• Previous Article
    The mathematical and theoretical biology institute - a model of mentorship through research
  • MBE Home
  • This Issue
  • Next Article
    Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations
2013, 10(5&6): 1335-1349. doi: 10.3934/mbe.2013.10.1335

Dynamics of an age-of-infection cholera model

1. 

Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2

2. 

Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., V8W 3R4, Canada

3. 

Department of Mathematics and Statistics, University of Victoria, Victoria B.C., Canada V8W 3P4

Received  July 2012 Revised  October 2012 Published  August 2013

A new model for the dynamics of cholera is formulated that incorporates both the infection age of infectious individuals and biological age of pathogen in the environment. The basic reproduction number is defined and proved to be a sharp threshold determining whether or not cholera dies out. Final size relations for cholera outbreaks are derived for simplified models when input and death are neglected.
Citation: Fred Brauer, Zhisheng Shuai, P. van den Driessche. Dynamics of an age-of-infection cholera model. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1335-1349. doi: 10.3934/mbe.2013.10.1335
References:
[1]

A. Alexanderian, M. K. Gobbert, K. R. Fister, H. Gaff, S. Lenhart and E. Schaefer, An age-structured model for the spread of epidemic cholera: Analysis and simulation,, Nonlinear Anal. Real World Appl., 12 (2011), 3483.  doi: 10.1016/j.nonrwa.2011.06.009.  Google Scholar

[2]

F. Brauer, Age-of-infection and the final size relation,, Math. Biosci. Eng., 5 (2008), 681.  doi: 10.3934/mbe.2008.5.681.  Google Scholar

[3]

F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology,", Second edition, (2012).  doi: 10.1007/978-1-4614-1686-9.  Google Scholar

[4]

F. Brauer, C. Castillo-Chavez and Z. Feng, Discrete epidemic models,, Math. Biosc. Eng., 7 (2010), 1.  doi: 10.3934/mbe.2010.7.1.  Google Scholar

[5]

F. Brauer, P. van den Driessche and J. Wu, eds., "Mathematical Epidemiology,", Lecture Notes in Math., (1945).  doi: 10.1007/978-3-540-78911-6.  Google Scholar

[6]

D. L. Chao, M. E. Halloran and I. M. Longini, Jr., Vaccination strategies for epidemic cholera in Haiti with implications for the developing world,, Proc. Natl. Acad. Sci. USA, 108 (2011), 7081.  doi: 10.1073/pnas.1102149108.  Google Scholar

[7]

J. M. Cushing, "An Introduction to Structured Population Dynamics,", CBMS-NSF Regional Conference Series in Applied Mathematics, 71 (1998).  doi: 10.1137/1.9781611970005.  Google Scholar

[8]

M. Enserink, Haiti's outbreak is latest in cholera's new global assault,, Science, 330 (2010), 738.  doi: 10.1126/science.330.6005.738.  Google Scholar

[9]

D. M. Hartley, J. G. Morris, Jr. and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLOS Med., 3 (2006), 63.  doi: 10.1371/journal.pmed.0030007.  Google Scholar

[10]

G. Huang, E. Beretta and Y. Takeuchi, Global stability for epidemic model with constant latency and infectious periods,, Math. Biosci. Eng., 9 (2012), 297.  doi: 10.3934/mbe.2012.9.297.  Google Scholar

[11]

G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model,, SIAM J. Appl. Math., 72 (2012), 25.  doi: 10.1137/110826588.  Google Scholar

[12]

J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV,, Math. Biosci., 155 (1999), 77.  doi: 10.1016/S0025-5564(98)10057-3.  Google Scholar

[13]

R. Koenig, International groups battle cholera in Zimbabwe,, Science, 323 (2009), 860.  doi: 10.1126/science.323.5916.860.  Google Scholar

[14]

J. Ma and D. J. D. Earn, Generality of the final size formula for an epidemic of a newly invading infectious disease,, Bull. Math. Biol., 68 (2006), 679.  doi: 10.1007/s11538-005-9047-7.  Google Scholar

[15]

P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model,, Appl. Anal., 89 (2010), 1109.  doi: 10.1080/00036810903208122.  Google Scholar

[16]

C. C. McCluskey, Delay versus age-of-infection-global stability,, Appl. Math. Comput., 217 (2010), 3046.   Google Scholar

[17]

Z. Mukandavire, S. Liao, J. Wang, H. Gaff, D. Smith and J. Morris, Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe,, Proc. Natl. Acad. Sci. USA, 108 (2011), 8767.   Google Scholar

[18]

Z. Shuai and P. van den Driessche, Global dynamics of cholera models with differential infectivity,, Math. Biosci., 234 (2010), 118.  doi: 10.1016/j.mbs.2011.09.003.  Google Scholar

[19]

H. L. Smith and H. R. Thieme, "Dynamical Systems and Population Persistence,", Graduate Studies in Mathematics, (2011).   Google Scholar

[20]

H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM J. Appl. Math., 53 (1993), 1447.  doi: 10.1137/0153068.  Google Scholar

[21]

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.  doi: 10.1007/s11538-010-9507-6.  Google Scholar

[22]

A. R. Tuite, J. H. Tien, M. Eisenberg, D. J. D. Earn, J. 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.  doi: 10.7326/0003-4819-154-9-201105030-00334.  Google Scholar

[23]

G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,", Monographs and Textbooks in Pure and Applied Mathematics, 89 (1985).   Google Scholar

[24]

World Health Organization, Cholera annual report 2006,, Weekly Epidemiological Record, 82 (2007), 273.   Google Scholar

[25]

World Health Organization, Cholera: Global surveillance summary 2008,, Weekly Epidemiological Record, 84 (2008), 309.   Google Scholar

[26]

World Health Organization, Cholera fact sheets,, August 2011. Available from: , (2011).   Google Scholar

show all references

References:
[1]

A. Alexanderian, M. K. Gobbert, K. R. Fister, H. Gaff, S. Lenhart and E. Schaefer, An age-structured model for the spread of epidemic cholera: Analysis and simulation,, Nonlinear Anal. Real World Appl., 12 (2011), 3483.  doi: 10.1016/j.nonrwa.2011.06.009.  Google Scholar

[2]

F. Brauer, Age-of-infection and the final size relation,, Math. Biosci. Eng., 5 (2008), 681.  doi: 10.3934/mbe.2008.5.681.  Google Scholar

[3]

F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology,", Second edition, (2012).  doi: 10.1007/978-1-4614-1686-9.  Google Scholar

[4]

F. Brauer, C. Castillo-Chavez and Z. Feng, Discrete epidemic models,, Math. Biosc. Eng., 7 (2010), 1.  doi: 10.3934/mbe.2010.7.1.  Google Scholar

[5]

F. Brauer, P. van den Driessche and J. Wu, eds., "Mathematical Epidemiology,", Lecture Notes in Math., (1945).  doi: 10.1007/978-3-540-78911-6.  Google Scholar

[6]

D. L. Chao, M. E. Halloran and I. M. Longini, Jr., Vaccination strategies for epidemic cholera in Haiti with implications for the developing world,, Proc. Natl. Acad. Sci. USA, 108 (2011), 7081.  doi: 10.1073/pnas.1102149108.  Google Scholar

[7]

J. M. Cushing, "An Introduction to Structured Population Dynamics,", CBMS-NSF Regional Conference Series in Applied Mathematics, 71 (1998).  doi: 10.1137/1.9781611970005.  Google Scholar

[8]

M. Enserink, Haiti's outbreak is latest in cholera's new global assault,, Science, 330 (2010), 738.  doi: 10.1126/science.330.6005.738.  Google Scholar

[9]

D. M. Hartley, J. G. Morris, Jr. and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLOS Med., 3 (2006), 63.  doi: 10.1371/journal.pmed.0030007.  Google Scholar

[10]

G. Huang, E. Beretta and Y. Takeuchi, Global stability for epidemic model with constant latency and infectious periods,, Math. Biosci. Eng., 9 (2012), 297.  doi: 10.3934/mbe.2012.9.297.  Google Scholar

[11]

G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model,, SIAM J. Appl. Math., 72 (2012), 25.  doi: 10.1137/110826588.  Google Scholar

[12]

J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV,, Math. Biosci., 155 (1999), 77.  doi: 10.1016/S0025-5564(98)10057-3.  Google Scholar

[13]

R. Koenig, International groups battle cholera in Zimbabwe,, Science, 323 (2009), 860.  doi: 10.1126/science.323.5916.860.  Google Scholar

[14]

J. Ma and D. J. D. Earn, Generality of the final size formula for an epidemic of a newly invading infectious disease,, Bull. Math. Biol., 68 (2006), 679.  doi: 10.1007/s11538-005-9047-7.  Google Scholar

[15]

P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model,, Appl. Anal., 89 (2010), 1109.  doi: 10.1080/00036810903208122.  Google Scholar

[16]

C. C. McCluskey, Delay versus age-of-infection-global stability,, Appl. Math. Comput., 217 (2010), 3046.   Google Scholar

[17]

Z. Mukandavire, S. Liao, J. Wang, H. Gaff, D. Smith and J. Morris, Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe,, Proc. Natl. Acad. Sci. USA, 108 (2011), 8767.   Google Scholar

[18]

Z. Shuai and P. van den Driessche, Global dynamics of cholera models with differential infectivity,, Math. Biosci., 234 (2010), 118.  doi: 10.1016/j.mbs.2011.09.003.  Google Scholar

[19]

H. L. Smith and H. R. Thieme, "Dynamical Systems and Population Persistence,", Graduate Studies in Mathematics, (2011).   Google Scholar

[20]

H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM J. Appl. Math., 53 (1993), 1447.  doi: 10.1137/0153068.  Google Scholar

[21]

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.  doi: 10.1007/s11538-010-9507-6.  Google Scholar

[22]

A. R. Tuite, J. H. Tien, M. Eisenberg, D. J. D. Earn, J. 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.  doi: 10.7326/0003-4819-154-9-201105030-00334.  Google Scholar

[23]

G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,", Monographs and Textbooks in Pure and Applied Mathematics, 89 (1985).   Google Scholar

[24]

World Health Organization, Cholera annual report 2006,, Weekly Epidemiological Record, 82 (2007), 273.   Google Scholar

[25]

World Health Organization, Cholera: Global surveillance summary 2008,, Weekly Epidemiological Record, 84 (2008), 309.   Google Scholar

[26]

World Health Organization, Cholera fact sheets,, August 2011. Available from: , (2011).   Google Scholar

[1]

Fred Brauer. Age-of-infection and the final size relation. Mathematical Biosciences & Engineering, 2008, 5 (4) : 681-690. doi: 10.3934/mbe.2008.5.681

[2]

Jinliang Wang, Ran Zhang, Toshikazu Kuniya. A note on dynamics of an age-of-infection cholera model. Mathematical Biosciences & Engineering, 2016, 13 (1) : 227-247. doi: 10.3934/mbe.2016.13.227

[3]

Jianxin Yang, Zhipeng Qiu, Xue-Zhi Li. Global stability of an age-structured cholera model. Mathematical Biosciences & Engineering, 2014, 11 (3) : 641-665. doi: 10.3934/mbe.2014.11.641

[4]

Yuming Chen, Junyuan Yang, Fengqin Zhang. The global stability of an SIRS model with infection age. Mathematical Biosciences & Engineering, 2014, 11 (3) : 449-469. doi: 10.3934/mbe.2014.11.449

[5]

Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859

[6]

Christine K. Yang, Fred Brauer. Calculation of $R_0$ for age-of-infection models. Mathematical Biosciences & Engineering, 2008, 5 (3) : 585-599. doi: 10.3934/mbe.2008.5.585

[7]

Yan-Xia Dang, Zhi-Peng Qiu, Xue-Zhi Li, Maia Martcheva. Global dynamics of a vector-host epidemic model with age of infection. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1159-1186. doi: 10.3934/mbe.2017060

[8]

Kazuo Yamazaki, Xueying Wang. Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 559-579. doi: 10.3934/mbe.2017033

[9]

Jinliang Wang, Xiu Dong. Analysis of an HIV infection model incorporating latency age and infection age. Mathematical Biosciences & Engineering, 2018, 15 (3) : 569-594. doi: 10.3934/mbe.2018026

[10]

Andrey V. Melnik, Andrei Korobeinikov. Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility. Mathematical Biosciences & Engineering, 2013, 10 (2) : 369-378. doi: 10.3934/mbe.2013.10.369

[11]

Shu Liao, Jin Wang. Stability analysis and application of a mathematical cholera model. Mathematical Biosciences & Engineering, 2011, 8 (3) : 733-752. doi: 10.3934/mbe.2011.8.733

[12]

Jinliang Wang, Jiying Lang, Yuming Chen. Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3721-3747. doi: 10.3934/dcdsb.2017186

[13]

Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643

[14]

C. Connell McCluskey. Global stability for an $SEI$ model of infectious disease with age structure and immigration of infecteds. Mathematical Biosciences & Engineering, 2016, 13 (2) : 381-400. doi: 10.3934/mbe.2015008

[15]

Deqiong Ding, Wendi Qin, Xiaohua Ding. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1971-1981. doi: 10.3934/dcdsb.2015.20.1971

[16]

Jinliang Wang, Jiying Lang, Xianning Liu. Global dynamics for viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3215-3233. doi: 10.3934/dcdsb.2015.20.3215

[17]

Yinshu Wu, Wenzhang Huang. Global stability of the predator-prey model with a sigmoid functional response. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019214

[18]

Yu Ji. Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection. Mathematical Biosciences & Engineering, 2015, 12 (3) : 525-536. doi: 10.3934/mbe.2015.12.525

[19]

Julien Arino, Fred Brauer, P. van den Driessche, James Watmough, Jianhong Wu. A final size relation for epidemic models. Mathematical Biosciences & Engineering, 2007, 4 (2) : 159-175. doi: 10.3934/mbe.2007.4.159

[20]

Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with generalized nonlinear incidence and vaccination age. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 977-996. doi: 10.3934/dcdsb.2016.21.977

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (0)

[Back to Top]