Taking account of spatial heterogeneity, latency in infected individuals, and time for shed bacteria to the aquatic environment, we build a delayed nonlocal reaction-diffusion cholera model. A feature of this model is that the incidences are of general nonlinear forms. By using the theories of monotone dynamical systems and uniform persistence, we obtain a threshold dynamics determined by the basic reproduction number $ \mathcal {R}_0 $. Roughly speaking, the cholera will die out if $ \mathcal{R}_0<1 $ while it persists if $ \mathcal{R}_0>1 $. Moreover, we derive the explicit formulae of $ \mathcal{R}_0 $ for two concrete situations.
Citation: |
[1] |
L. J. S. Allen, B. M. Bolker, 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.![]() ![]() ![]() |
[2] |
T. Berge, S. Bowong and J. M.-S. Lubuma, Global stability of a two-patch cholera model with fast and slow transmissions, Math. Comput. Simul., 133 (2017), 142-164.
doi: 10.1016/j.matcom.2015.10.013.![]() ![]() ![]() |
[3] |
F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model, Math Biosci. Eng., 10 (2013), 1335-1349.
doi: 10.3934/mbe.2013.10.1335.![]() ![]() ![]() |
[4] |
F. Capone, V. De Cataldis and R. De Luca, Influence of diffusion on the stability of equilibria in a reaction-diffusion system modeling cholera dynamic, J. Math. Biol., 71 (2015), 1107-1131.
doi: 10.1007/s00285-014-0849-9.![]() ![]() ![]() |
[5] |
M. C. Eisenberg, Z. Shuai, J. H. Tien and P. van den driessche, A cholera model in a patchy environment with water and human movement, Math. Biosci., 246 (2013), 105-112.
doi: 10.1016/j.mbs.2013.08.003.![]() ![]() ![]() |
[6] |
S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, in Nonlinear Dynamics and Evolution Equations, Vol. 48, American Matehmatical Society, Province, 2006, pp. 137–200.
doi: 10.1007/s00285-006-0050-x.![]() ![]() ![]() |
[7] |
Z. 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.![]() ![]() ![]() |
[8] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988.
doi: 10.1090/surv/025.![]() ![]() ![]() |
[9] |
D. M. Hartley, J. G. Morris Jr and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidmeics?, PLOS Med., 3 (2006), 63-69.
doi: 10.1371/journal.pmed.0030007.![]() ![]() |
[10] |
H. Li, R. Peng and F.-B. Wang, Varying total population enhances disease persistence: qualitative analysis on a diffusive SIS epidemic model, J. Differential Equations, 262 (2017), 885-913.
doi: 10.1016/j.jde.2016.09.044.![]() ![]() ![]() |
[11] |
J. Lin, R. Xu and X. Tian, Global dynamics of an age-structured cholera model with both human-to-human and environment-to-human transmissions and saturation incidence, Appl. Math. Modelling, 63 (2018), 688-708.
doi: 10.1016/j.apm.2018.07.013.![]() ![]() ![]() |
[12] |
J. Lin, R. Xu and X. Tian, Global dynamics of an age-structured cholera model with multiple transmissions, saturation incidence and imperfect vaccination, J. Biol. Dyn., 13 (2019), 69-102.
doi: 10.1080/17513758.2019.1570362.![]() ![]() ![]() |
[13] |
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.![]() ![]() ![]() |
[14] |
R. H. Martin Jr. and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.2307/2001590.![]() ![]() ![]() |
[15] |
J. A. J. Metz and O. Diekmann, Age dependence, The dynamics of physiologically structured populations (Amsterdam, 1983), Lecture Notes in Biomath., 68, Springer, Berlin, 1986,136–184.
doi: 10.1007/978-3-662-13159-6_4.![]() ![]() ![]() |
[16] |
J. B. H. Njagarah and F. Nyabadza, A metapopulation model for cholera transmission dynamics between communities linked by migration, Appl. Math. Comput., 241 (2014), 317-331.
doi: 10.1016/j.amc.2014.05.036.![]() ![]() ![]() |
[17] |
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.![]() ![]() ![]() |
[18] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Math. Surveys and Monographs, vol. 41, 1995.
![]() ![]() |
[19] |
H. L. Smith and X. -Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal. TMA, 47 (2001), 6169-6179.
doi: 10.1016/S0362-546X(01)00678-2.![]() ![]() ![]() |
[20] |
H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.
doi: 10.1007/BF00173267.![]() ![]() ![]() |
[21] |
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.![]() ![]() ![]() |
[22] |
H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. Real World Appl., 2 (2001), 145-160.
doi: 10.1016/S0362-546X(00)00112-7.![]() ![]() ![]() |
[23] |
J. H. Tien and D. J. D. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bull. Math. Biol., 72 (2010), 1506-1533.
doi: 10.1007/s11538-010-9507-6.![]() ![]() ![]() |
[24] |
F.-B. Wang, J. Shi and X. Zou, Dynamics of a host-pathogen system on a bounded spatial domain, Commun. Pure Appl. Anal., 14 (2015), 2535-2560.
doi: 10.3934/cpaa.2015.14.2535.![]() ![]() ![]() |
[25] |
X. Wang and J. Wang, Analysis of cholera epidemics with bacterial growth and spatial movement, J. Biol. Dyn., 9 (2015), 233-261.
doi: 10.1080/17513758.2014.974696.![]() ![]() ![]() |
[26] |
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.![]() ![]() ![]() |
[27] |
X. Wang, X.-Q. Zhao and J. Wang, A cholera epidemic model in a spatiotemporally heterogeneous environemnt, J. Math. Aanal. Appl., 468 (2018), 893-912.
doi: 10.1016/j.jmaa.2018.08.039.![]() ![]() ![]() |
[28] |
World Health Organization, Cholera fact shettes, January 2019, available from http://www.who.int
![]() |
[29] |
J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Science, vol. 119, Springer, New York, 1996.
doi: 10.1007/978-1-4612-4050-1.![]() ![]() ![]() |
[30] |
Y. Wu and X. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Differential Equations, 261 (2016), 4424-4447.
doi: 10.1016/j.jde.2016.06.028.![]() ![]() ![]() |
[31] |
Y. Wu and X. Zou, Dynamics and profiles of a diffusive host-pathogen system with distinct dispersal rates, J. Differential Equations, 264 (2018), 4989-5024.
doi: 10.1016/j.jde.2017.12.027.![]() ![]() ![]() |
[32] |
K. Yamazaki and X. Wang, Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model, Discrete Cont. Dynam. Syst., 21 (2016), 1297-1316.
doi: 10.3934/dcdsb.2016.21.1297.![]() ![]() ![]() |
[33] |
K. Yamazaki and X. Wang, Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model, Math. Biosci. Eng., 14 (2017), 559-579.
doi: 10.3934/mbe.2017033.![]() ![]() ![]() |