# American Institute of Mathematical Sciences

August  2018, 23(6): 2625-2640. doi: 10.3934/dcdsb.2018124

## Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity

 1 School of Mathematics, Shandong University, Jinan, Shandong 250100, China 2 School of Mathematics, Tianjin University, Tianjin 300350, China

* Corresponding author: Yuxiang Zhang

Received  June 2017 Revised  October 2017 Published  April 2018

Fund Project: X. Zhang is partially supported by the NSF of China (No. 11571200,11425105), and Y. Zhang is supported in part by the NSF of China (No. 11701415,11601386)

This work is devoted to study the spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. In the case of the spatial domain is bounded and heterogeneous, we assume some key parameters in the model explicitly depend on spatial location. We first define the basic reproduction number $\mathcal{R}_0$ for the disease transmission, which generalizes the existing definition of $\mathcal{R}_0$ for the system in spatially homogeneous environment. Then we establish a threshold type result for the disease eradication ($\mathcal{R}_0 <1$) or uniform persistence ($\mathcal{R}_0>1)$. In the case of the domain is linear, unbounded, and spatially homogenerous, we further establish the existence of traveling wave solutions and the minimum wave speed $c^*$ for the disease transmission. At the end of this work, we characteristic the minimum wave speed $c^*$ and provide a method for the calculation of $c^*$.

Citation: Xiaoyan Zhang, Yuxiang Zhang. Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2625-2640. doi: 10.3934/dcdsb.2018124
##### References:
 [1] L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profile of the steady states for an SIS epidemic disease reaction-diffusion model, Dis. Cont. Dyn. Syst., 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1. Google Scholar [2] N. Bacaer and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436. doi: 10.1007/s00285-006-0015-0. Google Scholar [3] E. Bertuzzo, On spatially explicit models of cholera epidemics, Journal of the Royal Society Interface, 7 (2010), 321-333. doi: 10.1098/rsif.2009.0204. Google Scholar [4] M. J. Bouma and M. Pascual, Seasonal and interannual cycles of endemic cholera in Bengal 1891-1940 in relation to climate and geography, Hydrobiologia, 460 (2001), 147-156. doi: 10.1007/978-94-017-3284-0_13. Google Scholar [5] F. Capone, C. V. De and L. R. De, 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. Google Scholar [6] F. Capone, C. V. De and L. R. De, Erratum to: Influence of diffusion on the stability of equilibria in a reaction-diffusion system modeling cholera dynamic, J. Math. Biol., 71 (2015), 1267-1268. doi: 10.1007/s00285-015-0915-y. Google Scholar [7] C. T. Codeco, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infect Dis, 1 (2001), p1. doi: 10.1186/1471-2334-1-1. Google Scholar [8] R. R. Colwell and A. Huq, Environmental reservoir of Vibrio cholerae, the causative agent of cholera, Annals of the New York Academy of Sciences, 740 (1994), 44-54. doi: 10.1111/j.1749-6632.1994.tb19852.x. Google Scholar [9] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [10] Z. Du and R. Peng, A priori $L^∞$ estimates for solutions of a class of reaction-diffusion systems, J. Math. Biol., 72 (2016), 1429-1439. doi: 10.1007/s00285-015-0914-z. Google Scholar [11] C. H. Fung, Cholera transmission dynamic models for public health practitioners, Emerging Themes in Epidemiology, 11 (2014), p1. doi: 10.1186/1742-7622-11-1. Google Scholar [12] P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. PDEs, 5 (1980), 999-1030. doi: 10.1080/03605308008820162. Google Scholar [13] E. I. Jury and M. Mansour, Positivity and nonnegativity conditions of a quartic equation and related problems, IEEE Trans. Automat. Contr., 26 (1981), 444-451. doi: 10.1109/TAC.1981.1102589. Google Scholar [14] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, Heidelberg, 1976. Google Scholar [15] H. Li, R. Peng and F. B. Wang, Varying total population enhances disease persistence: Qualitative analysis on a diffusive SIS epidemic model, J. Diff. Equ., 262 (2016), 885-913. doi: 10.1016/j.jde.2016.09.044. Google Scholar [16] B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98. doi: 10.1016/j.mbs.2005.03.008. Google Scholar [17] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154. Google Scholar [18] 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 [19] R. Peng and X.-Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471. doi: 10.1088/0951-7715/25/5/1451. Google Scholar [20] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS, Providence, 1995. Google Scholar [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. Google Scholar [22] J. P. Tian and J. Wang, Global stability for cholera epidemic models, Math Biosci, 232 (2011), 31-41. doi: 10.1016/j.mbs.2011.04.001. Google Scholar [23] X. Wang, D. Posny and J. Wang, A reaction-convection-diffusion model for cholera spatial dynamics, Dis. Cont. Dyn. Syst. Ser. B, 21 (2016), 2785-2809. doi: 10.3934/dcdsb.2016073. Google Scholar [24] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equ., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. Google Scholar [25] W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Systems, 11 (2012), 1652-1673. doi: 10.1137/120872942. Google Scholar [26] 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. Google Scholar [27] T. Zhang, Minimal wave speed for a class of non-cooperative reaction-diffusion systems of three equations, J. Diff. Equ., 262 (2017), 4724-4770. doi: 10.1016/j.jde.2016.12.017. Google Scholar

show all references

##### References:
 [1] L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profile of the steady states for an SIS epidemic disease reaction-diffusion model, Dis. Cont. Dyn. Syst., 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1. Google Scholar [2] N. Bacaer and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436. doi: 10.1007/s00285-006-0015-0. Google Scholar [3] E. Bertuzzo, On spatially explicit models of cholera epidemics, Journal of the Royal Society Interface, 7 (2010), 321-333. doi: 10.1098/rsif.2009.0204. Google Scholar [4] M. J. Bouma and M. Pascual, Seasonal and interannual cycles of endemic cholera in Bengal 1891-1940 in relation to climate and geography, Hydrobiologia, 460 (2001), 147-156. doi: 10.1007/978-94-017-3284-0_13. Google Scholar [5] F. Capone, C. V. De and L. R. De, 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. Google Scholar [6] F. Capone, C. V. De and L. R. De, Erratum to: Influence of diffusion on the stability of equilibria in a reaction-diffusion system modeling cholera dynamic, J. Math. Biol., 71 (2015), 1267-1268. doi: 10.1007/s00285-015-0915-y. Google Scholar [7] C. T. Codeco, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infect Dis, 1 (2001), p1. doi: 10.1186/1471-2334-1-1. Google Scholar [8] R. R. Colwell and A. Huq, Environmental reservoir of Vibrio cholerae, the causative agent of cholera, Annals of the New York Academy of Sciences, 740 (1994), 44-54. doi: 10.1111/j.1749-6632.1994.tb19852.x. Google Scholar [9] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [10] Z. Du and R. Peng, A priori $L^∞$ estimates for solutions of a class of reaction-diffusion systems, J. Math. Biol., 72 (2016), 1429-1439. doi: 10.1007/s00285-015-0914-z. Google Scholar [11] C. H. Fung, Cholera transmission dynamic models for public health practitioners, Emerging Themes in Epidemiology, 11 (2014), p1. doi: 10.1186/1742-7622-11-1. Google Scholar [12] P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. PDEs, 5 (1980), 999-1030. doi: 10.1080/03605308008820162. Google Scholar [13] E. I. Jury and M. Mansour, Positivity and nonnegativity conditions of a quartic equation and related problems, IEEE Trans. Automat. Contr., 26 (1981), 444-451. doi: 10.1109/TAC.1981.1102589. Google Scholar [14] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, Heidelberg, 1976. Google Scholar [15] H. Li, R. Peng and F. B. Wang, Varying total population enhances disease persistence: Qualitative analysis on a diffusive SIS epidemic model, J. Diff. Equ., 262 (2016), 885-913. doi: 10.1016/j.jde.2016.09.044. Google Scholar [16] B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98. doi: 10.1016/j.mbs.2005.03.008. Google Scholar [17] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154. Google Scholar [18] 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 [19] R. Peng and X.-Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471. doi: 10.1088/0951-7715/25/5/1451. Google Scholar [20] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS, Providence, 1995. Google Scholar [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. Google Scholar [22] J. P. Tian and J. Wang, Global stability for cholera epidemic models, Math Biosci, 232 (2011), 31-41. doi: 10.1016/j.mbs.2011.04.001. Google Scholar [23] X. Wang, D. Posny and J. Wang, A reaction-convection-diffusion model for cholera spatial dynamics, Dis. Cont. Dyn. Syst. Ser. B, 21 (2016), 2785-2809. doi: 10.3934/dcdsb.2016073. Google Scholar [24] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equ., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. Google Scholar [25] W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Systems, 11 (2012), 1652-1673. doi: 10.1137/120872942. Google Scholar [26] 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. Google Scholar [27] T. Zhang, Minimal wave speed for a class of non-cooperative reaction-diffusion systems of three equations, J. Diff. Equ., 262 (2017), 4724-4770. doi: 10.1016/j.jde.2016.12.017. Google Scholar
Biological interpretations for parameters in model (2)
 Symbols Interpretations $N_0$ Total population size at time $t=0$ $d_i$ Diffusion coefficients for $i=1, 2, 3, 4$ $\mu$ Birth/death rate $\sigma$ Recovery rate $\mu_B$ Loss rate of bacteria $\pi_B$ Growth rate of bacteria $\beta(x)$ Contact rate with contaminated water at location $x$ $e(x)$ Contribution of each infected person to the population of V. cholerae
 Symbols Interpretations $N_0$ Total population size at time $t=0$ $d_i$ Diffusion coefficients for $i=1, 2, 3, 4$ $\mu$ Birth/death rate $\sigma$ Recovery rate $\mu_B$ Loss rate of bacteria $\pi_B$ Growth rate of bacteria $\beta(x)$ Contact rate with contaminated water at location $x$ $e(x)$ Contribution of each infected person to the population of V. cholerae
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