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  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, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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