are not suitable for simulations of epidemics or early cases when some of the parameters are subjective and empirical; and the model results may be realistic and reliable only for the reporting region and may not be generalizable due to regional variability.

    • By using population characteristics and case data and applying mathematical simulation technology to simulate transmission, agent-based models complement epidemiological evaluation of epidemic characteristics and prevention and control measures (65). A feature of agent-based models is that the results can reflect spatial and temporal characteristics of a disease in the region, as data-driven and ODE models do. A key advantage is that agent-based models are more useful for analyzing infectious diseases transmission mechanisms and estimating infection rates. Shortcomings include that agent-based individual random models require a large amount of well-measured data and require researchers to have geographic environment modeling experience and a strong foundation in programming.

    • For the COVID-19 pandemic, each type of model has strengths, weaknesses, and a spectrum of applicability to different scientific questions, and each requires varying degrees of background knowledge from the researchers. Infectious disease data-driven models provided information for action by policymakers, clinicians, and public health practitioners during times of great uncertainty about the COVID-19 pandemic. Although most COVID-19 modeling studies used ODE models, as ODE technology advances, future research will involve more cross-pollination between different disciplines. Agent-based models were not widely utilized in COVID-19 prevention and control efforts. Research with more complex and systematic methods has only just begun, and there is still a great deal of in-depth research work to be done in this area.

      When dealing with an outbreak of an emerging infectious disease, model selection is based on several points. 1) Stage of the transmission. During the COVID-19 pandemic, different types of models were used at different stages. In the early stages of an epidemic, there is less knowledge about the disease, and therefore only basic SIR or SEIAR models can be built. As the disease is studied and research reveals important insights, it is possible to build more complex models that can perform new and different functions. 2) Data availability. Data are fundamental for building models, and data completeness constrains model selection. When there are insufficient real-world data, researchers can only use basic data-driven models to predict the trajectory of an epidemic. As the amount of data increases and data sources diversify, the type, veracity, and complexity of built models can increase. 3) Applications of the model. When choosing a model, it’s purpose must be clear. For example, to evaluate the impact of prevention and control measures, one must consider the natural history of the disease, characteristics and settings of the intervention, and other factors. To study the influence of individual factors, an agent-based model should be considered. 4) Generalizability of the model. Most modelling studies are set in small areas, and a model appropriate for one setting may not be applicable elsewhere. Therefore, when selecting model parameters, it is important to consider variability of study sites and parameter values in a comprehensive manner.

      Our review has shown that many studies have been conducted with COVID-19 mathematical models, and that mathematical models are highly suitable for analyses of the pandemic. Although many types and applications of COVID-19 models exist, we found that the models can be divided into mechanism-driven models and data-driven models, and that model applications are mainly focused on predicting trends, evaluating the effects of interventions, and estimating severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) transmissibility. Models are playing a critically important role in COVID-19 prevention and control. In the future, mathematical models will be used to provide scientific support for evidence-based, innovative, and precise responses to many infectious disease outbreaks.

    • No conflicts of interest.

Reference (74)

Citation:

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return