# American Institute of Mathematical Sciences

2016, 13(1): 227-247. doi: 10.3934/mbe.2016.13.227

## A note on dynamics of an age-of-infection cholera model

 1 School of Mathematical Science, Heilongjiang University, Harbin 150080, China 2 Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501

Received  March 2014 Revised  July 2015 Published  October 2015

A recent paper [F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 10, 2013, 1335--1349.] presented a model for the dynamics of cholera transmission. The model is incorporated with both the infection age of infectious individuals and biological age of pathogen in the environment. The basic reproduction number is proved to be a sharp threshold determining whether or not cholera dies out. The global stability for disease-free equilibrium and endemic equilibrium is proved by constructing suitable Lyapunov functionals. However, for the proof of the global stability of endemic equilibrium, we have to show first the relative compactness of the orbit generated by model in order to make use of the invariance principle. Furthermore, uniform persistence of system must be shown since the Lyapunov functional is possible to be infinite if $i(a, t)/i^* (a) =0$ on some age interval. In this note, we give a supplement to above paper with necessary mathematical arguments.
Citation: 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
##### References:
 [1] F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model,, Math. Biosci. Eng., 10 (2013), 1335. doi: 10.3934/mbe.2013.10.1335. Google Scholar [2] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs 25, (1988). Google Scholar [3] 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 [4] 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 [5] C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes,, Math. Biosci. Eng., 9 (2012), 819. doi: 10.3934/mbe.2012.9.819. Google Scholar [6] H. L. Smith, Mathematics in Population Biology,, Princeton University Press, (2003). Google Scholar [7] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence,, Amer. Math. Soc., (2011). Google Scholar [8] J. A. Walker, Dynamical Systems and Evolution Equations,, Plenum Press, (1980). Google Scholar [9] J. Wang, R. Zhang and T. Kuniya, The stability analysis of an SVEIR model with continuous age-structure in the exposed and infectious classes,, J. Biol. Dyna., 9 (2015), 73. doi: 10.1080/17513758.2015.1006696. Google Scholar [10] J. Wang, R. Zhang and T. Kuniya, Mathematical analysis for an age-structured HIV infection model with saturation infection rate,, Electron. J. Diff. Equ., 2015 (2015), 1. Google Scholar [11] J. Wang, R. Zhang and T. Kuniya, Global dynamics for a class of age-infection HIV models with nonlinear infection rate,, J. Math. Anal. Appl., 432 (2015), 289. doi: 10.1016/j.jmaa.2015.06.040. Google Scholar [12] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, Marcel Dekker, (1985). Google Scholar [13] J. Yang, Z. Qiu and X. Li, Global stability of an age-structured cholera model,, Math. Biosci. Eng., 11 (2014), 641. doi: 10.3934/mbe.2014.11.641. Google Scholar

show all references

##### References:
 [1] F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model,, Math. Biosci. Eng., 10 (2013), 1335. doi: 10.3934/mbe.2013.10.1335. Google Scholar [2] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs 25, (1988). Google Scholar [3] 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 [4] 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 [5] C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes,, Math. Biosci. Eng., 9 (2012), 819. doi: 10.3934/mbe.2012.9.819. Google Scholar [6] H. L. Smith, Mathematics in Population Biology,, Princeton University Press, (2003). Google Scholar [7] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence,, Amer. Math. Soc., (2011). Google Scholar [8] J. A. Walker, Dynamical Systems and Evolution Equations,, Plenum Press, (1980). Google Scholar [9] J. Wang, R. Zhang and T. Kuniya, The stability analysis of an SVEIR model with continuous age-structure in the exposed and infectious classes,, J. Biol. Dyna., 9 (2015), 73. doi: 10.1080/17513758.2015.1006696. Google Scholar [10] J. Wang, R. Zhang and T. Kuniya, Mathematical analysis for an age-structured HIV infection model with saturation infection rate,, Electron. J. Diff. Equ., 2015 (2015), 1. Google Scholar [11] J. Wang, R. Zhang and T. Kuniya, Global dynamics for a class of age-infection HIV models with nonlinear infection rate,, J. Math. Anal. Appl., 432 (2015), 289. doi: 10.1016/j.jmaa.2015.06.040. Google Scholar [12] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, Marcel Dekker, (1985). Google Scholar [13] J. Yang, Z. Qiu and X. Li, Global stability of an age-structured cholera model,, Math. Biosci. Eng., 11 (2014), 641. doi: 10.3934/mbe.2014.11.641. Google Scholar
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