Global dynamics of an age-structured model with relapse
Density dependent replicator-mutator models in directed evolution
June 2020, 25(6): 2223-2243. doi: 10.3934/dcdsb.2019225
Slow convergence to equilibrium for an evolutionary epidemiology integro-differential system
Jean-Baptiste Burie 1,, , Ramsès Djidjou-Demasse 2, and Arnaud Ducrot 3,
Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France
MIVEGEC, IRD, CNRS, Univ. Montpellier, Montpellier, France
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.
Keywords: Nonlocal equation, asymptotic behaviour, transient behaviour, population genetics.
Mathematics Subject Classification: Primary: 35B40, 35R09, 35Q92; Secondary: 92D10.
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, 2020, 25 (6) : 2223-2243. doi: 10.3934/dcdsb.2019225
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Figure 1. 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)
Figure 2. 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)
Figure 3. 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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Jean-Baptiste Burie Ramsès Djidjou-Demasse Arnaud Ducrot