Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-16T16:18:28.121Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamics of an infection age-space structured cholera model with Neumann boundary condition

Part of: Partial differential equations Equations of mathematical physics and other areas of application Applications - Dynamical systems and ergodic theory

Published online by Cambridge University Press:  08 April 2021

WEIWEI LIU ,
JINLIANG WANG Open the ORCID record for JINLIANG WANG [Opens in a new window]  and
RAN ZHANG
WEIWEI LIU
Affiliation:
School of Mathematical Science, Heilongjiang University, Harbin 150080, P. R. China emails: [email protected]; [email protected]
JINLIANG WANG
Affiliation:
School of Mathematical Science, Heilongjiang University, Harbin 150080, P. R. China emails: [email protected]; [email protected]
RAN ZHANG
Affiliation:
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing210016, P. R. China email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper investigates global dynamics of an infection age-space structured cholera model. The model describes the vibrio cholerae transmission in human population, where infection-age structure of vibrio cholerae and infectious individuals are incorporated to measure the infectivity during the different stage of disease transmission. The model is described by reaction–diffusion models involving the spatial dispersal of vibrios and the mobility of human populations in the same domain Ω ⊂ ℝn. We first give the well-posedness of the model by converting the model to a reaction–diffusion model and two Volterra integral equations and obtain two constant equilibria. Our result suggest that the basic reproduction number determines the dichotomy of disease persistence and extinction, which is achieved by studying the local stability of equilibria, disease persistence and global attractivity of equilibria.

Keywords

Cholerainfection age-space structured modeluniform persistencebasic reproduction numberglobal stability

MSC classification

Primary: 35B35: Stability 35B40: Asymptotic behavior of solutions 35Q92: PDEs in connection with biology and other natural sciences
Secondary: 37N25: Dynamical systems in biology 92D30: Epidemiology
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 3 , June 2022 , pp. 393 - 422
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ali, M., Nelson, A. R., Lopez, A. L. & Sack, D. A. (2015) Updated global burden of cholera in endemic countries. PLoS Negl. Trop. Dis. 9, e0003832.CrossRefGoogle ScholarPubMed
Adimy, M., Chekroun, A. & Kuniya, T. (2017) Delayed nonlocal reaction diffusion model for hematopoietic stem cell dynamics with Dirichlet boundary conditions. Math. Model Nat. Phenom. 12, 1–22.CrossRefGoogle Scholar
Amann, H. (1976) Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18, 620–709.CrossRefGoogle Scholar
Alam, A., LaRocque, R. C., Harris, J. B., Vanderspurt, C., Ryan, E. T., Qadri, F. & Calderwood, S. B. (2005) Hyperinfectivity of human-passaged Vibrio cholerae can be modelled by growth in the infant mouse. Infect. Immun. 73, 6674–6679.Google Scholar
Brauer, F., Shuai, Z. & van den Driessche, P. (2013) Dynamics of an age-of-infection cholera model. Math. Biosci. Eng. 10, 1335–1349.Google ScholarPubMed
Bertuzzo, E., Maritan, A., Gatto, M., Rodriguez-Iturbe, I. & Rinaldo, A. (2007) River networks and ecological corridors: reactive transport on fractals, migration fronts, hydrochory. Water Resour. Res. 43, W04419.Google Scholar
Cantrell, R. S. & Cosner, C. (2003) Spatial Ecology via Reaction-Diffusion Equations. Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Chichester.CrossRefGoogle Scholar
Chekroun, A. & Kuniya, T. (2020) Global threshold dynamics of an infection age-structured SIR epidemic model with diffusion under the Dirichlet boundary condition. J. Differential Equations 269, 117–148.CrossRefGoogle Scholar
Chekroun, A. & Kuniya, T. (2020) An infection age-space structured SIR epidemic model with neumann boundary condition. Appl. Anal. 99, 1972–1985.CrossRefGoogle Scholar
Chao, D., Longini, I. M. & Morris, J. G. (2003) Modeling cholera outbreaks. In: Cholera Outbreaks. Current Topics in Microbiology and Immunology, 379. Springer, Berlin.Google Scholar
Capasso, V. & Paveri-Fontana, S. L. (1979) A mathematical model for the 1973 cholera epidemic in the European Mediterranean region. Rev. Epidemiol. Sante. 27, 121–132.Google ScholarPubMed
Capone, F., De Cataldis, V. & De Luca, R. (2015) Influence of diffusion on the stability of equilibria in a reaction-diffusion system modeling cholera dynamic. J. Math. Biol. 27, 1107–1131.CrossRefGoogle Scholar
Codeço, C. T. (2001) Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir. BMC Infect. Dis. 1, 1.CrossRefGoogle ScholarPubMed
Carfora, M. F. & Torcicollo, I. (2020) Identification of epidemiological models: the case study of Yemen cholera outbreak. Appl. Anal. DOI: 10.1080/00036811.2020.1738402.CrossRefGoogle Scholar
D’Agata, E. M. C., Magal, P., Ruan, S. & Webb, G. F. (2006) Asymptotic behavior in nosocomial epidemic models with antibiotic resistance. Differ. Int. Equ. 19, 573–600.Google Scholar
Ducrot, A. & Magal, P. (2009) Travelling wave solutions for an infection-age structured model with diffusion. Proc. R. Soc. Edinb. 139, 459–482.CrossRefGoogle Scholar
Ducrot, A. & Magal, P. (2011) Travelling wave solutions for an infection-age structured epidemic model with external supplies. Nonlinearity 24, 2891–2911.CrossRefGoogle Scholar
Diekmann, O., Heesterbeek, J. A. P. & Metz, J. A. J. (1990) On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28, 365–382.CrossRefGoogle ScholarPubMed
Eisenberg, M. C., Shuai, Z., Tien, J.H. & van den Driessche, P. (2013) A cholera model in a patchy environment with water and human movement. Math. Biosci. 246, 105–112.CrossRefGoogle Scholar
Freedman, H.I. & Moson, P. (1990) Persistence definitions and their connections. Proc. Am. Math. Soc. 109, 1025–1033.Google Scholar
Hartley, D. M., Morris, J. G. & Smith, D. L. (2006) Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics? PLoS Med. 3, 63–69.Google ScholarPubMed
Kokomo, E. & Emvudu, Y. (2019) Mathematical analysis and numerical simulation of an age-structured model of cholera with vaccination and demographic movements. Nonlinear Anal.-Real 45, 142–156.CrossRefGoogle Scholar
Mukandavire, Z., Liao, S., Wang, J., Gaff, H., Smith, D. L. & Morris, J. G. (2019) Estimating the reproductive numbers for the 2008–2009 cholera outbreaks in Zimbabwe. Proc. Natl. Acad. Sci. USA 108, 8767–8772.CrossRefGoogle Scholar
Magal, P., Webb, G. F. & Wu, Y. (2019) On the basic reproduction number of reaction-diffusion epidemic models. SIAM J. Appl. Math. 79, 284–304.CrossRefGoogle Scholar
Mendelsohn, J. & Dawson, T. (2008) Climate and cholera in KwaZulu-Natal, South Africa: The role of environmental factors and implications for epidemic preparedness. Int. J. Hyg. Environ. Health 211, 156–162.CrossRefGoogle ScholarPubMed
Nelson, E. J., Harris, J. B., Morris, J. G., Calderwood, S. B. & Camilli, A. (2009) Cholera transmission: The host, pathogen and bacteriophage dynamics. Nat. Rev. Microbiol. 7, 693–702.Google Scholar
Rinaldo, A., Bertuzzo, E., Mari, L., Righetto, L., Blokesch, M., Gatto, M., Casagrandi, R., Murray, M., Vesenbeckh, S.M. & Rodriguez-Iturbe, I. (2012) Reassessment of the 2010–2011 Haiti cholera outbreak and rainfall-driven multiseason projections. Proc. Natl. Acad. Sci. USA 109, 6602–6607.CrossRefGoogle Scholar
Rudin, W. (1976) Principles of Mathematical Analysis (3rd edn). International Series in Pure and Applied Mathematics. McGraw-Hill, New York.Google Scholar
Rudin, W. (1987) Real and Complex Analysis (3rd edn). McGraw-Hill, New York.Google Scholar
Smith, H. L. & Thieme, H. R. (2011) Dynamical Systems and Population Persistence. Graduate Studies in Mathematics, 118. Providence. American Mathematical Society.Google Scholar
Shuai, Z., Tien, J. H. & van den Driessche, P. (2012) Cholera models with hyperinfectivity and temporary immunity. Bull. Math. Biol. 74, 2423–2445.CrossRefGoogle ScholarPubMed
Shuai, Z. & van den Driessche, P. (2011) Global dynamics of cholera models with differential infectivity. Math. Biosci. 234, 118–126.CrossRefGoogle ScholarPubMed
Shuai, Z. & van den Driessche, P. (2015) Modeling and control of cholera on networks with a common water source. J. Biol. Dyn. 9, 90–103.CrossRefGoogle Scholar
Tuite, A. R., Tien, J. H., Eisenberg, M., Earn, D. J. D., Ma, J. & Fisman, D. N. (2011) Cholera epidemic in Haiti, 2010: Using a transmission model to explain spatial spread of disease and identify optimal control interventions. Ann. Internal. Med. 154, 593–601.Google ScholarPubMed
Thieme, H. R. & Castillo-Chavez, C. (1993) How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS? SIAM. J. Appl. Math. 53, 1447–1479.CrossRefGoogle Scholar
Tien, J. H. & Earn, D. J. D. (2010) Multiple transmission pathways and disease dynamics in a waterborne pathogen model. Bull. Math. Biol. 72, 1506–1533.CrossRefGoogle Scholar
Thieme, H. R. (2009) Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J. Appl. Math. 70, 188–211.CrossRefGoogle Scholar
van den Driessche, P. & Watmough, J. (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48.CrossRefGoogle ScholarPubMed
Wang, J., Zhang, R. & Kuniya, T. (2016) A note on dynamics of an age-of-infection cholera model. Math. Biosci. Eng. 13, 227–247.CrossRefGoogle ScholarPubMed
Wang, J., Xie, F. & Kuniya, T. (2020) Analysis of a reaction-diffusion cholera epidemic model in a spatially heterogeneous environment. Commun. Nonlinear Sci. Numer. Simulat. 80, 104951.CrossRefGoogle Scholar
Wang, J. & Wang, J. (2020) Analysis of a reaction-diffusion cholera model with distinct dispersal rates in the human population. J. Dyn. Diff. Equat. Doi: 10.1007/s10884-019-09820-8.Google Scholar
Wang, X., Zhao, X.-Q. & Wang, J. (2018) A cholera epidemic model in a spatiotemporally heterogeneous environment. J. Math. Anal. Appl. 468, 893–912.CrossRefGoogle Scholar
Wang, X., Posny, P. & Wang, J. (2016) A reaction-convection-diffusion model for Cholera spatial dynamics. Discrete Contin. Dyn. Syst. Ser. B 21, 2785–2809.Google Scholar
Wang, X. & Wang, J. (2015) Analysis of cholera epidemics with bacterial growth and spatial movement. J. Math. Anal. Appl. 9, 233–261.Google ScholarPubMed
Wang, X., Gao, D. & Wang, J. (2015) Influence of human behavior on cholera dynamics. Math. Biosci. 267, 41–52.CrossRefGoogle ScholarPubMed
Wang, X. & Wang, J. (2015) Analysis of cholera epidemics with bacterial growth and spatial movement. J. Biol. Dyn. 9, 233–261.CrossRefGoogle ScholarPubMed
Wang, J. & Liao, S. (2012) A generalized cholera model and epidemic/endemic analysis. J. Biol. Dyn. 6, 568–589.CrossRefGoogle ScholarPubMed
Walker, J. A. (1980) Dynamical Systems and Evolution Equations: Theory and Applications. Mathematical Concepts and Methods in Science and Engineering, 20. Plenum Press, New York.CrossRefGoogle Scholar
Yamazaki, K. & Wang, X. (2016) Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model. Discrete Contin. Dyn. Syst. Ser. B 21, 1297–1316.CrossRefGoogle Scholar
Yamazaki, K. & Wang, X. (2017) Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model. Math. Biosci. Eng. 14, 559–579.Google ScholarPubMed
Yang, J., Qiu, Z. & Li, X. (2014) Global stability of an age-structured cholera model. Math. Biosci. Eng. 11, 641–665.Google ScholarPubMed
Yang, J., Xu, R. & Li, J. (2019) Threshold dynamics of an age-space structured brucellosis disease model with Neumann boundary condition. Nonlinear Anal.-Real 50, 192–217.CrossRefGoogle Scholar
Zhang, L., Wang, Z. & Zhang, Y. (2016) Dynamics of a reaction-diffusion waterborne pathogen model with direct and indirect transmission. Comput. Math. Appl. 72, 202–215.CrossRefGoogle Scholar
Zhang, X. & Zhang, Y. (2018) Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. Discrete Contin. Dyn. Syst. Ser. B 23, 2625–2640.Google Scholar
Zhang, L. & Wang, Z. (2016) A time-periodic reaction-diffusion epidemic model with infection period. Z. Angew. Math. Phys. 67, 117.CrossRefGoogle Scholar