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Research Article

Building mean field ODE models using the generalized linear chain trick & Markov chain theory

Pages S248-S272 | Received 01 Jun 2020, Accepted 17 Mar 2021, Published online: 13 Apr 2021

References

  • O.O. Aalen, Phase type distributions in survival analysis, Scand. J. Stat. 22 (1995), pp. 447–463. Available at http://www.jstor.org/stable/4616373.
  • L.J.S. Allen, An Introduction to Stochastic Processes with Applications to Biology, 2nd ed., Chapman and Hall/CRC, 2010.
  • L.J. Allen, A primer on stochastic epidemic models: formulation, numerical simulation, and analysis, Infect. Dis. Model. 2 (2017), pp. 128–142.
  • L. Allen, An Introduction to Mathematical Biology, Pearson/Prentice Hall, Upper Saddle River, NJ, 2007.
  • T. Altiok, On the phase-type approximations of general distributions, IIE Trans. 17 (1985), pp. 110–116.
  • D. Anderson and R. Watson, On the spread of a disease with gamma distributed latent and infectious periods, Biometrika 67 (1980), pp. 191–198.
  • R.M. Anderson, R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, New York, 1992.
  • B. Armbruster and E. Beck, Elementary proof of convergence to the mean-field model for the SIR process, J. Math. Biol. 75 (2017), pp. 327–339.
  • D.K. Arrowsmith, C.M. Place, An Introduction to Dynamical Systems, Cambridge University Press, New York, 1990.
  • M.C. Ausin, M.P. Wiper, and R.E. Lillo, Bayesian estimation of finite time ruin probabilities, Appl. Stoch. Models Bus. Ind. 25 (2009), pp. 787–805.
  • N.T.J. Bailey, The Elements of Stochastic Processes with Applications to the Natural Sciences, John Wiley & Sons, New York, 1990.
  • A.L. Bertozzi, E. Franco, G. Mohler, M.B. Short, and D. Sledge, The Challenges of Modeling and Forecasting the Spread of COVID-19, Proceedings of the National Academy of Sciences Vol. 117, 2020, pp. 16732–16738
  • A. Beuter, L. Glass, M.C. Mackey, and M.S. Titcombe, eds., Nonlinear Dynamics in Physiology and Medicine, Interdisciplinary Applied Mathematics (Book 25), Springer, 2003.
  • M. Bladt, B.F. Nielsen, Matrix-Exponential Distributions in Applied Probability, Springer, New York, 2017.
  • M. Bladt and B.F. Nielsen, Phase-Type Distributions, in Matrix-Exponential Distributions in Applied Probability, USA, Springer, 2017, pp. 125–197
  • D. Champredon, J. Dushoff, and D.J.D. Earn, Equivalence of the Erlang-distributed SEIR epidemic model and the renewal equation, SIAM J. Appl. Math. 78 (2018), pp. 3258–3278.
  • G. Chowell, J.M. Hyman, L.M.A. Bettencourt, and C. Castillo-Chavez, eds., Mathematical and Statistical Estimation Approaches in Epidemiology, Springer, Netherlands, 2009.
  • G. Clapp and D. Levy, A review of mathematical models for leukemia and lymphoma, Drug Dis. Today: Dis. Models 16 (2015), pp. 1–6.
  • D.R. Cox, The analysis of non-markovian stochastic processes by the inclusion of supplementary variables, Math. Proc. Cambridge Philos. Soc. 51 (1955), pp. 433–441.
  • D.R. Cox, A use of complex probabilities in the theory of stochastic processes, Math. Proc. Cambridge Philos. Soc. 51 (1955), pp. 313–319.
  • A. Cumani, On the canonical representation of homogeneous markov processes modelling failure -- time distributions, Microelectron. Reliab. 22 (1982), pp. 583–602.
  • J.M. Cushing, An Introduction to Structured Population Dynamics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1998.
  • J.M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Springer, Berlin Heidelberg, 1977.
  • J. Dawes and M. Souza, A derivation of holling's type i, II and III functional responses in predator–prey systems, J. Theor. Biol. 327 (2013), pp. 11–22.
  • P. Dayan and L.F. Abbott, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, Computational Neuroscience, The MIT Press, 2005.
  • L. de Pillis, K.R. Fister, W. Gu, C. Collins, M. Daub, D. Gross, J. Moore, and B. Preskill, Mathematical model creation for cancer chemo-immunotherapy, Comput. Math. Methods Med. 10 (2009), pp. 165–184.
  • M. Dehon and G. Latouche, A geometric interpretation of the relations between the exponential and generalized Erlang distributions, Adv. Appl. Probabil. 14 (1982), pp. 885–897.
  • O. Diekmann and J. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, LTD, New York, 2000.
  • J. Drake, A. Handel, E. O'Dea, and A. Tredennick, A Stochastic Model for the Transmission of SARS-CoV-2 in Georgia, USA, 2020. Available at https://www.covid19.uga.edu/stochastic-fitting-georgia-suplement.html. Accessed: 2020-01-20
  • L. Edelstein-Keshet, Mathematical Models in Biology, Classics in Applied Mathematics (Book 46), Society for Industrial and Applied Mathematics, 2005.
  • S.P. Ellner, J. Guckenheimer, Dynamic Models in Biology, Princeton University Press, Princeton, NJ, 2006.
  • M. Fackrell, Modelling healthcare systems with phase-type distributions, Health Care Manage. Sci.12 (2008), pp. 11–26.
  • D. Fargue, Réductibilité des systèmes héréditaires à des systèmes dynamiques (régis par des équations différentielles ou aux dérivées partielles), C. R. Acad. Sci. Paris Sér. A-B 277 (1973), pp. B471–B473MR 0333630
  • Z. Feng, D. Xu, and H. Zhao, Epidemiological models with non-exponentially distributed disease stages and applications to disease control, Bull. Math. Biol. 69 (2007), pp. 1511–1536.
  • Z. Feng, Y. Zheng, N. Hernandez-Ceron, H. Zhao, J.W. Glasser, and A.N. Hill, Mathematical models of Ebola-Consequences of underlying assumptions, Math. Biosci. 277 (2016), pp. 89–107.
  • W.M. Getz, C.R. Marshall, C.J. Carlson, L. Giuggioli, S.J. Ryan, S.S. Romañach, C. Boettiger, S.D. Chamberlain, L. Larsen, P. D'Odorico, and D. O'Sullivan, Making ecological models adequate, Ecol. Lett. 21 (2018), pp. 153–166.
  • Q.M. He and H. Zhang, PH-invariant polytopes and coxian representations of phase type distributions, Stoch. Models 22 (2006), pp. 383–409.
  • Q.M. He and H. Zhang, Coxian approximations of matrix-exponential distributions, Calcolo 44 (2007), pp. 235–264.
  • M.W. Hirsch, S. Smale, and R.L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, 3rd ed., Elsevier, 2013.
  • A. Hobolth, A. Siri-Jégousse, and M. Bladt, Phase-type distributions in population genetics, Theor. Popul. Biol. 127 (2019), pp. 16–32.
  • C.S. Holling, The components of predation as revealed by a study of small-mammal predation of the european pine sawfly, Can. Entomol. 91 (1959), pp. 293–320.
  • C.S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol. 91 (1959), pp. 385–398.
  • S. Hoops, R. Hontecillas, V. Abedi, A. Leber, C. Philipson, A. Carbo, J. Bassaganya-Riera, Ordinary Differential Equations (ODEs) Based Modeling, in Computational Immunology, Josep Bassaganya-Riera, eds., Elsevier, San Diego, CA, 2016. pp. 63–78.
  • A. Horváth, M. Scarpa, and M. Telek, Phase Type and Matrix Exponential Distributions in Stochastic Modeling, in Principles of Performance and Reliability Modeling and Evaluation: Essays in Honor of Kishor Trivedi on his 70th Birthday, L. Fiondella and A. Puliafito, eds., Springer International Publishing, Cham, 2016, pp. 3–25
  • G. Horváth, P. Reinecke, M. Telek, and K. Wolter, Efficient Generation of PH-Distributed Random Variates, in Proceedings of the 19th international conference on Analytical and Stochastic Modeling Techniques and Applications, K. Al-Begain, D. Fiems, and J.M. Vincent, eds., Berlin, Heidelberg, Springer, 2012, pp. 271–285.
  • G. Horváth and M. Telek, BuTools 2: A Rich Toolbox for Markovian Performance Evaluation, in ValueTools 2016 -- 10th EAI International Conference on Performance Evaluation Methodologies and Tools, 1. Association for Computing Machinery, 2017, pp. 137–142
  • G. Horváth and M. Telek, Butools v2.0. Available at http://webspn.hit.bme.hu/%7Etelek/tools/butools/doc/index.html (2020). Accessed: 2020-05-15
  • S.K. Howard and M. Taylor, An Introduction to Stochastic Modeling, 3rd ed., Academic Press, 1998.
  • P. Hurtado and C. Richards, A procedure for deriving new ODE models: using the generalized linear chain trick to incorporate phase-type distributed delay and dwell time assumptions, Math. Appl. Sci. Eng. 9999 (2020), pp. 1–14.
  • P.J. Hurtado, Building New Models: Rethinking and Revising ODE Model Assumptions, in Foundations for Undergraduate Research in Mathematics, Hannah Callender Highlander, Alex Capaldi, Carrie Diaz Eaton, ed., Springer International Publishing, Birkhäuser, Cham, 2020. pp. 1–86.
  • P.J. Hurtado and A.S. Kirosingh, Generalizations of the ‘linear chain trick’: incorporating more flexible dwell time distributions into mean field ODE models, J. Math. Bio. 79 (2019), pp. 1831–1883.
  • P.J. Hurtado and C. Richards, Finding reproduction numbers for epidemic models & predator-prey models of arbitrary finite dimension using the generalized linear chain trick (2020). arXiv:2008.06768
  • E.M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, Computational Neuroscience, MIT Press, 2010.
  • M.J. Keeling and B.T. Grenfell, Disease extinction and community size: modeling the persistence of measles, Science 275 (1997), pp. 65–67.
  • M.J. Keeling and B.T. Grenfell, Understanding the persistence of measles: reconciling theory, simulation and observation, Proc. R. Soc. Lond. Ser. B: Biol. Sci. 269 (2002), pp. 335–343.
  • J. Keener and J. Sneyd, Mathematical Physiology I: Cellular Physiology, 2nd ed., Springer, 2008.
  • J. Keener and J. Sneyd, Mathematical Physiology II: Systems Physiology, 2nd ed., Springer, 2008.
  • W.O. Kermack and A.G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. Ser. A, Containing Papers Math. Phys. Charact. 115 (1927), pp. 700–721.
  • W.O. Kermack and A.G. McKendrick, Contributions to the mathematical theory of epidemics. II. The problem of endemicity, Proc. R. Soc. Lond. Ser. A, Containing Papers Math. Phys. Character 138 (1932), pp. 55–83.
  • W.O. Kermack and A.G. McKendrick, Contributions to the mathematical theory of epidemics. III. Further studies of the problem of endemicity, Proc. R. Soc. Lond. Ser. A, Containing Papers Math. Phys. Character 141 (1933), pp. 94–122Available at http://www.jstor.org/stable/96207
  • W.O. Kermack and A.G. McKendrick, Contributions to the mathematical theory of epidemics – i, Bull. Math. Biol. 53 (1991), pp. 33–55.
  • W.O. Kermack and A.G. McKendrick, Contributions to the mathematical theory of epidemics – II. the problem of endemicity, Bull. Math. Biol. 53 (1991), pp. 57–87.
  • W.O. Kermack and A.G. McKendrick, Contributions to the mathematical theory of epidemics – III. further studies of the problem of endemicity, Bull. Math. Biol. 53 (1991), pp. 89–118.
  • S. Kim, J.H. Byun, and I.H. Jung, Global stability of an SEIR epidemic model where empirical distribution of incubation period is approximated by Coxian distribution, Adv. Diff. Equ. 2019 (2019).
  • D. Kirschner and J.C. Panetta, Modeling immunotherapy of the tumor -- immune interaction, J. Math. Biol. 37 (1998), pp. 235–252.
  • Z. Komárková, Phase-type approximation techniques, Bachelor's Thesis, Masaryk University, 2012. Available at https://is.muni.cz/th/ysfsq/
  • O. Krylova and D.J.D. Earn, Effects of the infectious period distribution on predicted transitions in childhood disease dynamics, J. R. Soc. Interface 10 (2013).
  • A.L. Lloyd, The dependence of viral parameter estimates on the assumed viral life cycle: limitations of studies of viral load data, Proc. R. Soc. Lond. B: Biol. Sci. 268 (2001), pp. 847–854.
  • A.L. Lloyd, Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods, Proc. R. Soc. Lond. B: Biol. Sci. 268 (2001), pp. 985–993.
  • A.L. Lloyd, Realistic distributions of infectious periods in epidemic models: changing patterns of persistence and dynamics, Theor. Popul. Biol. 60 (2001), pp. 59–71.
  • A.L. Lloyd, Sensitivity of Model-Based Epidemiological Parameter Estimation to Model Assumptions, in Mathematical and Statistical Estimation Approaches in Epidemiology, G. Chowell, J.M. Hyman, L.M.A. Bettencourt, and C. Castillo-Chavez, eds., Springer, Netherlands, Dordrecht, 2009, pp. 123–141
  • J. Ma and D.J.D. Earn, Generality of the final size formula for an epidemic of a newly invading infectious disease, Bull. Math. Biol. 68 (2006), pp. 679–702.
  • N. MacDonald, Time Lags in Biological Models, Lecture Notes in Biomathematics, Vol. 27, Springer, Verlag Berlin Heidelberg, 1978.
  • A.H. Marshall and S.I. McClean, Using coxian phase-Type distributions to identify patient characteristics for duration of stay in hospital, Health Care Manage. Sci. 7 (2004), pp. 285–289.
  • A.H. Marshall and M. Zenga, Experimenting with the coxian phase-Type distribution to uncover suitable fits, Methodol. Comput. Appl. Probab. 14 (2010), pp. 71–86.
  • K.S. McCann, Food Webs, Monographs in Population Biology (Book 57), Princeton University Press, 2011.
  • C. McGrory, A. Pettitt, and M. Faddy, A fully Bayesian approach to inference for Coxian phase-type distributions with covariate dependent mean, Comput. Stat. Data Anal. 53 (2009), pp. 4311–4321.
  • J.D. Meiss, Differential Dynamical Systems, Revised ed., Society for Industrial and Applied Mathematics, Philadelphia, PA, 2017.
  • J.A.J. Metz and O. Diekmann, eds., The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, Vol. 68, Springer, Berlin, Heidelberg, 1986.
  • J. Metz and O. Diekmann, Exact finite dimensional representations of models for physiologically structured populations. I: The abstract formulation of linear chain trickery, in Proceedings of Differential Equations With Applications in Biology, Physics, and Engineering 1989, J.A. Goldstein, F. Kappel, and W. Schappacher, eds., Vol. 133, 1991, pp. 269–289
  • W.W. Murdoch, C.J. Briggs, and R.M. Nisbet, Consumer–Resource Dynamics, Monographs in Population Biology , Vol. 36, Princeton University Press, Princeton, USA, 2003.
  • J.D. Murray, Mathematical Biology: I. An Introduction, Interdisciplinary Applied Mathematics (Book 17), Springer, 2007.
  • A. Nande, B. Adlam, J. Sheen, M.Z. Levy, and A.L. Hill, Dynamics of COVID-19 under social distancing measures are driven by transmission network structure, PLOS Comput. Biol. 17 (2021), p. e1008684.
  • R.M. Nisbet, W.S.C. Gurney, and J.A.J. Metz, Stage Structure Models Applied in Evolutionary Ecology, in Applied Mathematical Ecology, S.A. Levin, T.G. Hallam, and L.J. Gross, eds., Springer, Berlin, Heidelberg, 1989, pp. 428–449
  • C.A. O'Cinneide, On non-uniqueness of representations of phase-type distributions, Commun. Stat. Stoch. Models 5 (1989), pp. 247–259.
  • C.A. O'Cinneide, Characterization of phase-type distributions, Commun. Stat. Stoch. Models 6 (1990), pp. 1–57.
  • C.A. O'Cinneide, Phase-type distributions: open problems and a few properties, Commun. Stat. Stoch. Models 15 (1999), pp. 731–757.
  • H. Okamura and T. Dohi, Mapfit: An R-Based Tool for PH/MAP Parameter Estimation, in Proceedings of the 12th International Conference on Quantitative Evaluation of Systems, J. Campos and B.R. Haverkort, eds., QEST 2015 Vol. 9259, New York, NY, USA. Springer-Verlag, New York, Inc., 2015, pp. 105–112
  • T. Osogami and M. Harchol-Balter, Closed form solutions for mapping general distributions to quasi-minimal PH distributions, Perform. Eval. 63 (2006), pp. 524–552.
  • R Core Team, A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria , 2020. Available at https://www.R-project.org/.
  • P. Reinecke, L. Bodrog, and A. Danilkina, Phase-Type Distributions, in Resilience Assessment and Evaluation of Computing Systems, K. Wolter, A. Avritzer, M. Vieira, and A. van Moorsel, eds., Springer, Berlin, Heidelberg, 2012, pp. 85–113
  • P. Reinecke, T. Krauß, and K. Wolter, Cluster-based fitting of phase-type distributions to empirical data, Comput. Math. Appl. 64 (2012), pp. 3840–3851.
  • M. Renardy, M. Eisenberg, and D. Kirschner, Predicting the second wave of COVID-19 in washtenaw county, MI, J. Theor. Biol. 507 (2020), p. 110461.
  • S.I. Resnick, Adventures in Stochastic Processes, Birkhäuser, Boston, 2002.
  • J. Rizk, K. Burke, and C. Walsh, On the Non-uniqueness of Representations of Coxian Phase-Type Distributions (2019). arXiv:1901.03849v2
  • S.L. Robertson, S.M. Henson, T. Robertson, and J.M. Cushing, A matter of maturity: to delay or not to delay? continuous-time compartmental models of structured populations in the literature 2000-2016, Nat. Resour. Model. 31 (2018), pp. e12160.
  • M.L. Rosenzweig and R.H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Am. Nat. 97 (1963), pp. 209–223.
  • R. Sachs, L. Hlatky, and P. Hahnfeldt, Simple ode models of tumor growth and anti-angiogenic or radiation treatment, Math. Comput. Model. 33 (2001), pp. 1297–1305.
  • H.M.T. Samuel Karlin, A First Course in Stochastic Processes, 2nd ed., Academic Press, 1975.
  • H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Vol. 57, Springer Science & Business Media, 2010.
  • K. Soetaert, T. Petzoldt, and R.W. Setzer, Solving differential equations in R: package deSolve, J. Stat. Softw. 33 (2010), pp. 1–25Available at http://www.jstatsoft.org/v33/i09
  • W.J. Stewart, Introduction to the Numerical Solution of Markov Chains, Princeton University Press, Princeton, NJ, 1994.
  • S.H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Studies in Nonlinearity, 2nd ed., Westview Press, 2014.
  • X. Tang, Z. Luo, and J.C. Gardiner, Modeling hospital length of stay by Coxian phase-type regression with heterogeneity, Stat. Med. 31 (2012), pp. 1502–1516.
  • E. Vatamidou, I. Adan, M. Vlasiou, and B. Zwart, On the accuracy of phase-type approximations of heavy-tailed risk models, Scand. Actuar. J. 2014 (2012), pp. 510–534.
  • S. Venturini, F. Dominici, and G. Parmigiani, Gamma shape mixtures for heavy-tailed distributions, Ann. Appl. Stat. 2 (2008), pp. 756–776.
  • T. Vogel, Systèmes déferlants, systèmes héréditaires, systèmes dynamiques, in Proceedings of the International Symposium Nonlinear Vibrations, IUTAM, Kiev, 1961, pp. 123–130
  • T. Vogel, Théorie des Systèmes Évolutifs, no. 22 in Traité de physique théorique et de physique mathématique, Gauthier-Villars, Paris, 1965
  • X. Wang, Y. Shi, Z. Feng, and J. Cui, Evaluations of interventions using mathematical models with exponential and non-exponential distributions for disease stages: the case of ebola, Bull. Math. Biol. 79 (2017), pp. 2149–2173.
  • H.J. Wearing, P. Rohani, and M.J. Keeling, Appropriate models for the management of infectious diseases, PLOS Med. 2 (2005).
  • S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd ed., Texts in Applied Mathematics, Vol. 2, Springer-Verlag, New York, 2003.
  • J. Xia, Z. Liu, R. Yuan, and S. Ruan, The effects of harvesting and time delay on predator-prey systems with holling type II functional response, SIAM J. Appl. Math. 70 (2009), pp. 1178–1200.
  • C.A. Yates, M.J. Ford, and R.L. Mort, A multi-stage representation of cell proliferation as a Markov process, Bull. Math. Biol. 79 (2017), pp. 2905–2928.