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2013, 10(5&6): 1335-1349. doi: 10.3934/mbe.2013.10.1335

Dynamics of an age-of-infection cholera model

Fred Brauer 1, , Zhisheng Shuai 2, and  P. van den Driessche 3,

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.
Keywords: Lyapunov functional, Cholera model, final size., global stability, age-of-infection.
Mathematics Subject Classification: Primary: 92D30; Secondary: 35B35, 35Q9.
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), 3483-3498. doi: 10.1016/j.nonrwa.2011.06.009.

[2]

F. Brauer, Age-of-infection and the final size relation, Math. Biosci. Eng., 5 (2008), 681-690. doi: 10.3934/mbe.2008.5.681.

[3]

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

[4]

F. Brauer, C. Castillo-Chavez and Z. Feng, Discrete epidemic models, Math. Biosc. Eng., 7 (2010), 1-15. doi: 10.3934/mbe.2010.7.1.

[5]

F. Brauer, P. van den Driessche and J. Wu, eds., "Mathematical Epidemiology," Lecture Notes in Math., Vol. 1945, Springer, Berlin, 2008. doi: 10.1007/978-3-540-78911-6.

[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), 7081-7085. doi: 10.1073/pnas.1102149108.

[7]

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

[8]

M. Enserink, Haiti's outbreak is latest in cholera's new global assault, Science, 330 (2010), 738-739. doi: 10.1126/science.330.6005.738.

[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), 63-69. doi: 10.1371/journal.pmed.0030007.

[10]

G. Huang, E. Beretta and Y. Takeuchi, Global stability for epidemic model with constant latency and infectious periods, Math. Biosci. Eng., 9 (2012), 297-312. doi: 10.3934/mbe.2012.9.297.

[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), 25-38. doi: 10.1137/110826588.

[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), 77-109. doi: 10.1016/S0025-5564(98)10057-3.

[13]

R. Koenig, International groups battle cholera in Zimbabwe, Science, 323 (2009), 860-861. doi: 10.1126/science.323.5916.860.

[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), 679-702. doi: 10.1007/s11538-005-9047-7.

[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), 1109-1140. doi: 10.1080/00036810903208122.

[16]

C. C. McCluskey, Delay versus age-of-infection-global stability, Appl. Math. Comput., 217 (2010), 3046-3049.

[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), 8767-8772.

[18]

Z. Shuai and P. van den Driessche, Global dynamics of cholera models with differential infectivity, Math. Biosci., 234 (2010), 118-126. doi: 10.1016/j.mbs.2011.09.003.

[19]

H. L. Smith and H. R. Thieme, "Dynamical Systems and Population Persistence," Graduate Studies in Mathematics, Vol. 118, American Mathematical Society, Providence, 2011.

[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), 1447-1479. doi: 10.1137/0153068.

[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), 1506-1533. doi: 10.1007/s11538-010-9507-6.

[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), 593-601. doi: 10.7326/0003-4819-154-9-201105030-00334.

[23]

G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics," Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, New York, 1985.

[24]

World Health Organization, Cholera annual report 2006, Weekly Epidemiological Record, 82 (2007), 273-284.

[25]

World Health Organization, Cholera: Global surveillance summary 2008, Weekly Epidemiological Record, 84 (2008), 309-324.

[26]

World Health Organization, Cholera fact sheets, August 2011. Available from: http://www.who.int.

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), 3483-3498. doi: 10.1016/j.nonrwa.2011.06.009.

[2]

F. Brauer, Age-of-infection and the final size relation, Math. Biosci. Eng., 5 (2008), 681-690. doi: 10.3934/mbe.2008.5.681.

[3]

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

[4]

F. Brauer, C. Castillo-Chavez and Z. Feng, Discrete epidemic models, Math. Biosc. Eng., 7 (2010), 1-15. doi: 10.3934/mbe.2010.7.1.

[5]

F. Brauer, P. van den Driessche and J. Wu, eds., "Mathematical Epidemiology," Lecture Notes in Math., Vol. 1945, Springer, Berlin, 2008. doi: 10.1007/978-3-540-78911-6.

[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), 7081-7085. doi: 10.1073/pnas.1102149108.

[7]

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

[8]

M. Enserink, Haiti's outbreak is latest in cholera's new global assault, Science, 330 (2010), 738-739. doi: 10.1126/science.330.6005.738.

[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), 63-69. doi: 10.1371/journal.pmed.0030007.

[10]

G. Huang, E. Beretta and Y. Takeuchi, Global stability for epidemic model with constant latency and infectious periods, Math. Biosci. Eng., 9 (2012), 297-312. doi: 10.3934/mbe.2012.9.297.

[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), 25-38. doi: 10.1137/110826588.

[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), 77-109. doi: 10.1016/S0025-5564(98)10057-3.

[13]

R. Koenig, International groups battle cholera in Zimbabwe, Science, 323 (2009), 860-861. doi: 10.1126/science.323.5916.860.

[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), 679-702. doi: 10.1007/s11538-005-9047-7.

[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), 1109-1140. doi: 10.1080/00036810903208122.

[16]

C. C. McCluskey, Delay versus age-of-infection-global stability, Appl. Math. Comput., 217 (2010), 3046-3049.

[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), 8767-8772.

[18]

Z. Shuai and P. van den Driessche, Global dynamics of cholera models with differential infectivity, Math. Biosci., 234 (2010), 118-126. doi: 10.1016/j.mbs.2011.09.003.

[19]

H. L. Smith and H. R. Thieme, "Dynamical Systems and Population Persistence," Graduate Studies in Mathematics, Vol. 118, American Mathematical Society, Providence, 2011.

[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), 1447-1479. doi: 10.1137/0153068.

[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), 1506-1533. doi: 10.1007/s11538-010-9507-6.

[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), 593-601. doi: 10.7326/0003-4819-154-9-201105030-00334.

[23]

G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics," Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, New York, 1985.

[24]

World Health Organization, Cholera annual report 2006, Weekly Epidemiological Record, 82 (2007), 273-284.

[25]

World Health Organization, Cholera: Global surveillance summary 2008, Weekly Epidemiological Record, 84 (2008), 309-324.

[26]

World Health Organization, Cholera fact sheets, August 2011. Available from: http://www.who.int.

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