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Global stability and uniform persistence for an infection load-structured SI model with exponential growth velocity

The author would like to thank the reviewers whose remarks and suggestions greatly improved the manuscript

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  • In this article is perfomed a global stability analysis of an infection load-structured epidemic model using tools of dynamical systems theory. An explicit Duhamel formulation of the semiflow allows us to prove the existence of a compact attractor for the trajectories of the system. Then, according to the sharp threshold $\mathcal R_0$, the basic reproduction number of the disease, we make explicit the basins of attractions of the equilibria of the system and prove their global stability with respect to these basins, the attractivness property being obtained using infinite dimensional Lyapunov functions.

    Mathematics Subject Classification: 34D23, 34D45, 35B40, 37C75, 92D30.

    Citation:

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  • Figure 1.  Three examples of shapes of function $\beta$ - left : $i_0 < \underline i $ and ${\bar i} = +\infty$; right : $i_0 < \underline i $ and ${\bar i} < +\infty$; down : $i_0 = \underline i $ and ${\bar i} < +\infty$

    Table 1.  Parameters involved in the model

    Parameter definition symbol
    recruitment flux $ \gamma $
    minimal infection load $i_0$
    basic mortality rate $\mu _0$
    disease mortality rate $\mu$
    horizontal transmission rate $\beta$
    infection load velocity $\nu$
    infection load distribution at contamination $ \Phi$
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