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J Math Biol. 2016 Apr 8. [Epub ahead of print]
Dynamics of two-strain influenza model with cross-immunity and no quarantine class.
Chung KW1, Lui R2.
Author information
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 (Nu?o 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 Nu?o 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.
KEYWORDS:
Coexistence; Cross-immunity; Hopf bifurcation; Influenza; Local stability; Periodic solution; Two-strain model
PMID: 27059490 [PubMed - as supplied by publisher]
Dynamics of two-strain influenza model with cross-immunity and no quarantine class.
Chung KW1, Lui R2.
Author information
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 (Nu?o 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 Nu?o 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.
KEYWORDS:
Coexistence; Cross-immunity; Hopf bifurcation; Influenza; Local stability; Periodic solution; Two-strain model
PMID: 27059490 [PubMed - as supplied by publisher]