Global dynamics of a vector-borne disease model with latency and saturating incidence rate
This paper deals with a vector-borne disease model containing latency and nonlinear incidence rates. Global analysis is completely determined by suitable Lyapunov functionals. We explicitely determine the basic reproduction number and find that if this number is less than one then disease dies out, but if the number is larger than one, the disease causing strain become endemic. The study shows that the latency delay explicitely in°uence the disease persistence.
Keywords: Latency, saturating incidence, basic reproduction number, global attractivity, Lyapunov functionals.
F. Beretta, V. Capasso, F. Rinaldi, Global stability results for a generalized Lotka-Volterra system with distributed delays, J. Math. Biol., 26(1988), 661-688.
M.C. Bruce, M.R. Galindki, J.W. Barnwell, C.A. Donnelly, M. Walmsley, M.P. Alpers, D. Walliker and K.P. Day, Vaccination in disease models with antibody-dependent enhancement, J. Math. Biosci., 211(1988),265-281.
V. Capasso, and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42(1978), 41-61.
L. Cai, X. Li and J. Yu, Analysis of a delayed HIV/AIDS epidemic model with saturation incidence, J. Appl. Math. Comput., 27(2008),365-377.
K.L. Cooke, Stability analysis for a vector disease model, Rocky Mountain J. Math., 9(1979), 31-42.
O. Diekmann, J.S.P. Heesterbeek and J.A.J. Metz, On the de¯nition and the computation of the basic reproductionratio R 0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28(1990), 365-382.
J. Hale and S.V. Lunel, Introduction to functional Differential Equations, Springer-Verlag, New York, 1993.
H. Guo, M.Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136(2008), 2793-2802.
J. Hou and Z. Teng, Continuous and Impulsive Vaccination of SEIR epidemic models with saturation incidence rates, Mathematics and Computers in Simulation, 79(2009), 3038-3054.
A. Korobeinikov, and P.K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1(2004), 57-30.
J.P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadephia, 1976.
W. Ma, Y. Takeuchi, T. Hara and E. Beretta,Permanence of an epidemic model with distributed time delays, Tohoku Math. J., 54(2002), 581-591.
M.J. Mackinnon and A.F. Read,Virulence in malaria: an evolutionary viewpoint, Phil. Trans. R. Soc. Lond. B, 359(2004), 965-986.
C.C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlin. Anal. RWA, 11(2010), 3106-3109.
N. Mideo and T. Day, On the evolution of reproductive restraint in malaria, Proc. R. Soc. B, 275(2008), 1217-1224.
J.M. Tchuenche, A. Nwagwo and R. Levins, Global behaviour of an SIR epidemic model with time delay, Math. Meth.Appl. Sci., 30(2007), 733-749.
R. Xu and X. Ma, Global stabailty of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlin. Anal. RWA, 10(2009), 3175-3189.
C. Wongsrichanalai, C.K. Murray, M. Gray, R.S. Miller, P. Mcdaniel, W.J. Liao, A.L. Picard and A. J. Magill, Coinfection with malaria and leptospirosis, Am. J. Trop. Med. Hyg., 68(2003), 583-585.
P. van den Driessche and J. Watmough, Reproduction Numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180(2002), 29-48.
Y. Zhou, D. Xiao and Y. Li, Bifurcations of an epidemic model with non-monotonic incidence rate of saturated mass action, Chaos, Solitons and Fractals, 32(2007), 1903-1915.