November  2015, 14(6): 2535-2560. doi: 10.3934/cpaa.2015.14.2535

Dynamics of a host-pathogen system on a bounded spatial domain

1. 

Department of Natural Science in the Center for General Education, Chang Gung University, Kwei-Shan, Taoyuan 333

2. 

Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795

3. 

Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7

Received  November 2014 Revised  July 2015 Published  September 2015

We study a host-pathogen system in a bounded spatial habitat where the environment is closed. Extinction and persistence of the disease are investigated by appealing to theories of monotone dynamical systems and uniform persistence. We also carry out a bifurcation analysis for steady state solutions, and the results suggest that a backward bifurcation may occur when the parameters in the system are spatially dependent.
Citation: Feng-Bin Wang, Junping Shi, Xingfu Zou. Dynamics of a host-pathogen system on a bounded spatial domain. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2535-2560. doi: 10.3934/cpaa.2015.14.2535
References:
[1]

N. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equation, Commun. Partial. Diff. Eqns., 4 (1979), 827-868. doi: 10.1080/03605307908820113.

[2]

L. J. S. Allen, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst., 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1.

[3]

R. M. Anderson and R. M. May, The population dynamics of microparasites and their invertebrate hosts, Phil. Trans. Royal Soc. London B, Biol. Sci., 291 (1981), 451-524.

[4]

V. Capasso and L. Maddalena, Convergence to equilibrium states for a reactiondiffusion system modelling the spatial spread of a class of bacterial and viral diseases, J. Math. Biol., 13 (1981), 173-184. doi: 10.1007/BF00275212.

[5]

V. Capasso and R. E. Wilson, Analysis of a reactiondiffusion system modelling manenvironmentman epidemics, SIAM J. Appl. Math., 57 (1997), 327-346. doi: 10.1137/S0036139995284681.

[6]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Func. Anal., 8 (1971), 321-340.

[7]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-662-00547-7.

[8]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in the models for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324.

[9]

G. Dwyer, Density Dependence and Spatial Structure in the Dynamics of Insect Pathogens, The American Naturalist, 143 (1994), 533-562.

[10]

Y. Du and J. Shi, Spatially heterogeneous predator-prey model with protect zone for prey, J. Diff. Eqns., 229 (2006), 63-91. doi: 10.1016/j.jde.2006.01.013.

[11]

Y. Du and J. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Am. Math. Soc., 359 (2007), 4557-4593. doi: 10.1090/S0002-9947-07-04262-6.

[12]

Z. M. Guo, F.-B. Wang and X. Zou, Threshold dynamics of an infective disease model with a fixed latent period and non-local infections, J. Math. Biol., 65 (2012), 1387-1410. doi: 10.1007/s00285-011-0500-y.

[13]

J. Hale, Asymptotic behavior of dissipative systems, American Mathematical Society Providence, RI, 1988.

[14]

S. B. Hsu, J. Jiang and F. B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Diff. Eqns., 248 (2010), 2470-2496. doi: 10.1016/j.jde.2009.12.014.

[15]

S. B. Hsu, F. B. Wang and X.-Q. Zhao, Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone, Jour. Dyna. Diff. Equa., 23 (2011), 817-842. doi: 10.1007/s10884-011-9224-3.

[16]

S. B. Hsu, F. B. Wang and X.-Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats, J. Diff. Eqns., 255 (2013), 265-297. doi: 10.1016/j.jde.2013.04.006.

[17]

D. Le, Dissipativity and global attractors for a class of quasilinear parabolic systems, Commun. Partial. Diff. Eqns., 22 (1997), 413-433. doi: 10.1080/03605309708821269.

[18]

J. Li and X. Zou, Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain, Bull. Math. Biol., 71 (2009), 2048-2079. doi: 10.1007/s11538-009-9457-z.

[19]

R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590.

[20]

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.

[21]

R. D. Nussbaum, Eigenvectors of nonlinear positive operator and the linear Krein-Rutman theorem, in: E. Fadell, G. Fournier (Eds.), Fixed Point Theory, Lecture Notes in Mathematics, Springer, New York/Berlin, 886 (1981), 309-331.

[22]

W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.

[23]

A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[24]

R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. I., J. Diff. Equa., 247 (2009), 1096-1119. doi: 10.1016/j.jde.2009.05.002.

[25]

R. Peng and F.-Q. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: effects of epidemic risk and population movement, Phys. D, 259 (2013), 8-25. doi: 10.1016/j.physd.2013.05.006.

[26]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984 doi: 10.1007/978-1-4612-5282-5.

[27]

Junping Shi, Persistence and bifurcation of degenerate solutions, Jour. Funct. Anal., 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483.

[28]

J.-P. Shi and X.-F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Diff. Equa., 246 (2009), 2788-2812.

[29]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs 41, American Mathematical Society, Providence, RI, 1995.

[30]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179. doi: 10.1016/S0362-546X(01)00678-2.

[31]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267.

[32]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM, J. Appl. Math., 70 (2009), 188-211. doi: 10.1137/080732870.

[33]

P. van den Driessche and James Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[34]

N. K. Vaidya and F.-B. Wang and X. Zou, Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment, Disc. Conti. Dynam Syst. B, 17 (2012), 2829-2848. doi: 10.3934/dcdsb.2012.17.2829.

[35]

F. B. Wang, A system of partial differential equations modeling the competition for two complementary resources in flowing habitats, J. Diff. Eqns., 249 (2010), 2866-2888. doi: 10.1016/j.jde.2010.07.031.

[36]

W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168. doi: 10.1137/090775890.

[37]

W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Applied Dynamical Systems, 11 (2012), 1652-1673. doi: 10.1137/120872942.

[38]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1.

show all references

References:
[1]

N. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equation, Commun. Partial. Diff. Eqns., 4 (1979), 827-868. doi: 10.1080/03605307908820113.

[2]

L. J. S. Allen, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst., 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1.

[3]

R. M. Anderson and R. M. May, The population dynamics of microparasites and their invertebrate hosts, Phil. Trans. Royal Soc. London B, Biol. Sci., 291 (1981), 451-524.

[4]

V. Capasso and L. Maddalena, Convergence to equilibrium states for a reactiondiffusion system modelling the spatial spread of a class of bacterial and viral diseases, J. Math. Biol., 13 (1981), 173-184. doi: 10.1007/BF00275212.

[5]

V. Capasso and R. E. Wilson, Analysis of a reactiondiffusion system modelling manenvironmentman epidemics, SIAM J. Appl. Math., 57 (1997), 327-346. doi: 10.1137/S0036139995284681.

[6]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Func. Anal., 8 (1971), 321-340.

[7]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-662-00547-7.

[8]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in the models for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324.

[9]

G. Dwyer, Density Dependence and Spatial Structure in the Dynamics of Insect Pathogens, The American Naturalist, 143 (1994), 533-562.

[10]

Y. Du and J. Shi, Spatially heterogeneous predator-prey model with protect zone for prey, J. Diff. Eqns., 229 (2006), 63-91. doi: 10.1016/j.jde.2006.01.013.

[11]

Y. Du and J. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Am. Math. Soc., 359 (2007), 4557-4593. doi: 10.1090/S0002-9947-07-04262-6.

[12]

Z. M. Guo, F.-B. Wang and X. Zou, Threshold dynamics of an infective disease model with a fixed latent period and non-local infections, J. Math. Biol., 65 (2012), 1387-1410. doi: 10.1007/s00285-011-0500-y.

[13]

J. Hale, Asymptotic behavior of dissipative systems, American Mathematical Society Providence, RI, 1988.

[14]

S. B. Hsu, J. Jiang and F. B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Diff. Eqns., 248 (2010), 2470-2496. doi: 10.1016/j.jde.2009.12.014.

[15]

S. B. Hsu, F. B. Wang and X.-Q. Zhao, Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone, Jour. Dyna. Diff. Equa., 23 (2011), 817-842. doi: 10.1007/s10884-011-9224-3.

[16]

S. B. Hsu, F. B. Wang and X.-Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats, J. Diff. Eqns., 255 (2013), 265-297. doi: 10.1016/j.jde.2013.04.006.

[17]

D. Le, Dissipativity and global attractors for a class of quasilinear parabolic systems, Commun. Partial. Diff. Eqns., 22 (1997), 413-433. doi: 10.1080/03605309708821269.

[18]

J. Li and X. Zou, Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain, Bull. Math. Biol., 71 (2009), 2048-2079. doi: 10.1007/s11538-009-9457-z.

[19]

R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590.

[20]

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.

[21]

R. D. Nussbaum, Eigenvectors of nonlinear positive operator and the linear Krein-Rutman theorem, in: E. Fadell, G. Fournier (Eds.), Fixed Point Theory, Lecture Notes in Mathematics, Springer, New York/Berlin, 886 (1981), 309-331.

[22]

W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.

[23]

A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[24]

R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. I., J. Diff. Equa., 247 (2009), 1096-1119. doi: 10.1016/j.jde.2009.05.002.

[25]

R. Peng and F.-Q. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: effects of epidemic risk and population movement, Phys. D, 259 (2013), 8-25. doi: 10.1016/j.physd.2013.05.006.

[26]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984 doi: 10.1007/978-1-4612-5282-5.

[27]

Junping Shi, Persistence and bifurcation of degenerate solutions, Jour. Funct. Anal., 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483.

[28]

J.-P. Shi and X.-F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Diff. Equa., 246 (2009), 2788-2812.

[29]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs 41, American Mathematical Society, Providence, RI, 1995.

[30]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179. doi: 10.1016/S0362-546X(01)00678-2.

[31]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267.

[32]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM, J. Appl. Math., 70 (2009), 188-211. doi: 10.1137/080732870.

[33]

P. van den Driessche and James Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[34]

N. K. Vaidya and F.-B. Wang and X. Zou, Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment, Disc. Conti. Dynam Syst. B, 17 (2012), 2829-2848. doi: 10.3934/dcdsb.2012.17.2829.

[35]

F. B. Wang, A system of partial differential equations modeling the competition for two complementary resources in flowing habitats, J. Diff. Eqns., 249 (2010), 2866-2888. doi: 10.1016/j.jde.2010.07.031.

[36]

W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168. doi: 10.1137/090775890.

[37]

W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Applied Dynamical Systems, 11 (2012), 1652-1673. doi: 10.1137/120872942.

[38]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1.

[1]

Kousuke Kuto, Tohru Tsujikawa. Bifurcation structure of steady-states for bistable equations with nonlocal constraint. Conference Publications, 2013, 2013 (special) : 467-476. doi: 10.3934/proc.2013.2013.467

[2]

Yan'e Wang, Jianhua Wu. Stability of positive constant steady states and their bifurcation in a biological depletion model. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 849-865. doi: 10.3934/dcdsb.2011.15.849

[3]

Inmaculada Antón, Julián López-Gómez. Global bifurcation diagrams of steady-states for a parabolic model related to a nuclear engineering problem. Conference Publications, 2013, 2013 (special) : 21-30. doi: 10.3934/proc.2013.2013.21

[4]

Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489

[5]

Sebastian J. Schreiber. On persistence and extinction for randomly perturbed dynamical systems. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 457-463. doi: 10.3934/dcdsb.2007.7.457

[6]

Wen Jin, Horst R. Thieme. Persistence and extinction of diffusing populations with two sexes and short reproductive season. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3209-3218. doi: 10.3934/dcdsb.2014.19.3209

[7]

Keng Deng, Yixiang Wu. Extinction and uniform strong persistence of a size-structured population model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 831-840. doi: 10.3934/dcdsb.2017041

[8]

Wen Jin, Horst R. Thieme. An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 447-470. doi: 10.3934/dcdsb.2016.21.447

[9]

Linda J. S. Allen, P. van den Driessche. Stochastic epidemic models with a backward bifurcation. Mathematical Biosciences & Engineering, 2006, 3 (3) : 445-458. doi: 10.3934/mbe.2006.3.445

[10]

József Z. Farkas, Peter Hinow. Steady states in hierarchical structured populations with distributed states at birth. Discrete and Continuous Dynamical Systems - B, 2012, 17 (8) : 2671-2689. doi: 10.3934/dcdsb.2012.17.2671

[11]

Anne Nouri, Christian Schmeiser. Aggregated steady states of a kinetic model for chemotaxis. Kinetic and Related Models, 2017, 10 (1) : 313-327. doi: 10.3934/krm.2017013

[12]

Àngel Calsina, József Z. Farkas. Boundary perturbations and steady states of structured populations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6675-6691. doi: 10.3934/dcdsb.2019162

[13]

Shangzhi Li, Shangjiang Guo. Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5101-5134. doi: 10.3934/dcdsb.2020335

[14]

Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1

[15]

Anton Arnold, Laurent Desvillettes, Céline Prévost. Existence of nontrivial steady states for populations structured with respect to space and a continuous trait. Communications on Pure and Applied Analysis, 2012, 11 (1) : 83-96. doi: 10.3934/cpaa.2012.11.83

[16]

Shanshan Chen. Nonexistence of nonconstant positive steady states of a diffusive predator-prey model. Communications on Pure and Applied Analysis, 2018, 17 (2) : 477-485. doi: 10.3934/cpaa.2018026

[17]

Qian Xu. The stability of bifurcating steady states of several classes of chemotaxis systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 231-248. doi: 10.3934/dcdsb.2015.20.231

[18]

Xinfu Chen, Yuanwei Qi, Mingxin Wang. Steady states of a strongly coupled prey-predator model. Conference Publications, 2005, 2005 (Special) : 173-180. doi: 10.3934/proc.2005.2005.173

[19]

Jörg Weber. Confined steady states of the relativistic Vlasov–Maxwell system in an infinitely long cylinder. Kinetic and Related Models, 2020, 13 (6) : 1135-1161. doi: 10.3934/krm.2020040

[20]

Tian Xiang. A study on the positive nonconstant steady states of nonlocal chemotaxis systems. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2457-2485. doi: 10.3934/dcdsb.2013.18.2457

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (141)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]