Data-driven optimization and parameter estimation for a metric graph epidemic model with applications to COVID-19 spread in Poland: A real-world example of optimization for a challenging Rosenbrock-type objective function

Abstract

In this paper, we apply data-driven optimization to estimate key parameters in a metric graph-based epidemiological model, with the aim of analyzing the effect of road networks on the geographic spread of epidemics. As a case study, we fit our model to data from the COVID-19 pandemic in Poland during 2021. Our dataset integrates county-level daily case reports, national census information, and traffic flow studies. This framework allows us to examine the relative contribution of specific travel routes over time and infer unobserved transmission patterns in the presence of incomplete or unreliable case reporting. The optimization problem that arises from the model fitting yields an objective function resembling the Rosenbrock “banana” or “valley” function, a classical difficult benchmark for optimization algorithms. To our knowledge, this represents the first appearance of a Rosenbrock-type function in a real-world epidemiological context. We demonstrate that such a structure can emerge naturally from a simple uncoupled SIR model under specific conditions: a low initial incidence rate and a prolonged infectious period. This suggests that the Rosenbrock behavior is an intrinsic feature of fitting compartmental models to approximately Gaussian epidemiological data, providing a realistic yet simple scenario with which to test optimization algorithms. We explore optimization strategies suited to the Rosenbrock-type structure and identify a feasible parameter set for modeling the spread of COVID-19 in Poland. We use this set of parameters to identify discrepancies between the model and the data, explore how reducing traffic flow into urban areas can help flatten the infection curve, and identify some patterns in the distribution of intra-versus inter-city incidence rates. While recognizing the complex interplay of social and behavioral elements that cannot be fully captured in a high-level geographic model, our findings highlight the usefulness of metric graph-based models for understanding large-scale disease transmission in structured transportation networks.

Keywords

SIR model / Rosenbrock function / Metric graph / Epidemiology / Parameter estimation

Funding

This research was partially supported by BN’s work funded through the NSF grant #136228.

Declaration of competing interest

The authors declare they have no competing interests.

References

[1] Nwabor OF, Ikechukwu NE, Emenike MP, Christiana AO. Water and waterborne diseases: a review. Int J Trop Dis Health. 2016; 12(4):1-14. http://dx.doi.org/10.9734/IJTDH/2016/21895

[2] Bertassello L, Levy MC, Müller MF. Socio-hydrology, ecohydrology, and the space-time dynamics of human-altered catchments. Hydrol Sci J. 2021; 66(9):1393-1408. http://dx.doi.org/10.1080/02626667.2021.1948550

[3] Gunderson AK, Kumar RE, Recalde-Coronel C, et al. Malaria transmission and spillover across the Peru-Ecuador border: a spatiotemporal analysis. Int J Environ Res Public Health. 2020; 17(20):7434. http://dx.doi.org/10.3390/ijerph17207434

[4] Antonio-Nkondjio C, Ndo C, Costantini C, Awono-Ambene P, Fontenille D, Simard F. Distribution and larval habitat characterization of Anopheles moucheti, Anopheles nili, and other malaria vectors in river networks of southern Cameroon. Acta Trop. 2009; 112(3):270-276. http://dx.doi.org/10.1016/j.actatropica.2009.08.009

[5] Li Q, Cao W, Ren H, Ji Z, Jiang H. Spatiotemporal responses of dengue fever transmission to the road network in an urban area. Acta Trop. 2018; 183:8-13. http://dx.doi.org/10.1016/j.actatropica.2018.03.026

[6] Roques L, Bonnefon O. Modelling population dynamics in realistic landscapes with linear elements: A mechanistic-statistical reaction-diffusion approach. PLoS One. 2016; 11(3):e0151217. http://dx.doi.org/10.1371/journal.pone.0151217

[7] Bennett KL, Gómez Martínez C, Almanza A, et al. High infestation of invasive Aedes mosquitoes in used tires along the local transport network of Panama. Parasites Vectors. 2019; 12(1):264. http://dx.doi.org/10.1186/s13071-019-3522-8

[8] Egizi A, Kiser J, Abadam C, Fonseca DM. The hitchhiker’s guide to becoming invasive: Exotic mosquitoes spread across a US state by human transport not autonomous flight. Mol Ecol. 2016; 25(13):3033-3047. http://dx.doi.org/10.1111/mec.13653

[9] Boonman N, Chutrtong J, Wanna C, Boonsilp S, Chunchob S. Detection of Staphylococcus aureus from contact surfaces of public buses in Bangkok and metropolitan area, Thailand. Biodiversitas. 2022; 23(7). http://dx.doi.org/10.13057/biodiv/d230711

[10] Lutz JK, Van Balen J, Crawford JM, et al. Methicillin-resistant Staphylococcus aureus in public transportation vehicles (buses): Another piece to the epidemiologic puzzle. Am J Infect Control. 2014; 42(12):1285-1290. http://dx.doi.org/10.1016/j.ajic.2014.08.016

[11] Tan ASB, Erdoğdu G. Microbiological burden of public transport vehicles. Istanbul J Pharm. 2017; 47(2):52-56. http://dx.doi.org/10.5152/IstanbulJPharm.2017.008

[12] Roline CE, Crumpecker C, Dunn TM. Can methicillin-resistant Staphylococcus aureus be found in an ambulance fleet? Prehosp Emerg Care. 2007; 11(2):241-244. http://dx.doi.org/10.1080/10903120701205125

[13] Pavia AT. Germs on a plane: Aircraft, international travel, and the global spread of disease. J Infect Dis. 2007; 195(5):621-622. http://dx.doi.org/10.1086/511439

[14] Hamidi S, Hamidi I. Subway ridership, crowding, or population density: Determinants of COVID-19 infection rates in New York City. Am J Prev Med. 2021; 60(5):614-620. http://dx.doi.org/10.1016/j.amepre.2020.11.016

[15] Harris JE. The subways seeded the massive coronavirus epidemic in New York City. National Bureau of Economic Research Working Paper 27021; 2020. http://dx.doi.org/10.3386/w27021

[16] Jones RC. The spatial diffusion of COVID-19 in Texas. Soc Sci J. 2021:1-14. http://dx.doi.org/10.1080/03623319.2021.1926148

[17] Sun X, Wandelt S, Zhang A. On the degree of synchronization between air transport connectivity and COVID-19 cases at worldwide level. Transp Policy. 2021; 105:115-123. http://dx.doi.org/10.1016/j.tranpol.2021.03.005

[18] Sun X, Wandelt S, Zheng C, Zhang A. COVID-19 pandemic and air transportation: Successfully navigating the paper hurricane. J Air Transp Manag. 2021;94:102062. http://dx.doi.org/10.1016/j.jairtraman.2021.102062

[19] Nicolelis MA, Raimundo RL, Peixoto PS, Andreazzi CS. The impact of super-spreader cities, highways, and intensive care availability in the early stages of the COVID-19 epidemic in Brazil. Sci Rep. 2021; 11(1):92263. http://dx.doi.org/10.1038/s41598-021-92263-3

[20] Gatto M, Bertuzzo E, Mari L, et al. Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures. Proc Natl Acad Sci USA. 2020; 117(19):10484-10491. http://dx.doi.org/10.1073/pnas.2004978117

[21] Cartenì A, Di Francesco L, Henke I, Marino TV, Falanga A. The role of public transport during the second COVID-19 wave in Italy. Sustainability. 2021; 13(21):11905. http://dx.doi.org/10.3390/su132111905

[22] Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proc R Soc A Math Phys Eng Sci. 1927; 115(772):700-721. http://dx.doi.org/10.1098/rspa.1927.0118

[23] Noble JV. Geographic and temporal development of plagues. Nature. 1974; 250(5469):726-729. http://dx.doi.org/10.1038/250726a0

[24] Weiss H. The SIR model and the foundations of public health. MATerials MATemàtics. 2013; 2013:1-17. Available from: https://ddd.uab.cat/pub/matmat/matmat_a2013/matmat_a2013a3.pdf

[25] Cooper I, Mondal A, Antonopoulos CG. A SIR model assumption for the spread of COVID-19 in different communities. Chaos Solitons Fractals. 2020;139:110057. http://dx.doi.org/10.1016/j.chaos.2020.110057

[26] Saber S, Solouma E, Althubyani M, Messaoudi M. Statistical insights into zoonotic disease dynamics: simulation and control strategy evaluation. Symmetry. 2025; 17(5):733. http://dx.doi.org/10.3390/sym17050733

[27] Alazman I, Mishra MN, Alkahtani BS, Dubey RS. Analysis of infection and diffusion coefficient in an SIR model by using generalized fractional derivative. Fractal Fract. 2024; 8(9):537. http://dx.doi.org/10.3390/fractalfract8090537

[28] Althubyani M, Adam HD, Alalyani A, et al. Understanding zoonotic disease spread with a fractional order epidemic model. Sci Rep. 2025; 15(1):13921. http://dx.doi.org/10.1038/s41598-025-95943-6

[29] Berestycki H, Roquejoffre JM, Rossi L. Propagation of epidemics along lines with fast diffusion. Bull Math Biol. 2021; 83(1):1-34. http://dx.doi.org/10.1007/s11538-020-00826-8

[30] Kravitz H, Durón C, Brio M. A coupled spatial-network model: a mathematical framework for applications in epidemiology. Bull Math Biol. 2024; 86(11):132. http://dx.doi.org/10.1007/s11538-024-01364-3

[31] Besse C, Faye G. Dynamics of epidemic spreading on connected graphs. J Math Biol. 2021; 82(6):1-52. http://dx.doi.org/10.1007/s00285-021-01602-5

[32] Fitzgibbon WE, Morgan JJ, Webb GF, Wu Y. Modelling the aqueous transport of an infectious pathogen in regional communities: application to the cholera outbreak in Haiti. J R Soc Interface. 2020; 17(169):20200429. http://dx.doi.org/10.1098/rsif.2020.0429

[33] Wang X, Gao D, Wang J. Influence of human behavior on cholera dynamics. Math Biosci. 2015; 267:41-52. http://dx.doi.org/10.1016/j.mbs.2015.06.009

[34] Wang H, Yamamoto N. Using a partial differential equation with Google Mobility data to predict COVID-19 in Arizona. Math Biosci Eng. 2020; 17(5):4891-4904. http://dx.doi.org/10.3934/mbe.2020266

[35] Aristov VV, Stroganov AV, Yastrebov AD. Simulation of spatial spread of the COVID-19 pandemic on the basis of the kinetic-advection model. Physics. 2021; 3(1):85-102. http://dx.doi.org/10.3390/physics3010008

[36] Ódor G, Czifra D, Komjáthy J, Lovász L, Karsai M. Switchover phenomenon induced by epidemic seeding on geometric networks. Proc Natl Acad Sci USA. 2021; 118(41):e2112607118. http://dx.doi.org/10.1073/pnas.2112607118

[37] Lang JC, De Sterck H, Kaiser JL, Miller JC. Analytic models for SIR disease spread on random spatial networks. J Complex Netw. 2018; 6(6):948-970. http://dx.doi.org/10.1093/comnet/cny004

[38] Bärwolff G. A local and time resolution of the COVID-19 propagation-a two-dimensional approach for Germany including diffusion phenomena to describe the spatial spread of the COVID-19 pandemic. Physics. 2021; 3(3):536-548. http://dx.doi.org/10.3390/physics3030033

[39] Wang L, Zhang Y, Wang Z, Li X. The impact of human location-specific contact pattern on the SIR epidemic transmission between populations. Int J Bifurcation Chaos. 2013; 23(5):1350095. http://dx.doi.org/10.1142/S0218127413500958

[40] Regis S, Nuiro SP, Merat W, Doncescu A. A data-based approach using a multi-group SIR model with fuzzy subsets: Application to the COVID-19 simulation in the islands of Guadeloupe. Biology. 2021; 10(10):991. http://dx.doi.org/10.3390/biology10100991

[41] Shushtari ZJ, Salimi Y, Ahmadi S, et al. Social determinants of adherence to COVID-19 preventive guidelines: A comprehensive review. Osong Public Health Res Perspect. 2021; 12(6):346. http://dx.doi.org/10.24171/j.phrp.2021.0180

[42] Khanijahani A, Iezadi S, Gholipour K, Azami-Aghdash S, Naghibi D. A systematic review of racial/ethnic and socioeconomic disparities in COVID-19. Int J Equity Health. 2021;20:248. http://dx.doi.org/10.1186/s12939-021-01582-4

[43] Gorges RJ, Konetzka RT. Factors associated with racial differences in deaths among nursing home residents with COVID-19 infection in the US. JAMA Netw Open. 2021; 4(2):e2037431. http://dx.doi.org/10.1001/jamanetworkopen.2020.37431

[44] Bonnasse-Gahot L, Berestycki H, Depuiset MA, et al. Epidemiological modelling of the 2005 French riots: A spreading wave and the role of contagion. Sci Rep. 2018; 8(1):107. http://dx.doi.org/10.1038/s41598-017-18093-4

[45] Capaldi A, Behrend S, Berman B, Smith J, Wright J, Lloyd AL. Parameter estimation and uncertainty quantification for an epidemic model. Math Biosci Eng. 2012; 9(3):553-576. http://dx.doi.org/10.3934/mbe.2012.9.553

[46] Barber RM, Sorensen RJ, Pigott DM, et al. Estimating global, regional, and national daily and cumulative infections with SARS-CoV-2 through Nov 14, 2021: a statistical analysis. Lancet. 2022; 399(10344):2351-2380. http://dx.doi.org/10.1016/S0140-6736(22)00484-6

[47] Rahmandad H, Lim TY, Sterman J. Estimating COVID-19 underreporting across 86 nations: Implications for projections and control. medRxiv. Preprint. Published online 2020. http://dx.doi.org/10.1101/2020.06.24.20139451

[48] Hakki S, Zhou J, Jonnerby J, et al. Onset and window of SARS-CoV-2 infectiousness and temporal correlation with symptom onset: A prospective, longitudinal, community cohort study. Lancet Respir Med. 2022; 10(11):1061-1073. http://dx.doi.org/10.1016/S2213-2600(22)00226-0

[49] Czerwiński M, Stepień M, Juszczyk G, et al. Reversed urban-rural gradient in COVID-19 seroprevalence and related factors in a nationally representative survey, Poland, 29 March to 14 May 2021. Euro Surveill. 2023; 28(35):2200745. http://dx.doi.org/10.2807/1560-7917.ES.2023.28.35.2200745

[50] Prodanov D. Analytical parameter estimation of the SIR epidemic model: Applications to the COVID-19 pandemic. Entropy. 2020; 23(1):59. http://dx.doi.org/10.3390/e23010059

[51] Prodanov D. Analytical solutions and parameter estimation of the SIR epidemic model. In: Mathematical Analysis of Infectious Diseases. 2022:163-189. http://dx.doi.org/10.1016/B978-0-32-390504-6.00015-2

[52] Clémençon S, Chi Tran V, De Arazoza H. A stochastic SIR model with contact-tracing: Large population limits and statistical inference. J Biol Dyn. 2008; 2(4):392-414. http://dx.doi.org/10.1080/17513750801993266

[53] Raissi M, Ramezani N, Seshaiyer P. On parameter estimation approaches for predicting disease transmission through optimization, deep learning and statistical inference methods. Lett Biomathematics. 2019; 6(2):1-26. http://dx.doi.org/10.30707/LiB6.2Raissi

[54] Han S, Stelz L, Stoecker H, Wang L, Zhou K. Approaching epidemiological dynamics of COVID-19 with physics-informed neural networks. J Franklin Inst. 2024; 361(6):106671. http://dx.doi.org/10.1016/j.jfranklin.2024.106671

[55] Vega R, Flores L, Greiner R. SIMLR: Machine learning inside the SIR. Forecasting. 2022; 4(1):72-94. http://dx.doi.org/10.3390/forecast4010005

[56] Zhuang L, Cressie N, Pomeroy L, Janies D. Multi-species SIR models from a dynamical Bayesian perspective. Theor Ecol. 2013; 6(4):457-473. http://dx.doi.org/10.1007/s12080-013-0180-x

[57] El Maroufy H, Kernane T, Becheket S, Ouddadj A. Bayesian inference for nonlinear stochastic SIR epidemic model. J Stat Comput Simul. 2016; 86(11):2229-2240. http://dx.doi.org/10.1080/00949655.2015.1107561

[58] Susyanto N, Arcede JP. Unveiling SIR model parameters: Empirical parameter approach for explicit estimation and confidence interval construction. Jambura J Biomathematics. 2024; 5(1):54-62. http://dx.doi.org/10.37905/jjbm.v5i1.26287

[59] Schmitt FG. An algorithm for the direct estimation of the parameters of the SIR epidemic model from the I(t) dynamics. Eur Phys J Plus. 2022; 137(1):57. http://dx.doi.org/10.1140/epjp/s13360-021-02237-7

[60] Medvedeva M, Simos TE, Tsitouras C, Katsikis V. Direct estimation of SIR model parameters through second-order finite differences. Math Methods Appl Sci. 2021; 44(5):3819-3826. http://dx.doi.org/10.1002/mma.6985

[61] Cantó B, Coll C, Sánchez E. Estimation of parameters in a structured SIR model. Adv Differ Equ. 2017; 2017(1): 33. http://dx.doi.org/10.1186/s13662-017-1078-5

[62] Magal P, Webb G. The parameter identification problem for SIR epidemic models: Identifying unreported cases. J Math Biol. 2018; 77(6):1629-1648. http://dx.doi.org/10.1007/s00285-017-1203-9

[63] Amiri Mehra AH, Shafieirad M, Abbasi Z, Zamani I. Parameter estimation and prediction of COVID-19 epidemic turning point and ending time of a case study on SIR/SQAIR epidemic models. Comput Math Methods Med. 2020; 2020(1): 1465923. http://dx.doi.org/10.1155/2020/1465923

[64] Ministerstwo Zdrowia (Ministry of Health).Wykaz zakażeń koronawirusem SARS-CoV-2. Accessed December 10, 2024. https://www.gov.pl/web/koronawirus/wykaz-zarazen-koronawirusem-sars-cov-2

[65] Krzysztofik R, Kantor-Pietraga I, Spórna T. Spatial and functional dimensions of the COVID-19 epidemic in Poland. Eurasian Geogr Econ. 2020; 61(4-5):573-586. http://dx.doi.org/10.1080/15387216.2020.1783337

[66] Werner PA, Kesik-Brodacka M, Nowak K, et al. Modeling the spatial and temporal spread of COVID-19 in Poland based on a spatial interaction model. ISPRS Int J Geo Inf. 2022; 11(3):195. http://dx.doi.org/10.3390/ijgi11030195

[67] Berkolaiko G, Kuchment P. Introduction to Quantum Graphs. Providence, RI: American Mathematical Society; 2013. (Mathematical Surveys and Monographs 186).

[68] Kurasov P. Spectral Geometry of Graphs. Cham: Birkhäuser; 2024. (Operator Theory: Advances and Applications, Vol 293). http://dx.doi.org/10.1007/978-3-662-67872-5

[69] Acioli PH. Diffusion as a first model of spread of viral infection. Am J Phys. 2020; 88(8):600-604. http://dx.doi.org/10.1119/10.0001464

[70] Alqarni M, Nasir A, Alyami MA, et al. A SEIR epidemic model of whooping cough-like infections and its dynamically consistent approximation. Complexity. 2022; 2022(1): 3642444. http://dx.doi.org/10.1155/2022/3642444

[71] Källén A. Thresholds and travelling waves in an epidemic model for rabies. Nonlinear Anal Theory Methods Appl. 1984; 8(8):851-856. http://dx.doi.org/10.1016/0362-546X(84)90107-X

[72] Källén A, Arcuri P, Murray J. A simple model for the spatial spread and control of rabies. J Theor Biol. 1985; 116(3):377-393. http://dx.doi.org/10.1016/S0022-5193(85)80276-9

[73] Kuniya T, Wang J. Global dynamics of an SIR epidemic model with nonlocal diffusion. Nonlinear Anal Real World Appl. 2018; 43:262-282. http://dx.doi.org/10.1016/j.nonrwa.2018.03.001

[74] Miyaoka TY, Lenhart S, Meyer JF. Optimal control of vaccination in a vector-borne reaction-diffusion model applied to Zika virus. J Math Biol. 2019; 79(3):1077-1104. http://dx.doi.org/10.1007/s00285-019-01390-z

[75] Aguilar-Madera CG, Espinosa-Paredes G, Herrera-Hernández E, et al. The spreading of COVID-19 in Mexico: A diffusional approach. Results Phys. 2021;27:104555. http://dx.doi.org/10.1016/j.rinp.2021.104555

[76] Geng X, Katul GG, Gerges F, et al. A kernel-modulated SIR model for COVID-19 contagious spread from county to continent. Proc Natl Acad Sci USA. 2021; 118(21):e2023321118. http://dx.doi.org/10.1073/pnas.2023321118

[77] Deng K. Asymptotic behavior of an SIR reaction-diffusion model with a linear source. Discrete Contin Dyn Syst Ser B. 2019; 25(11). http://dx.doi.org/10.3934/dcdsb.2019114

[78] Kravitz H, Brio M, Caputo J-G. Localized eigenvectors on metric graphs. Math Comput Simul. 2023; 214:352-372. http://dx.doi.org/10.1016/j.matcom.2023.07.011

[79] Long Z, Lu Y, Dong B. PDENet 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network. J Comput Phys. 2019;399:108925. http://dx.doi.org/10.1016/j.jcp.2019.108925

[80] Rudy SH, Brunton SL, Proctor JL, Kutz JN. Data-driven discovery of partial differential equations. Sci Adv. 2017; 3(4):e1602614. http://dx.doi.org/10.1126/sciadv.1602614

[81] Walkowiak MP, Walkowiak D. Underestimation in reporting excess COVID-19 death data in Poland during the first three pandemic waves. Int J Environ Res Public Health. 2022; 19(6):3692. http://dx.doi.org/10.3390/ijerph19063692

[82] Muller CP. Do asymptomatic carriers of SARS-CoV-2 transmit the virus? Lancet Reg Health Eur. 2021;4:100082. http://dx.doi.org/10.1016/j.lanepe.2021.100082

[83] Embrett M, Sim SM, Caldwell HA, et al. Barriers to and strategies to address COVID-19 testing hesitancy: A rapid scoping review. BMC Public Health. 2022; 22(1):750. http://dx.doi.org/10.1186/s12889-022-13127-7

[84] Ciacchini B, Tonioli F, Marciano C, et al. Reluctance to seek pediatric care during the COVID-19 pandemic and the risks of delayed diagnosis. Ital J Pediatr. 2020; 46:1-4. http://dx.doi.org/10.1186/s13052-020-00849-w

[85] The MathWorks, Inc. MATLAB R2023b. Natick, MA: The MathWorks, Inc; 2023. Available from: https://www.mathworks.com/

[86] Guilmeau T, Chouzenoux E, Elvira V. Simulated annealing: A review and a new scheme. In: 2021 IEEE Statistical Signal Processing Workshop (SSP). IEEE; 2021:101-105. http://dx.doi.org/10.1109/SSP49050.2021.9513782

[87] Kirkpatrick S, Gelatt CD Jr, Vecchi MP. Optimization by simulated annealing. Science. 1983; 220(4598):671-680. http://dx.doi.org/10.1126/science.220.4598.671

[88] Gunantara N. A review of multiobjective optimization: Methods and its applications. Cogent Eng. 2018; 5(1):1502242. http://dx.doi.org/10.1080/23311916.2018.1502242

[89] Rosenbrock H. An automatic method for finding the greatest or least value of a function. Comput J. 1960; 3(3):175-184. http://dx.doi.org/10.1093/comjnl/3.3.175

[90] Emiola I, Adem R. Comparison of minimization methods for Rosenbrock functions. In: 2021 29th Mediterranean Conference on Control and Automation (MED). IEEE; 2021:837-842. http://dx.doi.org/10.1109/MED51440.2021.9480200

[91] Fridovich-Keil S, Recht B. Choosing the step size:Intuitive line search algorithms with efficient convergence. In: 11th Annual Workshop on Optimization.

[92] Possibly missing in your input—starts with “Newton methods for function minimization,” Math Comput 1970; 24(111):647-656.

[93] Berzi P. Convergence and stability improvement of quasi-Newton methods by full-rank update of the Jacobian approximates. Appl Math. 2024; 4(1):143-181. http://dx.doi.org/10.3390/appliedmath4010008

[94] Gebreslassie BH, Diwekar UM. Heterogeneous multi-agent optimization framework with application to synthesizing optimal nuclear waste blends. Clean Technol Environ Policy. 2018; 20(1):137-157. http://dx.doi.org/10.1007/s10098-017-1464-4

[95] Opawale S. Comparing and Contrasting Differential Evolution (DE) and Covariance Matrix Adaptation Evolution Strategies [dissertation]. Guelph, ON: University of Guelph; 2018.

[96] Kaya M, Ozkan O. UAV routing with genetic algorithm based matheuristic for border security missions. Int J Optim Control Theories Appl (IJOCTA). 2021; 11(2):128-138. http://dx.doi.org/10.11121/ijocta.01.2021.001023

[97] Granville V, Krivánek M, Rasson J-P. Simulated annealing: A proof of convergence. IEEE Trans Pattern Anal Mach Intell. 1994; 16(6):652-656. http://dx.doi.org/10.1109/34.295910

[98] Karabulut Türkseven E, Genç E, Şafak I. End of day process optimization through multi-mode resource constrained project scheduling - a banking case study. Int J Optim Control Theories Appl (IJOCTA). 2025; 15(2):330-342. http://dx.doi.org/10.36922/ijocta.1694

[99] Kirkcaldy RD, Weston E, Segurado AC, Hughes G. Epidemiology of gonorrhoea: A global perspective. Sex Health. 2019; 16(5):401-411. http://dx.doi.org/10.1071/SH19061

[100] Elder H, Platt L, Leach D, et al. Factors associated with delays in presentation and treatment of gonorrhea, Massachusetts 2015-2019. Sex Transm Dis. 2023; 50(10):1097. http://dx.doi.org/10.1097/OLQ.0000000000001917

[101] Ruão T, Silva SCM. Strategic science communication: The “flatten the curve” metaphor in COVID-19 public risk messaging. In: Balonas S, Ruão T, Carrillo-Durán M-V, eds. Strategic Communication in Context: Theoretical Debates and Applied Research. UMinho Editora/Centro de Estudos de Comunicação e Sociedade; 2021:175-211. http://dx.doi.org/10.21814/uminho.ed.46.9

[102] Amidon TR, Nielsen AC, Pflugfelder EH, Richards DP, Stephens SH. Visual risk literacy in “flatten the curve” COVID-19 visualizations. J Bus Tech Commun. 2021; 35(1):101-109. http://dx.doi.org/10.1177/1050651920963439

[103] Pana TA, Bhattacharya S, Gamble DT, et al. Country-level determinants of the severity of the first global wave of the COVID-19 pandemic: An ecological study. BMJ Open. 2021; 11(2):e042034. http://dx.doi.org/10.1136/bmjopen-2020-042034

[104] Baird CE, Lake D, Panagiotou OA, Gozalo P. County-level mandates were generally effective at slowing COVID-19 transmission: Study examines county-level public health mandates and COVID-19 transmission. Health Aff (Millwood). 2024; 43(3):433-442. http://dx.doi.org/10.1377/hlthaff.2023.00431

[105] Sleator RD, Smith N. COVID-19: did the masks work? Future Microbiol. 2024; 19(11):997-1002. http://dx.doi.org/10.1080/17460913.2024.2343558

[106] Wielechowski M, Czech K, Grzeda Ł. Decline in mobility: public transport in Poland in the time of the COVID-19 pandemic. Economies. 2020; 8(4):78. http://dx.doi.org/10.3390/economies8040078

[107] Durón C, Kravitz H, Brio M. Kolmogorov-Smirnov-based edge centrality measure for metric graphs. Dynamics. 2025; 5(2):16. http://dx.doi.org/10.3390/dynamics5020016

[108] Oke M, Ogunmiloro O, Akinwumi C, Raji R. Mathematical modeling and stability analysis of a SIRV epidemic model with non-linear force of infection and treatment. Commun Math Appl. 2019; 10(4):717-731. http://dx.doi.org/10.26713/cma.v10i4.1172

[109] Zareba A, Widawski K, Kołodziejczyk K, Krzemińska A, Marek A, Rozenkiewicz A. City profile: Poznań - one of the “normals” in the centre of Europe. Cities. 2021;111:103095. http://dx.doi.org/10.1016/j.cities.2020.103095

[110] Li KK, Jarvis SA, Minhas F. Elementary effects analysis of factors controlling COVID-19 infections in computational simulation reveals the importance of social distancing and mask usage. Comput Biol Med. 2021;134:104369. http://dx.doi.org/10.1016/j.compbiomed.2021.104369

[111] Gogolewski K, Miasojedow B, Sadkowska-Todys M, et al. Data-driven case fatality rate estimation for the primary lineage of SARS-CoV-2 in Poland. Methods. 2022; 203:584-593. http://dx.doi.org/10.1016/j.ymeth.2022.01.006

[112] Rosińska M, Czarkowski MP, Sadkowska-Todys M. Infectious diseases in Poland in 2021. Epidemiol Rev/Przegl Epidemiol. 2023; 77(4):411. http://dx.doi.org/10.32394/pe.77.36

[113] Generalna Dyrekcja Dróg Krajowych i Autostrad.Ruch Drogowy 2020/2021. Warszawa: Heller Consult; 2022. Available from: https://www.gov.pl/web/gddkia/generalny-pomiar-ruchu-20202021

[114] McMahon T, Havlin S, Gallos LK. Effect of cities and distance on COVID-19 spreading in the United States. Phys Rev E. 2023; 107(3):034302. http://dx.doi.org/10.1103/PhysRevE.107.034302

[115] Główny Urząd Statystyczny (National Statistics Office of Poland). Ludność w rejonach statystycznych i obwodach spisowych według płci, ekonomicznych i 10-letnich grup wieku; 2021.

[116] Morris MD. Factorial sampling plans for preliminary computational experiments. Technometrics. 1991; 33(2):161-174. http://dx.doi.org/10.1080/00401706.1991.10484804

[117] Qian G, Mahdi A. Sensitivity analysis methods in the biomedical sciences. Math Biosci. 2020;323:108306. http://dx.doi.org/10.1016/j.mbs.2020.108306

[118] Herman JD, Kollat JB, Reed PM, Wagener T. Technical note: Method of Morris effectively reduces the computational demands of global sensitivity analysis for distributed watershed models. Hydrol Earth Syst Sci. 2013; 17(7):2893-2903. http://dx.doi.org/10.5194/hess-17-2893-2013

[119] Sobol IM. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math Comput Simul. 2001; 55(1-3):271-280. http://dx.doi.org/10.1016/S0378-4754(00)00270-6

[120] Helton JC, Davis FJ. Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems. Reliab Eng Syst Saf. 2003; 81(1):23-69. http://dx.doi.org/10.1016/S0951-8320(03)00058-9

[121] Fung IC-H, Antia R, Handel A. How to minimize the attack rate during multiple influenza outbreaks in a heterogeneous population. PLoS One. 2012; 7(6):e36573. http://dx.doi.org/10.1371/journal.pone.0036573

[122] Rocha Filho T, Moret M, Chow C, et al. A data-driven model for COVID-19 pandemic-evolution of the attack rate and prognosis for Brazil. Chaos Solitons Fractals. 2021;152:111359. http://dx.doi.org/10.1016/j.chaos.2021.111359

[123] Kuhl E. The classical SIR model. In: Computational Epidemiology: Data-Driven Modeling of COVID-19. Cham: Springer International Publishing; 2021:41-59. http://dx.doi.org/10.1007/978-3-030-82890-5_3

[124] Lunelli A, Pugliese A. Final attack ratio in SIR epidemic models for multigroup populations. Ric Mat. 2018; 67(1):49-68. http://dx.doi.org/10.1007/s11587-017-0349-5

[125] Ridenhour B, Kowalik JM, Shay DK. Unraveling R0: Considerations for public health applications. Am J Public Health. 2014; 104(2):e32-e41. http://dx.doi.org/10.2105/AJPH.2013.301704