Research Article
Global dynamics of multi-group dengue disease model with latency distributions
Gang Huang,
Jinliang Wang,
Jian Zu,
Gang Huang
School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074 China
Search for more papers by this authorCorresponding Author
Jinliang Wang
School of Mathematical Science, Heilongjiang University, Harbin, 150080 China
Correspondence to: Jinliang Wang, School of Mathematical Science, Heilongjiang University, Harbin 150080, China.
E-mail: [email protected]
Search for more papers by this authorJian Zu
Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an, 710049 China
Search for more papers by this authorGang Huang,
Jinliang Wang,
Jian Zu,
Gang Huang
School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074 China
Search for more papers by this authorCorresponding Author
Jinliang Wang
School of Mathematical Science, Heilongjiang University, Harbin, 150080 China
Correspondence to: Jinliang Wang, School of Mathematical Science, Heilongjiang University, Harbin 150080, China.
E-mail: [email protected]
Search for more papers by this authorJian Zu
Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an, 710049 China
Search for more papers by this authorAbstract
In this paper, by incorporating latencies for both human beings and female mosquitoes to the mosquito-borne diseases model, we investigate a class of multi-group dengue disease model and study the impacts of heterogeneity and latencies on the spread of infectious disease. Dynamical properties of the multi-group model with distributed delays are established. The results showthat the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium depends only on the basic reproduction number. Our proofs for global stability of equilibria use the classical method of Lyapunov functions and the graph-theoretic approach for large-scale delay systems. Copyright © 2014 John Wiley & Sons, Ltd.
References
- 1 World Health Organization. Dengue and dengue haemorrhagic fever: fact sheet # 117. Geneva: World Health Organization, 2009. Available from: http://www.who.int/mediacentre/factsheets/fs117/en/ [Accessed on March 2014].
- 2 TDR/World Health Organization. Dengue guidelines for diagnosis, treatment, prevention, and control. WHO Press: Switzerland, 2009. Available from: http://www.who.int/rpc/guidelines/9789241547871/en/.
- 3 Brady OJ, Gething PW, Bhatt S, Messina JP, Brownstein JS, Hoen AG, Moyes CL, Farlow AW, Scott TW, Hay SI. Refining the global spatial limits of dengue virus transmission by evidence-based consensus. PLOS Neglected Tropical Diseases 2012; 6(8): e1760.
- 4 Ranjit S, Kissoon N. Dengue hemorrhagic fever and shock syndromes. Pediatric Critical Care Medicine 2011; 12: 90–100.
- 5 Bärnighausen T, Bloom DE, Cafiero ET, OBrien JC. Valuing the broader benefits of dengue vaccination, with a preliminary application to Brazil, seminars in immunology.
- 6 Bowman C, Gumel AB, van den Driessche P, Wu J, Zhu H. A mathematical model for assessing control strategies against West Nile virus. Bulletin of Mathematical Biology 2005; 67: 1107–1133.
- 7 Esteva L, Vargas C. A model for dengue disease with variable human population. Journal of Mathematical Biology 1999; 38: 220–240.
- 8 Feng Z, Velasco-Hernandez JX. Competitive exclusion in a vector-host model for the dengue fever. Journal of Mathematical Biology 1997; 35: 523–544.
- 9 Wan H, Zhu H. The backward bifurcation in compartmental models for West Nile virus. Mathematical Biosciences 2010; 227: 20–28.
- 10 Wan H, Cui J. A model for the transmission of malaria. Discrete and Continuous Dynamical Systems - Series B 2009; 11: 479–496.
- 11 Xiao Y, Zou X. On latencies in malaria infections and their impact on the disease dynamics. Mathematical Biosciences and Engineering 2013; 10: 463–481.
- 12 van den Driessche P, Wang L, Zou X. Modeling diseases with latency and relapse. Mathematical Biosciences and Engineering 2007; 4: 205–219.
- 13 Wang J, Zu J, Liu X, Huang G, Zhang J. Global dynamics of a multi-group epidemic model with general relapse distribution and nonlinear incidence rate. Journal of Biological Systems 2012; 20: 235–258.
- 14 Yuan Z, Zou X. Global threshold property in an epidemic models for disease with latency spreading in a heterogeneous host population. Nonlinear Analysis RWA 2010; 11: 3479–3490.
- 15 Andersson H, Britton T. Heterogeneity in epidemic models and its effect on the spread of infection. Journal of Applied Probability 1998; 35: 651–661.
- 16 Anderson RM, May RM. Spatial, temporal, and genetic heterogeneity in host populations and the design of immunization programmes. IMA Journal of Mathematics Applied in Medicine and Biology 1984; 1: 233–266.
- 17
Lajmanovich A,
Yorke JA. A deterministic model for gonorrhea in a nonhomogeneous population. Mathematical Biosciences 1976; 28(3/4): 221–0236.
10.1016/0025-5564(76)90125-5 Google Scholar
- 18 Ding D, Wang X, Ding X. Global stability of multigroup dengue disease transmission model. Journal of Applied Mathematics 2012; 11. Article ID 342472.
- 19 Berman A, Plemmons RJ. Nonnegative Matrices in the Mathematical Sciences. Academic Press: New York, 1979.
- 20 Guo HB, Li MY, Shuai ZS. Global stability of the endemic equilibrium of multigroup SIR epidemic models. Canadian Applied Mathematics Quarterly 2006; 14: 259–284.
- 21 Guo HB, Li MY, Shuai ZS. A graph-theoretic approach to the method of global Lyapunov functions. Proceedings of the American Mathematical Society 2008; 136: 2793–2802.
- 22 Li MY, Shuai ZS, Wang CC. Global stability of multi-group epidemic models with distributed delays. Journal of Mathematical Analysis and Applications 2010; 361: 38–47.
- 23 Li MY, Shuai ZS. Global-stability problem for coupled systems of differential equations on networks. Journal of Differential Equations 2010; 248: 1–20.
- 24 Sun R. Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence. Computers & Mathematics with Applications 2010; 60: 2286–2291.
- 25 Yuan Z, Wang L. Global stability of epidemiological models with group mixing and nonlinear incidence rates. Nonlinear Analysis RWA 2010; 11: 995–1004.
- 26 Miller RK. Nonlinear Volterra Integral Equations. W. A. Benjamin Inc.: New York, 1971.
- 27 Diekmann O, Heesterbeek JAP, Metz JAJ. On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology 1990; 28: 365–382.
- 28 van den Driessche P, Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences 2002; 180: 29–48.
- 29 Hale JK, Lunel SMV. Introduction to Functional Differential Equations, Appl Math Sci, vol. 99. Springer: New York, 1993.
- 30
Hino Y,
Murakami S,
Naito T. Functional-Differential Equations with Infinite Delay, Lecture Notes in Mathematics, vol. 1473. Springer-Verlag: Berlin, 1991.
10.1007/BFb0084432 Google Scholar
- 31 Horn RA, Johnson CR. Matrix Analysis. Cambridge University Press: Cambridge, UK, 1990.
- 32 Lasalle JP. The stability of dynamical systems. Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, USA, 1976; 76.
- 33
Freedman HI,
Ruan S,
Tang M. Uniform persistence and flows near a closed positively invariant set. Journal of Dynamics and Differential Equations 1994; 6: 583–600.
10.1007/BF02218848 Google Scholar
- 34 Li MY, Graef JR, Wang L, Karsai J. Global dynamics of a SEIR model with varying total population size. Mathematical Biosciences 1999; 160: 191–213.
- 35
Bhatia NP,
Szego GP. Dynamical Systems: Stability Theory and Applications. Springer-Verlag: Berlin, 1967.
10.1007/BFb0080630 Google Scholar
- 36
Smith HL,
Waltman P. The Theory of the Chemostat: Dynamics of Microbial Competition. Cambridge University Press: Cambridge, 1995.
10.1017/CBO9780511530043 Google Scholar
- 37 Kuniya T. Global stability of a multi-group SVIR epidemic model. Nonlinear Analysis RWA 2013; 14: 1135–1143.
- 38 Moon JW. Counting Labelled Trees. Canadian Mathematical Congress: Montreal, 1970.
- 39 Korobeinikov A. Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bulletin of Mathematical Biology 2006; 68: 615–626.
- 40 Korobeinikov A. Global properties of infectious disease models with nonlinear incidence. Bulletin of Mathematical Biology 2007; 69: 1871–1886.
- 41 McCluskey CC. Complete global stability for an SIR epidemic model with delay-distributed or discrete. Nonlinear Analysis RWA 2010; 11: 55–59.
- 42 McCluskey CC. Global stability for an SIER epidemiological model with varying infectivity and infinite delay. Mathematical Biosciences and Engineering 2009; 6: 603–610.
- 43 Huang G, Takeuchi Y. Global analysis on delay epidemiological dynamic models with nonlinear incidence. Journal of Mathematical Biology 2011; 63: 125–139.
- 44 Huang G, Takeuchi Y, Ma W, Wei D. Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate. Bulletin of Mathematical Biology 2010; 7: 1192–1207.
- 45 Wang J, Takeuchi Y, Liu S. A multi-group SVEIR epidemic model with distributed delay and vaccination. International Journal of Biomathematics 2012; 5(3):1260001.
