Research article Special Issues

Global dynamics of an age-structured malaria model with prevention

  • Received: 18 November 2018 Accepted: 30 January 2019 Published: 26 February 2019
  • In this paper, we formulate a new age-structured malaria model, which incorporates the age of prevention period of susceptible people, the age of latent period of human and the age of latent period of female Anopheles mosquitoes. We show that there exists a compact global attractor and obtain a sufficient condition for uniform persistence of the solution semiflow. We obtain the basic reproduction number $\mathcal{R}_{0}$ and show that $\mathcal{R}_{0}$ completely determines the global dynamics of the model, that is, if $\mathcal{R}_{0} < 1$, the disease-free equilibrium is globally asymptotically stable, if $\mathcal{R}_{0}>1$, there exists a unique endemic equilibrium that attracts all solutions for which malaria transmission occurs. Finally, we perform some numerical simulations to illustrate our theoretical results and give a brief discussion.

    Citation: Zhong-Kai Guo, Hai-Feng Huo, Hong Xiang. Global dynamics of an age-structured malaria model with prevention[J]. Mathematical Biosciences and Engineering, 2019, 16(3): 1625-1653. doi: 10.3934/mbe.2019078

    Related Papers:

  • In this paper, we formulate a new age-structured malaria model, which incorporates the age of prevention period of susceptible people, the age of latent period of human and the age of latent period of female Anopheles mosquitoes. We show that there exists a compact global attractor and obtain a sufficient condition for uniform persistence of the solution semiflow. We obtain the basic reproduction number $\mathcal{R}_{0}$ and show that $\mathcal{R}_{0}$ completely determines the global dynamics of the model, that is, if $\mathcal{R}_{0} < 1$, the disease-free equilibrium is globally asymptotically stable, if $\mathcal{R}_{0}>1$, there exists a unique endemic equilibrium that attracts all solutions for which malaria transmission occurs. Finally, we perform some numerical simulations to illustrate our theoretical results and give a brief discussion.


    加载中


    [1] World Health Organization,World malaria report, 2016. Available from: http://www.who.int/ malaria/publications/world-malaria-report-2016/en/.
    [2] CDC. malaria, malaria worldwide, 2018. Available from: https://www.cdc.gov/malaria/ about/biology/index.html.
    [3] R. Ross, The Prevention of Malaria, 2nd edition, Murray, London, 1911.
    [4] G. Macdonald, The epidemiology and control of malaria, Oxford University press, London, 1957.
    [5] S. A. Gourley, R. Liu and J. Wu, Some vector borne diseases with structured host populations: Extinction and spatial spread, SIAM J. Appl. Math., 67 (2007), 408–433.
    [6] Fatmawati and H. Tasman, An optimal control strategy to reduce the spread of malaria resistance, Math. Biosci., 262 (2015), 73–79.
    [7] S. Ruan, D. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for malaria transmission, B. Math. Biol., 70 (2008), 1098–1114.
    [8] L. M. Cai, X. Z. Li, B. Fang, et al., Global properties of vector–host disease models with time delays, J. Math. Biol., 74 (2017), 1397–1423.
    [9] J. Xu and Y. Zhou, Hopf bifurcation and its stability for a vector-borne disease model with delay and reinfection, Appl. Math. Model., 40 (2016), 1685–1702.
    [10] Y. Lou and X. Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM J. Appl. Math., 70 (2010), 2023–2044.
    [11] Y. X. Dang, Z. P. Qiu, X. Z. Li, et al., Global dynamics of a vector-host epidemic model with age of infection, Math. Biosci. Eng., 14 (2017), 1159–1186.
    [12] D. Gao and S. Ruan, A multipatch malaria model with logistic growth populations, SIAM J. Appl. Math., 72 (2012), 819–841.
    [13] J. Li, Y. Zhao and S. Li, Fast and slow dynamics of malaria model with relapse, Math. Biosci., 246 (2013), 94–104.
    [14] B. Cao, H. F. Huo and H. Xiang, Global stability of an age-structure epidemic model with imperfect vaccination and relapse, Physica A, 486 (2017), 638–655.
    [15] X. Duan, S. Yuan and X. Li, Global stability of an SVIR model with age of vaccination, Appl. Math. Comput., 226 (2014), 528–540.
    [16] H. Inaba and H. Sekine, A mathematical model for chagas disease with infection-age-dependent infectivity, Math. Biosci., 190 (2004), 39–69.
    [17] L.Wang, Z. Liu and X. Zhang, Global dynamics for an age-structured epidemic model with media impact and incomplete vaccination, Nonlinear Anal. RWA, 32 (2016), 136–158.
    [18] H. Chen, W. Wang, R. Fu, et al., Global analysis of a mathematical model on malaria with competitive strains and immune responses, Appl. Math. Comput., 259 (2015), 132–152.
    [19] H. Xiang, Y. L. Tang and H. F. Huo, A viral model with intracellular delay and humoral immunity, B. Malays. Math. Sci. So., 40 (2017), 1011–1023.
    [20] H. F. Huo, F. F. Cui and H. Xiang, Dynamics of an SAITS alcoholism model on unweighted and weighted networks, Physica A, 496 (2018), 249–262.
    [21] H. Xiang, Y. Y. Wang and H. F. Huo, Analysis of the binge drinking models with demographics and nonlinear infectivity on networks, J. Appl. Anal. Comput., 8 (2018), 1535–1554.
    [22] Y. Cai, J. Jiao, Z. Gui, et al., Environmental variability in a stochastic epidemic model, Appl. Math. Comput., 329 (2018), 210–226.
    [23] X. B. Zhang, Q. H. Shi, S. H. Ma, et al., Dynamic behavior of a stochastic SIQS epidemic model with Lévy jumps, Nonlinear Dynam., 93 (2018), 1481–1493.
    [24] Z. Du and Z. Feng, Existence and asymptotic behaviors of traveling waves of a modified vectordisease model, Commun. pur. appl. anal., 17 (2018), 1899–1920.
    [25] Malaria vaccine. Available from: https://en.wikipedia.org/wiki/Malaria_vaccine.
    [26] G. Webb, Theory of nonlinear age-dependent population dynamics, marcel dekker, New York, 1985.
    [27] M. Iannelli, Mathematical theory of age-structured population dynamics, Giadini Editori e Stampatori, Pisa, 1994.
    [28] H. L. Smith and H. R. Thieme, Dynamical systems and population persistence, American Mathematical Society, Providence, 2011.
    [29] P. Magal and X. Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Appl. Math., 37 (2005), 251–275.
    [30] E. M. C. D'Agata, P. Magal, S. Ruan, et al., Asymptotic behavior in nosocomial epidemic models with antibiotic resistance, Differ. Integral Equ., 19 (2006), 573–600.
    [31] C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model, Discrete Cont. Dyn. S. B, 18 (2013), 1999–2017.
    [32] H. M. Wei, X. Z. Li and M. Martcheva, An epidemic model of a vector-borne disease with direct transmission and time delay, J. Math. Anal. Appl., 342 (2008), 895–908.
    [33] J. Wang, J. Lang and X. Zou, Analysis of an age structured hiv infection model with virus-to-cell infection and cell-to-cell transmission, Nonlinear Anal. RWA, 34 (2017), 75–96.
    [34] J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Appl. Math., 20 (1989), 388–395.
    [35] Z. K. Guo, H. F. Huo, X. Hong, Bifurcation analysis of an age-structured alcoholism model, J. Biol. Dynam., 12 (2018), 1009–1033.
  • Reader Comments
  • © 2019 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3665) PDF downloads(820) Cited by(7)

Article outline

Figures and Tables

Figures(5)  /  Tables(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog