November  2021, 20(11): 3937-3957. doi: 10.3934/cpaa.2021138

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 Published  November 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 and Applied Analysis, 2021, 20 (11) : 3937-3957. 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. 

[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.

[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.

[4]

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

[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.

[6]

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

[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.

[8]

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

[9]

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

[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.

[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.

[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.

[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.

[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. 

[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.

[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. 

[17]

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

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

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. 

[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.

[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.

[4]

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

[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.

[6]

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

[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.

[8]

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

[9]

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

[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.

[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.

[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.

[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.

[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. 

[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.

[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. 

[17]

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

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

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