Likelihood Estimation for Stochastic Epidemics with Heterogeneous Mixing Populations
We consider a heterogeneously mixing SIR stochastic
epidemic process in populations described by a general graph.
Likelihood theory is developed to facilitate statistic inference for the
parameters of the model under complete observation. We show that
these estimators are asymptotically Gaussian unbiased estimates by
using a martingale central limit theorem.
[1] P. K. Andersen, ├ÿ. Borgan, R. D. Gill and N. Keiding, Statistical Models
Based on Counting Processes. Springer, New York, 1993
[2] H. Andersson, Epidemic models and social networks. Math. Scientist,
24(1999) 128-147
[3] H. Andersson and T. Britton, Stochastic Epidemic Models and Their
Statistical Analysis. Springer-Verlag, New York, 2000
[4] N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and Its
Application. Griffin, London, 1975
[5] F. G. Ball and O. D. Lyne, Stochastic multitype SIR epidemics among
a population partitioned into households. Adv. Appl. Prob., 33(2001)
99-123
[6] F. Ball, D. Mollison and G. Scalia-Tombra, Epidemics with two levels
of mixing. Ann. Appl. Probab., 7(1997) 46-89
[7] F. Brauer and J. Watmough, Age of infection epidemic models with
heterogeneous mixing. J. Biol. Dyn., 3(2009) 324-330
[8] T. Britton, T. Kypraios and P. D. O-Neill, Statistical inference
for stochastic epidemic models with three levels of mixing.
arXiv:0908.2066v1 [stat.AP], 2009
[9] T. Britton and P. D. O-Neill, Bayesian inference for stochastic epidemics
in popluations with random social structure. Scand. J. Statist., 29(2002)
375-390
[10] N. Demiris and P. D. O-Neill, Bayesian inference for epidemics with
two levels of mixing. Scand. J. Statist., 32(2005) 265-280
[11] N. Demiris and P. D. O-Neill, Bayesian inference for stochastic multitype
epidemics in structured populations via random graphs. J. Roy.
Statist. Soc. Ser. B, 67(2005) 731-746
[12] S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and
Convergence. Wiley, New York, 1986
[13] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical
theory of epidemics. Proc. Roy. Soc. London Ser. A, 115(1927)
700-721
[14] R. Rebolledo, Central limit theorems for local martingales. Z. Wahrsch.
Verw. Gebiete., 51(1980) 269-286
[15] W. N. Rida, Asymptotic properties of some estimators for the infection
rate in the general stochastic epidemic model. J. R. Statist. Soc. B,
53(1991) 269-283
[1] P. K. Andersen, ├ÿ. Borgan, R. D. Gill and N. Keiding, Statistical Models
Based on Counting Processes. Springer, New York, 1993
[2] H. Andersson, Epidemic models and social networks. Math. Scientist,
24(1999) 128-147
[3] H. Andersson and T. Britton, Stochastic Epidemic Models and Their
Statistical Analysis. Springer-Verlag, New York, 2000
[4] N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and Its
Application. Griffin, London, 1975
[5] F. G. Ball and O. D. Lyne, Stochastic multitype SIR epidemics among
a population partitioned into households. Adv. Appl. Prob., 33(2001)
99-123
[6] F. Ball, D. Mollison and G. Scalia-Tombra, Epidemics with two levels
of mixing. Ann. Appl. Probab., 7(1997) 46-89
[7] F. Brauer and J. Watmough, Age of infection epidemic models with
heterogeneous mixing. J. Biol. Dyn., 3(2009) 324-330
[8] T. Britton, T. Kypraios and P. D. O-Neill, Statistical inference
for stochastic epidemic models with three levels of mixing.
arXiv:0908.2066v1 [stat.AP], 2009
[9] T. Britton and P. D. O-Neill, Bayesian inference for stochastic epidemics
in popluations with random social structure. Scand. J. Statist., 29(2002)
375-390
[10] N. Demiris and P. D. O-Neill, Bayesian inference for epidemics with
two levels of mixing. Scand. J. Statist., 32(2005) 265-280
[11] N. Demiris and P. D. O-Neill, Bayesian inference for stochastic multitype
epidemics in structured populations via random graphs. J. Roy.
Statist. Soc. Ser. B, 67(2005) 731-746
[12] S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and
Convergence. Wiley, New York, 1986
[13] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical
theory of epidemics. Proc. Roy. Soc. London Ser. A, 115(1927)
700-721
[14] R. Rebolledo, Central limit theorems for local martingales. Z. Wahrsch.
Verw. Gebiete., 51(1980) 269-286
[15] W. N. Rida, Asymptotic properties of some estimators for the infection
rate in the general stochastic epidemic model. J. R. Statist. Soc. B,
53(1991) 269-283
@article{"International Journal of Engineering, Mathematical and Physical Sciences:56407", author = "Yilun Shang", title = "Likelihood Estimation for Stochastic Epidemics with Heterogeneous Mixing Populations", abstract = "We consider a heterogeneously mixing SIR stochastic
epidemic process in populations described by a general graph.
Likelihood theory is developed to facilitate statistic inference for the
parameters of the model under complete observation. We show that
these estimators are asymptotically Gaussian unbiased estimates by
using a martingale central limit theorem.", keywords = "statistic inference, maximum likelihood, epidemicmodel, heterogeneous mixing.", volume = "5", number = "7", pages = "981-5", }