# American Institute of Mathematical Sciences

December  2017, 14(5&6): 1159-1186. doi: 10.3934/mbe.2017060

## Global dynamics of a vector-host epidemic model with age of infection

 1 Department of Public Education, Zhumadian Vocational and Technical College, Zhumadian 463000, China 2 School of Science, Nanjing University of Science and Technology, Nanjing 210094, China 3 Department of Mathematics and Physics, Anyang Institute of Technology, Anyang 455000, China 4 Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105, Gainesville, FL 32611-8105, USA

* Corresponding author: Xue-Zhi Li

Received  July 2016 Accepted  December 2016 Published  May 2017

In this paper, a partial differential equation (PDE) model is proposed to explore the transmission dynamics of vector-borne diseases. The model includes both incubation age of the exposed hosts and infection age of the infectious hosts which describe incubation-age dependent removal rates in the latent period and the variable infectiousness in the infectious period, respectively. The reproductive number $\mathcal R_0$ is derived. By using the method of Lyapunov function, the global dynamics of the PDE model is further established, and the results show that the basic reproduction number $\mathcal R_0$ determines the transmission dynamics of vector-borne diseases: the disease-free equilibrium is globally asymptotically stable if $\mathcal R_0≤ 1$, and the endemic equilibrium is globally asymptotically stable if $\mathcal{R}_0>1$. The results suggest that an effective strategy to contain vector-borne diseases is decreasing the basic reproduction number $\mathcal{R}_0$ below one.

Citation: 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
##### References:
 [1] [2] [3] R. M. Anderson and R. M. May, Infectious Disease of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991. [4] C. Bowman, A. B. Gumel, J. Wu, P. V. Driessche and H. Zhu, A mathematical model for assessing control strategies against West Nile virus, Bull. Math. Biol., 67 (2005), 1107-1133.  doi: 10.1016/j.bulm.2005.01.002. [5] F. Brauer, P. V. Driessche and J. Wu, eds., Mathematical Epidemiology, Lecture Notes in Math, Springer, Berlin, 2008. doi: 10.1007/978-3-540-78911-6. [6] F. Brauer, Z. S. Shuai and P. V. Driessche, Dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 10 (2013), 1335-1349.  doi: 10.3934/mbe.2013.10.1335. [7] S. Busenberg, K. Cooke and M. Iannelli, Endemic threshold and stability in a class of age-structured epidemics, SIAM J. Appl. Math., 48 (1988), 1379-1395.  doi: 10.1137/0148085. [8] S. N. Busenberg, M. Iannelli and H. R. Thieme, Global behavior of an age-structured epdiemic model, SIAM J. Math. Anal., 22 (1991), 1065-1080.  doi: 10.1137/0522069. [9] S. Busenberg, M. Iannelli and H. Thieme, Dynamics of an age structured epidemic model, in: S.T. Liao, Y.Q. Ye, T.R. Ding (Eds.), Dynamical Systems, Nankai Series in Pure, Applied Mathematics and Theoretical Physics, 4 (1993), 1-19.  doi: 10.1007/978-3-642-75301-5_1. [10] C. Castillo-Chavez and Z. Feng, Global stability of an age-structure model for TB and its applications to optimal vaccination strategies, Math. Biosci., 151 (1998), 135-154. [11] Y. Cha, M. Iannelli and F. A. Milner, Existence and uniqueness of endemic states for the age-structured SIR epidemic model, Math. Biosci., 150 (1998), 177-190.  doi: 10.1016/S0025-5564(98)10006-8. [12] Z. L. Feng and J. X. Velasco-Hernandez, Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol., 35 (1997), 523-544.  doi: 10.1007/s002850050064. [13] Z. Feng, W. 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. [14] H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup sir epidemic models, Can. Appl. Math. Q., 14 (2006), 259-284. [15] H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802.  doi: 10.1090/S0002-9939-08-09341-6. [16] H. Inaba and H. Sekine, A mathematical model for chagas disease with infection-age-dependent infectivity, Math, Biosci., 190 (2004), 39-69.  doi: 10.1016/j.mbs.2004.02.004. [17] H. Inaba, Mathematical analysis of an age-structured SIR epidemic model with vertical transmission, Dis. Con. Dyn. Sys. B, 6 (2006), 69-96.  doi: 10.3934/dcdsb.2006.6.69. [18] M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Diff. Equs., 248 (2010), 1-20.  doi: 10.1016/j.jde.2009.09.003. [19] 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. [20] 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. [21] 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. [22] 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. [23] M. Martcheva and F. Hoppensteadt, India's approach to eliminating plasmodium falciparum malaria: A Modeling perspective, J. Biol. Systems, 18 (2010), 867-891.  doi: 10.1142/S0218339010003706. [24] M. Martcheva and X. Z. Li, Competitive exclusion in an infection-age structured model with environmental transmission, Math. Anal. Appl., 408 (2013), 225-246.  doi: 10.1016/j.jmaa.2013.05.064. [25] M. Martcheva and H. R. Thieme, Progression age enhanced backward bifurcation in an epidemic model with super-infection, J. Math. Biol., 46 (2003), 385-424.  doi: 10.1007/s00285-002-0181-7. [26] G. C. Pachecoa, L. Estevab, J. A. Montano-Hirosec and C. Vargasd, Modelling the dynamics of West Nile Virus, Bull. Math. Biol., 67 (2005), 1157-1172.  doi: 10.1016/j.bulm.2004.11.008. [27] Z. P. Qiu, Q. K. Kong, X. Z. Li and M. M. Martcheva, The vector-host epidemic model with multiple strains in a patchy environment, J. Math. Anal. Appl., 405 (2013), 12-36.  doi: 10.1016/j.jmaa.2013.03.042. [28] Z. P. Qiu and Z. L. Feng, Transmission dynamics of an influenza model with age of infection and antiviral treatment, J. Dyn. Diff. Equat., 22 (2010), 823-851.  doi: 10.1007/s10884-010-9178-x. [29] 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. [30] J. Tumwiine, J. Y. Mugisha and L. S. Luboobi, A mathematical model for the dynamics of malaria in a human host and mosquito vector with temporary immunity, Appl. Math. Comput., 189 (2007), 1953-1965.  doi: 10.1016/j.amc.2006.12.084. [31] J. X. Yang, Z. P. Qiu and X. Z. Li, Global stability of an age-structured cholera model, Math. Biol. Enger., 11 (2014), 641-665.  doi: 10.3934/mbe.2014.11.641. [32] P. Zhang, Z. Feng and F. Milner, A schistosomiasis model with an age-structure in human hosts and its application to treatment strategies, Math. Biosci., 205 (2007), 83-107.  doi: 10.1016/j.mbs.2006.06.006.

show all references

##### References:
 [1] [2] [3] R. M. Anderson and R. M. May, Infectious Disease of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991. [4] C. Bowman, A. B. Gumel, J. Wu, P. V. Driessche and H. Zhu, A mathematical model for assessing control strategies against West Nile virus, Bull. Math. Biol., 67 (2005), 1107-1133.  doi: 10.1016/j.bulm.2005.01.002. [5] F. Brauer, P. V. Driessche and J. Wu, eds., Mathematical Epidemiology, Lecture Notes in Math, Springer, Berlin, 2008. doi: 10.1007/978-3-540-78911-6. [6] F. Brauer, Z. S. Shuai and P. V. Driessche, Dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 10 (2013), 1335-1349.  doi: 10.3934/mbe.2013.10.1335. [7] S. Busenberg, K. Cooke and M. Iannelli, Endemic threshold and stability in a class of age-structured epidemics, SIAM J. Appl. Math., 48 (1988), 1379-1395.  doi: 10.1137/0148085. [8] S. N. Busenberg, M. Iannelli and H. R. Thieme, Global behavior of an age-structured epdiemic model, SIAM J. Math. Anal., 22 (1991), 1065-1080.  doi: 10.1137/0522069. [9] S. Busenberg, M. Iannelli and H. Thieme, Dynamics of an age structured epidemic model, in: S.T. Liao, Y.Q. Ye, T.R. Ding (Eds.), Dynamical Systems, Nankai Series in Pure, Applied Mathematics and Theoretical Physics, 4 (1993), 1-19.  doi: 10.1007/978-3-642-75301-5_1. [10] C. Castillo-Chavez and Z. Feng, Global stability of an age-structure model for TB and its applications to optimal vaccination strategies, Math. Biosci., 151 (1998), 135-154. [11] Y. Cha, M. Iannelli and F. A. Milner, Existence and uniqueness of endemic states for the age-structured SIR epidemic model, Math. Biosci., 150 (1998), 177-190.  doi: 10.1016/S0025-5564(98)10006-8. [12] Z. L. Feng and J. X. Velasco-Hernandez, Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol., 35 (1997), 523-544.  doi: 10.1007/s002850050064. [13] Z. Feng, W. 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. [14] H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup sir epidemic models, Can. Appl. Math. Q., 14 (2006), 259-284. [15] H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802.  doi: 10.1090/S0002-9939-08-09341-6. [16] H. Inaba and H. Sekine, A mathematical model for chagas disease with infection-age-dependent infectivity, Math, Biosci., 190 (2004), 39-69.  doi: 10.1016/j.mbs.2004.02.004. [17] H. Inaba, Mathematical analysis of an age-structured SIR epidemic model with vertical transmission, Dis. Con. Dyn. Sys. B, 6 (2006), 69-96.  doi: 10.3934/dcdsb.2006.6.69. [18] M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Diff. Equs., 248 (2010), 1-20.  doi: 10.1016/j.jde.2009.09.003. [19] 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. [20] 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. [21] 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. [22] 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. [23] M. Martcheva and F. Hoppensteadt, India's approach to eliminating plasmodium falciparum malaria: A Modeling perspective, J. Biol. Systems, 18 (2010), 867-891.  doi: 10.1142/S0218339010003706. [24] M. Martcheva and X. Z. Li, Competitive exclusion in an infection-age structured model with environmental transmission, Math. Anal. Appl., 408 (2013), 225-246.  doi: 10.1016/j.jmaa.2013.05.064. [25] M. Martcheva and H. R. Thieme, Progression age enhanced backward bifurcation in an epidemic model with super-infection, J. Math. Biol., 46 (2003), 385-424.  doi: 10.1007/s00285-002-0181-7. [26] G. C. Pachecoa, L. Estevab, J. A. Montano-Hirosec and C. Vargasd, Modelling the dynamics of West Nile Virus, Bull. Math. Biol., 67 (2005), 1157-1172.  doi: 10.1016/j.bulm.2004.11.008. [27] Z. P. Qiu, Q. K. Kong, X. Z. Li and M. M. Martcheva, The vector-host epidemic model with multiple strains in a patchy environment, J. Math. Anal. Appl., 405 (2013), 12-36.  doi: 10.1016/j.jmaa.2013.03.042. [28] Z. P. Qiu and Z. L. Feng, Transmission dynamics of an influenza model with age of infection and antiviral treatment, J. Dyn. Diff. Equat., 22 (2010), 823-851.  doi: 10.1007/s10884-010-9178-x. [29] 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. [30] J. Tumwiine, J. Y. Mugisha and L. S. Luboobi, A mathematical model for the dynamics of malaria in a human host and mosquito vector with temporary immunity, Appl. Math. Comput., 189 (2007), 1953-1965.  doi: 10.1016/j.amc.2006.12.084. [31] J. X. Yang, Z. P. Qiu and X. Z. Li, Global stability of an age-structured cholera model, Math. Biol. Enger., 11 (2014), 641-665.  doi: 10.3934/mbe.2014.11.641. [32] P. Zhang, Z. Feng and F. Milner, A schistosomiasis model with an age-structure in human hosts and its application to treatment strategies, Math. Biosci., 205 (2007), 83-107.  doi: 10.1016/j.mbs.2006.06.006.
 [1] Ling Xue, Caterina Scoglio. Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences & Engineering, 2015, 12 (3) : 565-584. doi: 10.3934/mbe.2015.12.565 [2] Xia Wang, Yuming Chen. An age-structured vector-borne disease model with horizontal transmission in the host. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1099-1116. doi: 10.3934/mbe.2018049 [3] 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 [4] Shangbing Ai. Global stability of equilibria in a tick-borne disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 567-572. doi: 10.3934/mbe.2007.4.567 [5] Derdei Mahamat Bichara. Effects of migration on vector-borne diseases with forward and backward stage progression. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6297-6323. doi: 10.3934/dcdsb.2019140 [6] Xinli Hu, Yansheng Liu, Jianhong Wu. Culling structured hosts to eradicate vector-borne diseases. Mathematical Biosciences & Engineering, 2009, 6 (2) : 301-319. doi: 10.3934/mbe.2009.6.301 [7] Kbenesh Blayneh, Yanzhao Cao, Hee-Dae Kwon. Optimal control of vector-borne diseases: Treatment and prevention. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 587-611. doi: 10.3934/dcdsb.2009.11.587 [8] Cruz Vargas-De-León, Alberto d'Onofrio. Global stability of infectious disease models with contact rate as a function of prevalence index. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1019-1033. doi: 10.3934/mbe.2017053 [9] 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 [10] 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 [11] C. Connell McCluskey. Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes. Mathematical Biosciences & Engineering, 2012, 9 (4) : 819-841. doi: 10.3934/mbe.2012.9.819 [12] Jing-Jing Xiang, Juan Wang, Li-Ming Cai. Global stability of the dengue disease transmission models. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2217-2232. doi: 10.3934/dcdsb.2015.20.2217 [13] 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 [14] 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 [15] Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101 [16] Andrei Korobeinikov, Philip K. Maini. A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Mathematical Biosciences & Engineering, 2004, 1 (1) : 57-60. doi: 10.3934/mbe.2004.1.57 [17] Jing Feng, Bin-Guo Wang. An almost periodic Dengue transmission model with age structure and time-delayed input of vector in a patchy environment. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3069-3096. doi: 10.3934/dcdsb.2020220 [18] Tongtong Chen, Jixun Chu. Hopf bifurcation for a predator-prey model with age structure and ratio-dependent response function incorporating a prey refuge. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022082 [19] Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313 [20] Holly Gaff. Preliminary analysis of an agent-based model for a tick-borne disease. Mathematical Biosciences & Engineering, 2011, 8 (2) : 463-473. doi: 10.3934/mbe.2011.8.463

2018 Impact Factor: 1.313