title: Seasonal dynamics in an SIR epidemic system.
abstract: We consider a seasonally forced SIR epidemic model where periodicity occurs in the contact rate. This periodical forcing represents successions of school terms and holidays. The epidemic dynamics are described by a switched system. Numerical studies in such a model have shown the existence of periodic solutions. First, we analytically prove the existence of an invariant domain D containing all periodic (harmonic and subharmonic) orbits. Then, using different scales in time and variables, we rewrite the SIR model as a slow-fast dynamical system and we establish the existence of a macroscopic attractor domain K, included in D, for the switched dynamics. The existence of a unique harmonic solution is also proved for any value of the magnitude of the seasonal forcing term which can be interpreted as an annual infection. Subharmonic solutions can be seen as epidemic outbreaks. Our theoretical results allow us to exhibit quantitative characteristics about epidemics, such as the maximal period between major outbreaks and maximal prevalence.
authors: Laboratoire de Mathématiques, Image et Applications (MIA), Université de La Rochelle, 17042 , La Rochelle, France, eaugerau@univ-lr.fr.
authors: E
authors: E
authors: Augeraud-Véron
authors: N
authors: N
authors: Sari
mesh_terms: D003141:Communicable Diseases / Q000453:epidemiology*; D058872:Epidemics; D006801:Humans; D008954:Models, Biological; D009716:Numerical Analysis, Computer-Assisted; D010507:Periodicity; D012621:Seasons
references: IMA J Math Appl Med Biol. 1984;1(2):169-91
references: 6600102
references: J Math Biol. 1992;30(5):473-91
references: 1578191
references: Am Nat. 2002 May;159(5):469-81
references: 18707430
references: Proc Biol Sci. 2006 Oct 7;273(1600):2541-50
references: 16959647
references: J Theor Biol. 2010 Nov 7;267(1):85-94
references: 20723547
references: J Math Biol. 2009 Dec;59(6):841-80
references: 19266170
references: Theor Popul Biol. 2007 Sep;72(2):274-91
references: 17588629
references: J Math Biol. 1983;18(3):233-53
references: 6663207
references: J Math Biol. 1985;21(3):347-61
references: 4031698
references: J Math Biol. 2008 Jun;56(6):827-39
references: 17989980
references: J Math Biol. 1990;28(3):257-70
references: 2332704
references: Emerg Infect Dis. 2001 May-Jun;7(3):369-74
references: 11384511
references: Science. 2000 Jan 28;287(5453):667-70
references: 10650003
references: Am J Epidemiol. 1973 Dec;98(6):453-68
references: 4767622
references: Proc Biol Sci. 1993 Jan 22;251(1330):75-81
references: 8094567
references: J Theor Biol. 1984 Oct 21;110(4):665-79
references: 6521486
references: Int J Epidemiol. 1982 Mar;11(1):5-14
references: 7085179
references: Ecol Lett. 2006 Apr;9(4):467-84
references: 16623732
references: Math Biosci. 2002 Nov-Dec;180:187-205
references: 12387923
references: Math Med Biol. 2007 Mar;24(1):1-15
references: 17317756
references: J Math Biol. 1983;17(2):179-90
references: 6886570