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Analysis of a simple vector-host epidemic model with direct transmission. (English) Zbl 1190.92029
Summary: Vector-host epidemic models with direct transmission are proposed and analyzed. It is shown that the stability of the equilibria in the proposed models can be controlled by the basic reproduction number of the disease transmission. One model considers that the dynamics of human hosts and vectors are described by SIS and SI models, respectively, where the global asymptotical stability for the equilibria of the models is analyzed by constructing Lyapunov functions, respectively. The other model considers that the dynamics of the human hosts and vectors are described by SIRS and SI models, respectively, where the global stability of the disease-free equilibrium and the persistence of the disease in the model are also analyzed, respectively.

34D20Stability of ODE
37N25Dynamical systems in biology
Full Text: DOI EuDML
[1] R. M. Anderson and R. M. May, Infectious Diseases of Humans, Oxford University Press, London, UK, 1991.
[2] Z. Ma, Y. Zhou, W. Wang, and Z. Jin, Mathematical Models and Dynamics of Infectious Diseases, China Sciences Press, Beijing, China, 2004.
[3] Z. Lu and Y. Zhou, Advance in Mathematic Biology, China Sciences Press, Beijing, China, 2006.
[4] H. W. Hethcote, “The mathematics of infectious diseases,” SIAM Review, vol. 42, no. 4, pp. 599-653, 2000. · Zbl 0993.92033 · doi:10.1137/S0036144500371907
[5] P. De Leenheer and H. L. Smith, “Virus dynamics: a global analysis,” SIAM Journal on Applied Mathematics, vol. 63, no. 4, pp. 1313-1327, 2003. · Zbl 1035.34045 · doi:10.1137/S0036139902406905
[6] H. W. Hethcote and J. W. Van Ark, Modelling HIV Transmission and AIDS in the United States, vol. 95 of Lecture Notes in Biomathematics, Springer, Berlin, Germany, 1992. · Zbl 0805.92026
[7] R. V. Culshaw and S. Ruan, “A delay-differential equation model of HIV infection of CD4+ T-cells,” Mathematical Biosciences, vol. 165, no. 1, pp. 27-39, 2000. · Zbl 0981.92009 · doi:10.1016/S0025-5564(00)00006-7
[8] J. Li and Z. Ma, “Qualitative analyses of SIS epidemic model with vaccination and varying total population size,” Mathematical and Computer Modelling, vol. 35, no. 11-12, pp. 1235-1243, 2002. · Zbl 1045.92039 · doi:10.1016/S0895-7177(02)00082-1
[9] S. Ruan and W. Wang, “Dynamical behavior of an epidemic model with a nonlinear incidence rate,” Journal of Differential Equations, vol. 188, no. 1, pp. 135-163, 2003. · Zbl 1028.34046 · doi:10.1016/S0022-0396(02)00089-X
[10] L. Esteva and C. Vargas, “A model for dengue disease with variable human population,” Journal of Mathematical Biology, vol. 38, no. 3, pp. 220-240, 1999. · Zbl 0981.92016 · doi:10.1007/s002850050147
[11] Z. Qiu, “Dynamical behavior of a vector-host epidemic model with demographic structure,” Computers & Mathematics with Applications, vol. 56, no. 12, pp. 3118-3129, 2008. · Zbl 1165.34382 · doi:10.1016/j.camwa.2008.09.002
[12] Y. Takeuchi, W. Ma, and E. Beretta, “Global asymptotic properties of a delay SIR epidemic model with finite incubation times,” Nonlinear Analysis: Theory, Methods & Applications, vol. 42, no. 6, pp. 931-947, 2000. · Zbl 0967.34070 · doi:10.1016/S0362-546X(99)00138-8
[13] G. R. Hosack, P. A. Rossignol, and P. van den Driessche, “The control of vector-borne disease epidemics,” Journal of Theoretical Biology, vol. 255, no. 1, pp. 16-25, 2008. · doi:10.1016/j.jtbi.2008.07.033
[14] H. Wan and J.-A. Cui, “A model for the transmission of malaria,” Discrete and Continuous Dynamical Systems. Series B, vol. 11, no. 2, pp. 479-496, 2009. · Zbl 1153.92028 · doi:10.3934/dcdsb.2009.11.479
[15] L. Cai, S. Guo, X. Li, and M. Ghosh, “Global dynamics of a dengue epidemic mathematical model,” Chaos, Solitons & Fractals, vol. 42, no. 4, pp. 2297-2304, 2009. · Zbl 1198.34075 · doi:10.1016/j.chaos.2009.03.130
[16] L. Cai and Q. Luo, “Stability analysis of a kind of vector-host epidemic model,” Journal of Xinyang Normal University. Natural Science Edition, vol. 23, no. 4, pp. 1-6, 2010. · Zbl 1221.92068
[17] H.-M. Wei, X.-Z. Li, and M. Martcheva, “An epidemic model of a vector-borne disease with direct transmission and time delay,” Journal of Mathematical Analysis and Applications, vol. 342, no. 2, pp. 895-908, 2008. · Zbl 1146.34059 · doi:10.1016/j.jmaa.2007.12.058
[18] H. Inaba and H. Sekine, “A mathematical model for Chagas disease with infection-age-dependent infectivity,” Mathematical Biosciences, vol. 190, no. 1, pp. 39-69, 2004. · Zbl 1049.92033 · doi:10.1016/j.mbs.2004.02.004
[19] L. Perko, Differential Equations and Dynamical Systems, vol. 7 of Texts in Applied Mathematics, Springer, New York, NY, USA, 2nd edition, 1996. · Zbl 0854.34001
[20] O. D. Makinde, “Adomian decomposition approach to a SIR epidemic model with constant vaccination strategy,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 842-848, 2007. · Zbl 1109.92041 · doi:10.1016/j.amc.2006.06.074
[21] O. D. Makinde, “On non-perturbative approach to transmission dynamics of infectious diseases with waning immunity,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 4, pp. 451-458, 2009.
[22] M. Y. Li and J. S. Muldowney, “A geometric approach to global-stability problems,” SIAM Journal on Mathematical Analysis, vol. 27, no. 4, pp. 1070-1083, 1996. · Zbl 0873.34041 · doi:10.1137/S0036141094266449
[23] H. R. Thieme, “Persistence under relaxed point-dissipativity (with application to an endemic model),” SIAM Journal on Mathematical Analysis, vol. 24, no. 2, pp. 407-435, 1993. · Zbl 0774.34030 · doi:10.1137/0524026
[24] P. van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, pp. 29-48, 2002. · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6
[25] http://www.cdc.gov/.