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The global stability of SEIRS models with \nnonlinear incidence rates was conjectured in [W. M. Liu, H. W. Hethcote and S. A. Levin, J. Math. Biol. 25 (1987), \n359-380.] and has been \nstated as an outstanding open question for classical bilinear \nmodels in [M. Y. Li, J. S. Muldowney and P. van den Driessche, Canad. Appl. Math. Q. 7 (1999), 409-425.]. By applying the Poincar e-Bendixson property \nof dynamic systems in space, the authors in [M. Y. Li and J. S. Muldowney, \nSIAM J. Math. Anal. 27 (1996), 1070-1083.] have proven \nthe conjecture for the bilinear model with a su fficiently long \naverage immunity period, and in [M. Y. Li, J. S. Muldowney and P. van den Driessche, Canad. Appl. Math. Q. 7 (1999), 409-425.] the authors have shown \nthe case with a suffi ciently long average infection period. In \nthis paper, we solve the open problem for the bilinear case \ncompletely, and furthermore have relaxed the constraint on the \ngen
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The global stability of SEIRS models with \nnonlinear incidence rates was conjectured in [W. M. Liu, H. W. Hethcote and S. A. Levin, J. Math. Biol. 25 (1987), \n359-380.] and has been \nstated as an outstanding open question for classical bilinear \nmodels in [M. Y. Li, J. S. Muldowney and P. van den Driessche, Canad. Appl. Math. Q. 7 (1999), 409-425.]. By applying the Poincar e-Bendixson property \nof dynamic systems in space, the authors in [M. Y. Li and J. S. Muldowney, \nSIAM J. Math. Anal. 27 (1996), 1070-1083.] have proven \nthe conjecture for the bilinear model with a su fficiently long \naverage immunity period, and in [M. Y. Li, J. S. Muldowney and P. van den Driessche, Canad. Appl. Math. Q. 7 (1999), 409-425.] the authors have shown \nthe case with a suffi ciently long average infection period. In \nthis paper, we solve the open problem for the bilinear case \ncompletely, and furthermore have relaxed the constraint on the \ngen
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