September  2017, 22(7): 2879-2905. doi: 10.3934/dcdsb.2017155

Asymptotic behaviour of an age and infection age structured model for the propagation of fungal diseases in plants

1. 

UMR CNRS 5251 IMB, Université de Bordeaux, 3ter Place de la Victoire, 33076 Bordeaux, France

2. 

UMI-IRD-209 UMMISCO and LANI, UFR de Sciences Appliquées et de Technologie, Université Gaston Berger, B.P. 234 Saint-Louis, Sénégal

* Corresponding author: J.-B. Burie

Received  June 2016 Revised  March 2017 Published  May 2017

A mathematical model describing the propagation of fungal diseases in plants is proposed. The model takes into account both chronological age and age since infection. We investigate and fully characterize the large time behaviour of the solutions. Existence of a unique endemic stationary state is ensured by a threshold condition: $\mathcal R_0>1$. Then using Lyapounov arguments, we prove that if $\mathcal R_0 ≤ 1$ the disease free stationary state is globally stable while when $\mathcal R_0>1$, the unique endemic stationary state is globally stable with respect to a suitable set of initial data.

Citation: Jean-Baptiste Burie, Arnaud Ducrot, Abdoul Aziz Mbengue. Asymptotic behaviour of an age and infection age structured model for the propagation of fungal diseases in plants. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2879-2905. doi: 10.3934/dcdsb.2017155
References:
[1]

W. Arendt, Resolvent positive operators and integrated semigroups, Proc. London Math. Soc., 54 (1987), 321-349.  doi: 10.1112/plms/s3-54.2.321.  Google Scholar

[2]

W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math., 59 (1987), 327-352.  doi: 10.1007/BF02774144.  Google Scholar

[3]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems Monographs in Mathematics, 96, Birkhauser, 2001. doi: 10.1007/978-3-0348-5075-9.  Google Scholar

[4]

J. B. BurieM. Langlais and A. Calonnec, Switching from a mechanistic model to a continuous model to study at different scales the effect of vine growth on the dynamic of a powdery mildew epidemic, Annals of Botany, 107 (2011), 885-895.  doi: 10.1093/aob/mcq233.  Google Scholar

[5]

A. CalonnecP. CartolaroJ. M. NaulinD. Bailey and M. Langlais, A host-pathogen simulation model: Powdery mildew of grapevine, Plant Pathology, 57 (2008), 493-508.  doi: 10.1111/j.1365-3059.2007.01783.x.  Google Scholar

[6]

G. Da Prato and P. Grisvard, Somme d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Appl., 54 (1975), 305-387.   Google Scholar

[7]

G. Da Prato and E. Sinestrari, Differential operators with nondense domain, Ann. Sc. Norm. Pisa, 14 (1987), 285-344.   Google Scholar

[8]

K. Dietz and D. Schenzle, Proportionate mixing models for age-dependent infection transmission, J. Math. Biol., 22 (1985), 117-120.  doi: 10.1007/BF00276550.  Google Scholar

[9]

A. Ducrot and P. Magal, A center manifold for second order semi-linear differential equations on the real line and application to the existence of wave trains for the Gurtin-McCamy equation, submitted. Google Scholar

[10]

K. -J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.  Google Scholar

[11]

W. E. Fitzgibbon and M. Langlais, Weakly coupled hyperbolic systems modeling the circulation of FeLV in structured feline populations, Math. Biosci., 165 (2000), 79-95.  doi: 10.1016/S0025-5564(00)00011-0.  Google Scholar

[12]

M. A. Flaishman and P. E. Kolattukudy, Timing of fungal invasion using host's ripening hormone as a signal, PNAS, 91 (1994), 6579-6583.  doi: 10.1073/pnas.91.14.6579.  Google Scholar

[13]

D. GadouryR. SeemA. Ficke and W. Wilcox, Ontogenic resistance to powdery mildew in grape berries, Phytopathology, 93 (2003), 547-555.  doi: 10.1094/PHYTO.2003.93.5.547.  Google Scholar

[14]

G. Gripenberg, On a nonlinear integral equation modelling an epidemic in an age-structured population, J. Reine Angew. Math., 341 (1983), 54-67.  doi: 10.1515/crll.1983.341.54.  Google Scholar

[15]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical surveys and monographs, 25 American Mathematical Society, Providence, RI, 1988.  Google Scholar

[16]

J. K. Hale and P. Waltman, Persistence in infinite dimensional systems, SIAM J. Math. Anal., 20 (1989), 288-395.  doi: 10.1137/0520025.  Google Scholar

[17]

F. Hoppensteadt, An age dependent epidemic model, J. Franklin Inst., 197 (1974), 325-333.   Google Scholar

[18]

R. HorbachA. R. Navarro-QuesadaW. Knogge and H. B. Deising, When and how to kill a plant cell: Infection strategies of plant pathogenic fungi, J. Plant Physiol., 168 (2011), 51-62.  doi: 10.1016/j.jplph.2010.06.014.  Google Scholar

[19]

M. Iannelli, Mathematical Theory of Age-structured Population Dynamics Applied Mathematics Monographs CNR Giadini Editori e Stampatori, Pisa, 1994. Google Scholar

[20]

H. Inaba, Endemic threshold results in an age-duration-structured population model for HIV infection, Math. Biosci., 201 (2006), 15-47.  doi: 10.1016/j.mbs.2005.12.017.  Google Scholar

[21]

H. Inaba and H. Nishiura, The basic reproduction number of an infectious disease in a stable population: The impact of population growth rate on the eradication threshold, Math. Model. Nat. Phenom., 3 (2008), 194-228.  doi: 10.1051/mmnp:2008050.  Google Scholar

[22]

H. Kellermann and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180.  doi: 10.1016/0022-1236(89)90116-X.  Google Scholar

[23]

W. D. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. Ser. A, 115 (1927), 700-721.   Google Scholar

[24]

L. Madden, G. Hughes and F. Van Den Bosch, Study of Plant Disease Epidemics American Phytopathological Society, 2007. Google Scholar

[25]

P. Magal, Compact attractors for time-periodic age structured population models, Electron. J. Differential Equations, 65 (2001), 1-35.   Google Scholar

[26]

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.  Google Scholar

[27]

P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to hopf bifurcation in age structured models Mem. Amer. Math. Soc. 202 (2009), vi+71 pp. doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar

[28]

P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. in Differential Equations, 14 (2009), 1041-1084.   Google Scholar

[29]

P. Magal and C. C. McCluskey, Two group infection age model: An application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095.  doi: 10.1137/120882056.  Google Scholar

[30]

P. MagalC. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.  doi: 10.1080/00036810903208122.  Google Scholar

[31]

R. Nagel and E. Sinestrari, Extrapolation spaces and minimal regularity for evolution equations, J. Evol. Equ., 6 (2006), 287-303.  doi: 10.1007/s00028-006-0246-y.  Google Scholar

[32]

F. Neubrander, Integrated semigroups and their application to the abstract Cauchy problem, Pac. J. Math., 135 (1988), 111-155.  doi: 10.2140/pjm.1988.135.111.  Google Scholar

[33]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[34]

K. SegarraM. J. Jeger and F. van den Bosch, Epidemic dynamics and patterns of plant diseases, Phytopathology, 91 (2001), 1001-1010.  doi: 10.1094/PHYTO.2001.91.10.1001.  Google Scholar

[35]

H. L. Smith and H. Thieme, Dynamical Systems and Population Persistence American Mathematical Society, 2011.  Google Scholar

[36]

H. R. Thieme, "Integrated Semigroups" and Integrated solutions to Abstract Cauchy Problems, J. Math. Anal. Appl., 152 (1990), 416-447.  doi: 10.1016/0022-247X(90)90074-P.  Google Scholar

[37]

H. R. Thieme, Analysis of age-structured population models with an additional structure, Mathematical population dynamics (New Brunswick, NJ, 1989), Lecture Notes in Pure and Appl. Math., 131 (1991), 115-126.   Google Scholar

[38]

H. R. Thieme, On commutative sums of generators, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 421-451.   Google Scholar

[39]

H. R. Thieme, Quasi-compact semigroups via bounded perturbation, in Series in Mathematical Biology and Medecine, (World Scientific, Singapore, London), 6 (1997), 691-711.   Google Scholar

[40]

H. R. Thieme, 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.  Google Scholar

[41]

J. E. Vanderplank, Plant Diseases: Epidemics and Control Academic Press, 1963. Google Scholar

[42]

G. F. Webb, Theory of nonlinear age-dependent population dynamics, Monographs and Textbooks in Pure and Applied Mathematics, 89. Marcel Dekker, Inc. , New York, 1985.  Google Scholar

[43]

G. F. Webb, Population models structured by age, size, and spatial position, Structured Population Models in Biology and Epidemiology, Lecture Notes in Math., 1936 (2008), 1-49.  doi: 10.1007/978-3-540-78273-5_1.  Google Scholar

[44]

Y. ZhouB. Song and Z. Ma, The global stability analysis for an sis model with age and infection age structures, Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory, The IMA Volumes in Mathematics and its Applications, 126 (2002), 313-335.  doi: 10.1007/978-1-4613-0065-6_18.  Google Scholar

show all references

References:
[1]

W. Arendt, Resolvent positive operators and integrated semigroups, Proc. London Math. Soc., 54 (1987), 321-349.  doi: 10.1112/plms/s3-54.2.321.  Google Scholar

[2]

W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math., 59 (1987), 327-352.  doi: 10.1007/BF02774144.  Google Scholar

[3]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems Monographs in Mathematics, 96, Birkhauser, 2001. doi: 10.1007/978-3-0348-5075-9.  Google Scholar

[4]

J. B. BurieM. Langlais and A. Calonnec, Switching from a mechanistic model to a continuous model to study at different scales the effect of vine growth on the dynamic of a powdery mildew epidemic, Annals of Botany, 107 (2011), 885-895.  doi: 10.1093/aob/mcq233.  Google Scholar

[5]

A. CalonnecP. CartolaroJ. M. NaulinD. Bailey and M. Langlais, A host-pathogen simulation model: Powdery mildew of grapevine, Plant Pathology, 57 (2008), 493-508.  doi: 10.1111/j.1365-3059.2007.01783.x.  Google Scholar

[6]

G. Da Prato and P. Grisvard, Somme d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Appl., 54 (1975), 305-387.   Google Scholar

[7]

G. Da Prato and E. Sinestrari, Differential operators with nondense domain, Ann. Sc. Norm. Pisa, 14 (1987), 285-344.   Google Scholar

[8]

K. Dietz and D. Schenzle, Proportionate mixing models for age-dependent infection transmission, J. Math. Biol., 22 (1985), 117-120.  doi: 10.1007/BF00276550.  Google Scholar

[9]

A. Ducrot and P. Magal, A center manifold for second order semi-linear differential equations on the real line and application to the existence of wave trains for the Gurtin-McCamy equation, submitted. Google Scholar

[10]

K. -J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.  Google Scholar

[11]

W. E. Fitzgibbon and M. Langlais, Weakly coupled hyperbolic systems modeling the circulation of FeLV in structured feline populations, Math. Biosci., 165 (2000), 79-95.  doi: 10.1016/S0025-5564(00)00011-0.  Google Scholar

[12]

M. A. Flaishman and P. E. Kolattukudy, Timing of fungal invasion using host's ripening hormone as a signal, PNAS, 91 (1994), 6579-6583.  doi: 10.1073/pnas.91.14.6579.  Google Scholar

[13]

D. GadouryR. SeemA. Ficke and W. Wilcox, Ontogenic resistance to powdery mildew in grape berries, Phytopathology, 93 (2003), 547-555.  doi: 10.1094/PHYTO.2003.93.5.547.  Google Scholar

[14]

G. Gripenberg, On a nonlinear integral equation modelling an epidemic in an age-structured population, J. Reine Angew. Math., 341 (1983), 54-67.  doi: 10.1515/crll.1983.341.54.  Google Scholar

[15]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical surveys and monographs, 25 American Mathematical Society, Providence, RI, 1988.  Google Scholar

[16]

J. K. Hale and P. Waltman, Persistence in infinite dimensional systems, SIAM J. Math. Anal., 20 (1989), 288-395.  doi: 10.1137/0520025.  Google Scholar

[17]

F. Hoppensteadt, An age dependent epidemic model, J. Franklin Inst., 197 (1974), 325-333.   Google Scholar

[18]

R. HorbachA. R. Navarro-QuesadaW. Knogge and H. B. Deising, When and how to kill a plant cell: Infection strategies of plant pathogenic fungi, J. Plant Physiol., 168 (2011), 51-62.  doi: 10.1016/j.jplph.2010.06.014.  Google Scholar

[19]

M. Iannelli, Mathematical Theory of Age-structured Population Dynamics Applied Mathematics Monographs CNR Giadini Editori e Stampatori, Pisa, 1994. Google Scholar

[20]

H. Inaba, Endemic threshold results in an age-duration-structured population model for HIV infection, Math. Biosci., 201 (2006), 15-47.  doi: 10.1016/j.mbs.2005.12.017.  Google Scholar

[21]

H. Inaba and H. Nishiura, The basic reproduction number of an infectious disease in a stable population: The impact of population growth rate on the eradication threshold, Math. Model. Nat. Phenom., 3 (2008), 194-228.  doi: 10.1051/mmnp:2008050.  Google Scholar

[22]

H. Kellermann and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180.  doi: 10.1016/0022-1236(89)90116-X.  Google Scholar

[23]

W. D. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. Ser. A, 115 (1927), 700-721.   Google Scholar

[24]

L. Madden, G. Hughes and F. Van Den Bosch, Study of Plant Disease Epidemics American Phytopathological Society, 2007. Google Scholar

[25]

P. Magal, Compact attractors for time-periodic age structured population models, Electron. J. Differential Equations, 65 (2001), 1-35.   Google Scholar

[26]

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.  Google Scholar

[27]

P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to hopf bifurcation in age structured models Mem. Amer. Math. Soc. 202 (2009), vi+71 pp. doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar

[28]

P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. in Differential Equations, 14 (2009), 1041-1084.   Google Scholar

[29]

P. Magal and C. C. McCluskey, Two group infection age model: An application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095.  doi: 10.1137/120882056.  Google Scholar

[30]

P. MagalC. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.  doi: 10.1080/00036810903208122.  Google Scholar

[31]

R. Nagel and E. Sinestrari, Extrapolation spaces and minimal regularity for evolution equations, J. Evol. Equ., 6 (2006), 287-303.  doi: 10.1007/s00028-006-0246-y.  Google Scholar

[32]

F. Neubrander, Integrated semigroups and their application to the abstract Cauchy problem, Pac. J. Math., 135 (1988), 111-155.  doi: 10.2140/pjm.1988.135.111.  Google Scholar

[33]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[34]

K. SegarraM. J. Jeger and F. van den Bosch, Epidemic dynamics and patterns of plant diseases, Phytopathology, 91 (2001), 1001-1010.  doi: 10.1094/PHYTO.2001.91.10.1001.  Google Scholar

[35]

H. L. Smith and H. Thieme, Dynamical Systems and Population Persistence American Mathematical Society, 2011.  Google Scholar

[36]

H. R. Thieme, "Integrated Semigroups" and Integrated solutions to Abstract Cauchy Problems, J. Math. Anal. Appl., 152 (1990), 416-447.  doi: 10.1016/0022-247X(90)90074-P.  Google Scholar

[37]

H. R. Thieme, Analysis of age-structured population models with an additional structure, Mathematical population dynamics (New Brunswick, NJ, 1989), Lecture Notes in Pure and Appl. Math., 131 (1991), 115-126.   Google Scholar

[38]

H. R. Thieme, On commutative sums of generators, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 421-451.   Google Scholar

[39]

H. R. Thieme, Quasi-compact semigroups via bounded perturbation, in Series in Mathematical Biology and Medecine, (World Scientific, Singapore, London), 6 (1997), 691-711.   Google Scholar

[40]

H. R. Thieme, 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.  Google Scholar

[41]

J. E. Vanderplank, Plant Diseases: Epidemics and Control Academic Press, 1963. Google Scholar

[42]

G. F. Webb, Theory of nonlinear age-dependent population dynamics, Monographs and Textbooks in Pure and Applied Mathematics, 89. Marcel Dekker, Inc. , New York, 1985.  Google Scholar

[43]

G. F. Webb, Population models structured by age, size, and spatial position, Structured Population Models in Biology and Epidemiology, Lecture Notes in Math., 1936 (2008), 1-49.  doi: 10.1007/978-3-540-78273-5_1.  Google Scholar

[44]

Y. ZhouB. Song and Z. Ma, The global stability analysis for an sis model with age and infection age structures, Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory, The IMA Volumes in Mathematics and its Applications, 126 (2002), 313-335.  doi: 10.1007/978-1-4613-0065-6_18.  Google Scholar

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