Uniform weak implies uniform strong persistence for non-autonomous semiflows

It is shown that, under two additional assumptions, uniformly weakly persistent semiflows are also uniformly strongly persistent even if they are non-autonomous. This result is applied to a time-heterogeneous model of S-I-R-S type for the spread of infectious childhood diseases. If some of the parameter functions are almost periodic, an almost sharp threshold result is obtained for uniform strong endemicity versus extinction.

[1]  H J Bremermann,et al.  A competitive exclusion principle for pathogen virulence , 1989, Journal of mathematical biology.

[2]  S. Levin,et al.  Periodicity in Epidemiological Models , 1989 .

[3]  H L Smith,et al.  Subharmonic bifurcation in an S-I-R epidemic model , 1983, Journal of mathematical biology.

[4]  J. So,et al.  Predator-prey models with almost periodic coefficients , 1988 .

[5]  H. Thieme,et al.  Recurrent outbreaks of childhood diseases revisited: the impact of isolation. , 1995, Mathematical biosciences.

[6]  Paul Waltman,et al.  Perturbation of a globally stable steady state , 1999 .

[7]  D. Schenzle.  An age-structured model of pre- and post-vaccination measles transmission. , 1984, IMA journal of mathematics applied in medicine and biology.

[8]  Xiao-Qiang Zhao.  Asymptotic Behavior for Asymptotically Periodic Semiflows with Applications , 1996 .

[9]  Mary Ballyk,et al.  Effects of Random Motility on Microbial Growth and Competition in a Flow Reactor , 1998, SIAM J. Appl. Math..

[10]  Horst R. Thieme,et al.  Persistence under relaxed point-dissipativity (with application to an endemic model) , 1993 .

[11]  Jack K. Hale,et al.  Persistence in infinite-dimensional systems , 1989 .

[12]  Carlos Castillo-Chavez,et al.  Asymptotically Autonomous Epidemic Models , 1994 .

[13]  Uniform repellers for processes with applications to periodic differential systems , 1990 .

[14]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[15]  S. Ruan,et al.  A generalization of the Butler-McGehee lemma and its applications in persistence theory , 1996, Differential and Integral Equations.

[16]  Klaus Dietz,et al.  Mathematical Models for Infectious Disease Statistics , 1985 .

[17]  Yang Kuang,et al.  Permanence in Kolmogorov-Type Systems of Nonautonomous Functional Differential Equations , 1996 .

[18]  V. Hutson,et al.  Permanence for nonautonomous predator-prey systems , 1991 .

[19]  K. Schmitt,et al.  Permanence and the dynamics of biological systems. , 1992, Mathematical biosciences.

[20]  H. I. Freedman,et al.  Persistence definitions and their connections , 1990 .

[21]  Xiaodong Lin,et al.  Global stability of the endemic equilibrium and uniform persistence in epidemic models with subpopulations , 1993, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[22]  Paul Waltman,et al.  A brief survey of persistence in dynamical systems , 1991 .

[23]  Horst R. Thieme,et al.  Asymptotically Autonomous Differential Equations in the Plane , 1993 .

[24]  Global asymptotic behavior in a periodic competitor-competitor-mutualist parabolic system , 1997 .

[25]  Shigui Ruan,et al.  Uniform persistence and flows near a closed positively invariant set , 1994 .

[26]  J. Hale.  Asymptotic Behavior of Dissipative Systems , 1988 .

[27]  H L Smith,et al.  Multiple stable subharmonics for a periodic epidemic model , 1983, Journal of mathematical biology.

[28]  Chris Cosner,et al.  Ecological models, permanence and spatial heterogeneity , 1996 .

[29]  H. Smith.  Subharmonic Bifurcation in an SIR Epidemic Model , 2022 .

[30]  Horst R. Thieme,et al.  Demographic change and persistence of HIV/AIDS in heterogeneous population , 1991 .