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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.  Google Scholar

[2]

Y. ChenS. 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.  Google Scholar

[3]

K. Dietz, L. Molineaux and A. Thomas, A malaria model tested in the African savannah, Bull. World Health Organ., 50(1974), 347-357. Google Scholar

[4]

X. FengS. RuanZ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Am. Math. Soc., Providence, RI, 1988.  Google Scholar

[8]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[9]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori E Stampatori, Pisa, 1995. Google Scholar

[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.  Google Scholar

[11]

Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, Boston, MA, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

G. Macdonald, The analysis of equilibrium in malaria, Trop. Dis. Bull., 49 (1952), 813-829.   Google Scholar

[15]

P. MagalC. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[19]

R. Ross, The Prevention of Malaria, J. Murray, London, 1910. Google Scholar

[20]

R. Ross, Some quantitative studies in epidemiology, Nature, 87 (1911), 466-467.  doi: 10.1038/087466a0.  Google Scholar

[21]

S. RuanD. 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.  Google Scholar

[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.  Google Scholar

[23]

J. TumwiineJ. 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.  Google Scholar

[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.  Google Scholar

[25]

C. Vargas-de-LeónL. 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.  Google Scholar

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.  Google Scholar

[2]

Y. ChenS. 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.  Google Scholar

[3]

K. Dietz, L. Molineaux and A. Thomas, A malaria model tested in the African savannah, Bull. World Health Organ., 50(1974), 347-357. Google Scholar

[4]

X. FengS. RuanZ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Am. Math. Soc., Providence, RI, 1988.  Google Scholar

[8]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[9]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori E Stampatori, Pisa, 1995. Google Scholar

[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.  Google Scholar

[11]

Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, Boston, MA, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

G. Macdonald, The analysis of equilibrium in malaria, Trop. Dis. Bull., 49 (1952), 813-829.   Google Scholar

[15]

P. MagalC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

R. Ross, The Prevention of Malaria, J. Murray, London, 1910. Google Scholar

[20]

R. Ross, Some quantitative studies in epidemiology, Nature, 87 (1911), 466-467.  doi: 10.1038/087466a0.  Google Scholar

[21]

S. RuanD. 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.  Google Scholar

[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.  Google Scholar

[23]

J. TumwiineJ. 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.  Google Scholar

[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.  Google Scholar

[25]

C. Vargas-de-LeónL. 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.  Google Scholar

Figure 1.  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
Figure 2.  When $R_0>1$, the infected equilibrium $E^{\ast}$ of (2) is globally asymptotically stable
Table 1.  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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