June  2020, 25(6): 2223-2243. doi: 10.3934/dcdsb.2019225

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  June 2020 Early access  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 and Continuous Dynamical Systems - B, 2020, 25 (6) : 2223-2243. doi: 10.3934/dcdsb.2019225
References:
[1]

M. Alfaro and M. Veruete, Evolutionary branching via replicator mutator equations, Journal of Dynamics and Differential Equations, (2018), 1–24. doi: 10.1007/s10884-018-9692-9.

[2]

O. BonnefonJ. Coville and G. Legendre, Concentration phenomenon in some non-local equation, Discrete Continuous Dynam. Systems - B, 22 (2017), 763-781.  doi: 10.3934/dcdsb.2017037.

[3]

J.-B. Burie, R. Djidjou-Demasse and A. Ducrot, Asymptotic and transient behaviour for a nonlocal problem arising in population genetics, European Journal of Applied Mathematics, 2018, 1–27. doi: 10.1017/S0956792518000487.

[4]

À. CalsinaS. CuadradoL. Desvillettes and G. Raoul, Asymptotics of steady states of a selection-mutation equation for small mutation rate, Proc. Math. Roy. Soc. Edinb., 143 (2013), 1123-1146.  doi: 10.1017/S0308210510001629.

[5]

À. Calsina and S. Cuadrado, Stationary solutions of a selection mutation model: The pure mutation case, Mathematical Models and Methods in Applied Sciences, 15 (2005), 1091-1117.  doi: 10.1142/S0218202505000637.

[6]

S. Cuadrado, Equilibria of a predator prey model of phenotype evolution, J. Math. Anal. Appl., 354 (2009), 286-294.  doi: 10.1016/j.jmaa.2009.01.002.

[7]

T. Day and S. Gandon, Applying population-genetic models in theoretical evolutionary epidemiology, Ecology Letters, 10 (2007), 876-888.  doi: 10.1111/j.1461-0248.2007.01091.x.

[8]

T. Day and S. R. Proulx, A general theory for the evolutionary dynamics of virulence, The American Naturalist, 163 (2004), E40–E63. doi: 10.1086/382548.

[9]

O. DiekmannP. E. JabinS. Mischler and B. Perthame, The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach, Theor. Popul. Biol., 67 (2005), 257-271.  doi: 10.1016/j.tpb.2004.12.003.

[10]

R. Djidjou-DemasseA. Ducrot and F. Fabre, Steady state concentration for a phenotypic structured problem modeling the evolutionary epidemiology of spore producing pathogens, Mathematical Models and Methods in Applied Sciences, 27 (2017), 385-426.  doi: 10.1142/S0218202517500051.

[11]

F. FabreE. RousseauL. Mailleret and B. Moury, Durable strategies to deploy plant resistance in agricultural landscapes, New Phytologist, 193 (2012), 1064-1075.  doi: 10.1111/j.1469-8137.2011.04019.x.

[12]

F. FabreE. RousseauL. Mailleret and B. Moury, Epidemiological and evolutionary management of plant resistance: optimizing the deployment of cultivar mixtures in time and space in agricultural landscapes, Evol. Appl., 8 (2015), 919-932.  doi: 10.1111/eva.12304.

[13]

S. A. GeritzJ. A. MetzÉ. Kisdi and G. Meszéna, Dynamics of adaptation and evolutionary branching, Phys. Rev. Lett., 78 (1997), 2024-2027.  doi: 10.1103/PhysRevLett.78.2024.

[14]

Q. Griette, Singular measure traveling waves in an epidemiological model with continuous phenotypes, Transactions of the American Mathematical Society, 371 (2019), 4411-4458.  doi: 10.1090/tran/7700.

[15]

G. L. IaconoF. van den Bosch and N. Paveley, The evolution of plant pathogens in response to host resistance: factors affecting the gain from deployment of qualitative and quantitative resistance, J. Theo. Biol., 304 (2012), 152-163.  doi: 10.1016/j.jtbi.2012.03.033.

[16]

C. Lannou, Variation and selection of quantitative traits in plant pathogens, Annu. Rev. Phytopathol., 50 (2012), 319-338.  doi: 10.1146/annurev-phyto-081211-173031.

[17]

S. Lion and S. Gandon, Spatial evolutionary epidemiology of spreading epidemics, Proc. R. Soc. B, 283 (2016). doi: 10.1098/rspb.2016.1170.

[18]

J. A. J. Metz, S. A. H. Geritz, G. Meszéna, F. J. A. Jacobs and J. S. van Heerwaarden, Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction, in Stochastic and Spatial Structures of Dynamical System (eds. S. J. van Strien, S. M. Verduyn Lunel), North-Holland, Amsterdam, 45 (1996), 183–231.

[19]

P. Meyer-Nieberg, Banach Lattices, Springer-Verlag Berlin Heidelberg, 1991. doi: 10.1007/978-3-642-76724-1.

[20]

S. Mirrahimi, B. Perthame, E. Bouin and P. Millien, Population formulation of adaptative meso-evolution: Theory and numerics, in The Mathematics of Darwins Legacy, Springer Basel, Basel, (2011), 159–174. doi: 10.1007/978-3-0348-0122-5_9.

[21]

S. MirrahimiB. Perthame and J. Y. Wakano, Evolution of species trait through resource competition, J. Math. Biol., 64 (2012), 1189-1223.  doi: 10.1007/s00285-011-0447-z.

[22]

S. NordmannB. Perthame and C. Taing, Dynamics of concentration in a population model structured by age and a phenotypical trait, Acta Appl. Math., 155 (2018), 197-225.  doi: 10.1007/s10440-017-0151-0.

[23]

M. Zerner, Quelques propriétés spectrales des opérateurs positifs, Journal of Functional Analysis, 72 (1987), 381-417.  doi: 10.1016/0022-1236(87)90094-2.

[24]

J. ZhanP. H. ThrallJ. PapaïL. Xie and J. J. Burdon, Playing on a pathogen's weakness: Using evolution to guide sustainable plant disease control strategies, Annu. Rev. Phytopathol., 53 (2015), 19-43.  doi: 10.1146/annurev-phyto-080614-120040.

show all references

References:
[1]

M. Alfaro and M. Veruete, Evolutionary branching via replicator mutator equations, Journal of Dynamics and Differential Equations, (2018), 1–24. doi: 10.1007/s10884-018-9692-9.

[2]

O. BonnefonJ. Coville and G. Legendre, Concentration phenomenon in some non-local equation, Discrete Continuous Dynam. Systems - B, 22 (2017), 763-781.  doi: 10.3934/dcdsb.2017037.

[3]

J.-B. Burie, R. Djidjou-Demasse and A. Ducrot, Asymptotic and transient behaviour for a nonlocal problem arising in population genetics, European Journal of Applied Mathematics, 2018, 1–27. doi: 10.1017/S0956792518000487.

[4]

À. CalsinaS. CuadradoL. Desvillettes and G. Raoul, Asymptotics of steady states of a selection-mutation equation for small mutation rate, Proc. Math. Roy. Soc. Edinb., 143 (2013), 1123-1146.  doi: 10.1017/S0308210510001629.

[5]

À. Calsina and S. Cuadrado, Stationary solutions of a selection mutation model: The pure mutation case, Mathematical Models and Methods in Applied Sciences, 15 (2005), 1091-1117.  doi: 10.1142/S0218202505000637.

[6]

S. Cuadrado, Equilibria of a predator prey model of phenotype evolution, J. Math. Anal. Appl., 354 (2009), 286-294.  doi: 10.1016/j.jmaa.2009.01.002.

[7]

T. Day and S. Gandon, Applying population-genetic models in theoretical evolutionary epidemiology, Ecology Letters, 10 (2007), 876-888.  doi: 10.1111/j.1461-0248.2007.01091.x.

[8]

T. Day and S. R. Proulx, A general theory for the evolutionary dynamics of virulence, The American Naturalist, 163 (2004), E40–E63. doi: 10.1086/382548.

[9]

O. DiekmannP. E. JabinS. Mischler and B. Perthame, The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach, Theor. Popul. Biol., 67 (2005), 257-271.  doi: 10.1016/j.tpb.2004.12.003.

[10]

R. Djidjou-DemasseA. Ducrot and F. Fabre, Steady state concentration for a phenotypic structured problem modeling the evolutionary epidemiology of spore producing pathogens, Mathematical Models and Methods in Applied Sciences, 27 (2017), 385-426.  doi: 10.1142/S0218202517500051.

[11]

F. FabreE. RousseauL. Mailleret and B. Moury, Durable strategies to deploy plant resistance in agricultural landscapes, New Phytologist, 193 (2012), 1064-1075.  doi: 10.1111/j.1469-8137.2011.04019.x.

[12]

F. FabreE. RousseauL. Mailleret and B. Moury, Epidemiological and evolutionary management of plant resistance: optimizing the deployment of cultivar mixtures in time and space in agricultural landscapes, Evol. Appl., 8 (2015), 919-932.  doi: 10.1111/eva.12304.

[13]

S. A. GeritzJ. A. MetzÉ. Kisdi and G. Meszéna, Dynamics of adaptation and evolutionary branching, Phys. Rev. Lett., 78 (1997), 2024-2027.  doi: 10.1103/PhysRevLett.78.2024.

[14]

Q. Griette, Singular measure traveling waves in an epidemiological model with continuous phenotypes, Transactions of the American Mathematical Society, 371 (2019), 4411-4458.  doi: 10.1090/tran/7700.

[15]

G. L. IaconoF. van den Bosch and N. Paveley, The evolution of plant pathogens in response to host resistance: factors affecting the gain from deployment of qualitative and quantitative resistance, J. Theo. Biol., 304 (2012), 152-163.  doi: 10.1016/j.jtbi.2012.03.033.

[16]

C. Lannou, Variation and selection of quantitative traits in plant pathogens, Annu. Rev. Phytopathol., 50 (2012), 319-338.  doi: 10.1146/annurev-phyto-081211-173031.

[17]

S. Lion and S. Gandon, Spatial evolutionary epidemiology of spreading epidemics, Proc. R. Soc. B, 283 (2016). doi: 10.1098/rspb.2016.1170.

[18]

J. A. J. Metz, S. A. H. Geritz, G. Meszéna, F. J. A. Jacobs and J. S. van Heerwaarden, Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction, in Stochastic and Spatial Structures of Dynamical System (eds. S. J. van Strien, S. M. Verduyn Lunel), North-Holland, Amsterdam, 45 (1996), 183–231.

[19]

P. Meyer-Nieberg, Banach Lattices, Springer-Verlag Berlin Heidelberg, 1991. doi: 10.1007/978-3-642-76724-1.

[20]

S. Mirrahimi, B. Perthame, E. Bouin and P. Millien, Population formulation of adaptative meso-evolution: Theory and numerics, in The Mathematics of Darwins Legacy, Springer Basel, Basel, (2011), 159–174. doi: 10.1007/978-3-0348-0122-5_9.

[21]

S. MirrahimiB. Perthame and J. Y. Wakano, Evolution of species trait through resource competition, J. Math. Biol., 64 (2012), 1189-1223.  doi: 10.1007/s00285-011-0447-z.

[22]

S. NordmannB. Perthame and C. Taing, Dynamics of concentration in a population model structured by age and a phenotypical trait, Acta Appl. Math., 155 (2018), 197-225.  doi: 10.1007/s10440-017-0151-0.

[23]

M. Zerner, Quelques propriétés spectrales des opérateurs positifs, Journal of Functional Analysis, 72 (1987), 381-417.  doi: 10.1016/0022-1236(87)90094-2.

[24]

J. ZhanP. H. ThrallJ. PapaïL. Xie and J. J. Burdon, Playing on a pathogen's weakness: Using evolution to guide sustainable plant disease control strategies, Annu. Rev. Phytopathol., 53 (2015), 19-43.  doi: 10.1146/annurev-phyto-080614-120040.

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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