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Moment Equations and Dynamics of a Household SIS Epidemiological Model

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Abstract

An SIS epidemiological model of individuals partitioned into households is studied, where infections take place either within or between households, the latter generally happening much less frequently. The model is explored using stochastic spatial simulations, as well as mathematical models which consist of an infinite system of ordinary differential equations for the moments of the distribution describing the proportions of individuals who are infectious among households. Various moment-closure approximations are used to truncate the system of ODEs to finite systems of equations. These approximations can sometimes lead to a system of ill-behaved ODEs which predict moments which become negative or unbounded. A reparametrization of the ODEs is then developed, which forces all moments to satisfy necessary constraints.

Changing the proportion of contacts within and between households does not change the endemic equilibrium, but does affect the amount of time it takes to approach the fixed point; increasing the proportion of contacts within households slows the spread of the infection toward endemic equilibrium. The system of moment equations does describe this phenomenon, although less accurately in the limit as the proportion of between-household contacts approaches zero. The results indicate that although controlling the movement of individuals does not affect the long-term frequency of an infection with SIS dynamics, it can have a large effect on the time-scale of the dynamics, which may provide an opportunity for other controls such as immunizations to be applied.

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References

  • Anderson, R.M., May, R.M., 1979. Population biology of infection diseases: Part I. Nature 280, 361–367.

    Article  Google Scholar 

  • Andersson, H., Britton, T., 1998. Heterogeneity in epidemic models and its effect on the spread of infection. J. Appl. Prob. 35, 651–661.

    Article  MathSciNet  MATH  Google Scholar 

  • Ball, F., 1999. Stochastic and deterministic models for SIS epidemics among a population partitioned into households. Math. Biosci. 156, 41–67.

    Article  MathSciNet  MATH  Google Scholar 

  • Ball, F., Mollison, D., Scalia-Tomba, G., 1997. Epidemics with two levels of mixing. Ann. Appl. Prob. 7(1), 46–89.

    Article  MathSciNet  MATH  Google Scholar 

  • Ball, F., Neal, P., 2004. Poisson approximations for epidemics with two levels of mixing. Ann. Prob. 32(1B), 1168–1200.

    Google Scholar 

  • Barthélemy, M., Barrat, A., Pastor-Satorras, R., Vespignani, A., 2005. Dynamical patterns of epidemic outbreaks in complex heterogeneous networks. J. Theor. Biol. 235, 275–288.

    Article  Google Scholar 

  • Bolker, B.M., 1999. Analytic models for the patchy spread of plant disease. Bull. Math. Biol. 61, 849–874.

    Article  Google Scholar 

  • Boots, M., Sasaki, A., 1999. ‘Small worlds’ and the evolution of virulence: Infection occurs locally and at a distance. Proc. R. Soc. Lond. B 266, 1933–1938.

    Article  Google Scholar 

  • Brown, D.H., Bolker, B.M., 2004. The effects of disease dispersal and host clustering on the epidemic threshold in plants. Bull. Math. Biol. 66, 341–371.

    Article  MathSciNet  Google Scholar 

  • de Aguiar, M.A.M., Rauch, E.M., Bar-Yam, Y., 2003. Mean-field approximation to a spatial host-pathogen model. Phys. Rev. E 67, 047102.

    Google Scholar 

  • Dieckmann, U., Law, R., 2000. Relaxation projections and the method of moments. In: Dieckmann, U., Law, R., Metz, J.A. (Eds.), The Geometry of Ecological Interactions, Ch. 21. Cambridge University Press, Cambridge, UK, pp. 412–455.

  • Ellner, S.P., Sasaki, A., Haraguchi, Y., Matsuda, H., 1998. Speed of invasion in lattice population models: Pair-edge approximations. J. Math. Biol. 36, 469–484.

    Article  MathSciNet  MATH  Google Scholar 

  • Filipe, J.A.N., Gibson, G.J., 1998. Studying and approximating spatio-temporal models for epidemic spread and control. Phil. Trans. R. Soc. Lond. B 353, 2153–2162.

    Article  Google Scholar 

  • Filipe, J.A.N., Gibson, G.J., 2001. Comparing approximations to spatio-temporal models for epidemics with local spread. Bull. Math. Biol. 63, 603–624.

    Article  Google Scholar 

  • Filipe, J.A.N., Maule, M.M., 2004. Effects of dispersal mechanisms on spatio-temporal development of epidemics. J. Theor. Biol. 226, 125–141.

    Article  MathSciNet  Google Scholar 

  • Filipe, J.A.N., Maule, M.M., Gilligan, C.A., 2004. On Analytical models for the patchy spread of plant disease. Bull. Math. Biol. 66, 1027–1037.

    Article  Google Scholar 

  • Ghoshal, G., Sander, L.M., Sokolov, I.M., 2004. SIS epidemics with household structure: The self-consistent field method. Math. Biosci. 190, 71–85.

    Article  MathSciNet  MATH  Google Scholar 

  • Gutowitz, H.A., Victor, J.D., 1987. Local structure theory in more than one dimension. Complex Syst. 1, 57–68.

    MathSciNet  MATH  Google Scholar 

  • Gutowitz, H.A., Victor, J.D., Knight, B.W., 1987. Local structure theory for cellular automata. Physica D 28, 18–48.

    Article  MathSciNet  MATH  Google Scholar 

  • Harada, Y., 1999. Short- vs. long-range disperser: The evolutionarily stable allocation in a lattice-structured habitat. J. Theor. Biol. 201, 171–187.

    Article  Google Scholar 

  • Harada, Y., Iwasa, Y., 1994. Lattice population dynamics for plants with dispersing seeds and vegetative propagation. Res. Popul. Ecol. 36(2), 237–249.

    Article  Google Scholar 

  • Hethcote, H.W., 1978. An immunization model for a heterogeneous population. Theor. Popul. Biol. 14, 338–349.

    Article  MathSciNet  Google Scholar 

  • Hiebeler, D., 1997. Stochastic spatial models: From simulations to mean field and local structure approximations. J. Theor. Biol. 187, 307–319.

    Article  Google Scholar 

  • Hiebeler, D., 2000. Populations on fragmented landscapes with spatially structured heterogeneities: Landscape generation and local dispersal. Ecology 81(6), 1629–1641.

    Article  Google Scholar 

  • Hiebeler, D., 2004. Competition between near and far dispersers in spatially structured habitats. Theor. Popul. Biol. 66(3), 205–218.

    Article  Google Scholar 

  • Hiebeler, D., 2005a. A cellular automaton SIS epidemiological model with spatially clustered recoveries. Lec. Notes Comp. Sci. 3515, 360–367.

    Article  Google Scholar 

  • Hiebeler, D., 2005b. Spatially correlated disturbances in a locally dispersing population model. J. Theor. Biol. 232(1), 143–149.

    Article  MathSciNet  Google Scholar 

  • Hwang, D.-U., Boccaletti, S., Moreno, Y., López-Ruiz, R., Apr. 2005. Thresholds for epidemic outbreaks in finite scale-free networks. Math. Biosci. Eng 2(2), 317–327.

    MathSciNet  Google Scholar 

  • Keeling, M., 2005. The implications of network structure for epidemic dynamics. Theor. Popul. Biol. 67, 1–8.

    Article  MATH  Google Scholar 

  • Keeling, M.J., 2000. Multiplicative moments and measures of persistence in ecology. J. Theor. Biol. 205, 269–281.

    Article  Google Scholar 

  • Kermack, W.O., McKendrick, A.G., 1927. A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A 115, 700–721.

    Article  Google Scholar 

  • Kirkwood, J.G., 1935. Statistical mechanics of fluid mixtures. J. Chem. Phys. 3, 300–313.

    Article  Google Scholar 

  • Levin, S.A., Durrett, R., 1996. From individuals to epidemics. Phil. Trans.: Biol. Sci. 351, 1615–1621.

    Article  Google Scholar 

  • Lloyd, A.L., 2004. Estimating variability in models for recurrent epidemics: Assessing the use of moment closure techniques. Theor. Popul. Biol. 65, 49–65.

    Article  MATH  Google Scholar 

  • Lloyd, A.L., Jansen, V.A., 2004. Spatiotemporal dynamics of epidemics: Synchrony in metapopulation models. Math. Biosci. 188, 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  • Matsuda, H., Ogita, N., Sasaki, A., Sato, K., 1992. Statistical mechanics of population. Prog. Theor. Phys. 88(6), 1035–1049.

    Article  Google Scholar 

  • May, R.M., Anderson, R.M., 1984. Spatial heterogeneity and the design of immunization programs. Math. Biosci. 72, 83–111.

    Article  MathSciNet  MATH  Google Scholar 

  • Murrell, D.J., Dieckmann, U., Law, R., 2004. On moment closures for population dynamics in continuous space. J. Theor. Biol. 229, 421–432.

    Article  Google Scholar 

  • Newman, M.E.J., 2002. The spread of epidemic disease on networks. Phys. Rev. E 66, 016128.

    Google Scholar 

  • Newman, M.E.J., Jensen, I., Ziff, R.M., 2002. Percolation and epidemics in a two-dimensional small world. Phys. Rev. E 65, 021904.

    Google Scholar 

  • Pastor-Satorras, R., Vespignani, A., 2001. Epidemic spreading in scale-free networks. Phys. Rev. Let. 86(14), 3200–3203.

    Article  Google Scholar 

  • Petermann, T., De Los Rios, P., 2004a. Cluster approximations for epidemic processes: A systematic description of correlations beyond the pair level. J. Theor. Biol. 229, 1–11.

    Article  MathSciNet  Google Scholar 

  • Petermann, T., De Los Rios, P., 2004b. The role of clustering and gridlike ordering in epidemic spreading. Phys. Rev. E 69, 066116. Doi: 10.1103/PhysRevE.69.066116.

  • Post, W.M., DeAngelis, D.L., Travis, C.C., 1983. Endemic disease in environments with spatially heterogeneous host populations. Math. Biosci. 63, 289–302.

    Article  MathSciNet  MATH  Google Scholar 

  • Read, J.M., Keeling, M.J., 2003. Disease evolution on networks: The role of contact structure. Proc. R. Soc. Lond. B 270, 699–708.

    Article  Google Scholar 

  • Rushton, S., Mautner, A.J., 1955. The deterministic model of a simple epidemic for more than one community. Biometrika 42, 126–132.

    MathSciNet  MATH  Google Scholar 

  • Saramäki, J., Kaski, K., 2005. Modelling development of epidemics with dynamic small-world networks. J. Theor. Biol. 234, 413–421.

    Article  Google Scholar 

  • Watson, R.K., 1972. On an epidemic in a stratified population. J. Appl. Prob. 9, 659–666.

    Article  MATH  Google Scholar 

  • Watts, D.J., Muhamad, R., Medina, D.C., Dodds, P.S., 2005. Multiscale, resurgent epidemics in a hierarchical metapopulation model. Proc. Nat. Acad. Sci. 102, 11157–11162.

    Article  Google Scholar 

Download references

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  1. Department of Mathematics and Statistics, University of Maine, Orono, ME, 04469-5752, USA

    David Hiebeler

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Hiebeler, D. Moment Equations and Dynamics of a Household SIS Epidemiological Model. Bull. Math. Biol. 68, 1315–1333 (2006). https://doi.org/10.1007/s11538-006-9080-1

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  • DOI: https://doi.org/10.1007/s11538-006-9080-1

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