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**Optimal vaccination policies for stochastic epidemics among a population of households.**
*(English)*
Zbl 0996.92032

Summary: This paper considers stochastic epidemics among a population partitioned into households, with mixing locally within households and globally throughout the population. The two levels of mixing have important implications for the threshold behaviour of the epidemic and consequently for the form and construction of optimal vaccination policies. Optimality is considered in terms of the cost of the vaccination program, the form of which is general enough to include costs of the vaccine itself, its administration, travel to and/or contact with the households. New explicit results are obtained by a constructive method which explain the form of optimal vaccination policies. Numerical studies are presented which exemplify the results discussed.

### MSC:

92D30 | Epidemiology |

60J20 | Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) |

93E20 | Optimal stochastic control |

### Keywords:

SIR; SIS; SIRS epidemics; threshold parameter; households epidemic model; optimal vaccination policies
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\textit{F. G. Ball} and \textit{O. D. Lyne}, Math. Biosci. 177--178, 333--354 (2002; Zbl 0996.92032)

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### References:

[1] | Becker, N.G.; Dietz, K., The effect of the household distribution on transmission and control of highly infectious diseases, Math. biosci., 127, 207, (1995) · Zbl 0824.92025 |

[2] | Ball, F.G.; Mollison, D.; Scalia-Tomba, G., Epidemics with two levels of mixing, Ann. appl. prob., 7, 46, (1997) · Zbl 0909.92028 |

[3] | Britton, T.; Becker, N.G., Estimating the immunity coverage required to prevent epidemics in a community of households, Biostatistics, 1, 389, (2000) · Zbl 1089.62525 |

[4] | Andersson, H., Epidemics in a population with social structures, Math. biosci., 140, 79, (1997) · Zbl 0881.92027 |

[5] | Becker, N.G.; Starczak, D.N., Optimal vaccination strategies for a community of households, Math. biosci., 139, 117, (1997) · Zbl 0881.92028 |

[6] | F.G. Ball, Threshold behaviour in stochastic epidemics among households, in: C.C. Heyde, Y.V. Prohorov, R. Pyke and S.T. Rachev (Eds.) Athens Conference on Applied Probability and Time Series, Volume I: Applied Probability, Lecture Notes in Statistics 114, 1996, p. 253 · Zbl 0854.92015 |

[7] | Ball, F.G., Stochastic and deterministic models for SIS epidemics among a population partitioned into households, Math. biosci., 156, 41, (1999) · Zbl 0979.92033 |

[8] | Ball, F.G., A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models, Adv. appl. prob., 18, 289, (1986) · Zbl 0606.92018 |

[9] | Ludwig, D., Final size distributions for epidemics, Math. biosci., 23, 33, (1975) · Zbl 0318.92025 |

[10] | McNeil, D.R., Integral functionals of birth and death processes and related limiting distributions, Ann. math. statist., 41, 480, (1970) · Zbl 0196.20403 |

[11] | Watson, R.K., A useful random time-scale transformation for the standard epidemic model, J. appl. prob., 17, 324, (1980) · Zbl 0446.92021 |

[12] | Hérnandez-Suárez, C.M.; Castillo-Chavez, C., A basic result on the integral for birth – death Markov processes, Math. biosci., 161, 95, (1999) · Zbl 0938.60056 |

[13] | Kelly, F.P., Reversibility and stochastic networks, (1979), Wiley Chichester · Zbl 0422.60001 |

[14] | Heesterbeek, J.A.P.; Dietz, K., The concept of R0 in epidemic theory, Statistica neerlandica, 50, 89, (1996) · Zbl 0854.92018 |

[15] | Becker, N.G.; Starczak, D.N., The effect of random vaccine response on the vaccination coverage required to prevent epidemics, Math. biosci., 154, 117, (1998) · Zbl 0930.92016 |

[16] | Monto, A.S.; Koopman, J.S.; Longini, I.M., Tecumseh study of illness, XIII, influenza infection and disease, 1976-1981, Am. J. epidemiol., 121, 811, (1985) |

[17] | Longini, I.M.; Koopman, J.S.; Haber, M.; Cotsonis, G.A., Statistical inference for infectious diseases: risk-specific household and community transmission parameters, Am. J. epidemiol., 128, 845, (1988) |

[18] | F.G. Ball, O.D. Lyne, Estimation for stochastic multitype SIR epidemics among a population partitioned into households, 2001, in preparation · Zbl 0978.92025 |

[19] | Becker, N.G.; Hall, R., Immunization levels for preventing epidemics in a community of households made up of individuals of various types, Math. biosci., 132, 205, (1996) · Zbl 0844.92020 |

[20] | Ball, F.G.; Lyne, O.D., Stochastic multitype SIR epidemics among a population partitioned into households, Adv. appl. prob., 33, 99, (2001) · Zbl 0978.92025 |

[21] | Addy, C.L.; Longini, I.M.; Haber, M., A generalized stochastic model for the analysis of infectious disease final size data, Biometrics, 47, 961, (1991) · Zbl 0729.62564 |

[22] | O.D. Lyne, F.G. Ball, Parameter estimation for SIR epidemics in households, Bull. Int. Statist. Inst. 52nd Session, Contributed Papers, Vol. LVIII, Book 2, p. 251 · Zbl 1122.92303 |

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