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

Xia Wang; Yuming Chen; Xia Wang; Yuming Chen

doi:10.3934/mbe.2018049

Mathematical Biosciences and Engineering

2018, Volume 15, Issue 5: 1099-1116. doi: 10.3934/mbe.2018049

Previous Article Next Article

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

Xia Wang ^{1
,
,},
Yuming Chen ²

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

Received: 04 April 2017 Accepted: 22 March 2018 Published: 01 October 2018
MSC : Primary: 34K20, 92D25; Secondary: 35A24

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.
- Vector-borne disease,
- infection age,
- global stability,
- global attractor,
- Lyapunov functional
Citation: Xia Wang, Yuming Chen. An age-structured vector-borne disease model with horizontal transmission in the host[J]. Mathematical Biosciences and Engineering, 2018, 15(5): 1099-1116. doi: 10.3934/mbe.2018049

Related Papers:

Abstract

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.

References

[1]	[ C. J. Browne,S. S. Pilyugin, Global analysis of age-structured within-host virus model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013): 1999-2017.
[2]	[ Y. Chen,S. Zou,J. Yang, Global analysis of an SIR epidemic model with infection age and saturated incidence, Nonlinear Anal. Real World Appl., 30 (2016): 16-31.
[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,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.
[5]	[ Z. Feng,J. X. Velasco-HerNández, Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol., 35 (1997): 523-544.
[6]	[ F. Forouzannia,A. B. Gumel, Mathematical analysis of an age-structured model for malaria transmission dynamics, Math. Biosci., 247 (2014): 80-94.
[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.
[9]	[ M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori E Stampatori, Pisa, 1995.
[10]	[ H. Inaba,H. Sekine, A mathematical model for Chagas disease with infection-age-dependent infectivity, Math. Biosci., 190 (2004): 39-69.
[11]	[ Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, Boston, MA, 1993.
[12]	[ A. A. Lashari,G. Zaman, Global dynamics of vector-borne diseases with horizontal transmission in host population, Comput. Math. Appl., 61 (2011): 745-754.
[13]	[ Y. Lou,X.-Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM J. Appl. Math., 70 (2010): 2023-2044.
[14]	[ G. Macdonald, The analysis of equilibrium in malaria, Trop. Dis. Bull., 49 (1952): 813-829.
[15]	[ P. Magal,C. C. McCluskey,G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010): 1109-1140.
[16]	[ A. V. Melnik,A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility, Math. Biosci. Eng., 10 (2013): 369-378.
[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.
[18]	[ Z. Qiu, Dynamical behavior of a vector-host epidemic model with demographic structure, Comput. Math. Appl., 56 (2008): 3118-3129.
[19]	[ R. Ross, The Prevention of Malaria, J. Murray, London, 1910.
[20]	[ R. Ross, Some quantitative studies in epidemiology, Nature, 87 (1911): 466-467.
[21]	[ S. Ruan,D. Xiao,J. C. Beier, On the delayed Ross-Macdonald model for malaria transmission, Bull. Math. Biol., 70 (2008): 1098-1114.
[22]	[ H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci., 166 (2000): 173-201.
[23]	[ J. Tumwiine,J. Y. T. Mugisha,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.
[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.
[25]	[ C. Vargas-de-León,L. Esteva,A. Korobeinikov, Age-dependency in host-vector models: The global analysis, Appl. Math. Comput., 243 (2014): 969-981.

Reader Comments

Your name:*

Email:*
© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)