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October 2018, 15(5): 1099-1116. doi: 10.3934/mbe.2018049

## An age-structured vector-borne disease model with horizontal transmission in the host

 1 College of Mathematics and Information Science, Xinyang Normal University, Xinyang 464000, Henan, China 2 Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada

1Correspondence email: xywangxia@163.com(X. Wang)

Received  April 04, 2017 Accepted  March 22, 2018 Published  May 2018

We concern with a vector-borne disease model with horizontal transmission and infection age in the host population. With the approach of Lyapunov functionals, we establish a threshold dynamics, which is completely determined by the basic reproduction number. Roughly speaking, if the basic reproduction number is less than one then the infection-free equilibrium is globally asymptotically stable while if the basic reproduction number is larger than one then the infected equilibrium attracts all solutions with initial infection. These theoretical results are illustrated with numerical simulations.

Citation: 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
##### References:
 [1] C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1999-2017. doi: 10.3934/dcdsb.2013.18.1999. [2] Y. Chen, S. Zou and J. Yang, Global analysis of an SIR epidemic model with infection age and saturated incidence, Nonlinear Anal. Real World Appl., 30 (2016), 16-31. doi: 10.1016/j.nonrwa.2015.11.001. [3] K. Dietz, L. Molineaux and A. Thomas, A malaria model tested in the African savannah, Bull. World Health Organ., 50(1974), 347-357. [4] X. Feng, S. Ruan, Z. Teng and K. Wang, Stability and backward bifurcation in a malaria transmission model with applications to the control of malaria in China, Math. Biosci., 266 (2015), 52-64. doi: 10.1016/j.mbs.2015.05.005. [5] Z. Feng and J. X. Velasco-HerNández, Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol., 35 (1997), 523-544. doi: 10.1007/s002850050064. [6] F. Forouzannia and A. B. Gumel, Mathematical analysis of an age-structured model for malaria transmission dynamics, Math. Biosci., 247 (2014), 80-94. doi: 10.1016/j.mbs.2013.10.011. [7] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Am. Math. Soc., Providence, RI, 1988. [8] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [9] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori E Stampatori, Pisa, 1995. [10] 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. [11] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, Boston, MA, 1993. [12] A. A. Lashari and G. Zaman, Global dynamics of vector-borne diseases with horizontal transmission in host population, Comput. Math. Appl., 61 (2011), 745-754. doi: 10.1016/j.camwa.2010.12.018. [13] Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM J. Appl. Math., 70 (2010), 2023-2044. doi: 10.1137/080744438. [14] G. Macdonald, The analysis of equilibrium in malaria, Trop. Dis. Bull., 49 (1952), 813-829. [15] P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140. doi: 10.1080/00036810903208122. [16] A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility, Math. Biosci. Eng., 10 (2013), 369-378. doi: 10.3934/mbe.2013.10.369. [17] V. N. Novosltsev, A. I. Michalski, J. A. Novoseltsevam A. I. Tashin, J. R. Carey and A. M. Ellis, An age-structured extension to the vectorial capacity model, PloS ONE, 7 (2012), e39479. doi: 10.1371/journal.pone.0039479. [18] Z. Qiu, Dynamical behavior of a vector-host epidemic model with demographic structure, Comput. Math. Appl., 56 (2008), 3118-3129. doi: 10.1016/j.camwa.2008.09.002. [19] R. Ross, The Prevention of Malaria, J. Murray, London, 1910. [20] R. Ross, Some quantitative studies in epidemiology, Nature, 87 (1911), 466-467. doi: 10.1038/087466a0. [21] S. Ruan, D. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for malaria transmission, Bull. Math. Biol., 70 (2008), 1098-1114. doi: 10.1007/s11538-007-9292-z. [22] H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci., 166 (2000), 173-201. doi: 10.1016/S0025-5564(00)00018-3. [23] J. Tumwiine, J. Y. T. 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. [24] C. Vargas-de-León, Global analysis of a delayed vector-bias model for malaria transmission with incubation period in mosquitoes, Math. Biosci. Eng., 9 (2012), 165-174. doi: 10.3934/mbe.2012.9.165. [25] C. Vargas-de-León, L. Esteva and A. Korobeinikov, Age-dependency in host-vector models: The global analysis, Appl. Math. Comput., 243 (2014), 969-981. doi: 10.1016/j.amc.2014.06.042.

show all references

##### References:
 [1] C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1999-2017. doi: 10.3934/dcdsb.2013.18.1999. [2] Y. Chen, S. Zou and J. Yang, Global analysis of an SIR epidemic model with infection age and saturated incidence, Nonlinear Anal. Real World Appl., 30 (2016), 16-31. doi: 10.1016/j.nonrwa.2015.11.001. [3] K. Dietz, L. Molineaux and A. Thomas, A malaria model tested in the African savannah, Bull. World Health Organ., 50(1974), 347-357. [4] X. Feng, S. Ruan, Z. Teng and K. Wang, Stability and backward bifurcation in a malaria transmission model with applications to the control of malaria in China, Math. Biosci., 266 (2015), 52-64. doi: 10.1016/j.mbs.2015.05.005. [5] Z. Feng and J. X. Velasco-HerNández, Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol., 35 (1997), 523-544. doi: 10.1007/s002850050064. [6] F. Forouzannia and A. B. Gumel, Mathematical analysis of an age-structured model for malaria transmission dynamics, Math. Biosci., 247 (2014), 80-94. doi: 10.1016/j.mbs.2013.10.011. [7] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Am. Math. Soc., Providence, RI, 1988. [8] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [9] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori E Stampatori, Pisa, 1995. [10] 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. [11] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, Boston, MA, 1993. [12] A. A. Lashari and G. Zaman, Global dynamics of vector-borne diseases with horizontal transmission in host population, Comput. Math. Appl., 61 (2011), 745-754. doi: 10.1016/j.camwa.2010.12.018. [13] Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM J. Appl. Math., 70 (2010), 2023-2044. doi: 10.1137/080744438. [14] G. Macdonald, The analysis of equilibrium in malaria, Trop. Dis. Bull., 49 (1952), 813-829. [15] P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140. doi: 10.1080/00036810903208122. [16] A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility, Math. Biosci. Eng., 10 (2013), 369-378. doi: 10.3934/mbe.2013.10.369. [17] V. N. Novosltsev, A. I. Michalski, J. A. Novoseltsevam A. I. Tashin, J. R. Carey and A. M. Ellis, An age-structured extension to the vectorial capacity model, PloS ONE, 7 (2012), e39479. doi: 10.1371/journal.pone.0039479. [18] Z. Qiu, Dynamical behavior of a vector-host epidemic model with demographic structure, Comput. Math. Appl., 56 (2008), 3118-3129. doi: 10.1016/j.camwa.2008.09.002. [19] R. Ross, The Prevention of Malaria, J. Murray, London, 1910. [20] R. Ross, Some quantitative studies in epidemiology, Nature, 87 (1911), 466-467. doi: 10.1038/087466a0. [21] S. Ruan, D. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for malaria transmission, Bull. Math. Biol., 70 (2008), 1098-1114. doi: 10.1007/s11538-007-9292-z. [22] H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci., 166 (2000), 173-201. doi: 10.1016/S0025-5564(00)00018-3. [23] J. Tumwiine, J. Y. T. 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. [24] C. Vargas-de-León, Global analysis of a delayed vector-bias model for malaria transmission with incubation period in mosquitoes, Math. Biosci. Eng., 9 (2012), 165-174. doi: 10.3934/mbe.2012.9.165. [25] C. Vargas-de-León, L. Esteva and A. Korobeinikov, Age-dependency in host-vector models: The global analysis, Appl. Math. Comput., 243 (2014), 969-981. doi: 10.1016/j.amc.2014.06.042.
When $R_0<1$, the infection-free equilibrium $E^0$ of (2) is globally asymptotically stable. Here since $E_h(t)$ converges to $0$ very fast, we use the time interval $[0, 100]$ different from the interval $[0, 1000]$ for other components
When $R_0>1$, the infected equilibrium $E^{\ast}$ of (2) is globally asymptotically stable
Biological meanings of parameters in (1)
 Parameter Meaning $\lambda_h$ Per capita host birth rate $\mu_h$ Host death rate $\beta_1$ Rate of horizontal transmission of the disease $\beta_2$ Rate of a pathogen carrying mosquito biting susceptible host $\alpha_h$ Inverse of host latent period $\delta_h$ Disease related death rate of host $\gamma_h$ Recovery rate of host $\lambda_v$ Per capita vector birth rate $k$ Biting rate of per susceptible vector per host per unit time $\mu_v$ Vector death rate $\alpha_v$ Inverse of vector latent period $\delta_v$ Disease related death rate of vectors
 Parameter Meaning $\lambda_h$ Per capita host birth rate $\mu_h$ Host death rate $\beta_1$ Rate of horizontal transmission of the disease $\beta_2$ Rate of a pathogen carrying mosquito biting susceptible host $\alpha_h$ Inverse of host latent period $\delta_h$ Disease related death rate of host $\gamma_h$ Recovery rate of host $\lambda_v$ Per capita vector birth rate $k$ Biting rate of per susceptible vector per host per unit time $\mu_v$ Vector death rate $\alpha_v$ Inverse of vector latent period $\delta_v$ Disease related death rate of vectors
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