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January  2019, 18(1): 15-32. doi: 10.3934/cpaa.2019002

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

 UMR CNRS 6249 Chrono-environnement, 16 route de Gray, F-25030 Besançon cedex FRANCE, University Bourgogne Franche-Comté

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

Received  June 2017 Revised  January 2018 Published  August 2018

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.

Citation: Antoine Perasso. Global stability and uniform persistence for an infection load-structured SI model with exponential growth velocity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 15-32. doi: 10.3934/cpaa.2019002
##### References:

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##### References:
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 ${\bar i} < +\infty$
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 $\Phi$
 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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