The COVID-19 pandemic highlighted the need to quickly respond, via public policy, to the onset of an infectious disease breakout. Deciding the type and level of interventions a population must consider to mitigate risk and keep the disease under control could mean saving thousands of lives. Many models were quickly introduced highlighting lockdowns, testing, contact tracing, travel policies, later on vaccination, and other intervention strategies along with costs of implementation. Here, we provided a framework for capturing population heterogeneity whose consideration may be crucial when developing a mitigation strategy based on non-pharmaceutical interventions. Precisely, we used age-stratified data to segment our population into groups with unique interactions that policy can affect such as school children or the oldest of the population, and formulated a corresponding optimal control problem considering the economic cost of lockdowns and deaths. We applied our model and numerical methods to census data for the state of New Jersey and determined the most important factors contributing to the cost and the optimal strategies to contained the pandemic impact.
Citation: Ryan Weightman, Temitope Akinode, Benedetto Piccoli. Optimal control of pandemics via a sociodemographic model of non-pharmaceutical interventions[J]. Networks and Heterogeneous Media, 2024, 19(2): 500-525. doi: 10.3934/nhm.2024022
The COVID-19 pandemic highlighted the need to quickly respond, via public policy, to the onset of an infectious disease breakout. Deciding the type and level of interventions a population must consider to mitigate risk and keep the disease under control could mean saving thousands of lives. Many models were quickly introduced highlighting lockdowns, testing, contact tracing, travel policies, later on vaccination, and other intervention strategies along with costs of implementation. Here, we provided a framework for capturing population heterogeneity whose consideration may be crucial when developing a mitigation strategy based on non-pharmaceutical interventions. Precisely, we used age-stratified data to segment our population into groups with unique interactions that policy can affect such as school children or the oldest of the population, and formulated a corresponding optimal control problem considering the economic cost of lockdowns and deaths. We applied our model and numerical methods to census data for the state of New Jersey and determined the most important factors contributing to the cost and the optimal strategies to contained the pandemic impact.
| [1] |
K Aabed, M M.A. Lashin, An analytical study of the factors that influence Covid-19 spread, Saudi J Biol Sci, 28 (2021), 1177–1195. https://doi.org/10.1016/j.sjbs.2020.11.067 doi: 10.1016/j.sjbs.2020.11.067
|
| [2] |
J A E Andersson, J Gillis, G Horn, J B Rawlings, M Diehl, Casadi–A software framework for nonlinear optimization and optimal control, Math Program Comput, 11 (2019), 1–36. https://doi.org/10.1007/s12532-018-0139-4 doi: 10.1007/s12532-018-0139-4
|
| [3] | C Baunez, M Degoulet, S Luchini, P A Pintus, M Teschl, An early assessment of curfew and second Covid-19 lock-down on virus propagation in france, medRxiv [Preprint], (2020), [cited 2024 May 14]. Available from: https://doi.org/10.1101/2020.11.11.20230243 |
| [4] |
N Bellomo, R Bingham, M A J Chaplain, G Dosi, G Forni, D A Knopoff, et al., A multiscale model of virus pandemic: heterogeneous interactive entities in a globally connected world, Math. Mod. Meth. Appl. S, 30 (2020), 1591–1651. https://doi.org/10.1142/S0218202520500323 doi: 10.1142/S0218202520500323
|
| [5] | M Bicher, N Popper, Agent-based derivation of the sir-differential equations, 2013 8th EUROSIM Congress on Modelling and Simulation, 2013,306–311. |
| [6] |
L Bolzoni, E Bonacini, C Soresina, M Groppi, Time-optimal control strategies in sir epidemic models, Math Biosci, 292 (2017), 86–96. https://doi.org/10.1016/j.mbs.2017.07.011 doi: 10.1016/j.mbs.2017.07.011
|
| [7] | A Bressan, B Piccoli, Introduction to the mathematical theory of control, Springfield: American institute of mathematical sciences, 2007. |
| [8] |
T Britton, F Ball, P Trapman, A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2, Science, 369 (2020), 846–849. https://doi.org/10.1126/science.abc6810 doi: 10.1126/science.abc6810
|
| [9] | US Census Bureau, Survey of income and program participation (SIPP), 2022. https://www.census.gov/programs-surveys/sipp.html. |
| [10] | C Conover, How economists calculate the costs and benefits of Covid-19 lockdowns, Forbes, 2020. Available from: https://www.forbes.com/sites/theapothecary/2020/03/27/how-economists-calculate-the-costs-and-benefits-of-covid-19-lockdowns/?sh = 44a20e846f63. |
| [11] | Y Chen, P Lu, C Chang, T Liu, A time-dependent SIR model for Covid-19 with undetectable infected persons, IEEE Trans. Network Sci. Eng., 7 (2020), 3279–3294. |
| [12] |
G Cho, Y J Kim, S Seo, G Jang, H Lee, Cost-effectiveness analysis of Covid-19 variants effects in an age-structured model, Sci. Rep., 13 (2023), 15844. https://doi.org/10.1038/s41598-023-41876-x doi: 10.1038/s41598-023-41876-x
|
| [13] | M Chyba, T Klotz, Y Mileyko, C Shanbrom, A look at endemic equilibria of compartmental epidemiological models and model control via vaccination and mitigation, Math. Control Signals Syst., (2023), 1–31. https://doi.org/10.1007/s00498-023-00365-2 |
| [14] |
A T Ciota, L D Kramer, Vector-virus interactions and transmission dynamics of west nile virus, Viruses, 5 (2013), 3021–3047. https://doi.org/10.3390/v5123021 doi: 10.3390/v5123021
|
| [15] | K Dietz, The estimation of the basic reproduction number for infectious diseases, Stat Methods Med Res, 2 (1993), 23–41. |
| [16] |
R Donnelly, H A Patrinos, Learning loss during Covid-19: An early systematic review, Prospects, 51 (2021), 601–609. https://doi.org/10.1007/s11125-021-09582-6 doi: 10.1007/s11125-021-09582-6
|
| [17] | E Dorn, B Hancock, J Sarakatsannis, E Viruleg, Covid-19 and learning loss—disparities grow and students need help. McKinsey & Company, 2020. Available from: https://www.mckinsey.com/industries/public-sector/our-insights/covid-19-and-learning-loss-disparities-grow-and-students-need-help. |
| [18] |
D Fanelli, F Piazza, Analysis and forecast of Covid-19 spreading in china, Italy and france, Chaos Solitons Fractals, 134 (2020), 109761. https://doi.org/10.1016/j.chaos.2020.109761 doi: 10.1016/j.chaos.2020.109761
|
| [19] |
A S Fauci, The aids epidemic—considerations for the 21st century, New Engl J Med., 341 (1999), 1046–1050. https://doi.org/10.1056/NEJM199909303411406 doi: 10.1056/NEJM199909303411406
|
| [20] |
G. Giordano, F. Blanchini, R. Bruno, P. Colaneri, A. D. Filippo, A. D. Matteo, et al., Modelling the Covid-19 epidemic and implementation of population-wide interventions in Italy, Nat Med, 26 (2020), 855–860. https://doi.org/10.1038/s41591-020-0883-7 doi: 10.1038/s41591-020-0883-7
|
| [21] | L. O Gostin, D Lucey, A Phelan, The ebola epidemic: a global health emergency, JAMA, 312 (2014), 1095–1096. |
| [22] |
C. Gunaratne, R. Reyes, E. Hemberg, UM O'Reilly, Evaluating efficacy of indoor non-pharmaceutical interventions against Covid-19 outbreaks with a coupled spatial-SIR agent-based simulation framework, Sci. Rep., 12 (2022), 6202. https://doi.org/10.1038/s41598-022-09942-y doi: 10.1038/s41598-022-09942-y
|
| [23] |
S Hammerstein, C König, T Dreisörner, A Frey, Effects of Covid-19-related school closures on student achievement-a systematic review, Front. Psychol., 12 (2021), 746289. https://doi.org/10.3389/fpsyg.2021.746289 doi: 10.3389/fpsyg.2021.746289
|
| [24] |
J K Hammitt, Valuing mortality risk in the time of Covid-19, J Risk Uncertain, 61 (2020), 129–154. https://doi.org/10.1007/s11166-020-09338-1 doi: 10.1007/s11166-020-09338-1
|
| [25] |
X Huang, X Shao, L Xing, Y Hu, D D. Sin, X Zhang, The impact of lockdown timing on Covid-19 transmission across us counties, EClinicalMedicine, 38 (2021), 101035. https://doi.org/10.1016/j.eclinm.2021.101035 doi: 10.1016/j.eclinm.2021.101035
|
| [26] | V Kala, K Guo, E Swantek, A Tong, M. Chyba, Y Mileyko, et al., Pandemics in hawaii: 1918 influenza and Covid-19, The Ninth International Conference on Global Health Challenges, 2020. |
| [27] |
W. O. Kermack, A. G. McKendrick, Contributions to the mathematical theory of epidemics. Ⅱ.—The problem of endemicity, Proc. R. Soc. London, Ser. A, 138 (1932), 55–83. https://doi.org/10.1098/rspa.1932.0171 doi: 10.1098/rspa.1932.0171
|
| [28] |
M. L. King, M. Tertilt, IPUMS-CPS: An integrated version of the march current population survey, 1962–2002, Hist. Methods, 36 (2003), 35–40. https://doi.org/10.1080/01615440309601213 doi: 10.1080/01615440309601213
|
| [29] |
J. M. Kirigia, R. N. D. K. Muthuri, The fiscal value of human lives lost from coronavirus disease (Covid-19) in China, BMC Res. Notes, 13 (2020), 198. https://doi.org/10.1186/s13104-020-05044-y doi: 10.1186/s13104-020-05044-y
|
| [30] | N. Kousar, R. Mahmood, M. Ghalib, A numerical study of SIR epidemic model, Int. J. Sci.: Basic Appl. Res., 25 (2016), 354–363. |
| [31] |
A. J. Krener, H. Schättler, The structure of small-time reachable sets in low dimensions, SIAM J. Control Optim., 27 (1989), 120–147. https://doi.org/10.1137/0327008 doi: 10.1137/0327008
|
| [32] |
J. S. Kutter, M. I. Spronken, P. L. Fraaij, R. A. M. Fouchier, S. Herfst, Transmission routes of respiratory viruses among humans, Curr. Opin. Virol., 28 (2018), 142–151. https://doi.org/10.1016/j.coviro.2018.01.001 doi: 10.1016/j.coviro.2018.01.001
|
| [33] |
K. Y. Leung, P. Trapman, T. Britton, Who is the infector? Epidemic models with symptomatic and asymptomatic cases, Math. Biosci., 301 (2018), 190–198. https://doi.org/10.1016/j.mbs.2018.04.002 doi: 10.1016/j.mbs.2018.04.002
|
| [34] |
C. Lobry, Contro labilite des syste mes non lineaires, SIAM J. Control Optim., 8 (1970), 573. https://doi.org/10.1137/0308042 doi: 10.1137/0308042
|
| [35] |
Q. Luo, R. Weightman, S. T. McQuade, M. DÍaz, E. Trélat, W. Barbour, et al., Optimization of vaccination for Covid-19 in the midst of a pandemic, Netw. Heterog. Media, 17 (2022), 443–466. https://doi.org/10.3934/nhm.2022016 doi: 10.3934/nhm.2022016
|
| [36] |
S. T. McQuade, R. Weightman, N. J. Merrill, A. Yadav, E. Trélat, S. R. Allred, et al., Control of Covid-19 outbreak using an extended seir model, Math. Mod. Meth. Appl. S., 31 (2021), 2399–2424. https://doi.org/10.1142/S0218202521500512 doi: 10.1142/S0218202521500512
|
| [37] |
M. Miller, 2019 novel coronavirus Covid-19 (2019-nCoV) data repository: Johns hopkins university center for systems science and engineering, Assoc. Can. Map Lib. Arch. Bull., 164 (2020), 47–51. https://doi.org/10.15353/acmla.n164.1730 doi: 10.15353/acmla.n164.1730
|
| [38] |
B. W. Mol, J. Karnon, Strict lockdown versus flexible social distance strategy for Covid-19 disease: A cost-effectiveness analysis, Arch. Clin. Biomed. Res., 7 (2023), 58–63. https://doi.org/10.26502/acbr.50170319 doi: 10.26502/acbr.50170319
|
| [39] | Centers for Disease Control and Prevention, Weekly updates by select demographic and geographic characteristics, National Center for Health Statistics, 2021. Available from: file:///C:/Users/mabin/Desktop/cdc_105341_DS1.pdf. |
| [40] | World Health Organization, Statement on the second meeting of the International Health Regulations (2005) Emergency Committee regarding the outbreak of novel coronavirus (2019-nCoV), 2020. Available from: https://www.who.int/zh/news/item/30-01-2020-statement-on-the-second-meeting-of-the-international-health-regulations-(2005)-emergency-committee-regarding-the-outbreak-of-novel-coronavirus-(2019-ncov). |
| [41] |
K. Prem, A. R. Cook, M. Jit, Projecting social contact matrices in 152 countries using contact surveys and demographic data, PLoS Comput. Biol., 13 (2017), e1005697. https://doi.org/10.1371/journal.pcbi.1005697 doi: 10.1371/journal.pcbi.1005697
|
| [42] |
H. E. Rodriguez, Collecting Covid-19: Documenting the CDC response, Collections, 17 (2021), 102–111. https://doi.org/10.1177/1550190620980411 doi: 10.1177/1550190620980411
|
| [43] |
M. Roy, R. D. Holt, Effects of predation on host–pathogen dynamics in SIR models, Theor. Popul. Biol., 73 (2008), 319–331. https://doi.org/10.1016/j.tpb.2007.12.008 doi: 10.1016/j.tpb.2007.12.008
|
| [44] |
C. A. Stafford, G. P. Walker, D. E. Ullman, Hitching a ride: Vector feeding and virus transmission, Commun. Integr. Biol., 5 (2012), 43–49. https://doi.org/10.4161/cib.18640 doi: 10.4161/cib.18640
|
| [45] |
R. M. Viner, S. J. Russell, H. Croker, J. Packer, J. Ward, C. Stansfield, et al., School closure and management practices during coronavirus outbreaks including Covid-19: A rapid systematic review, Lancet Child Adolesc., 4 (2020), 397–404. https://doi.org/10.1016/S2352-4642(20)30095-X doi: 10.1016/S2352-4642(20)30095-X
|
| [46] |
A. Wächter, L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25–57. https://doi.org/10.1007/s10107-004-0559-y doi: 10.1007/s10107-004-0559-y
|
| [47] | S. H. Waterman, D. J. Gubler, Dengue fever, Clin. Dermatol., 7 (1989), 117–122. https://doi.org/10.1016/0738-081x(89)90034-5 |
| [48] | R. Weightman, Age based control github repository, PLabCOVIDModeling, 2023. Available from: https://t.ly/Ms8C_. |
| [49] | R. Weightman, B. Piccoli, Optimization of non-pharmaceutical interventions for a mutating virus, 2023 American Control Conference (ACC), San Diego: IEEE, (2023), 307–312. https://doi.org/10.23919/ACC55779.2023.10156293 |
| [50] |
Q. Wen, J. Yang, T. Luo, First case of Covid-19 in the united states, N. Engl. J. Med., 382 (2020), e53. https://doi.org/10.1056/NEJMc2004794 doi: 10.1056/NEJMc2004794
|
| [51] | Worldometer, Cases and deaths from Covid-19 virus pandemic, 2020. Available from: https://www.worldometers.info/coronavirus/country/us/. |
| [52] |
O. Zakary, M. Rachik, I. Elmouki, On the impact of awareness programs in HIV/AIDS prevention: An SIR model with optimal control, Int. J. Comput. Appl., 133 (2016), 1–6. https://doi.org/10.5120/ijca2016908030 doi: 10.5120/ijca2016908030
|
| [53] |
J. Zhang, M. Litvinova, W. Wang, Y. Wang, X. Deng, X. Chen, et al., Evolving epidemiology and transmission dynamics of coronavirus disease 2019 outside Hubei province, China: A descriptive and modelling study, Lancet Infect. Dis., 20 (2020), 793–802. https://doi.org/10.1016/S1473-3099(20)30230-9 doi: 10.1016/S1473-3099(20)30230-9
|
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Ryan Weightman, Temitope Akinode, Benedetto Piccoli. Optimal control of pandemics via a sociodemographic model of non-pharmaceutical interventions[J]. Networks and Heterogeneous Media, 2024, 19(2): 500-525. doi: 10.3934/nhm.2024022
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