Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-16T07:28:00.821Z Has data issue: false hasContentIssue false
  • English
  • Français

Epidemiological Models With Parametric Heterogeneity :Deterministic Theory for Closed Populations

Published online by Cambridge University Press:  06 June 2012

A.S. Novozhilov
A.S. Novozhilov*
Affiliation:
Applied Mathematics–1, Moscow State University of Railway Engineering Obraztsova 9, bldg. 9, Moscow 127994, Russia
*
⋆ Corresponding author. E-mail: [email protected]
Get access

Abstract

We present a unified mathematical approach to epidemiological models with parametricheterogeneity, i.e., to the models that describe individuals in the population as havingspecific parameter (trait) values that vary from one individuals to another. This is anatural framework to model, e.g., heterogeneity in susceptibility or infectivity ofindividuals. We review, along with the necessary theory, the results obtained using thediscussed approach. In particular, we formulate and analyze an SIR model with distributedsusceptibility and infectivity, showing that the epidemiological models for closedpopulations are well suited to the suggested framework. A number of known results from theliterature is derived, including the final epidemic size equation for an SIR model withdistributed susceptibility. It is proved that the bottom up approach of the theory ofheterogeneous populations with parametric heterogeneity allows to infer the populationlevel description, which was previously used without a firm mechanistic basis; inparticular, the power law transmission function is shown to be a consequence of theinitial gamma distributed susceptibility and infectivity. We discuss how the generaltheory can be applied to the modeling goals to include the heterogeneous contactpopulation structure and provide analysis of an SI model with heterogeneous contacts. Weconclude with a number of open questions and promising directions, where the theory ofheterogeneous populations can lead to important simplifications and generalizations.

Keywords

SIR modelheterogeneous populationsdistributed susceptibilityfinal epidemic sizeheterogeneous contact structurepower law transmission function
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 3: Epidemiology , 2012 , pp. 147 - 167
Copyright
© EDP Sciences, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aalen, O. O.. Effects of frailty in survival analysis. Statistical Methods in Medical Research, 3 (1994), No. 3, 227. CrossRefGoogle ScholarPubMed
O. O. Aalen, Ø. Borgan, H. K. Gjessing. Survival and event history analysis : a process point of view. Springer Verlag, 2008.
Ackleh, A. S.. Estimation of rate distributions in generalized Kolmogorov community models. Non-Linear Analysis, 33 (1998), No. 7, 729–745. CrossRefGoogle Scholar
Ackleh, A. S., Marshall, D. F., Heatherly, H. E.. Extinction in a generalized Lotka – Volterra predator–prey model. Journal of Applied Mathematics and Stochastic Analysis, 13 (2000), No. 3, 287–297. CrossRefGoogle Scholar
Ackleh, A. S., Marshall, D. F., Heatherly, H. E., Fitzpatrick, B. G.. Survival of the fittest in a generalized logistic model. Mathematical Models and Methods in Applied Sciences, 9 (1999), No. 9, 1379–1391. CrossRefGoogle Scholar
R. M. Anderson, R. M. C. May. Infectious diseases of humans : Dynamics and control. Oxford University Press, New York, 1991.
H. Andersson and T. Britton. Stochastic epidemic models and their statistical analysis. Springer Verlag, 2000.
V. Andreasen. The final size of an epidemic and its relation to the basic reproduction number. Bulletin of Mathematical Biology, (2011) in press.
Ball, F.. Deterministic and stochastic epidemics with several kinds of susceptibles. Advances in Applied Probability, 17 (1985), No. 1, 1–22. CrossRefGoogle Scholar
Ball, F. and Clancy, D.. The final size and severity of a generalised stochastic multitype epidemic model. Advances in Applied Probability, 25 (1993), No. 4, 721–736. CrossRefGoogle Scholar
Bansal, S., Grenfell, B. T., and Meyers, L. A.. When individual behaviour matters : homogeneous and network models in epidemiology. Journal of Royal Sosciety Interface, 4 (2007), No. 16, 879–891. CrossRefGoogle Scholar
Berezovskaya, F. S., Novozhilov, A. S., Karev, G. P.. Population models with singular equilibrium. Mathematical Biosciences, 208 (2007), No. 1, 270–299. CrossRefGoogle ScholarPubMed
Bonzi, B., Fall, A. A., Iggidr, A., Sallet, G.. Stability of differential susceptibility and infectivity epidemic models. Journal of Mathematical Biology, 62 (2011), No. 1, 39–64. CrossRefGoogle ScholarPubMed
Boylan, R. D.. A note on epidemics in heterogeneous populations. Mathematical Biosciences, 105 (1991), No. 1, 133–137. CrossRefGoogle ScholarPubMed
A. S. Bratus, A. S. Novozhilov, Platonov A. P. Dynamical systems and models in biology. Fizmatlit, 2010.
Coutinho, F. A. B., Massad, E., Lopez, L. F., Burattini, M. N., Struchiner, C. J., Azevedo-Neto, R. S.. Modelling heterogeneities in individual frailties in epidemic models. Mathematical and Computer Modelling, 30 (1999), No. 1, 97–115. CrossRefGoogle Scholar
L. Danon, A. P. Ford, T. House, C. P. Jewell, M. J. Keeling, G. O. Roberts, J. V. Ross, M. C. Vernon. Networks and the epidemiology of infectious disease. Interdisciplinary Perspectives on Infectious Diseases, 28 (2011).
A. M. De Roos, L. Persson. Unstructured population models : Do population-level assumptions yield general theory ? In K. Cuddington and B. Beisner, editors, Ecological paradigms lost : Routes of theory change, pages 31–62. Academic Press, 2005.
O. Diekmann, J. A. P. Heesterbeek. Mathematical epidemiology of infectious diseases : Model building, analysis and interpretation, John Wiley, 2000.
Diekmann, O., Heesterbeek, J. A. P., Metz, J. A. J.. On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology, 28 (1990), No. 4, 365–382. CrossRefGoogle Scholar
O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz. The legacy of Kermack and McKendrick. In D. Mollison, editor, Epidemic models : Their structure and relation to data, 95–115. Cambridge University Press, 1995.
Duffy, M.A., Sivars-Becker, L.. Rapid evolution and ecological host–parasite dynamics. Ecology Letters, 10 (2007), No. 1, 44–53. CrossRefGoogle ScholarPubMed
Dushoff, J.. Host heterogeneity and disease endemicity : A moment-based approach. Theoretical Population Biology, 56 (1999), No. 3, 325–335. CrossRefGoogle ScholarPubMed
G. Dwyer, J. Dushoff, J. S. Elkinton, J. P. Burand, S. A. Levin. Variation in susceptibility : Lessons from an insect virus. In U. Diekmann, H. Metz, M. Sabelis, and K. Sigmund, editors, Adaptive dynamics of infectious diseases : In pursuit of virulence management, 74–84. Cambridge Univercity Press, 2002.
Dwyer, G., Dushoff, J., Elkinton, J. S., Levin, S. A.. Pathogen-driven outbreaks in forest defoliators revisited : Building models from experimental data. The American Naturalist, 156 (2000), No. 2, 105–120. Google ScholarPubMed
Dwyer, G., Elkinton, J. S., Buonaccorsi, J. P.. Host heterogeneity in susceptibility and disease dynamics : Tests of a mathematical model. The American Naturalist, 150 (1997), No. 6, 685–707. CrossRefGoogle ScholarPubMed
Gart, J. J.. The statistical analysis of chain-binomial epidemic models with several kinds of susceptibles. Biometrics, 28 (1972), No. 4, 921–930. CrossRefGoogle ScholarPubMed
Gorban, A. N.. Selection theorem for systems with inheritance. Mathematical Modelling of Natural Phenomena, 2 (2007), No. 4, 1–45. CrossRefGoogle Scholar
A. N. Gorban. Self-simplification in Darwin’s systems. In A. N. Gorban and D. Roose, editors, Coping with complexity : Model reduction and data analysis, pages 311–340. Springer Verlag, 2010.
A. Hastings. Unstructured models in ecology : past, present, and future. In K. Cuddington and B. E. Beisner, editors, Ecological paradigms lost : Routes of theory change, pages 9–30. Academic Press, 2005.
J. A. P. Heesterbeek. The law of mass-action in epidemiology : a historical perspective. In K. Cuddington and B. E. Beisner, editors, Ecological paradigms lost : Routes of theory change, pages 81–104. Academic Press, 2005.
S. Hsu Schmitz. Effects of genetic heterogeneity on HIV transmission in homosexual populations. In C. Castillo-Chavez, editor, Mathematical approaches for emerging and reemerging infectious diseases : Models, methods, and theory, pages 245–260. IMA, 2002.
Hyman, J. M., Li, J.. Differential susceptibility epidemic models. Journal of Mathematical Biology, 50 (2005), No. 6, 626–644. CrossRefGoogle Scholar
Karev, G. P.. Heterogeneity effects in population dynamics. Doklady Mathematics, 62 (2000), No. 1, 141–144. Google Scholar
Karev, G. P.. Inhomogeneous models of tree stand self-thinning. Ecological Modelling, 160 (2003), No. 1-2, 23–37. CrossRefGoogle Scholar
Karev, G. P.. Dynamics of inhomogeneous populations and global demography models. Journal of Biological Systems, 13 (2005), No. 1, 83–104. CrossRefGoogle Scholar
Karev, G. P.. On mathematical theory of selection : continuous time population dynamics. Journal of Mathematical Biology, 60 (2010), No. 1, 107–129. CrossRefGoogle ScholarPubMed
Karev, G. P.. Replicator equations and the principle of minimal production of information. Bulletin of Mathematical Biology, 72 (2010), No. 5, 1124–1142. CrossRefGoogle Scholar
Karev, G. P., Novozhilov, A. S., Berezovskaya, F. S.. On the asymptotic behavior of the solutions to the replicator equation. Mathematical Medicine and Biology, 28 (2011), No. 2, 89–110. CrossRefGoogle ScholarPubMed
Karev, G. P., Novozhilov, A. S., Koonin, E. V.. Mathematical modeling of tumor therapy with oncolytic viruses : Effects of parametric heterogeneity on cell dynamics. Biology Direct, 1 (2006), No. 30, 19. CrossRefGoogle ScholarPubMed
G. Katriel. The size of epidemics in populations with heterogeneous susceptibility. Journal of Mathematical Biology, (2011), in press.
M. J. Keeling, P. Rohani. Modeling infectious diseases in humans and animals. Princeton University Press, 2008.
Kermack, W. O., McKendrick, A. G.. A Contribution to themathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, 115 (1927), No. 772, 700–721. CrossRefGoogle Scholar
Liu, W. M., Hethcote, H. W., Levin, S. A.. Dynamical behavior of epidemiological models with nonlinear incidence rates. Journal of Mathematical Biology, 25 (1987), No. 4, 359–380. CrossRefGoogle Scholar
Liu, W. M., Levin, S. A., Iwasa, Y.. Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. Journal of Mathematical Biology, 23 (1986), No. 2, 187–204. CrossRefGoogle ScholarPubMed
May, R. M., Anderson, R. M.. The transmission dynamics of human immunodeficiency virus. Proceedings of the Royal Society of London. Series B : Biological Sciences, 321 (1988), 565–607. CrossRefGoogle ScholarPubMed
McCallum, H., Barlow, N., Hone, J.. How should pathogen transmission be modelled ? Trends in Ecology & Evolution, 16 (2001), No. 6, 295–300. CrossRefGoogle Scholar
Meyers, L.A.. Contact network epidemiology : Bond percolation applied to infectious disease prediction and control. Bulletin of American Mathematical Society, 44 (2007), 63–86. CrossRefGoogle Scholar
Nikolaou, M., Tam, V. H.. A New Modeling approach to the effect of antimicrobial agents on heterogeneous microbial populations. Journal of Mathematical Biology, 52 (2006), No. 2, 154–182. CrossRefGoogle ScholarPubMed
Novozhilov, A. S.. Analysis of a generalized population predator–prey model with a parameter distributed normally over the individuals in the predator population. Journal of Computer and System Sciences International, 43 (2004), No. 3, 378–382. Google Scholar
Novozhilov, A. S.. On the spread of epidemics in a closed heterogeneous population. Mathematical Biosciences, 215 (2008), No. 2, 177–185. CrossRefGoogle Scholar
Novozhilov, A. S.. Heterogeneous Susceptibles–Infectives model : Mechanistic derivation of the power law transmission function. Dynamics of Continuous, Discrete and Impulsive Systems (Series A, Mathematical Analysis), 16 (2009), No. S1, 136–140. Google Scholar
A. S. Novozhilov. On the stochastic SIR model with heterogeneous susceptibility. (2012), in preparation.
Novozhilov, A. S., Berezovskaya, F. S., Koonin, E. V., Karev, G. P.. Mathematical modeling of tumor therapy with oncolytic viruses : Regimes with complete tumor elimination within the framework of deterministic models. Biology Direct, 1 (2006), No. 6, 18. CrossRefGoogle Scholar
Rodrigues, P., Margheri, A., Rebelo, C., Gomes, M. G. M.. Heterogeneity in susceptibility to infection can explain high reinfection rates. Journal of Theoretical Biology, 259 (2009), No. 2, 280–290. CrossRefGoogle Scholar
Roy, M., Pascual, M.. On representing network heterogeneities in the incidence rate of simple epidemic models. Ecological Complexity, 3 (2006), No. 1, 80–90. CrossRefGoogle Scholar
Scalia-Tomba, G.. Asymptotic final size distribution of the multitype reed- frost process. Journal of Applied Probability, 23 (1986), No. 3, 563–584. Google Scholar
Severo, N. C.. Generalizations of some stochastic epidemic models. Mathematical Biosciences, 4 (1969), 395–402. CrossRefGoogle Scholar
Stroud, P. D., Sydoriak, S. J., Riese, J. M., Smith, J. P., Mniszewski, S. M., Romero, P. R.. Semi-empirical power-law scaling of new infection rate to model epidemic dynamics with inhomogeneous mixing. Mathematical Biosciences, 203 (2006), No. 2, 301–318. CrossRefGoogle ScholarPubMed
Veliov, V. M.. On the effect of population heterogeneity on dynamics of epidemic diseases. Journal of Mathematical Biology, 51 (2005), No. 2, 123–143. CrossRefGoogle ScholarPubMed
Wilson, E. B., Worcester, J.. The law of mass action in epidemiology. Proceedings of the National Academy of Sciences of the USA, 31 (1945), No. 1, 24–34. CrossRefGoogle ScholarPubMed
Wilson, E. B., Worcester, J.. The law of mass action in epidemiology, II. Proceedings of the National Academy of Sciences of the USA, 31 (1945), No. 4, 109–116.CrossRefGoogle Scholar