# American Institute of Mathematical Sciences

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 & Applied Analysis, 2015, 14 (6) : 2535-2560. doi: 10.3934/cpaa.2015.14.2535
##### References:
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Math., 57 (1997), 327-346. doi: 10.1137/S0036139995284681.  Google Scholar [6] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Func. Anal., 8 (1971), 321-340.  Google Scholar [7] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-662-00547-7.  Google Scholar [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.  Google Scholar [9] G. Dwyer, Density Dependence and Spatial Structure in the Dynamics of Insect Pathogens, The American Naturalist, 143 (1994), 533-562. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] J. Hale, Asymptotic behavior of dissipative systems, American Mathematical Society Providence, RI, 1988.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] Junping Shi, Persistence and bifurcation of degenerate solutions, Jour. Funct. Anal., 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [38] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [6] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Func. Anal., 8 (1971), 321-340.  Google Scholar [7] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-662-00547-7.  Google Scholar [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.  Google Scholar [9] G. Dwyer, Density Dependence and Spatial Structure in the Dynamics of Insect Pathogens, The American Naturalist, 143 (1994), 533-562. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] J. Hale, Asymptotic behavior of dissipative systems, American Mathematical Society Providence, RI, 1988.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] Junping Shi, Persistence and bifurcation of degenerate solutions, Jour. Funct. Anal., 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [38] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

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