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Mathematical Biosciences and Engineering

2022, Volume 19, Issue 10: 9853-9886. doi: 10.3934/mbe.2022459
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The COVID-19 pandemic as inspiration to reconsider epidemic models: A novel approach to spatially homogeneous epidemic spread modeling

  • 1.
    Laboratory of Chemical and Environmental Technology, Department of Chemistry, Aristotle University of Thessaloniki, 54 124 Thessaloniki, 54124, Greece
  • 2.
    Laboratory of Environmental Engineering & Planning, Department of Civil Engineering, Aristotle University of Thessaloniki, Thessaloniki, 54124, Greece
  • 3.
    Infectious Diseases Unit and 3rd Department of Pediatrics, Aristotle University School of Health Sciences, Hippokration Hospital, Thessaloniki, 54642, Greece
  • 4.
    Faculty of Veterinary Medicine, School of Health Sciences, Aristotle University of Thessaloniki, Thessaloniki, 54124, Greece
  • 5.
    Department of Microbiology, Medical School, Aristotle University of Thessaloniki, Thessaloniki, 54124, Greece
  • 6.
    Department of Haematology, First Department of Internal Medicine, Faculty of Medicine, AHEPA General Hospital, Aristotle University of Thessaloniki, Thessaloniki, 54636, Greece
  • 7.
    School of Spatial Planning and Development, Faculty of Engineering, Aristotle University of Thessaloniki, 54124, Greece
  • 8.
    National Public Health Organization, Athens, Greece
  • 9.
    European University Cyprus, Nicosia, Cyprus
  • 10.
    National and Kapodistrian University of Athens, Athens, Greece
  • 11.
    Department of Environmental Health, Directory of Epidemiology and Prevention of Non-Communicable Diseases and Injuries, National Public Health Organization, Athens, Greece
  • Received: 05 May 2022 Revised: 29 June 2022 Accepted: 30 June 2022 Published: 11 July 2022

  • Epidemic spread models are useful tools to study the spread and the effectiveness of the interventions at a population level, to an epidemic. The workhorse of spatially homogeneous class models is the SIR-type ones comprising ordinary differential equations for the unknown state variables. The transition between different states is expressed through rate functions. Inspired by -but not restricted to- features of the COVID-19 pandemic, a new framework for modeling a disease spread is proposed. The main concept refers to the assignment of properties to each individual person as regards his response to the disease. A multidimensional distribution of these properties represents the whole population. The temporal evolution of this distribution is the only dependent variable of the problem. All other variables can be extracted by post-processing of this distribution. It is noteworthy that the new concept allows an improved consideration of vaccination modeling because it recognizes vaccination as a modifier of individuals response to the disease and not as a means for individuals to totally defeat the disease. At the heart of the new approach is an infection age model engaging a sharp cut-off. This model is analyzed in detail, and it is shown to admit self-similar solutions. A hierarchy of models based on the new approach, from a generalized one to a specific one with three dominant properties, is derived. The latter is implemented as an example and indicative results are presented and discussed. It appears that the new framework is general and versatile enough to simulate disease spread processes and to predict the evolution of several variables of the population during this spread.

    Citation: Margaritis Kostoglou, Thodoris Karapantsios, Maria Petala, Emmanuel Roilides, Chrysostomos I. Dovas, Anna Papa, Simeon Metallidis, Efstratios Stylianidis, Theodoros Lytras, Dimitrios Paraskevis, Anastasia Koutsolioutsou-Benaki, Georgios Panagiotakopoulos, Sotirios Tsiodras, Nikolaos Papaioannou. The COVID-19 pandemic as inspiration to reconsider epidemic models: A novel approach to spatially homogeneous epidemic spread modeling[J]. Mathematical Biosciences and Engineering, 2022, 19(10): 9853-9886. doi: 10.3934/mbe.2022459

    Related Papers:

  • Epidemic spread models are useful tools to study the spread and the effectiveness of the interventions at a population level, to an epidemic. The workhorse of spatially homogeneous class models is the SIR-type ones comprising ordinary differential equations for the unknown state variables. The transition between different states is expressed through rate functions. Inspired by -but not restricted to- features of the COVID-19 pandemic, a new framework for modeling a disease spread is proposed. The main concept refers to the assignment of properties to each individual person as regards his response to the disease. A multidimensional distribution of these properties represents the whole population. The temporal evolution of this distribution is the only dependent variable of the problem. All other variables can be extracted by post-processing of this distribution. It is noteworthy that the new concept allows an improved consideration of vaccination modeling because it recognizes vaccination as a modifier of individuals response to the disease and not as a means for individuals to totally defeat the disease. At the heart of the new approach is an infection age model engaging a sharp cut-off. This model is analyzed in detail, and it is shown to admit self-similar solutions. A hierarchy of models based on the new approach, from a generalized one to a specific one with three dominant properties, is derived. The latter is implemented as an example and indicative results are presented and discussed. It appears that the new framework is general and versatile enough to simulate disease spread processes and to predict the evolution of several variables of the population during this spread.


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    [1] M. J. Keeling, K. T. D. Eames, Networks and epidemic models, J. R. Soc. Interface, 2 (2005), 295-307. https://doi.org/10.1098/rsif.2005.0051 doi: 10.1098/rsif.2005.0051
    [2] N.L. Komarova, L.M. Schang, D. Wodarz, Patterns of the COVID-19 pandemic spread around the world: Exponential versus power laws, J. R. Soc. Interface, 17 (2020), 20200518. https://doi.org/10.1098/rsif.2020.0518 doi: 10.1098/rsif.2020.0518
    [3] M.G. Hâncean, J. Lerner, M. Perc, M.C. Ghiţǎ, D.A. Bunaciu, A.A. Stoica, B.E. Mihǎilǎ, The role of age in the spreading of COVID-19 across a social network in Bucharest, J. Complex Netw, 9 (2021), 1-20. https://doi.org/10.1093/comnet/cnab026 doi: 10.1093/comnet/cnab026
    [4] C. Gai, D. Iron, T. Kolokolnikov, Localized outbreaks in an S-I-R model with diffusion, J. Math. Biol., 80 (2020), 1389-1411. https://doi.org/10.1007/s00285-020-01466-1 doi: 10.1007/s00285-020-01466-1
    [5] V. Capasso, Mathematical Structures of Epidemic Systems, in Lecture Notes in Biomathematics, Springer, (1993). https://doi.org/10.1007/978-3-540-70514-7
    [6] H. W. Hethcote, The Mathematics of Infectious Diseases, SIAM Rev., 42, (2000), 599-653. https://doi.org/10.1137/S0036144500371907 doi: 10.1137/S0036144500371907
    [7] P. G. Kevrekidis, J. Cuevas-Maraver, Y. Drossinos, Z. Rapti, G. A. Kevrekidis, Reaction-diffusion spatial modeling of COVID-19: Greece and Andalusia as case examples, Phys. Rev. E., 104 (2021), 024412. https://doi.org/10.1103/PhysRevE.104.024412 doi: 10.1103/PhysRevE.104.024412
    [8] W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. A Math. Phys. Eng. Sci., 115 (1927), 700-721. https://doi.org/10.1098/rspa.1927.0118 doi: 10.1098/rspa.1927.0118
    [9] N. C. Grassly, C. Fraser, Mathematical models of infectious disease transmission, Nat. Rev. Microbiol., 6 (2008), 477-487. https://doi.org/10.1038/nrmicro1845 doi: 10.1038/nrmicro1845
    [10] A. Danchin, G. Turinici, Immunity after COVID-19: Protection or sensitization? Math. Biosci., 331 (2021), 108499. https://doi.org/10.1016/j.mbs.2020.108499 doi: 10.1016/j.mbs.2020.108499
    [11] O. N. Bjørnstad, K. Shea, M. Krzywinski, N. Altman, The SEIRS model for infectious disease dynamics, Nat. Methods, 17 (2020), 557-558. https://doi.org/10.1038/s41592-020-0856-2 doi: 10.1038/s41592-020-0856-2
    [12] H. W. Hethcote, P. van den Driessche, An SIS epidemic model with variable population size and a delay, J. Math. Biol., 34 (1995), 177-194. https://doi.org/10.1007/BF00178772 doi: 10.1007/BF00178772
    [13] F. A. Rihan, M. N. Anwar, Qualitative analysis of delayed SIR epidemic model with a saturated incidence rate, Int. J. Differ. Equ., 2012 (2012), 1-13. https://doi.org/10.1155/2012/408637 doi: 10.1155/2012/408637
    [14] V. Ram, L. P. Schaposnik, A modified age-structured SIR model for COVID-19 type viruses, Sci. Rep., 11 (2021), 15194. https://doi.org/10.1038/s41598-021-94609-3 doi: 10.1038/s41598-021-94609-3
    [15] F. M. G. Magpantay, A. A. King, P. Rohani, Age-structure and transient dynamics in epidemiological systems, J. R. Soc. Interface, 16 (2019), 20190151. https://doi.org/10.1098/rsif.2019.0151 doi: 10.1098/rsif.2019.0151
    [16] G. F. Webb, Population Models Structured by Age, Size, and Spatial Position in Structured Population Models in Biology and Epidemiology, in Lecture Notes in Mathematics, Springer, (2008). https://doi.org/10.1007/978-3-540-78273-5_1
    [17] J. M. Hyman, J. Li, Infection-age structured epidemic models with behavior change or treatment, J. Biol. Dyn., 1 (2007), 109-131. https://doi.org/10.1080/17513750601040383 doi: 10.1080/17513750601040383
    [18] M. Iannelli, F. Milner, The Basic Approach to Age-structured Population Dynamics, in Models, Methods and Numerics, Springer, (2017).
    [19] I. J. Rao, M. L. Brandeau, Optimal allocation of limited vaccine to minimize the effective reproduction number, Math. Biosci., 339 (2021), 108654. https://doi.org/10.1016/j.mbs.2021.108654 doi: 10.1016/j.mbs.2021.108654
    [20] S. Aniţa, M. Banerjee, S. Ghosh, V. Volpert, Vaccination in a two-group epidemic model, Appl. Math. Lett., 119 (2021), 107197. https://doi.org/10.1016/j.aml.2021.107197 doi: 10.1016/j.aml.2021.107197
    [21] F. Brauer, J. Watmough, Age of infection epidemic models with heterogeneous mixing, J. Biol. Dyn., 3, (2009), 324-330. https://doi.org/10.1080/17513750802415822 doi: 10.1080/17513750802415822
    [22] T. Karapantsios, M. X. Loukidou, K. A. Matis, Sorption kinetics, in Oceanography, Meteorology, Physics and Chemistry, Water Law and Water History, Art and Culture, Water Encyclopedia, Wiley, (2005). https://doi.org/10.1002/047147844X.pc487
    [23] I. Area, F. J. Fernández, J. J. Nieto, F. A. F. Tojo, Concept and solution of digital twin based on a Stieltjes differential equation, Math. Methods Appl. Sci., (2022), 1-15. https://doi.org/10.1002/mma.8252 doi: 10.1002/mma.8252
    [24] N. Perra, Non-pharmaceutical interventions during the COVID-19 pandemic: A review, Phys. Rep., 913 (2021), 1-52. https://doi.org/10.1016/j.physrep.2021.02.001 doi: 10.1016/j.physrep.2021.02.001
    [25] E. Estrada, COVID-19 and SARS-CoV-2. Modeling the present, looking at the future, Phys. Rep., 869 (2020), 1-51. https://doi.org/10.1016/j.physrep.2020.07.005 doi: 10.1016/j.physrep.2020.07.005
    [26] A. Vespignani, H. Tian, C. Dye, J. O. Lloyd-Smith, R. M. Eggo, M. Shrestha, et al., Modelling COVID-19, Nat. Rev. Phys., 2 (2020), 279-281. https://doi.org/10.1038/s42254-020-0178-4 doi: 10.1038/s42254-020-0178-4
    [27] D. Baleanu, M. Hassan Abadi, A. Jajarmi, K. Zarghami Vahid, J. J. Nieto, A new comparative study on the general fractional model of COVID-19 with isolation and quarantine effects, AEJ Alex. Eng. J., 61 (2022), 4779-4791. https://doi.org/10.1016/j.aej.2021.10.030 doi: 10.1016/j.aej.2021.10.030
    [28] P. Samui, J. Mondal, S. Khajanchi, A mathematical model for COVID-19 transmission dynamics with a case study of India, Chaos Solit. Fract., 140 (2020), 110173. https://doi.org/10.1016/j.chaos.2020.110173 doi: 10.1016/j.chaos.2020.110173
    [29] S. Khajanchi, K. Sarkar, Forecasting the daily and cumulative number of cases for the COVID-19 pandemic in India, Chaos, 30 (2020), 1-16. https://doi.org/10.1063/5.0016240 doi: 10.1063/5.0016240
    [30] S. Khajanchi, K. Sarkar, J. Mondal, K. S. Nisar, S. F. Abdelwahab, Mathematical modeling of the COVID-19 pandemic with intervention strategies, Results Phys., 25 (2021), 104285. https://doi.org/10.1016/j.rinp.2021.104285 doi: 10.1016/j.rinp.2021.104285
    [31] K. Sarkar, S. Khajanchi, J. J. Nieto, Modeling and forecasting the COVID-19 pandemic in India, Chaos Solit. Fract., 139 (2020), 110049. https://doi.org/10.1016/j.chaos.2020.110049 doi: 10.1016/j.chaos.2020.110049
    [32] P. K. Tiwari, R. K. Rai, S. Khajanchi, R. K. Gupta, A. K. Misra, Dynamics of coronavirus pandemic: effects of community awareness and global information campaigns, Eur. Phys. J. Plus., 136 (2021), 994. https://doi.org/10.1140/epjp/s13360-021-01997-6 doi: 10.1140/epjp/s13360-021-01997-6
    [33] S. Khajanchi, K. Sarkar, J. Mondal, Dynamics of the COVID-19 pandemic in India, arXiv, (2020). https://doi.org/10.21203/rs.3.rs-27112/v1
    [34] R. K. Rai, S. Khajanchi, P. K. Tiwari, E. Venturino, A. K. Misra, Impact of social media advertisements on the transmission dynamics of COVID-19 pandemic in India, J. Appl. Math. Comput., 68 (2022), 19-44. https://doi.org/10.1007/s12190-021-01507-y doi: 10.1007/s12190-021-01507-y
    [35] J. Mondal, S. Khajanchi, Mathematical modeling and optimal intervention strategies of the COVID-19 outbreak, Nonlinear Dyn., (2022), 1-26. https://doi.org/10.1007/s11071-022-07235-7 doi: 10.1007/s11071-022-07235-7
    [36] L. J. S. Allen, P. van de Driessche, Stochastic epidemic models with a backward bifurcation, Math. Biosci. Eng., 3 (2006), 445-458. https://doi.org/10.3934/mbe.2006.3.445 doi: 10.3934/mbe.2006.3.445
    [37] M. Z. Xin, B. G. Wang, Y. Wang, Stationary distribution and extinction of a stochastic influenza virus model with disease resistance, Math. Biosci. Eng., 19 (2022), 9125-9146. https://doi.org/10.1155/2017/6027509 doi: 10.1155/2017/6027509
    [38] O. Levenspiel, Chemical Reaction Engineering, Wiley, (1999).
    [39] S. Khajanchi, S. Bera, T. K. Roy, Mathematical analysis of the global dynamics of a HTLV-I infection model, considering the role of cytotoxic T-lymphocytes, Math. Comput. Simul., 180 (2021), 354-378. https://doi.org/10.1016/j.matcom.2020.09.009 doi: 10.1016/j.matcom.2020.09.009
    [40] E. N. Bird, R. B. Stewart, W. E. Lightfoot, Transport Phenomena, Wiley, (2001).
    [41] S. K. Friedlander, Smoke, Dust and Haze: Fundamentals of Aerosol Behavior, Wiley Interscience, (1977).
    [42] M. Petala, D. Dafou, M. Kostoglou, T. Karapantsios, E. Kanata, A. Chatziefstathiou, et al., A physicochemical model for rationalizing SARS-CoV-2 concentration in sewage, Case study: The city of Thessaloniki in Greece, Sci. Total Environ., 755 (2021), 142855. https://doi.org/10.1016/j.scitotenv.2020.142855 doi: 10.1016/j.scitotenv.2020.142855
    [43] M. Kostoglou, M. Petala, T. Karapantsios, C. Dovas, E. Roilides, S. Metallidis et al., SARS-CoV-2 adsorption on suspended solids along a sewerage network: mathematical model formulation, sensitivity analysis, and parametric study, Environ. Sci. Pollut. Res., 29 (2021), 11304-11319. https://doi.org/10.1007/s11356-021-16528-0 doi: 10.1007/s11356-021-16528-0
    [44] M. Petala, M. Kostoglou, T. Karapantsios, C. I. Dovas, T. Lytras, D. Paraskevis, et al., Relating SARS-CoV-2 shedding rate in wastewater to daily positive tests data: A consistent model based approach, Sci. Total Environ., 807 (2022), 150838. https://doi.org/10.1016/j.scitotenv.2021.150838 doi: 10.1016/j.scitotenv.2021.150838
    [45] F. Miura, M. Kitajima, R. Omori, Duration of SARS-CoV-2 viral shedding in faces as a parameter for wastewater-based epidemiology: Re-analysis of patient data using a shedding dynamics model, Sci. Total Environ., 769 (2021), 144549. https://doi.org/10.1016/j.scitotenv.2020.144549 doi: 10.1016/j.scitotenv.2020.144549
    [46] T. Hoffmann, J. Alsing, Faecal shedding models for SARS-CoV-2 RNA amongst hospitalised patients and implications for wastewater-based epidemiology, MedRxiv, (2021). https://doi.org/10.1101/2021.03.16.21253603
    [47] P. Jiménez-Rodríguez, G. A. Muñoz-Fernández, J. C. Rodrigo-Chocano, J. B. Seoane-Sepúlveda, A. Weber, A population structure-sensitive mathematical model assessing the effects of vaccination during the third surge of COVID-19 in Italy, J. Math. Anal. Appl., (2021), 125975. https://doi.org/10.1016/j.jmaa.2021.125975 doi: 10.1016/j.jmaa.2021.125975
    [48] M. Namiki, R. Yano, A numerical method to calculate multiple epidemic waves in COVID-19 with a realistic total number of people involved, J. Stat. Mech. Theory Exp., (2022), 033403. https://doi.org/10.1088/1742-5468/ac57bb doi: 10.1088/1742-5468/ac57bb
    [49] R. Markovič, M. Šterk, M. Marhl, M. Perc, M. Gosak, Socio-demographic and health factors drive the epidemic progression and should guide vaccination strategies for best COVID-19 containment, Results Phys., 26 (2021), 104433. https://doi.org/10.1016/j.rinp.2021.104433 doi: 10.1016/j.rinp.2021.104433
    [50] M. Kostoglou, A. J. Karabelas, Evaluation of numerical methods for simulating an evolving particle size distribution in growth processes, Chem. Eng. Commun., 136 (1995), 177-199. https://doi.org/10.1080/00986449508936360 doi: 10.1080/00986449508936360
    [51] M. Fuentes-Garí, R. Misener, D. García-Munzer, E. Velliou, M.C. Georgiadis, M. Kostoglou, et al., A mathematical model of subpopulation kinetics for the deconvolution of leukaemia heterogeneity, J. R. Soc. Interface., 12 (2015), 20150276. https://doi.org/10.1098/rsif.2015.0276 doi: 10.1098/rsif.2015.0276
    [52] M. Fuentes-Garí, R. Misener, M. C. Georgiadis, M. Kostoglou, N. Panoskaltsis, A. Mantalaris, et al., Selecting a differential equation cell cycle model for simulating leukemia treatment, Ind. Eng. Chem. Res., 54 (2015), 8847-8859. https://doi.org/10.1021/acs.iecr.5b01150 doi: 10.1021/acs.iecr.5b01150
    [53] M. Kostoglou, M. Fuentes-Garí, D. García-Münzer, M. C. Georgiadis, N. Panoskaltsis, E. N. Pistikopoulos, et al., A comprehensive mathematical analysis of a novel multistage population balance model for cell proliferation, Comput. Chem. Eng., 91 (2016), 157-166. https://doi.org/10.1016/j.compchemeng.2016.02.012 doi: 10.1016/j.compchemeng.2016.02.012
    [54] M. Kostoglou, J. Lioumbas, T. Karapantsios, A population balance treatment of bubble size evolution in free draining foams, Collo. Surf. A Physicochem. Eng. Asp., 473 (2015), 75-84. https://doi.org/10.1016/j.colsurfa.2014.11.036 doi: 10.1016/j.colsurfa.2014.11.036
    [55] M. Kostoglou, T. D. Karapantsios, On the adequacy of some low-order moments method to simulate certain particle removal processes, Collol. Interf., 5 (2021), 46. https://doi.org/10.3390/colloids5040046 doi: 10.3390/colloids5040046
    [56] J. D. Peterson, R. Adhikari, Efficient and flexible methods for simulating models of time since infection, Phys. Rev. E., 104 (2021), 024410. https://doi.org/10.1103/PhysRevE.104.024410 doi: 10.1103/PhysRevE.104.024410
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