October 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:
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http://www.who.int/mediacentre/factsheets/fs387/en/.

[2]

http://www.shanghaidaily.com/national/Guangdong-sees-1074-new-dengue-cases/shdaily.shtml.

[3]

R. M. Anderson and R. M. May, Infectious Disease of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.

[4]

C. BowmanA. B. GumelJ. WuP. 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. BrauerZ. 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. BusenbergK. 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. BusenbergM. 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. BusenbergM. 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. ChaM. 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. FengW. 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. GuoM. 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. GuoM. 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. MagalC. 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. PachecoaL. EstevabJ. 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. QiuQ. K. KongX. 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. TumwiineJ. 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. YangZ. 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. ZhangZ. 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]

http://www.who.int/mediacentre/factsheets/fs387/en/.

[2]

http://www.shanghaidaily.com/national/Guangdong-sees-1074-new-dengue-cases/shdaily.shtml.

[3]

R. M. Anderson and R. M. May, Infectious Disease of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.

[4]

C. BowmanA. B. GumelJ. WuP. 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. BrauerZ. 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. BusenbergK. 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. BusenbergM. 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. BusenbergM. 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. ChaM. 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. FengW. 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. GuoM. 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. GuoM. 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. MagalC. 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. PachecoaL. EstevabJ. 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. QiuQ. K. KongX. 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. TumwiineJ. 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. YangZ. 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. ZhangZ. 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.

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