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**Spatiotemporal dynamics of epidemics: Synchrony in metapopulation models.**
*(English)*
Zbl 1036.92029

Summary: Multi-patch models – also known as metapopulation models – provide a simple framework within which the role of spatial processes in disease transmission can be examined. An \(n\)-patch model which distinguishes between \(k\) different classes of individuals is considered. The linear stability of spatially homogeneous solutions of such models is studied using an extension of an analysis technique previously described for a population setting in which individuals migrate between patches according to a simple linear term. The technique considerably simplifies the analysis as it decouples the \(nk\) dimensional linearized system into \(n\) distinct \(k\)-dimensional systems.

An important feature of the spatial epidemiological model is that the spatial coupling may involve nonlinear terms. As an example of the use of this technique, the dynamical behavior in the vicinity of the endemic equilibrium of a symmetric SIR model is decomposed into spatial modes. For parameter values appropriate for childhood diseases, expressions for the eigenvalues corresponding to in-phase and out-of-phase modes are obtained, and it is shown that the dominant mode of the system is an in-phase mode. Furthermore, the out-of-phase modes are shown to decay much more rapidly than the in-phase mode for a broad range of coupling strengths.

An important feature of the spatial epidemiological model is that the spatial coupling may involve nonlinear terms. As an example of the use of this technique, the dynamical behavior in the vicinity of the endemic equilibrium of a symmetric SIR model is decomposed into spatial modes. For parameter values appropriate for childhood diseases, expressions for the eigenvalues corresponding to in-phase and out-of-phase modes are obtained, and it is shown that the dominant mode of the system is an in-phase mode. Furthermore, the out-of-phase modes are shown to decay much more rapidly than the in-phase mode for a broad range of coupling strengths.

### MSC:

92D30 | Epidemiology |

### Keywords:

Epidemiological model; Metapopulation; Spatial dynamics; Linear stability analysis; System of ordinary differential equations
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\textit{A. L. Lloyd} and \textit{V. A. A. Jansen}, Math. Biosci. 188, 1--16 (2004; Zbl 1036.92029)

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### References:

[1] | Anderson, R. M.; May, R. M.: Infectious diseases of humans. (1991) |

[2] | Aron, J. L.; Schwartz, I. B.: Seasonality and period-doubling bifurcations in an epidemic model. J. theor. Biol. 110, 665 (1984) |

[3] | Bailey, N. T. J: The mathematical theory of infectious diseases. (1975) · Zbl 0334.92024 |

[4] | Ball, F. G.: Dynamic population epidemic models. Math. biosci. 107, 229 (1991) · Zbl 0747.92025 |

[5] | Baroyan, O. V.; Rvachev, L. A.: Deterministic epidemic models for a territory with a transport network. Kibernetika 3, 67 (1967) |

[6] | Bartlett, M. S.: Deterministic and stochastic models for recurrent epidemics. Proc. third Berkeley symp. On mathematical statistics and probability 4, 81 (1956) · Zbl 0070.15004 |

[7] | Bolker, B. M.; Grenfell, B. T.: Impact of vaccination on the spatial correlation and persistence of measles dynamics. Proc. natl. Acad. sci. USA 93, 12648 (1996) |

[8] | Diekmann, O.; Heesterbeek, J. A. P; Metz, J. A. J: On the definition and computation of the basic reproduction ratio R0 in models of infectious diseases in heterogeneous populations. J. math. Biol. 28, 365 (1990) · Zbl 0726.92018 |

[9] | Dietz, K.: The incidence of infectious diseases under the influence of seasonal fluctuations. Lect. notes biomath. 11, 1 (1976) · Zbl 0333.92014 |

[10] | Grasman, J.: Stochastic epidemics: the expected duration of the endemic period in higher dimensional models. Math. biosci. 152, 13 (1998) · Zbl 0930.92021 |

[11] | Hethcote, H. W.; Ark, J. W. V: Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation, and immunization programs. Math. biosci. 84, 85 (1987) · Zbl 0619.92006 |

[12] | Huang, Y.; Diekmann, O.: Interspecific influence on mobility and Turing instability. Bull. math. Biol. 65, 143 (2003) · Zbl 1334.92341 |

[13] | Isham, V.: Assessing the variability of stochastic epidemics. Math. biosci. 107, 209 (1991) · Zbl 0739.92015 |

[14] | Jansen, V. A. A: The dynamics of two diffusively coupled predator–prey systems. Theor. popul. Biol. 59, 119 (2001) · Zbl 1036.92034 |

[15] | Jansen, V. A. A; Lloyd, A. L.: Local stability analysis of spatially homogeneous solutions of multi-patch systems. J. math. Biol. 41, 232 (2000) · Zbl 0982.92032 |

[16] | Keeling, M. J.; Rohani, P.: Estimating spatial coupling in epidemiological systems: a mechanistic approach. Ecol. lett. 5, 20 (2002) |

[17] | Lajmanovich, A.; Yorke, J. A.: A deterministic model for gonorrhea in a nonhomogeneous population. Math. biosci. 28, 221 (1976) · Zbl 0344.92016 |

[18] | Lloyd, A. L.: Estimating variability in models for recurrent epidemics. Theor. popul. Biol. 65, 49 (2004) · Zbl 1105.92031 |

[19] | Lloyd, A. L.; May, R. M.: Spatial heterogeneity in epidemic models. J. theor. Biol. 179, 1 (1996) |

[20] | London, W. P.; Yorke, J. A.: Recurrent outbreaks of measles, chickenpox and mumps. I. seasonal variation in contact rates. Am. J. Epidemiol. 98, 453 (1973) |

[21] | May, R. M.: Complexity in model ecosystems. (1974) |

[22] | Minc, H.: Nonnegative matrices. (1988) · Zbl 0638.15008 |

[23] | Othmer, H. G.; Scriven, L. E.: Instability and dynamic pattern in cellular networks. J. theor. Biol. 32, 507 (1971) |

[24] | Plahte, E.: Pattern formation in discrete cell lattices. J. math. Biol. 43, 411 (2001) · Zbl 0993.92004 |

[25] | Rohani, P.; Earn, D. J. D; Grenfell, B. T.: Opposite patterns of synchrony in sympatric disease metapopulations. Science 286, 968 (1999) |

[26] | Sattenspiel, L.; Herring, D. A.: Simulating the effect of quarantine on the spread of the 1918–19 flu in central Canada. Bull. math. Biol. 65, 1 (2003) · Zbl 1334.92427 |

[27] | Schwartz, I. B.: Small amplitude, long period outbreaks in seasonally driven epidemics. J. math. Biol. 30, 473 (1992) · Zbl 0745.92026 |

[28] | Smith, D. L.; Lucey, B.; Waller, L. A.; Childs, J. E.; Real, L. A.: Predicting the spatial dynamics of rabies epidemics on heterogeneous landscapes. Proc. nat. Acad. sci. USA 99, 3668 (2002) |

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