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Comparative analysis of phenomenological growth models applied to epidemic outbreaks

  • Received: 19 March 2019 Accepted: 12 May 2019 Published: 16 May 2019
  • Phenomenological models are particularly useful for characterizing epidemic trajectories because they often offer a simple mathematical form defined through ordinary differential equations (ODEs) that in many cases can be solved explicitly. Such models avoid the description of biological mechanisms that may be difficult to identify, are based on a small number of model parameters that can be calibrated easily, and can be utilized for efficient and rapid forecasts with quantified uncertainty. These advantages motivate an in-depth examination of 37 data sets of epidemic outbreaks, with the aim to identify for each case the best suited model to describe epidemiological growth. Four parametric ODE-based models are chosen for study, namely the logistic and Gompertz model with their respective generalizations that in each case consists in elevating the cumulative incidence function to a power $p\in [0, 1]$. This parameter within the generalized models provides a criterion on the early growth behavior of the epidemic between constant incidence for $p = 0$, sub-exponential growth for $0 < p < 1$ and exponential growth for $p = 1$. Our systematic comparison of a number of epidemic outbreaks using phenomenological growth models indicates that the GLM model outperformed the other models in describing the great majority of the epidemic trajectories. In contrast, the errors of the GoM and GGoM models stay fairly close to each other and the contribution of the adjustment of $p$ remains subtle in some cases. More generally, we also discuss how this methodology could be extended to assess the "distance" between models irrespective of their complexity.

    Citation: Raimund Bürger, Gerardo Chowell, Leidy Yissedt Lara-Díıaz. Comparative analysis of phenomenological growth models applied to epidemic outbreaks[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 4250-4273. doi: 10.3934/mbe.2019212

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  • Phenomenological models are particularly useful for characterizing epidemic trajectories because they often offer a simple mathematical form defined through ordinary differential equations (ODEs) that in many cases can be solved explicitly. Such models avoid the description of biological mechanisms that may be difficult to identify, are based on a small number of model parameters that can be calibrated easily, and can be utilized for efficient and rapid forecasts with quantified uncertainty. These advantages motivate an in-depth examination of 37 data sets of epidemic outbreaks, with the aim to identify for each case the best suited model to describe epidemiological growth. Four parametric ODE-based models are chosen for study, namely the logistic and Gompertz model with their respective generalizations that in each case consists in elevating the cumulative incidence function to a power $p\in [0, 1]$. This parameter within the generalized models provides a criterion on the early growth behavior of the epidemic between constant incidence for $p = 0$, sub-exponential growth for $0 < p < 1$ and exponential growth for $p = 1$. Our systematic comparison of a number of epidemic outbreaks using phenomenological growth models indicates that the GLM model outperformed the other models in describing the great majority of the epidemic trajectories. In contrast, the errors of the GoM and GGoM models stay fairly close to each other and the contribution of the adjustment of $p$ remains subtle in some cases. More generally, we also discuss how this methodology could be extended to assess the "distance" between models irrespective of their complexity.


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    [1] G. Chowell, Fitting dynamic models to epidemic outbreaks with quantified uncertainty: A primer for parameter uncertainty, identifiability, and forecast, Infect. Disease Model., 2 (2017), 379–398.
    [2] P. F. Verhulst, Notice sur la loi que la population suit dans son accroissement, Corresp. Math. Phys., 10 (1838), 113–121.
    [3] B. Gompertz, On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies, Phil. Trans. R. Soc. Lond., 115 (1825), 513–583.
    [4] F. J. Richards, A flexible growth function for empirical use, J. Exp. Bot., 10 (1959), 290–301.
    [5] X. S. Wang, J. Wu and Y. Yang, Richards model revisited: Validation by and application to infection dynamics, J. Theor. Biol., 313 (2012), 12–19.
    [6] J. D. Murray, Mathematical Biology: I. An Introduction, Springer-Verlag, New York, 2002.
    [7] D. S. Jones and B. D. Sleeman, Differential Equations and Mathematical Biology, Chapman & Hall/CRC, Boca Raton, FL, 2003.
    [8] N. F. Britton, Essential Mathematical Biology, Springer-Verlag, London, 2003.
    [9] F. Brauer and C. Castillo-Chavez, Mathematical models in population biology and epidemiology, Second Ed., Springer, New York, 2012.
    [10] O. Diekmann, J. Heesterbeek and T. Britton, Mathematical tools for understanding infectious dis-ease dynamics, Princeton Series in Theoretical and Computational Biology, Princeton University Press, 2012.
    [11] L. A. Segel and L. Edelstein-Keshet, A Primer on Mathematical Models in Biology, SIAM, Philadelphia, PA, 2013.
    [12] F. Brauer and C. Kribs, Dynamical Systems for Biological Modeling: An Introduction, CRC Press, Boca Raton, FL, USA, 2016.
    [13] L. von Bertalanffy, A quantitative theory of organic growth (Inquiries on growth laws. II), Human Biol., 10 (1938), 181–213.
    [14] L. von Bertalanffy, Quantitative laws in metabolism and growth, Quart. Rev. Biol., 32 (1957), 217–231.
    [15] E. Tjørve and K. M. C. Tjørve, A unified approach to the Richards-model family for use in growth analyses: Why we need only two model forms, J. Theor. Biol., 267 (2010), 417–425.
    [16] K. M. C. Tjørve and E. Tjørve, A proposed family of Unified models for sigmoidal growth, Ecol. Modelling, 359 (2017), 117–127.
    [17] K.M.C.TjørveandE.Tjørve, TheuseofGompertzmodelsingrowthanalyses, andnewGompertz-model approach: An addition to the Unified-Richards family, PLoS One, 12 (2017), e0178691.
    [18] C. Viboud, L. Simonsen and G. Chowell, A generalized-growth model to characterize the early ascending phase of infectious disease outbreaks, Epidemics, 15 (2016), 27–37.
    [19] G. Chowell and C. Viboud, Is it growing exponentially fast? - Impact of assuming exponential growth for characterizing and forecasting epidemics with initial near-exponential growth dynamics, Infect. Disease Model., 1 (2016), 71–78.
    [20] G. Chowell, L. Sattenspiel, S. Bansal, et al., Mathematical models to characterize early epidemic growth: A review, Physics Life Rev., 18 (2016), 66–97.
    [21] J. Ma, J. Dushoff, B.M. Bolker, et al., Estimating initial epidemic growth rates, Bull. Math. Biol., 76 (2014), 245–260.
    [22] G. Chowell and F. Brauer, The basic reproduction number of infectious diseases: Computation and estimation using compartmental epidemic models. In G. Chowell, J.M. Hyman, and L.M.A. Bette-nourt, et al. (eds.), Mathematical and Statistical Estimation Approaches in Epidemiology, Springer, Dordrecht, The Netherlands, (2009), 1–30.
    [23] G. Chowell, C. Viboud, L. Simonsen, et al., Characterizing the reproduction number for epidemics with sub-exponential growth dynamics, J. Roy. Soc. Interface, 13 (2016), 20160659.
    [24] H. Nishiura and G. Chowell, The effective reproduction number as a prelude to statistical estima-tion of time-dependent epidemic trends. In G. Chowell, J. M. Hyman, L. M. A. Bettenourt, et al., Mathematical and Statistical Estimation Approaches in Epidemiology, Springer, Dordrecht, The Netherlands, (2009), 103–121.
    [25] J. Wallinga and M. Lipsitch, How generation intervals shape the relationship between growth rates and reproductive numbers, Proc. Roy. Soc. B: Biol. Sci., 274 (2007), 599–604.
    [26] P. Román-Román, J. J. Serrano-Pérez and F. Torres-Ruiz, Some notes about inference for the log- normal diffusion process with exogeneous factors, Mathematics, 2018, 85–97.
    [27] P. Román-Román and F. Torres-Ruiz, The nonhomogeneous lognormal diffusion process as a gen-eral process to model particular types of growth patterns. In Recent Advances in Probability and Statistics, Lect. Notes Semin. Interdiscip. Mat., 12, Semin. Interdiscip. Mat. (S.I.M.), Potenza, Italy (2015), 201–219.
    [28] R. Gutiérrez-Jáimez, P. Román, D. Romero, et al., A new Gompertz-type diffusion process with application to random growth, Math. Biosci., 208 (2007), 147–165.
    [29] P. Román-Román, D. Romero and F. Torres-Ruiz, A diffusion process to model generalized von Bertalanffy growth patterns: Fitting to real data, J. Theor. Biol., 263 (2010), 59–69.
    [30] P. Román-Román and F. Torres-Ruiz, Modelling logistic growth by a new diffusion process: Appli-cation to biological systems, BioSystems, 110 (2012), 9–21.
    [31] P. Román-Román and F. Torres-Ruiz, A stochastic model related to the Richards-type growth curve. Estimation by means of simulated annealing and variable neighborhood search, Appl. Math. Com-put., 266 (2015), 579–598.
    [32] I. Luz-Sant'Ana, P. Román-Román and F. Torres-Ruiz, Modeling oil production and its peak by means of a stochastic diffusion process based on the Hubbert curve, Energy, 133 (2017), 455–470. 33. A. Barrera, P. P. Román-Román and F. Torres-Ruiz, A hyperbolastic type-I diffusion process: Pa-rameter estimation by means of the firefly algoritm, BioSystems, 163 (2018), 11–22.
    [33] 34. S. Ohnishi, T. Yamakawa and T. Akamine, On the analytical solution of the Pütter-Bertalanffy growth equation, J. Theor. Biol., 343 (2014), 174–177.
    [34] 35. G. Chowell, C. Viboud, J. M. Hyman, et al., The Western Africa Ebola virus disease epidemic exhibits both global exponential and local polynomial growth rates, PLOS Currents Outbreaks, 7 (2015).
    [35] 36. G. Chowell, C. Viboud, L. Simonsen, et al., Perspectives on model forecasts of the 2014–2015 Ebola epidemic in West Africa: lessons and the way forward, BMC Medicine, 15 (2017), 42–49.
    [36] 37. B. Pell, Y. Kuang, C. Viboud, et al., Using phenomenological models for forecasting the 2015 Ebola challenge, Epidemics, 22 (2018), 62–70.
    [37] 38. 2015 Ebola response roadmap-Situation report-14 October 2015. Available from: http://apps.who.int/ebola/current-situation/ebola-situation-report-14-october-2015 (accessed 17 October 2015).
    [38] 39. J. G. Breman, P. Piot, K.M. Johnson, et al., The epidemiology of Ebola hemorrhagic fever in Zaire, 1976. in Ebola Virus Haemorrhagic Fever. Proceedings of an International Colloquium on Ebola Virus Infection and Other Haemorrhagic Fevers held in Antwerp, Belgium, 6–8 December, 1977 (ed. S.R. Pattyn) Elsevier/North Holland Biomedical Press, Amsterdam, (1978), 103–124.
    [39] 40. A. Camacho, A. J. Kicharski, S. Funk, et al., Potential for large outbreaks of Ebola virus diease. Epidemics, 9 (2014), 70–78.
    [40] 41. G. Chowell, N. W. Hengartner, C. Castillo-Chavez, et al., The basic reproductive number of Ebola and effects of public health measure: the cases of Congo and Uganda, J. Theor. Biol., 229 (2004), 119–126.
    [41] 42. World Health Organization (WHO), Outbreak of Ebola hemorrhagic fever, Uganda, August 2000–January 2001, Weekly Epidemiol. Rec., 76 (2001), 48.
    [42] 43. B. Bolker, Measles times-series data. Professor B. Bolker's personal data repository at McMaster University. Available from: https://ms.mcmaster.ca/˜bolker/measdata.html.
    [43] 44. Anonymous, XXII. The epidemiological observations made by the commission in Bombay city, J. Hyg. (London), 7 (1907), 724–798.
    [44] 45. World Health Organization (WHO), Plague outbreak situation reports, Madagascar, October 2017–December 2017. Available from: http://www.afro.who.int/health-topics/plague/plague-outbreak-situation-reports.
    [45] 46. A. Sommer, The 1972 smallpox outbreak in Khulna Municipality, Bangledesh. II. Effectiveness of surveillance and continment in urban epidemic control, Am. J. Epidemiol., 99 (1974), 303–313.
    [46] 47. World Health Organization (WHO), Yellow fever situation reports, Angola, situation reports March 2016–July 2016. Available from: https://www.who.int/emergencies/yellow-fever/situation-reports/archive/en/.
    [47] 48. G. Chowell, A. L. Rivas, S. D. Smith, et al., Identification of case clusters and counties with high infective connectivity in the 2001 epidemicof foot-and-mouth disease in Uruguay, Am. J. Vet. Res.,67 (2006), 102–113.
    [48] 49. G. Chowell, A. L. Rivas, N.W. Hengartner, et al., The role of spatial mixing in the spread of foot-and-mouth disease, Prev. Vet. Med., 73 (2006), 297–314.
    [49] 50. G. Chowell, H. Nishiura, and L. M. A. Bettencourt, Comparative estimation of the reproduction number for pandemic influenza from daily case notification data, J. R. Soc. Interface, 4 (2007), 155–166.
    [50] 51. G. Chowell, D. Hincapie-Palacio, J. Ospina, et al., Using phenomenological models to characterize transmissibility and forecast patterns and final burden of Zika epidemics, PLoS Currents Outbreaks,Edition 1, (2016).
    [51] 52. Anonymous, HIV/AIDS in Japan, 2013, Infect. Agents Surv. Rep., 35 (2014), 203–204.
    [52] 53. Centers for Disease Control and Prevention (CDC). CDC Wonder-AIDS Public Information Dataset U.S. Surveillance. Available from: http://wonder.cdc.gov/aidsPublic.html (accessed 27 september 2016).
    [53] 54. Det Kongelige Sundhedskollegium Aarsberetning for 18 Uddrag fra Aalborg Physikat. Available from: http://docplayer.dk/11876516-uddrag-af-det-kongelige-sundhedskollegiums-aarsberetning-for-1853.html.
    [54] 55. M. Kuhn and K. Johnson, Applied Predictive Modeling, Springer, New York, 2013.
    [55] 56. B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap, CRC Press, Boca Raton, FL, USA, 1994.
    [56] 57. G. Chowell, C. E. Ammon, N. W. Hengartner, et al., Transmission dynamics of the great influenza pandemic of 1918 in Geneva, Switzerland: Assessing the effects of hypothetical interventions, J. Theor. Biol., 241 (2006), 193–204.
    [57] 58. G. Chowell, E. Shim, F. Brauer, et al., Modeling the transmission dynamics of Acute Hemorrhagic Conjunctivitis: Application to the 2003 outbreak in Mexico, Stat. Med., 25 (2006), 1840–18
    [58] 59. P. Román-Román, D. Romero, M.A. Rubio, et al., Estimating the parameters of a Gompertz-type diffusion process by means of Simulated Annealing, Appl. Math. Comput., 218 (2012), 5131–5131.
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