×

zbMATH — the first resource for mathematics

Susceptible-infectious-recovered models revisited: from the individual level to the population level. (English) Zbl 1315.92081
Summary: The classical susceptible-infectious-recovered (SIR) model, originated from the seminal papers of R. Ross [“An application of the theory of probabilities to the study of a priori pathometry. I”, Proc. R. Soc. London A, 92, No. 638, 204–230 (1916; doi:10.1098/rspa.1916.0007); with H. Hudson, “An application of the theory of probabilities to the study of a priori pathometry. II”, ibid. 92, No. 650, 212–225 (1917), http://www.jstor.org/stable/i206250; “An application of the theory of probabilities to the study of a priori pathometry. III”, ibid. 93, No. 650, 225–240 (1917; doi:10.1098/rspa.1917.0015)] in 1916–1917 and the fundamental contributions of W. O. Kermack and A. G. McKendrick [“A contribution to the mathematical theory of epidemics”, Proc. R. Soc. London A 115, No. 772, 700–721 (1927; doi:10.1098/rspa.1927.0118); “Contributions to the mathematical theory of epidemics. II: The problem of endemicity”, ibid. 138, No. 834, 55–83 (1932; doi:10.1098/rspa.1932.0171); “Contributions to the mathematical theory of epidemics. III: Further studies of the problem of endemicity”, ibid. 141, No. 843, 94–122 (1933; doi:10.1098/rspa.1933.0106)] in 1927–1932, describes the transmission of infectious diseases between susceptible and infective individuals and provides the basic framework for almost all later epidemic models, including stochastic epidemic models using Monte Carlo simulations or individual-based models (IBM). In this paper, by defining the rules of contacts between susceptible and infective individuals, the rules of transmission of diseases through these contacts, and the time of transmission during contacts, we provide detailed comparisons between the classical deterministic SIR model and the IBM stochastic simulations of the model. More specifically, for the purpose of numerical and stochastic simulations we distinguish two types of transmission processes: that initiated by susceptible individuals and that driven by infective individuals. Our analysis and simulations demonstrate that in both cases the IBM converges to the classical SIR model only in some particular situations. In general, the classical and individual-based SIR models are significantly different. Our study reveals that the timing of transmission in a contact at the individual level plays a crucial role in determining the transmission dynamics of an infectious disease at the population level.

MSC:
92D30 Epidemiology
92C60 Medical epidemiology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ajelli, M.; Gonalves, B.; Balcan, D.; Colizza, V.; Hu, H.; Ramasco, J. J.; Merler, S.; Vespignani, A., Comparing large-scale computational approaches to epidemic modeling: agent-based versus structured metapopulation models, BMC Inf. Dis., 10, 190, (2010)
[2] Allen, L. J.S., An introduction to stochastic processes with applications to biology, (2003), Prentice Hall NJ · Zbl 1205.60001
[3] Allen, L. J.S., An introduction to stochastic epidemic models, (Brauer, F.; van den Driessche, P.; Wu, J., Mathematical Epidemiology, Lecture Notes in Math., vol. 1945, (2008), Springer New York), 81-130 · Zbl 1206.92022
[4] Anderson, R. M.; May, R. M., Infective diseases of humans: dynamics and control, (1991), Oxford University Press Oxford
[5] Andersson, H.; Britton, T., Stochastic epidemic models and their statistical analysis, Lecture Notes in Stat., vol. 151, (2000), Springer-Verlag New York · Zbl 0951.92021
[6] Arino, J., Diseases in metapopulations, (Ma, Z.; Zhou, Y.; Wu, J., Modeling and Dynamics of Infectious Diseases, (2009), World Scientific Singapore)
[7] Bailey, N. T.J., The mathematical theory of epidemics, (1957), Charles Griffin London
[8] Barrat, A.; Bathélemy, M.; Vespignani, A., Dynamical processes on complex networks, (2008), Cambridge University Press Cambridge
[9] Bartlett, M., Stochastic population models in ecology and epidemiology, (1960), Methuen London · Zbl 0096.13702
[10] Bernoulli, D., Essai d’une nouvelle analyse de la mortalité causée par la petite vérole et des avantages de l’inoculation pour la prévenir, Mém. Math. Phys. Acad. Roy. Sci., Paris, 1, (1760)
[11] Brauer, F.; Castillo-Chavez, C., Mathematical models in population biology and epidemiology, (2000), Springer New York · Zbl 1302.92001
[12] Britton, T., Stochastic epidemic models: a survey, Math. Biosci., 225, 24, (2010) · Zbl 1188.92031
[13] Busenberg, S.; Cooke, K., Vertically transmitted diseases: models and dynamics, Lecture Notes in Biomath., vol. 23, (1993), Springer-Verlag Berlin · Zbl 0837.92021
[14] Capasso, V., Mathematical structures of epidemic systems, Lecture Notes in Biomath., vol. 97, (1993), Springer-Verlag Heidelberg · Zbl 0798.92024
[15] Centers for Disease Control and Prevention, Measles outbreak—Netherlands, April 1999-January 2000, MMWR, 49 2000, pp. 299-303.
[16] Daley, D. J.; Gani, J., Epidemic modelling an introduction, Cambridge Studies Math. Biol., vol. 15, (1999), Cambridge University Press Cambridge · Zbl 0922.92022
[17] D’Agata, E. M.C.; Magal, P.; Olivier, D.; Ruan, S.; Webb, G. F., Modeling antibiotic resistance in hospitals: the impact of minimizing treatment duration, J. Theor. Biol., 249, 487, (2007)
[18] DeAngelis, D. L.; Mooij, W. M., Individual-based modeling of ecological and evolutionary processes, Annu. Rev. Ecol. Evol. Syst., 36, 147, (2005)
[19] Diekmann, O.; Heesterbeek, J. A.P., Mathematical epidemiology of infectious diseases: model building, analysis and interpretation, (2000), Wiley Chichester · Zbl 0997.92505
[20] Dietz, K.; Heesterbeek, J. A.P., Bernoulli was ahead of modern epidemiology, Nature, 408, 513, (2000)
[21] Dietz, K.; Heesterbeek, J. A.P., Daniel bernoulli’s epidemiological model revisited, Math. Biosci., 180, 1, (2002) · Zbl 1019.92028
[22] Doob, J. L., Topics in the theory of markoff chains, Trans. Am. Math. Soc., 52, 37, (1942) · Zbl 0063.09001
[23] Doob, J. L., Markoff chains - denumerable case, Trans. Am. Math. Soc., 58, 455, (1945) · Zbl 0063.01146
[24] Durrett, R., Random graph dynamics, (2007), Cambridge University Press Cambridge · Zbl 1116.05001
[25] Durrett, R., Some features of the spread of epidemics and information on a random graph, Proc. Natl. Acad. Sci. USA, 107, 4491, (2010)
[26] Durrett, R.; Levin, S. A., The importance of being discrete (and spatial), Theor. Popul. Biol., 46, 363, (1994) · Zbl 0846.92027
[27] Gillespie, D. T., A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys., 22, 403, (1976)
[28] Gillespie, D. T., Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81, 2340, (1977)
[29] Grimm, V.; Railsback, S. F., Individual-based modeling and ecology, (2005), Princeton University Press Princeton · Zbl 1085.92043
[30] Hethcote, H. W., The mathematics of infectious diseases, SIAM Rev., 42, 599, (2000) · Zbl 0993.92033
[31] Higgins, J. A.; Hoffman, S.; Dworkin, S. L., Rethinking gender, heterosexual men, and women’s vulnerability to HIV/AIDS, Am. J. Public Health, 100, 435, (2010)
[32] Hinow, P.; Le Foll, F.; Magal, P.; Webb, G. F., Analysis of a model for transfer phenomena in biological populations, SIAM J. Appl. Math., 70, 40, (2009) · Zbl 1201.34092
[33] Iannelli, M., Mathematical theory of age-structured population dynamics, Applied Mathematics Monographs CNR, vol. 7, (1994), Giadini Editori e Stampatori Pisa
[34] Keeling, M. J.; Grenfell, B. T., Individual-based perspectives on \(R_0\), J. Theor. Biol., 203, 51, (2000)
[35] Keeling, M. J.; Rohani, P., Modeling infectious diseases in humans and animals, (2007), Princeton University Press Princeton · Zbl 1279.92038
[36] Kermack, W. O.; McKendrick, A. G., A contribution to the mathematical theory of epidemics, Proc. R. Soc. London A, 115, 700, (1927) · JFM 53.0517.01
[37] Kermack, W. O.; McKendrick, A. G., Contributions to the mathematical theory of epidemics: II, Proc. R. Soc. London A, 138, 55, (1932) · Zbl 0005.30501
[38] Kermack, W. O.; McKendrick, A. G., Contributions to the mathematical theory of epidemics: III, Proc. R. Soc. London A, 141, 94, (1933) · Zbl 0007.31502
[39] Kurtz, T. G., Approximation of population processes, (1981), Society for Industrial and Applied Mathematics Philadelphia · Zbl 0465.60078
[40] Levin, S. A.; Durrett, R., From individuals to epidemics, Phil. Trans. R. Soc. London B, 351, 1615, (1996)
[41] Magal, P.; McCluskey, C. C.; Webb, G. F., Liapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89, 1109, (2010) · Zbl 1208.34126
[42] Martcheva, M.; Thieme, H. R., Infinite ODE systems modeling size-structured metapopulations, macroparasitic diseases, and prion proliferation, (Magal, P.; Ruan, S., Structured Population Models in Biology and Epidemiology, Lecture Notes in Math., vol. 1936, (2008), Springer Berlin), 51
[43] Meyers, L. A., Contact network epidemiology: bond percolation applied to infectious disease prediction and control, Bull. Am. Math. Soc., 44, 63, (2007) · Zbl 1106.92061
[44] Meyers, L. A.; Newman, M. E.J.; Pourbohloul, B., Predicting epidemics on directed contact networks, J. Theor. Biol., 240, 400, (2006)
[45] Muench, H., Catalytic models in epidemiology, (1959), Harvard University Press Cambridge
[46] Murray, J. D., Mathematical biology, (1993), Springer Berlin · Zbl 0779.92001
[47] Mode, C. J.; Sleeman, C. K., Stochastic processes in epidemiology. HIV/AIDS, other infectious diseases and computers, (2000), World Scientific Singapore · Zbl 0984.92028
[48] Newman, M. E.J., The structure and function of complex networks, SIAM Rev., 45, 167, (2003) · Zbl 1029.68010
[49] Pascual, M.; Levin, S. A., From individuals to population densities: searching for the intermediate scale of nontrivial determinism, Ecology, 80, 2225, (1999)
[50] Rass, L.; Radcliffe, J., Spatial deterministic epidemics, Math. Surveys Monogr., vol. 102, (2003), Amer. Math. Soc. Providence, RI · Zbl 1018.92028
[51] Ross, R., An application of the theory of probabilities to the study of a priori pathometry: I, Proc. R. Soc. London A, 92, 204, (1916) · JFM 46.0789.01
[52] Ross, R.; Hudson, H. P., An application of the theory of probabilities to the study of a priori pathometry: II, Proc. R. Soc. London A, 93, 212, (1917) · JFM 46.0789.02
[53] Ross, R.; Hudson, H. P., An application of the theory of probabilities to the study of a priori pathometry: III, Proc. R. Soc. London A, 93, 225, (1917) · JFM 46.0789.02
[54] Ruan, S., Spatial-temporal dynamics in nonlocal epidemiological models, (Takeuchi, Y.; Sato, K.; Iwasa, Y., Mathematics for Life Science and Medicine, (2007), Springer-Verlag Berlin), 99
[55] Ruan, S.; Wu, J., Modeling spatial spread of communicable diseases involving animal hosts, (Cantrell, S.; Cosner, C.; Ruan, S., Spatial Ecology, (2009), Chapman & Hall/CRC Boca Raton, FL), 293 · Zbl 1183.92074
[56] Sharkey, K. J., Deterministic epidemiological models at the individual level, J. Math. Biol., 57, 311, (2008) · Zbl 1141.92039
[57] Smieszek, T.; Fiebig, L.; Scholz, R. W., Models of epidemics: when contact repetition and clustering should be included, Theor. Biol. Med. Model., 6, 11, (2009)
[58] Smith, H. L., Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Math. Surveys Monogr., vol. 41, (1995), Amer. Math. Soc. · Zbl 0821.34003
[59] Thieme, H. R., Mathematics in population biology, (2003), Princeton University Press Princeton · Zbl 1054.92042
[60] Thieme, H. R.; Castillo-Chavez, C., How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM. J. Appl. Math., 53, 1447, (1993) · Zbl 0811.92021
[61] Webb, G. F., Theory of nonlinear age-dependent population dynamics, (1985), Marcel Dekker New York · Zbl 0555.92014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.