Mathematical Biosciences and Engineering, 2014, 11(4): 929-945. doi: 10.3934/mbe.2014.11.929

Primary: 92D30, 35Q92; Secondary: 45P05.

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$R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission

1. Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914
2. Dipartimento di Mathematica, Università di Trento, 38050 Povo (Trento)

In this paper, we study an age-structured SIS epidemic model with periodicity and vertical transmission. We show that the spectral radius of the Fréchet derivative of a nonlinear integral operator plays the role of a threshold value for the global behavior of the model, that is, if the value is less than unity, then the disease-free steady state of the model is globally asymptotically stable, while if the value is greater than unity, then the model has a unique globally asymptotically stable endemic (nontrivial) periodic solution. We also show that the value can coincide with the well-know epidemiological threshold value, the basic reproduction number $\mathcal{R}_0$.
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