# American Institute of Mathematical Sciences

## Slow convergence to equilibrium for an evolutionary epidemiology integro-differential system

 1 Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France 2 MIVEGEC, IRD, CNRS, Univ. Montpellier, Montpellier, France 3 Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, 76600 Le Havre, France

* Corresponding author: Jean-Baptiste Burie

Received  January 2019 Revised  May 2019 Published  September 2019

This work is devoted to the study of an integro-differential system of equations modelling the genetic adaptation of a pathogen by taking into account both mutation and selection processes. Using the variance of the dispersion in the phenotype trait space as a small parameter we provide a complete picture of the dynamical behaviour of the solutions of the problem. We show that the dynamics exhibits two main and long regimes – those durations are estimated – before the solution finally reaches its long time configuration, the endemic equilibrium. The analysis provided in this work rigorously explains and justifies the complex behaviour observed through numerical simulations of the system.

Citation: Jean-Baptiste Burie, Ramsès Djidjou-Demasse, Arnaud Ducrot. Slow convergence to equilibrium for an evolutionary epidemiology integro-differential system. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019225
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##### References:
Fitness function $\Psi$ and density of infected population at time $t = 0$. (Left) Time evolution of the infected population $\sqrt{\varepsilon} v(t, x)$ for $\varepsilon = 0.02$. (Right)
Slow convergence of the infected population at $x = x_1$ and $x = x_2$ towards the asymptotic configuration for $\varepsilon = 0.02\times2^{1/3}$ (Left) and $\varepsilon = 0.02$ (Right)
Linear dependence in log-log scale with respect to $\varepsilon$ of the value of the "coexistence time" $t_{\varepsilon, c}$ for which $v(t_{\varepsilon, c}, x_1) = v(t_{\varepsilon, c}, x_2)$. The slope of the corresponding line is approximatively $-3.4$
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