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Dynamics of two-strain influenza model with cross-immunity and no quarantine class. (English) Zbl 1354.34080
Summary: The question about whether a periodic solution can exists for a given epidemiological model is a complicated one and has a long history H. W. Hethcote and S. A. Levin, “Periodicity in epidemiological models”, in: Simon A. Levin (ed)., et al., Applied mathematical ecology. Berlin: Springer. 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 [J. L. Aron and I. B. Schwartz, IMA J. Math. Appl. Med. Biol. 1, 267–276 (1984; Zbl 0609.92024); Yu. A. Kuznetsov and C. Piccardi, J. Math. Biol. 32, No. 2, 109–121 (1994; Zbl 0786.92022)], or a multiple-strain model with cross-immunity and quarantine class or age-structure [M. Nuño et al., Lect. Notes Math. 1945, 349–364 (2008; Zbl 1206.92016)]. 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 M. Nuño et al. [SIAM J. Appl. Math. 65, No. 3, 964–982 (2005; Zbl 1075.92041)]. This problem is important because understanding the reasons behind periodic outbreaks of seasonal flu is an important issue in public health.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
92D30 Epidemiology
34C23 Bifurcation theory for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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