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doi: 10.3934/cpaa.2021138
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Analysis of a diffusive cholera model incorporating latency and bacterial hyperinfectivity

School of Mathematical Science, Heilongjiang University, Harbin 150080, China

* Corresponding author

Received  February 2020 Revised  April 2021 Early access August 2021

Fund Project: J. Wang was supported by National Natural Science Foundation of China (nos. 12071115, 11871179) and Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems

In this paper, we are concerned with the threshold dynamics of a diffusive cholera model incorporating latency and bacterial hyperinfectivity. Our model takes the form of spatially nonlocal reaction-diffusion system associated with zero-flux boundary condition and time delay. By studying the associated eigenvalue problem, we establish the threshold dynamics that determines whether or not cholera will spread. We also confirm that the threshold dynamics can be determined by the basic reproduction number. By constructing Lyapunov functional, we address the global attractivity of the unique positive equilibrium whenever it exists. The theoretical results are still hold for the case when the constant parameters are replaced by strictly positive and spatial dependent functions.

Citation: Wei Yang, Jinliang Wang. Analysis of a diffusive cholera model incorporating latency and bacterial hyperinfectivity. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021138
References:
[1]

J. R. Andrews and S. Basu, Transmission dynamics and control of cholera in Haiti: an epidemic model, Lancet, 377 (2011), 1248-1255.   Google Scholar

[2]

F. BrauerZ. 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.  Google Scholar

[3]

F. CaponeV. 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.  Google Scholar

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C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infect. Dis., 1 (2001), 1. Google Scholar

[5]

M. C. EisenbergZ. ShuaiJ. 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.  Google Scholar

[6]

J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988. doi: 10.1090/surv/025.  Google Scholar

[7]

D. M. Hartley, J. G. Jr Morris and D. L. Smith, Hyperinfectivity: a critical element in the ability of V. cholerae to cause epidemics?, PloS Med., 3 (2006), e7. Google Scholar

[8]

S. It$ \hat o $, Diffusion Equations, American Mathematical Society, Providence, RI, 1992. doi: 10.1090/mmono/114.  Google Scholar

[9]

K. KoelleX. RodM. PascualM. Yunus and G. Mostafa, Refractory periods and climate forcing in cholera dynamics, Nature, 436 (2005), 696-700.   Google Scholar

[10]

F. Li and X. Q. Zhao, Global dynamics of a nonlocal periodic reaction-diffusion model of bluetongue disease, J. Differ. Equ., 272 (2021) 127–163. doi: 10.1016/j.jde.2020.09.019.  Google Scholar

[11]

Y. Lou and X. Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8.  Google Scholar

[12]

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

[13]

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

[14]

Z. MukandavireS. LiaoJ. WangH. GaffD. L. Smith and J. G. Morris, Estimating the reproductive numbers for the 2008–2009 cholera outbreaks in Zimbabwe, Proc. Nat. Acad. Sci. USA, 108 (2011), 8767-8772.   Google Scholar

[15]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer, New York, 1986. doi: 10.1007/978-3-642-93287-8_2.  Google Scholar

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E. J. NelsonJ. B. HarrisJ. G. MorrisS. B. Calderwood and A. Camilli, Cholera transmission: The host, pathogen and bacteriophage dynamics, Nat. Rev.: Microbiol., 7 (2009), 693-702.   Google Scholar

[17]

M. H. Protter, H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag (1984). doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[18]

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

[19]

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.  Google Scholar

[20]

A. R. TuiteJ. H. TienM. EisenbergD. J. D. EarnJ. Ma and D. N. Fisman, Cholera epidemic in Haiti, 2010: Using a transmission model to explain spatial spread of disease and identify optimal control interventions, Ann. Internal Med., 154 (2011), 593-601.   Google Scholar

[21]

H. R. Thieme and X. Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. RWA, 2 (2001), 145-160.  doi: 10.1016/S0362-546X(00)00112-7.  Google Scholar

[22]

J. WangR. Zhang and T. Kuniya, A note on dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 13 (2016), 227-247.  doi: 10.3934/mbe.2016.13.227.  Google Scholar

[23]

J. Wang and J. Wang, Analysis of a reaction-diffusion cholera model with distinct dispersal rates in the human population, J. Dyn. Differ. Equ., 33 (2021), 549-575.  doi: 10.1007/s10884-019-09820-8.  Google Scholar

[24]

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

[25]

X. Wang and F. B. Wang, Impact of bacterial hyperinfectivity on cholera epidemics in a spatially heterogeneous environment, J. Math. Anal. Appl., 480 (2019), 123407. doi: 10.1016/j.jmaa.2019.123407.  Google Scholar

[26]

J. Wu, Theory and applications of partial functional differential equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[27]

J. YangZ. Qiu and X. Li, Global stability of an age-structured cholera model, Math. Biosci. Eng., 11 (2014), 641-665.  doi: 10.3934/mbe.2014.11.641.  Google Scholar

show all references

References:
[1]

J. R. Andrews and S. Basu, Transmission dynamics and control of cholera in Haiti: an epidemic model, Lancet, 377 (2011), 1248-1255.   Google Scholar

[2]

F. BrauerZ. 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.  Google Scholar

[3]

F. CaponeV. 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.  Google Scholar

[4]

C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infect. Dis., 1 (2001), 1. Google Scholar

[5]

M. C. EisenbergZ. ShuaiJ. 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.  Google Scholar

[6]

J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988. doi: 10.1090/surv/025.  Google Scholar

[7]

D. M. Hartley, J. G. Jr Morris and D. L. Smith, Hyperinfectivity: a critical element in the ability of V. cholerae to cause epidemics?, PloS Med., 3 (2006), e7. Google Scholar

[8]

S. It$ \hat o $, Diffusion Equations, American Mathematical Society, Providence, RI, 1992. doi: 10.1090/mmono/114.  Google Scholar

[9]

K. KoelleX. RodM. PascualM. Yunus and G. Mostafa, Refractory periods and climate forcing in cholera dynamics, Nature, 436 (2005), 696-700.   Google Scholar

[10]

F. Li and X. Q. Zhao, Global dynamics of a nonlocal periodic reaction-diffusion model of bluetongue disease, J. Differ. Equ., 272 (2021) 127–163. doi: 10.1016/j.jde.2020.09.019.  Google Scholar

[11]

Y. Lou and X. Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8.  Google Scholar

[12]

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

[13]

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

[14]

Z. MukandavireS. LiaoJ. WangH. GaffD. L. Smith and J. G. Morris, Estimating the reproductive numbers for the 2008–2009 cholera outbreaks in Zimbabwe, Proc. Nat. Acad. Sci. USA, 108 (2011), 8767-8772.   Google Scholar

[15]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer, New York, 1986. doi: 10.1007/978-3-642-93287-8_2.  Google Scholar

[16]

E. J. NelsonJ. B. HarrisJ. G. MorrisS. B. Calderwood and A. Camilli, Cholera transmission: The host, pathogen and bacteriophage dynamics, Nat. Rev.: Microbiol., 7 (2009), 693-702.   Google Scholar

[17]

M. H. Protter, H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag (1984). doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[18]

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

[19]

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.  Google Scholar

[20]

A. R. TuiteJ. H. TienM. EisenbergD. J. D. EarnJ. Ma and D. N. Fisman, Cholera epidemic in Haiti, 2010: Using a transmission model to explain spatial spread of disease and identify optimal control interventions, Ann. Internal Med., 154 (2011), 593-601.   Google Scholar

[21]

H. R. Thieme and X. Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. RWA, 2 (2001), 145-160.  doi: 10.1016/S0362-546X(00)00112-7.  Google Scholar

[22]

J. WangR. Zhang and T. Kuniya, A note on dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 13 (2016), 227-247.  doi: 10.3934/mbe.2016.13.227.  Google Scholar

[23]

J. Wang and J. Wang, Analysis of a reaction-diffusion cholera model with distinct dispersal rates in the human population, J. Dyn. Differ. Equ., 33 (2021), 549-575.  doi: 10.1007/s10884-019-09820-8.  Google Scholar

[24]

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

[25]

X. Wang and F. B. Wang, Impact of bacterial hyperinfectivity on cholera epidemics in a spatially heterogeneous environment, J. Math. Anal. Appl., 480 (2019), 123407. doi: 10.1016/j.jmaa.2019.123407.  Google Scholar

[26]

J. Wu, Theory and applications of partial functional differential equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[27]

J. YangZ. Qiu and X. Li, Global stability of an age-structured cholera model, Math. Biosci. Eng., 11 (2014), 641-665.  doi: 10.3934/mbe.2014.11.641.  Google Scholar

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