Dynamics of two-strain influenza model with cross-immunity and no quarantine class.

*(English)*Zbl 1354.34080Summary: 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 |

##### Keywords:

influenza; two-strain model; cross-immunity; coexistence; local stability; Hopf bifurcation; periodic solution
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\textit{K. W. Chung} and \textit{R. Lui}, J. Math. Biol. 73, No. 6--7, 1467--1489 (2016; Zbl 1354.34080)

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##### References:

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