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October  2021, 41(10): 4959-4985. doi: 10.3934/dcds.2021064

Asymptotic speed of spread for a nonlocal evolutionary-epidemic system

Lara Abi Rizk 1,2, , Jean-Baptiste Burie 1,2,, and  Arnaud Ducrot 3,

1. 

Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France
2. 

CNRS, IMB, UMR 5251, F-33400 Talence, France
3. 

Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, 76600 Le Havre, France

* Corresponding author: Jean-Baptiste Burie

Received  October 2020 Revised  February 2021 Published  October 2021 Early access  March 2021

We investigate spreading properties of solutions for a spatially distributed system of equations modelling the evolutionary epidemiology of plant-pathogen interactions. In this work the mutation process is described using a non-local convolution operator in the phenotype space. Initially equipped with a localized amount of infection, we prove that spreading occurs with a definite spreading speed that coincides with the minimal speed of the travelling wave solutions discussed in [1]. Moreover, the solution of the Cauchy problem asymptotically converges to some specific function for which the moving frame variable and the phenotype one are separated.

Keywords: Population dynamics, long time behaviour, epidemiology, evolution, spreading speed, minimal speed, nonlocal interactions, diffusive epidemic system.
Mathematics Subject Classification: Primary: 35B40, 35R09, 35Q92, 92D10.
Citation: Lara Abi Rizk, Jean-Baptiste Burie, Arnaud Ducrot. Asymptotic speed of spread for a nonlocal evolutionary-epidemic system. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4959-4985. doi: 10.3934/dcds.2021064
References:
[1]

L. Abi Rizk, J. -B. Burie and A. Ducrot, Travelling wave solutions for a non-local evolutionary-epidemic system, J. Differential Equations, 267 (2019), 1467-1509. doi: 10.1016/j. jde. 2019.02.012.

[2]

O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differential Equations, 33 (1979), 58-73. doi: 10.1016/0022-0396(79)90080-9.

[3]

R. Djidjou-Demasse, A. Ducrot and F. Fabre, Steady state concentration for a phenotypic structured problem modeling the evolutionary epidemiology of spore producing pathogens, Math. Models Methods Appl. Sci., 27 (2017), 385-426. doi: 10.1142/S0218202517500051.

[4]

A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations, 260 (2016), 8316-8357. doi: 10.1016/j. jde. 2016.02.023.

[5]

A Ducrot, T. Giletti, J.-S. Guo and M. Shimojo, Asymptotic spreading speeds for a predator-prey system with two predators and one prey, Nonlinearity, 34 (2021), 669-705.  doi: 10.1088/1361-6544/abd289.

[6]

A. Ducrot, T. Giletti and H. Matano, Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 137, 34pp. doi: 10.1007/s00526-019-1576-2.

[7]

A. Ducrot, J. -S. Guo, G. Lin and S. Pan, The spreading speed and the minimal wave speed of a predator-prey system with nonlocal dispersal, Z. Angew. Math. Phys., 70 (2019), Paper No. 146, 25pp. doi: 10.1007/s00033-019-1188-x.

[8]

L. Girardin, Non-cooperative Fisher-KPP systems: Asymptotic behavior of traveling waves, Math. Models Methods Appl. Sci., 28 (2018), 1067-1104. doi: 10.1142/S0218202518500288.

[9]

—————, Non-cooperative Fisher-KPP systems: Traveling waves and long-time behavior, Nonlinearity, 31 (2018), 108-164. doi: 10.1088/1361-6544/aa8ca7.

[10]

G. L. Iacono, F. 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.

[11]

Q. Griette and G. Raoul, Existence and qualitative properties of travelling waves for an epidemiological model with mutations, J. Differential Equations, 260 (2016), 7115-7151. doi: 10.1016/j. jde. 2016.01.022.

[12]

J. -S. Guo, A. A. L. Poh and M. Shimojo, The spreading speed of an SIR epidemic model with nonlocal dispersal, Asymptotic Analysis, 120 (2020), 163-174. doi: 10.3233/ASY-191584.

[13]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025.

[14]

W. -T. Li, W. -B. Xu and L. Zhang, Traveling waves and entire solutions for an epidemic model with asymmetric dispersal, Discrete Contin. Dyn. Syst., 37 (2017), 2483-2512. doi: 10.3934/dcds. 2017107.

[15]

F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM J. Appl. Math., 65 (2005), 1305-1327. doi: 10.1137/S0036139904440400.

[16]

P. Magal and X. -Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173.

[17]

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

[18]

A. Morris, L. Borger and E. Crooks, Individual variability in dispersal and invasion speed, Mathematics, 7 (2019), p. 795.

[19]

S. Pan, Asymptotic spreading in a Lotka-Volterra predator-prey system, J. Math. Anal. Appl., 407 (2013), 230-236. doi: 10.1016/j. jmaa. 2013.05.031.

[20]

L. Rimbaud, J. Papaïx, J. -F. Rey, L. G. Barrett and P. H. Thrall, Assessing the durability and efficiency of landscape-based strategies to deploy plant resistance to pathogens, PLOS Computational Biology, 14 (2018), 1-33.

[21]

S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in Mathematics for life science and medicine, Biol. Med. Phys. Biomed. Eng., Springer, Berlin, 2007, 97-122.

[22]

H. H. Schaefer, Banach Lattices and Positive Operators, Die Grundlehren der mathematischen Wissenschaften, Band 215. Springer-Verlag, New York-Heidelberg, 1974.

[23]

H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci., 166 (2000), 173-201. doi: 10.1016/S0025-5564(00)00018-3.

[24]

—————, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, J. Differential Equations, 250 (2011), 3772-3801. doi: 10.1016/j. jde. 2011.01.007.

[25]

Z. -C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237-261. doi: 10.1098/rspa. 2009.0377.

[26]

C. Wu, The spreading speed for a predator-prey model with one predator and two preys, Appl. Math. Lett., 91 (2019), 9-14. doi: 10.1016/j. aml. 2018.11.022.

[27]

G. -B. Zhang and X. -Q. Zhao, Propagation phenomena for a two species Lotka-Volterra strong competition system with nonlocal dispersal, Calc. Var., 59 (2019), Paper No. 10, 34 pp. doi: 10.1007/s00526-019-1662-5.

[28]

M. Zhao, W. Li and Y. Du, The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries, Commun. Pure Appl. Anal., 19 (2020), 4599-4620. doi: 10.3934/cpaa. 2020208.

show all references

References:
[1]

L. Abi Rizk, J. -B. Burie and A. Ducrot, Travelling wave solutions for a non-local evolutionary-epidemic system, J. Differential Equations, 267 (2019), 1467-1509. doi: 10.1016/j. jde. 2019.02.012.

[2]

O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differential Equations, 33 (1979), 58-73. doi: 10.1016/0022-0396(79)90080-9.

[3]

R. Djidjou-Demasse, A. Ducrot and F. Fabre, Steady state concentration for a phenotypic structured problem modeling the evolutionary epidemiology of spore producing pathogens, Math. Models Methods Appl. Sci., 27 (2017), 385-426. doi: 10.1142/S0218202517500051.

[4]

A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations, 260 (2016), 8316-8357. doi: 10.1016/j. jde. 2016.02.023.

[5]

A Ducrot, T. Giletti, J.-S. Guo and M. Shimojo, Asymptotic spreading speeds for a predator-prey system with two predators and one prey, Nonlinearity, 34 (2021), 669-705.  doi: 10.1088/1361-6544/abd289.

[6]

A. Ducrot, T. Giletti and H. Matano, Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 137, 34pp. doi: 10.1007/s00526-019-1576-2.

[7]

A. Ducrot, J. -S. Guo, G. Lin and S. Pan, The spreading speed and the minimal wave speed of a predator-prey system with nonlocal dispersal, Z. Angew. Math. Phys., 70 (2019), Paper No. 146, 25pp. doi: 10.1007/s00033-019-1188-x.

[8]

L. Girardin, Non-cooperative Fisher-KPP systems: Asymptotic behavior of traveling waves, Math. Models Methods Appl. Sci., 28 (2018), 1067-1104. doi: 10.1142/S0218202518500288.

[9]

—————, Non-cooperative Fisher-KPP systems: Traveling waves and long-time behavior, Nonlinearity, 31 (2018), 108-164. doi: 10.1088/1361-6544/aa8ca7.

[10]

G. L. Iacono, F. 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.

[11]

Q. Griette and G. Raoul, Existence and qualitative properties of travelling waves for an epidemiological model with mutations, J. Differential Equations, 260 (2016), 7115-7151. doi: 10.1016/j. jde. 2016.01.022.

[12]

J. -S. Guo, A. A. L. Poh and M. Shimojo, The spreading speed of an SIR epidemic model with nonlocal dispersal, Asymptotic Analysis, 120 (2020), 163-174. doi: 10.3233/ASY-191584.

[13]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025.

[14]

W. -T. Li, W. -B. Xu and L. Zhang, Traveling waves and entire solutions for an epidemic model with asymmetric dispersal, Discrete Contin. Dyn. Syst., 37 (2017), 2483-2512. doi: 10.3934/dcds. 2017107.

[15]

F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM J. Appl. Math., 65 (2005), 1305-1327. doi: 10.1137/S0036139904440400.

[16]

P. Magal and X. -Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173.

[17]

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

[18]

A. Morris, L. Borger and E. Crooks, Individual variability in dispersal and invasion speed, Mathematics, 7 (2019), p. 795.

[19]

S. Pan, Asymptotic spreading in a Lotka-Volterra predator-prey system, J. Math. Anal. Appl., 407 (2013), 230-236. doi: 10.1016/j. jmaa. 2013.05.031.

[20]

L. Rimbaud, J. Papaïx, J. -F. Rey, L. G. Barrett and P. H. Thrall, Assessing the durability and efficiency of landscape-based strategies to deploy plant resistance to pathogens, PLOS Computational Biology, 14 (2018), 1-33.

[21]

S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in Mathematics for life science and medicine, Biol. Med. Phys. Biomed. Eng., Springer, Berlin, 2007, 97-122.

[22]

H. H. Schaefer, Banach Lattices and Positive Operators, Die Grundlehren der mathematischen Wissenschaften, Band 215. Springer-Verlag, New York-Heidelberg, 1974.

[23]

H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci., 166 (2000), 173-201. doi: 10.1016/S0025-5564(00)00018-3.

[24]

—————, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, J. Differential Equations, 250 (2011), 3772-3801. doi: 10.1016/j. jde. 2011.01.007.

[25]

Z. -C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237-261. doi: 10.1098/rspa. 2009.0377.

[26]

C. Wu, The spreading speed for a predator-prey model with one predator and two preys, Appl. Math. Lett., 91 (2019), 9-14. doi: 10.1016/j. aml. 2018.11.022.

[27]

G. -B. Zhang and X. -Q. Zhao, Propagation phenomena for a two species Lotka-Volterra strong competition system with nonlocal dispersal, Calc. Var., 59 (2019), Paper No. 10, 34 pp. doi: 10.1007/s00526-019-1662-5.

[28]

M. Zhao, W. Li and Y. Du, The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries, Commun. Pure Appl. Anal., 19 (2020), 4599-4620. doi: 10.3934/cpaa. 2020208.

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