# American Institute of Mathematical Sciences

2015, 12(3): 473-490. doi: 10.3934/mbe.2015.12.473

## The effect of time delay in plant--pathogen interactions with host demography

 1 Department of Mathematics and Applications, University of Naples Federico II, via Cintia, I-80126 Naples 2 Department of Mathematics, University of Portsmouth, Portsmouth, PO1 3HF, United Kingdom

Received  April 2014 Revised  November 2014 Published  January 2015

Botanical epidemic models are very important tools to study invasion, persistence and control of diseases. It is well known that limitations arise from considering constant infection rates. We replace this hypothesis in the framework of delay differential equations by proposing a delayed epidemic model for plant--pathogen interactions with host demography. Sufficient conditions for the global stability of the pathogen-free equilibrium and the permanence of the system are among the results obtained through qualitative analysis. We also show that the delay can cause stability switches of the coexistence equilibrium. In the undelayed case, we prove that the onset of oscillations may occur through Hopf bifurcation.
Citation: Bruno Buonomo, Marianna Cerasuolo. The effect of time delay in plant--pathogen interactions with host demography. Mathematical Biosciences & Engineering, 2015, 12 (3) : 473-490. doi: 10.3934/mbe.2015.12.473
##### References:
 [1] R. M. Anderson and R. M. May, Population biology of infectious diseases: Part I,, Nature, 280 (1979), 361. doi: 10.1038/280361a0. Google Scholar [2] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control,, Oxford University Press, (1991). Google Scholar [3] E. Beretta and D. Breda, An SEIR epidemic model with constant latency time and infectious period,, Math. Biosci. Eng., 8 (2011), 931. doi: 10.3934/mbe.2011.8.931. Google Scholar [4] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters,, SIAM J. Math. Anal., 33 (2002), 1144. doi: 10.1137/S0036141000376086. Google Scholar [5] E. Beretta and Y. Takeuchi, Global stability of an SIR epidemic model with time delays,, J. Math. Biol., 33 (1995), 250. doi: 10.1007/BF00169563. Google Scholar [6] G. Birkhoff and G. Rota, Ordinary Differential Equations,, John Wiley and Sons, (1982). Google Scholar [7] B. Buonomo and M. Cerasuolo, Stability and bifurcation in plant-pathogens interactions,, Appl. Math. Comput., 232 (2014), 858. doi: 10.1016/j.amc.2014.01.127. Google Scholar [8] V. Capasso, Mathematical Structures of Epidemic Systems,, Springer, (1993). doi: 10.1007/978-3-540-70514-7. Google Scholar [9] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications,, Math. Biosci. Engin., 1 (2004), 361. doi: 10.3934/mbe.2004.1.361. Google Scholar [10] N. J. Cunniffe and C. A. Gilligan, nvasion, persistence and control in epidemic models for plant pathogens: The effect of host demography,, J. Royal Soc. Interface, 7 (2010), 439. doi: 10.1098/rsif.2009.0226. Google Scholar [11] N. J. Cunniffe and C. A. Gilligan, A theoretical framework for biological control of soil-borne plant pathogens: Identifying effective strategies,, J. Theor. Biol., 278 (2011), 32. doi: 10.1016/j.jtbi.2011.02.023. Google Scholar [12] N. J. Cunniffe, R. O. J. H. Stutt, F. van den Bosch and C. A. Gilligan, Time-dependent infectivity and flexible latent and infectious periods in compartmental models of plant disease,, Phytopathology, 102 (2012), 365. doi: 10.1094/PHYTO-12-10-0338. Google Scholar [13] J. Dushoff, W. Huang and C. Castillo-Chavez, Backward bifurcations and catastrophe in simple models of fatal diseases,, J. Math. Biol., 36 (1998), 227. doi: 10.1007/s002850050099. Google Scholar [14] C. A. Gilligan, An epidemiological framework for disease management,, Adv. Bot. Res., 38 (2002), 1. doi: 10.1016/S0065-2296(02)38027-3. Google Scholar [15] C. A. Gilligan, Sustainable agriculture and plant diseases: An epidemiological perspective,, Philos. T. Roy. Soc. B, 363 (2008), 741. doi: 10.1098/rstb.2007.2181. Google Scholar [16] C. A. Gilligan and F. van den Bosch, Epidemiological models for invasion and persistence of pathogens,, Annu. Rev. Phytopathol., 46 (2008), 385. doi: 10.1146/annurev.phyto.45.062806.094357. Google Scholar [17] S. Gubbins, C. A. Gilligan and A. Kleczkowski., Population dynamics of plant-parasite interactions: thresholds for invasion., Theor. Pop. Biol., 57 (2000), 219. doi: 10.1006/tpbi.1999.1441. Google Scholar [18] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-1140-2. Google Scholar [19] J. Hale and P. Waltman, Persistence in infinite-dimensional systems,, SIAM J. Math. Anal., 20 (1989), 388. doi: 10.1137/0520025. Google Scholar [20] G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence,, J. Math. Biol., 63 (2011), 125. doi: 10.1007/s00285-010-0368-2. Google Scholar [21] M. J. Jeger, Asymptotic behaviour and threshold criteria in model plant disease epidemics,, Plant Pathol., 35 (1986), 355. doi: 10.1111/j.1365-3059.1986.tb02026.x. Google Scholar [22] M. J. Jeger, J. Holt, F. Van Den Bosch and L. V. Madden, Epidemiology of insect-transmitted plant viruses: Modelling disease dynamics and control interventions,, Physiol. Entomol., 29 (2004), 291. doi: 10.1111/j.0307-6962.2004.00394.x. Google Scholar [23] M. J. Jeger, P. Jeffries, Y. Elad and X-M Xu, A generic theoretical model for biological control of foliar plant diseases,, J. Theor. Biol., 256 (2009), 201. doi: 10.1016/j.jtbi.2008.09.036. Google Scholar [24] M. J. Jeger and F. van den Bosch, Threshold criteria for model plant disease epidemics. I. asymptotic results,, Phytopathology, 84 (1994), 24. doi: 10.1094/Phyto-84-24. Google Scholar [25] M. J. Jeger, F. van den Bosch and L. V. Madden, Modelling virus-and host-limitation in vectored plant disease epidemics,, Virus Res., 159 (2011), 215. doi: 10.1016/j.virusres.2011.05.012. Google Scholar [26] M. J. Jeger, F. Van Den Bosch, L. V. Madden and J. Holt, A model for analysing plant-virus transmission characteristics and epidemic development,, Math. Med. Biol., 15 (1998), 1. doi: 10.1093/imammb15.1.1. Google Scholar [27] Z. Jin and Z. Ma, The stability of an SIR epidemic model with time delays,, Math. Biosci. Eng., 3 (2006), 101. Google Scholar [28] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics,, Academic Press, (1993). Google Scholar [29] M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size,, Math. Biosci., 160 (1999), 191. doi: 10.1016/S0025-5564(99)00030-9. Google Scholar [30] W. M. Liu, Criterion of Hopf bifurcations without using eigenvalues,, J. Math. Anal. Appl., 182 (1994), 250. doi: 10.1006/jmaa.1994.1079. Google Scholar [31] W. Ma, M. Song and Y. Takeuchi, Global stability of an SIR epidemic model with time delay,, Appl. Math. Lett., 17 (2004), 1141. doi: 10.1016/j.aml.2003.11.005. Google Scholar [32] W. Ma, Y. Takeuchi, T. Hara and E. Beretta, Permanence of an SIR epidemic model with distributed time delays,, Tohoku Math. J., 54 (2002), 581. doi: 10.2748/tmj/1113247650. Google Scholar [33] L. V. Madden, Botanical epidemiology: Some key advances and its continuing role in disease management,, Eur. J. Plant Path., 115 (2006), 3. doi: 10.1007/s10658-005-1229-5. Google Scholar [34] L. V. Madden, G. Hughes and F. Van den Bosch, The Study of Plant Disease Epidemics,, American Phytopathological Society, (2007). Google Scholar [35] R. M. May and R. M. Anderson, Population biology of infectious diseases: Part II,, Nature, 280 (1979), 455. doi: 10.1038/280455a0. Google Scholar [36] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete,, Nonlinear Anal. RWA, 11 (2010), 55. doi: 10.1016/j.nonrwa.2008.10.014. Google Scholar [37] M. T. McGrath, N. Shishkoff, C. Bornt and D. D. Moyer, First occurrence of powdery mildew caused by Leveillula taurica on pepper in New York,, Plant Disease, 85 (2001), 1122. Google Scholar [38] H. L. Smith, L. Wang and M. Y. Li, Global dynamics of an SEIR epidemic model with vertical transmission,, SIAM J. Appl. Math., 62 (2001), 58. doi: 10.1137/S0036139999359860. Google Scholar [39] R. N. Strange, Introduction to Plant Pathology,, John Wiley & Sons, (2006). Google Scholar [40] Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times,, Nonlinear Anal., 42 (2000), 931. doi: 10.1016/S0362-546X(99)00138-8. Google Scholar [41] J. M. Tchuenche and C. Chiyaka, Global dynamics of a time delayed SIR model with varying population size,, Dynamical Systems, 27 (2012), 145. doi: 10.1080/14689367.2011.607798. Google Scholar [42] J. M. Tchuenche, A. Nwagwo and R. Levins, Global behaviour of an SIR epidemic model with time delay,, Math. Methods Appl. Sci., 30 (2007), 733. doi: 10.1002/mma.810. Google Scholar [43] F. Van Den Bosch, G. Akudibilah, S. Seal and M. Jeger, Host resistance and the evolutionary response of plant viruses,, J. Appl. Ecol., 43 (2006), 506. Google Scholar [44] F. Van den Bosch, M. J. Jeger and C. A. Gilligan, Disease control and its selection for damaging plant virus strains in vegetatively propagated staple food crops; a theoretical assessment,, Proc. Royal Soc. Lond. B Biol., 274 (2007), 11. Google Scholar [45] F. Van den Bosch, N. McRoberts, F. van den Bergh and L. V. Madden, The basic reproduction number of plant pathogens: Matrix approaches to complex dynamics,, Phytopathology, 98 (2008), 239. Google Scholar [46] P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [47] J. E. Van der Plank, Plant Diseases: Epidemics and Control,, Academic Press, (1963). Google Scholar [48] R. Xu and Y. Du, A delayed SIR epidemic model with saturation incidence and a constant infectious period,, J. Appl. Math. Comput., 35 (2011), 229. doi: 10.1007/s12190-009-0353-3. Google Scholar [49] R. Xu and Z. Ma, Global stability of a delayed SEIRS epidemic model with saturation incidence rate,, Nonlinear Dynam., 61 (2010), 229. doi: 10.1007/s11071-009-9644-3. Google Scholar [50] J. C. Zadoks, Systems analysis and the dynamics of epidemics,, Phytopathology, 61 (1971), 600. Google Scholar [51] H. Zhang, L. Chen and J. J. Nieto, A delayed epidemic model with stage-structure and pulses for pest management strategy,, Nonlinear Anal. RWA, 9 (2008), 1714. doi: 10.1016/j.nonrwa.2007.05.004. Google Scholar [52] J. Z. Zhang, Z. Jin, Q. X. Liu and Z. Y. Zhang, Analysis of a delayed SIR model with nonlinear incidence rate,, Discrete Dyn. Nat. Soc., 2008 (2008). doi: 10.1155/2008/636153. Google Scholar

show all references

##### References:
 [1] R. M. Anderson and R. M. May, Population biology of infectious diseases: Part I,, Nature, 280 (1979), 361. doi: 10.1038/280361a0. Google Scholar [2] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control,, Oxford University Press, (1991). Google Scholar [3] E. Beretta and D. Breda, An SEIR epidemic model with constant latency time and infectious period,, Math. Biosci. Eng., 8 (2011), 931. doi: 10.3934/mbe.2011.8.931. Google Scholar [4] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters,, SIAM J. Math. Anal., 33 (2002), 1144. doi: 10.1137/S0036141000376086. Google Scholar [5] E. Beretta and Y. Takeuchi, Global stability of an SIR epidemic model with time delays,, J. Math. Biol., 33 (1995), 250. doi: 10.1007/BF00169563. Google Scholar [6] G. Birkhoff and G. Rota, Ordinary Differential Equations,, John Wiley and Sons, (1982). Google Scholar [7] B. Buonomo and M. Cerasuolo, Stability and bifurcation in plant-pathogens interactions,, Appl. Math. Comput., 232 (2014), 858. doi: 10.1016/j.amc.2014.01.127. Google Scholar [8] V. Capasso, Mathematical Structures of Epidemic Systems,, Springer, (1993). doi: 10.1007/978-3-540-70514-7. Google Scholar [9] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications,, Math. Biosci. Engin., 1 (2004), 361. doi: 10.3934/mbe.2004.1.361. Google Scholar [10] N. J. Cunniffe and C. A. Gilligan, nvasion, persistence and control in epidemic models for plant pathogens: The effect of host demography,, J. Royal Soc. Interface, 7 (2010), 439. doi: 10.1098/rsif.2009.0226. Google Scholar [11] N. J. Cunniffe and C. A. Gilligan, A theoretical framework for biological control of soil-borne plant pathogens: Identifying effective strategies,, J. Theor. Biol., 278 (2011), 32. doi: 10.1016/j.jtbi.2011.02.023. Google Scholar [12] N. J. Cunniffe, R. O. J. H. Stutt, F. van den Bosch and C. A. Gilligan, Time-dependent infectivity and flexible latent and infectious periods in compartmental models of plant disease,, Phytopathology, 102 (2012), 365. doi: 10.1094/PHYTO-12-10-0338. Google Scholar [13] J. Dushoff, W. Huang and C. Castillo-Chavez, Backward bifurcations and catastrophe in simple models of fatal diseases,, J. Math. Biol., 36 (1998), 227. doi: 10.1007/s002850050099. Google Scholar [14] C. A. Gilligan, An epidemiological framework for disease management,, Adv. Bot. Res., 38 (2002), 1. doi: 10.1016/S0065-2296(02)38027-3. Google Scholar [15] C. A. Gilligan, Sustainable agriculture and plant diseases: An epidemiological perspective,, Philos. T. Roy. Soc. B, 363 (2008), 741. doi: 10.1098/rstb.2007.2181. Google Scholar [16] C. A. Gilligan and F. van den Bosch, Epidemiological models for invasion and persistence of pathogens,, Annu. Rev. Phytopathol., 46 (2008), 385. doi: 10.1146/annurev.phyto.45.062806.094357. Google Scholar [17] S. Gubbins, C. A. Gilligan and A. Kleczkowski., Population dynamics of plant-parasite interactions: thresholds for invasion., Theor. Pop. Biol., 57 (2000), 219. doi: 10.1006/tpbi.1999.1441. Google Scholar [18] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-1140-2. Google Scholar [19] J. Hale and P. Waltman, Persistence in infinite-dimensional systems,, SIAM J. Math. Anal., 20 (1989), 388. doi: 10.1137/0520025. Google Scholar [20] G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence,, J. Math. Biol., 63 (2011), 125. doi: 10.1007/s00285-010-0368-2. Google Scholar [21] M. J. Jeger, Asymptotic behaviour and threshold criteria in model plant disease epidemics,, Plant Pathol., 35 (1986), 355. doi: 10.1111/j.1365-3059.1986.tb02026.x. Google Scholar [22] M. J. Jeger, J. Holt, F. Van Den Bosch and L. V. Madden, Epidemiology of insect-transmitted plant viruses: Modelling disease dynamics and control interventions,, Physiol. Entomol., 29 (2004), 291. doi: 10.1111/j.0307-6962.2004.00394.x. Google Scholar [23] M. J. Jeger, P. Jeffries, Y. Elad and X-M Xu, A generic theoretical model for biological control of foliar plant diseases,, J. Theor. Biol., 256 (2009), 201. doi: 10.1016/j.jtbi.2008.09.036. Google Scholar [24] M. J. Jeger and F. van den Bosch, Threshold criteria for model plant disease epidemics. I. asymptotic results,, Phytopathology, 84 (1994), 24. doi: 10.1094/Phyto-84-24. Google Scholar [25] M. J. Jeger, F. van den Bosch and L. V. Madden, Modelling virus-and host-limitation in vectored plant disease epidemics,, Virus Res., 159 (2011), 215. doi: 10.1016/j.virusres.2011.05.012. Google Scholar [26] M. J. Jeger, F. Van Den Bosch, L. V. Madden and J. Holt, A model for analysing plant-virus transmission characteristics and epidemic development,, Math. Med. Biol., 15 (1998), 1. doi: 10.1093/imammb15.1.1. Google Scholar [27] Z. Jin and Z. Ma, The stability of an SIR epidemic model with time delays,, Math. Biosci. Eng., 3 (2006), 101. Google Scholar [28] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics,, Academic Press, (1993). Google Scholar [29] M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size,, Math. Biosci., 160 (1999), 191. doi: 10.1016/S0025-5564(99)00030-9. Google Scholar [30] W. M. Liu, Criterion of Hopf bifurcations without using eigenvalues,, J. Math. Anal. Appl., 182 (1994), 250. doi: 10.1006/jmaa.1994.1079. Google Scholar [31] W. Ma, M. Song and Y. Takeuchi, Global stability of an SIR epidemic model with time delay,, Appl. Math. Lett., 17 (2004), 1141. doi: 10.1016/j.aml.2003.11.005. Google Scholar [32] W. Ma, Y. Takeuchi, T. Hara and E. Beretta, Permanence of an SIR epidemic model with distributed time delays,, Tohoku Math. J., 54 (2002), 581. doi: 10.2748/tmj/1113247650. Google Scholar [33] L. V. Madden, Botanical epidemiology: Some key advances and its continuing role in disease management,, Eur. J. Plant Path., 115 (2006), 3. doi: 10.1007/s10658-005-1229-5. Google Scholar [34] L. V. Madden, G. Hughes and F. Van den Bosch, The Study of Plant Disease Epidemics,, American Phytopathological Society, (2007). Google Scholar [35] R. M. May and R. M. Anderson, Population biology of infectious diseases: Part II,, Nature, 280 (1979), 455. doi: 10.1038/280455a0. Google Scholar [36] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete,, Nonlinear Anal. RWA, 11 (2010), 55. doi: 10.1016/j.nonrwa.2008.10.014. Google Scholar [37] M. T. McGrath, N. Shishkoff, C. Bornt and D. D. Moyer, First occurrence of powdery mildew caused by Leveillula taurica on pepper in New York,, Plant Disease, 85 (2001), 1122. Google Scholar [38] H. L. Smith, L. Wang and M. Y. Li, Global dynamics of an SEIR epidemic model with vertical transmission,, SIAM J. Appl. Math., 62 (2001), 58. doi: 10.1137/S0036139999359860. Google Scholar [39] R. N. Strange, Introduction to Plant Pathology,, John Wiley & Sons, (2006). Google Scholar [40] Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times,, Nonlinear Anal., 42 (2000), 931. doi: 10.1016/S0362-546X(99)00138-8. Google Scholar [41] J. M. Tchuenche and C. Chiyaka, Global dynamics of a time delayed SIR model with varying population size,, Dynamical Systems, 27 (2012), 145. doi: 10.1080/14689367.2011.607798. Google Scholar [42] J. M. Tchuenche, A. Nwagwo and R. Levins, Global behaviour of an SIR epidemic model with time delay,, Math. Methods Appl. Sci., 30 (2007), 733. doi: 10.1002/mma.810. Google Scholar [43] F. Van Den Bosch, G. Akudibilah, S. Seal and M. Jeger, Host resistance and the evolutionary response of plant viruses,, J. Appl. Ecol., 43 (2006), 506. Google Scholar [44] F. Van den Bosch, M. J. Jeger and C. A. Gilligan, Disease control and its selection for damaging plant virus strains in vegetatively propagated staple food crops; a theoretical assessment,, Proc. Royal Soc. Lond. B Biol., 274 (2007), 11. Google Scholar [45] F. Van den Bosch, N. McRoberts, F. van den Bergh and L. V. Madden, The basic reproduction number of plant pathogens: Matrix approaches to complex dynamics,, Phytopathology, 98 (2008), 239. Google Scholar [46] P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [47] J. E. Van der Plank, Plant Diseases: Epidemics and Control,, Academic Press, (1963). Google Scholar [48] R. Xu and Y. Du, A delayed SIR epidemic model with saturation incidence and a constant infectious period,, J. Appl. Math. Comput., 35 (2011), 229. doi: 10.1007/s12190-009-0353-3. Google Scholar [49] R. Xu and Z. Ma, Global stability of a delayed SEIRS epidemic model with saturation incidence rate,, Nonlinear Dynam., 61 (2010), 229. doi: 10.1007/s11071-009-9644-3. Google Scholar [50] J. C. Zadoks, Systems analysis and the dynamics of epidemics,, Phytopathology, 61 (1971), 600. Google Scholar [51] H. Zhang, L. Chen and J. J. Nieto, A delayed epidemic model with stage-structure and pulses for pest management strategy,, Nonlinear Anal. RWA, 9 (2008), 1714. doi: 10.1016/j.nonrwa.2007.05.004. Google Scholar [52] J. Z. Zhang, Z. Jin, Q. X. Liu and Z. Y. Zhang, Analysis of a delayed SIR model with nonlinear incidence rate,, Discrete Dyn. Nat. Soc., 2008 (2008). doi: 10.1155/2008/636153. Google Scholar
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