# American Institute of Mathematical Sciences

2014, 11(3): 641-665. doi: 10.3934/mbe.2014.11.641

## Global stability of an age-structured cholera model

 1 Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, China, China 2 Department of Mathematics, Xinyang Normal University, Xinyang 464000

Received  November 2012 Revised  August 2013 Published  January 2013

In this paper, an age-structured epidemic model is formulated to describe the transmission dynamics of cholera. The PDE model incorporates direct and indirect transmission pathways, infection-age-dependent infectivity and variable periods of infectiousness. Under some suitable assumptions, the PDE model can be reduced to the multi-stage models investigated in the literature. By using the method of Lyapunov function, we established the dynamical properties of the PDE model, and the results show that the global dynamics of the model is completely determined by the basic reproduction number $\mathcal R_0$: if $\mathcal R_0 < 1$ the cholera dies out, and if $\mathcal R_0 >1$ the disease will persist at the endemic equilibrium. Then the global results obtained for multi-stage models are extended to the general continuous age model.
Citation: 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
##### References:
 [1] J. R. Andrews and S. Basu, Transmission dynamics and control of cholera in Haiti: An epidemic model, Lancet, 377 (2011), 1248-1255. doi: 10.1016/S0140-6736(11)60273-0. [2] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and computation of the basic reproduction ratio $R_0$ in models for infections diseases in hetereogeneous populations, J. Math. Biol., 28 (1998), 365-382. doi: 10.1007/BF00178324. [3] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infections Disease: Model Building, Analysis and Interpretation, Wiley, New York, 2000. [4] Z. L. Feng, W. Z. Huang and C. Castillo-Chavez, Global behavior of a multi-group SIS epidemic model with age structure, J. Diff. Equs., 218 (2005), 292-324. doi: 10.1016/j.jde.2004.10.009. [5] H. I. Freedman and J. W. H. So, Global stability and persistence of simple food chains, Math. Biosci., 76 (1985), 69-86. doi: 10.1016/0025-5564(85)90047-1. [6] B. S. Goh, Global stability in many-species systems, Amer. Natur., 111 (1977), 135-143. doi: 10.1086/283144. [7] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, American Mathematical Society, Providence, RI, 1988. [8] J. K. Hale and P. Waltman, Persistence in infinite dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025. [9] J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems: Mathematical Aspects of Selection, Cambridge University Press, Cambridge, 1988. [10] G. Huang, X. N. Liu and Y. Takeuchi, Lyapunov function and global stability for age-structured HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-38. doi: 10.1137/110826588. [11] M. Iannelli, Mathematical Theory of Age-structured Population Dynamics, Applied Mathematics Monographs 7, comitato nazionale per le scienze matematiche, Consiglio Nazionale delle Ricerche (C. N. R), Giardini, Pisa, 1995. [12] H. Inaba, A semigroup approach to the strong ergodic theorem of the multistate stable population process, Math. Popul. Studi., 17 (1988), 47-77. doi: 10.1080/08898488809525260. [13] A. L. Lloyd, Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods, Proc. Roy. Soc. Lond. B, 268 (2001), 985-993. doi: 10.1098/rspb.2001.1599. [14] A. L. Lloyd, Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics, Theor. Popul. Biol., 60 (2001), 59-71. doi: 10.1006/tpbi.2001.1525. [15] P. Magal, Compact attractors for time periodic age-structured population models, Electron. J. Diff. Equs., 65 (2001), 1-35. [16] P. Magal and H. R. Thieme, Eventual compactness for a semiflow generated by an age-structured models, Communications on Pure and Applied Analysis, 3 (2004), 695-727. doi: 10.3934/cpaa.2004.3.695. [17] P. Magal and X. Q. Zhao, Global attracotor in uniformly persistence dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173. [18] E. D'Agata, P. Magal, S. Ruan and G. F. Webb, Asymptotical behavior in nosocomial epidemic model with antibiotic resistance, Diff. Integr. Equs., 19 (2006), 573-600. [19] P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140. doi: 10.1080/00036810903208122. [20] P. Magal and C. McCluskey, Two group infection age model: an application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095. doi: 10.1137/120882056. [21] F. A. Milner and A. Pugliese, Periodic solutions: a robust numerical method for an SIR model of epidemics, J. Math. Biol., 39 (1999), 471-492. doi: 10.1007/s002850050175. [22] Z. S. Shuai and P. Van den Driessche, Global dynamics of cholera models with differential infectivity, Math. Biosci., 234 (2011), 118-126. doi: 10.1016/j.mbs.2011.09.003. [23] Z. S. Shuai, J. H. Tien and P. van den Driessche, Cholera models with hyperinfectivity and temporary immunity, Bull. Math. Biol., 74 (2010), 2423-2445. doi: 10.1007/s11538-012-9759-4. [24] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Diff. Integr. Equs., 3 (1990), 1035-1066. [25] H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics if HIV/AIDs? SIAM J. Appl. Math., 53 (1993), 1447-1479. doi: 10.1137/0153068. [26] J. P. Tian, S. Liao and J. Wang, Dynamical Analysis and Control Strategies in Modeling Cholera, 2010. Available from: http://www.math.wm.edu/~jptian/preprints/pr-7-ode-cholera.pdf. [27] J. P. Tian and J. Wang, Global stability for cholera epidemic model, Math. Biosci., 232 (2011), 31-41. doi: 10.1016/j.mbs.2011.04.001. [28] 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. [29] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [30] G. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.

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. doi: 10.1016/S0140-6736(11)60273-0. [2] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and computation of the basic reproduction ratio $R_0$ in models for infections diseases in hetereogeneous populations, J. Math. Biol., 28 (1998), 365-382. doi: 10.1007/BF00178324. [3] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infections Disease: Model Building, Analysis and Interpretation, Wiley, New York, 2000. [4] Z. L. Feng, W. Z. Huang and C. Castillo-Chavez, Global behavior of a multi-group SIS epidemic model with age structure, J. Diff. Equs., 218 (2005), 292-324. doi: 10.1016/j.jde.2004.10.009. [5] H. I. Freedman and J. W. H. So, Global stability and persistence of simple food chains, Math. Biosci., 76 (1985), 69-86. doi: 10.1016/0025-5564(85)90047-1. [6] B. S. Goh, Global stability in many-species systems, Amer. Natur., 111 (1977), 135-143. doi: 10.1086/283144. [7] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, American Mathematical Society, Providence, RI, 1988. [8] J. K. Hale and P. Waltman, Persistence in infinite dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025. [9] J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems: Mathematical Aspects of Selection, Cambridge University Press, Cambridge, 1988. [10] G. Huang, X. N. Liu and Y. Takeuchi, Lyapunov function and global stability for age-structured HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-38. doi: 10.1137/110826588. [11] M. Iannelli, Mathematical Theory of Age-structured Population Dynamics, Applied Mathematics Monographs 7, comitato nazionale per le scienze matematiche, Consiglio Nazionale delle Ricerche (C. N. R), Giardini, Pisa, 1995. [12] H. Inaba, A semigroup approach to the strong ergodic theorem of the multistate stable population process, Math. Popul. Studi., 17 (1988), 47-77. doi: 10.1080/08898488809525260. [13] A. L. Lloyd, Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods, Proc. Roy. Soc. Lond. B, 268 (2001), 985-993. doi: 10.1098/rspb.2001.1599. [14] A. L. Lloyd, Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics, Theor. Popul. Biol., 60 (2001), 59-71. doi: 10.1006/tpbi.2001.1525. [15] P. Magal, Compact attractors for time periodic age-structured population models, Electron. J. Diff. Equs., 65 (2001), 1-35. [16] P. Magal and H. R. Thieme, Eventual compactness for a semiflow generated by an age-structured models, Communications on Pure and Applied Analysis, 3 (2004), 695-727. doi: 10.3934/cpaa.2004.3.695. [17] P. Magal and X. Q. Zhao, Global attracotor in uniformly persistence dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173. [18] E. D'Agata, P. Magal, S. Ruan and G. F. Webb, Asymptotical behavior in nosocomial epidemic model with antibiotic resistance, Diff. Integr. Equs., 19 (2006), 573-600. [19] P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140. doi: 10.1080/00036810903208122. [20] P. Magal and C. McCluskey, Two group infection age model: an application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095. doi: 10.1137/120882056. [21] F. A. Milner and A. Pugliese, Periodic solutions: a robust numerical method for an SIR model of epidemics, J. Math. Biol., 39 (1999), 471-492. doi: 10.1007/s002850050175. [22] Z. S. Shuai and P. Van den Driessche, Global dynamics of cholera models with differential infectivity, Math. Biosci., 234 (2011), 118-126. doi: 10.1016/j.mbs.2011.09.003. [23] Z. S. Shuai, J. H. Tien and P. van den Driessche, Cholera models with hyperinfectivity and temporary immunity, Bull. Math. Biol., 74 (2010), 2423-2445. doi: 10.1007/s11538-012-9759-4. [24] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Diff. Integr. Equs., 3 (1990), 1035-1066. [25] H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics if HIV/AIDs? SIAM J. Appl. Math., 53 (1993), 1447-1479. doi: 10.1137/0153068. [26] J. P. Tian, S. Liao and J. Wang, Dynamical Analysis and Control Strategies in Modeling Cholera, 2010. Available from: http://www.math.wm.edu/~jptian/preprints/pr-7-ode-cholera.pdf. [27] J. P. Tian and J. Wang, Global stability for cholera epidemic model, Math. Biosci., 232 (2011), 31-41. doi: 10.1016/j.mbs.2011.04.001. [28] 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. [29] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [30] G. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.
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