title: Dynamics of two-strain influenza model with cross-immunity and no quarantine class.
abstract: The question about whether a periodic solution can exists for a given epidemiological model is a complicated one and has a long history (Hethcote and Levin, Applied math. ecology, biomathematics, vol 18. Springer, Berlin, pp 193-211, 1989). For influenza models, it is well known that a periodic solution can exists for a single-strain model with periodic contact rate (Aron and Schwartz, J Math Biol 110:665-679, 1984; Kuznetsov and Piccardi, J Math Biol 32:109-121, 1994), or a multiple-strain model with cross-immunity and quarantine class or age-structure (Nuño et al., Mathematical epidemiology. Lecture notes in mathematics, vol 1945. Springer, Berlin, 2008, chapter 13). In this paper, we prove the local asymptotic stability of the interior steady-state of a two-strain influenza model with sufficiently close cross-immunity and no quarantine class or age-structure. We also show that if the cross-immunity between two strains are far apart; then it is possible for the interior steady-state to lose its stability and bifurcation of periodic solutions can occur. Our results extend those obtained by Nuño et.al. (SIAM J Appl Math 65:964-982, 2005). This problem is important because understanding the reasons behind periodic outbreaks of seasonal flu is an important issue in public health.
authors.affiliation: Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong.
authors.forename: K W
authors.initials: KW
authors.lastname: Chung
authors.affiliation: Department of Mathematical Sciences, WPI, Worcester, MA, 01609, USA. rlui@wpi.edu.
authors.forename: Roger
authors.initials: R
authors.lastname: Lui
mesh_terms: D056738:Cross Protection / Q000276:immunology*; D004196:Disease Outbreaks / Q000517:prevention & control / Q000706:statistics & numerical data; D004812:Epidemiologic Methods; D006801:Humans; D009980:Influenza A virus / Q000145:classification; D007251:Influenza, Human / Q000453:epidemiology* / Q000821:virology*; D008954:Models, Biological; D011790:Quarantine
keywords: Coexistence; Cross-immunity; Hopf bifurcation; Influenza; Local stability; Periodic solution; Two-strain model