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:
[1]

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

[2]

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

[3]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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