SIS epidemic attractors in periodic environments

References

• Anderson , R. M. and May , R. M. 1992 . Infectious Diseases of Humans: Dynamics and Control , Oxford : Oxford University Press .
   Google Scholar
• Bailey , N. T.J. 1975 . The Mathematical Theory of Infectious Diseases and Its Applications , London : Griffin .
   Google Scholar
• Costantino , R. F. , Cushing , J. M. , Dennis , B. , Desharnais , R. A. and Henson , S. M. 1998 . Resonant population cycles in temporarily fluctuating habitats . Bulletin of Mathematical Biology , 60 : 247 – 273 .
   PubMed Web of Science ®Google Scholar
• Cushing , J. M. and Henson , S. M. 2001 . Global dynamics of some periodically forced, monotone difference equations . Journal of Difference Equations and Applications , 7 : 859 – 872 .
   Web of Science ®Google Scholar
• Elaydi , S. N. and Sacker , R. J. 2005 . Global stability of periodic orbits of nonautonomous difference equations and population biology . Journal of Difference Equations , 208 : 258 – 273 .
   Web of Science ®Google Scholar
• Elaydi , S. N. and Sacker , R. J. Global stability of periodic orbits of nonautonomous difference equations in population biology and Cushing–Henson conjectures . Proceedings of the 8th International Conference on Difference Equations and Applications . 2005 . pp. 113 – 126 . Boca Raton, FL : Chapman & Hall/CRC .
   Google Scholar
• Elaydi , S. N. and Sacker , R. J. 2005 . Nonautonomous Beverton–Holt equations and the Cushing-Henson conjectures . Journal of Difference Equations and Applications , 11 : 336 – 346 .
   Web of Science ®Google Scholar
• Elaydi , S. N. and Sacker , R. J. 2006 . Periodic difference equations, populations biology and the Cushing–Henson conjectures . Mathematical Biosciences , 201 : 195 – 207 .
   PubMed Web of Science ®Google Scholar
• Franke , J. E. and Selgrade , J. F. 2003 . Attractor for periodic dynamical systems . Journal of Mathematics Analysis and Applications , 286 : 64 – 79 .
   Web of Science ®Google Scholar
• Franke , J. E. and Yakubu , A.-A. 2005 . Periodic dynamical systems in unidirectional metapopulation models . Journal of Difference Equations and Applications , 11 : 687 – 700 .
   Web of Science ®Google Scholar
• Franke , J. E. and Yakubu , A.-A. 2005 . Multiple attractors via cusp bifurcation in periodically varying environments . Journal of Difference Equations and Applications , 11 : 365 – 377 .
   Web of Science ®Google Scholar
• Franke , J. E. and Yakubu , A.-A. 2005 . Population models with periodic recruitment functions and survival rates . Journal of Difference Equations and Applications , 11 : 1169 – 1184 .
   Web of Science ®Google Scholar
• Franke , J. E. and Yakubu , A.-A. 2006 . Discrete-time SIS Epidemic Model In a Seasonal Environment . SIAM Journal on Applied Mathematics , 66 : 1563 – 1587 .
   Web of Science ®Google Scholar
• Fretwell , S. D. 1972 . Populations in a Seasonal Environment , Princeton, NJ : Princeton University Press .
   Google Scholar
• Henson , S. M. 2000 . Multiple attractors and resonance in periodically forced population models . Physica D , 140 : 33 – 49 .
   Web of Science ®Google Scholar
• Henson , S. M. 1999 . The effect of periodicity in maps . Journal of Difference Equations and Applications , 5 : 31 – 56 .
   Web of Science ®Google Scholar
• Henson , S. M. , Costantino , R. F. , Cushing , J. M. , Dennis , B. and Desharnais , R. A. 1999 . Multiple attractors, saddles, and population dynamics in periodic habitats . Bulletin of Mathematical Biology , 61 : 1121 – 1149 .
   PubMed Web of Science ®Google Scholar
• Henson , S. M. and Cushing , J. M. 1997 . The effect of periodic habitat fluctuations on a nonlinear insect population model . Journal of Mathematical Biology , 36 : 201 – 226 .
   Web of Science ®Google Scholar
• Jillson , D. 1980 . Insect populations respond to fluctuating environments . Nature , 288 : 699 – 700 .
   Web of Science ®Google Scholar
• Kocic , V. L. 2005 . A note on nonautonomous Beverton–Holt model . Journal of Difference Equations and Applications , 11 : 415 – 422 .
   Web of Science ®Google Scholar
• Kon , R. 2004 . A note on attenuant cycles of population models with periodic carrying capacity . Journal of Difference Equations and Applications , 10 : 791 – 793 .
   Web of Science ®Google Scholar
• Kon , R. 2005 . Attenuant cycles of population models with periodic carrying capacity . Journal of Difference Equations and Applications , 11 : 423 – 430 .
   Web of Science ®Google Scholar
• Li , J. 1992 . Periodic solutions of population models in a periodically fluctuating environment . Mathematics and Bioscience , 110 : 17 – 25 .
   PubMed Web of Science ®Google Scholar
• Nisbet , R. M. and Gurney , W. S.C. 1982 . Modelling Fluctuating Populations , New York : Wiley & Sons .
   Google Scholar
• Rosenblat , S. 1980 . Population models in a periodically fluctuating environment . Journal of Mathematical Biology , 9 : 23 – 36 .
   Web of Science ®Google Scholar
• Selgrade , J. F. and Roberds , H. D. 2001 . On the structure of attractors for discrete, periodically forced systems with applications to population models . Physica D , 158 : 69 – 82 .
   Web of Science ®Google Scholar
• Yakubu , A. -A. Periodically forced nonlinear difference equations with delay . Difference Equations and Discrete Dynamical Systems, Proceedings of the 9th International Conferenc . Edited by: Allen , L. , Aulbach , B. , Elaydi , S. and Sacker , R. pp. 217 – 231 . River Edge, NJ : World Scientific . University of Southern California
   Google Scholar
• Castillo-Chavez , C. and Yakubu , A. 2001 . Dispersal, disease and life-history evolution . Mathematics and Bioscience , 173 : 35 – 53 .
   PubMed Web of Science ®Google Scholar
• Castillo-Chavez , C. and Yakubu , A. 2001 . Discrete-time S-I-S models with complex dynamics . Nonlinear Analysis , 47 : 4753 – 4762 .
   Web of Science ®Google Scholar
• Castillo-Chavez , C. and Yakubu , A. A. 2002 . “ Intraspecific competition, dispersal and disease dynamics in discrete-time patchy environments ” . In Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction to Models, Methods and Theory , Edited by: Blower , S , Castillo-Chavez , C. , van den Driessche , P. , Kirschner , D. and Yakubu , A.-A. 165 – 181 . New York : Springer-Verlag .
   Google Scholar
• May , R. M. and Oster , G. F. 1976 . Bifurcations and dynamic complexity in simple ecological models . American Naturalist , 110 : 573 – 579 .
   Web of Science ®Google Scholar
• May , R. M. 1977 . Simple mathematical models with very complicated dynamics . Nature , 261 : 459 – 469 .
   Web of Science ®Google Scholar
• May , R. M. 1974 . Stability and Complexity in Model Ecosystems , Princeton, NJ : Princeton University Press .
   Google Scholar
• Nicholson , A. J. 1954 . Compensatory reactions of populations to stresses, and their evolutionary significance . Australian Journal of Zoology , 2 : 1 – 65 .
   Web of Science ®Google Scholar
• Allen , L. J.S. and Burgin , A. M. 2000 . Comparison of deterministic and stochastic SIS and SIR models in discrete-time . Mathematics and Bioscience , 163 : 1 – 33 .
   PubMed Web of Science ®Google Scholar
• Allen , L. J.S. 1994 . Some discrete-time SI, SIR and SIS epidemic models . Mathematics and Bioscience , 124 : 83 – 105 .
   PubMed Web of Science ®Google Scholar
• Beverton , R. J.H. and Holt , S. J. 1957 . On the Dynamics of Exploited Fish Populations, Fish. Invest. Ser. II , London : H. M. Stationery Office .
   Google Scholar
• Elaydi , S. N. and Yakubu , A.-A. 2002 . Global stability of cycles: Lotka–Volterra competition model with stocking . Journal of Difference Equations and Applications , 8 : 537 – 549 .
   Web of Science ®Google Scholar
• Alligood , K. , Sauer , T. and Yorke , J. A. 1996 . Chaos: An Introduction to Dynamical Systems , New York : Springer-Verlag .
   Google Scholar
• Yakubu , A.-A. and Fogarty , M. 2006 . Spatially discrete metapopulation models with directional dispersal . Mathematical Bioscience , 204 : 68 – 101 .
   PubMed Web of Science ®Google Scholar
• Hadeler , K. P. and van den Driessche , P. 1997 . Backward bifurcation in epidemic control . Mathematics and Bioscience , 146 : 15 – 35 .
   PubMed Web of Science ®Google Scholar
• Hassell , M. P. , Lawton , J. H. and May , R. M. 1976 . Patterns of dynamical behavior in single species populations . Journal of Animal Ecology , 45 : 471 – 486 .
   Web of Science ®Google Scholar
• van den Driessche , P. and Watmough , J. 2000 . A simple SIS epidemic model with a backward bifurcation . Journal of Mathematical Biology , 40 : 525 – 540 .
   PubMed Web of Science ®Google Scholar