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Global dynamics of vector-borne diseases with horizontal transmission in host population. (English) Zbl 1217.34064
Summary: The paper presents the dynamical features of a vector-host epidemic model with direct transmission. First, we extended the model by taking into account the exposed individuals in both human and vector population with the impact of disease related deaths and total time dependent population size. Using Lyapunov function theory some sufficient conditions for global stability of both the disease-free equilibrium and the endemic equilibrium are obtained. For the basic reproductive number ${R}_{0}>1$, a unique endemic equilibrium exists and is globally asymptotically stable. Furthermore, it is found that the model exhibits the phenomenon of backward bifurcation, where the stable disease-free equilibria coexists with a stable endemic equilibrium. Finally, numerical simulations are carried out to investigate the influence of the key parameters on the spread of the vector-borne disease, to support the analytical conclusion and illustrate possible behavioral scenarios of the model.
##### MSC:
 34C23 Bifurcation (ODE) 92D30 Epidemiology 34D20 Stability of ODE 37N25 Dynamical systems in biology
##### References:
 [1] The World Health Report 2004: Changing History, WHO, Geneva, Switzerland. [2] Ross, R.: The prevention of malaria, (1911) [3] Macdonald, G.: The analysis of equilibrium in malaria, Trop. dis. Bull. 49, 813-828 (1952) [4] Aron, J. L.; May, R. M.: The population dynamics of infectious diseases, (1982) [5] Dietz, K.; Molineaux, L.; Thomas, A.: A malaria model tested in the african savannah, Bull. world health org. 50, 347-357 (1974) [6] Wei, H. M.; Li, X. Z.; Martcheva, M.: An epidemic model of a vector-borne disease with direct transmission and time delay, J. math. Anal. appl. 342, 895-908 (2008) · Zbl 1146.34059 · doi:10.1016/j.jmaa.2007.12.058 [7] Hethcote, H. W.: The mathematics of infectious diseases, SIAM rev. 42, No. 4, 599-653 (2000) · Zbl 0993.92033 · doi:10.1137/S0036144500371907 [8] Feng, Z.; Hernandez, V.: Competitive exclusion in a vector-host model for the dengue fever, J. math. Biol. 35, 523-544 (1997) · Zbl 0878.92025 · doi:10.1007/s002850050064 [9] Qiu, Z.: Dynamical behavior of a vector-host epidemic model with demographic structure, Comput. math. Appl. 56, 3118-3129 (2008) · Zbl 1165.34382 · doi:10.1016/j.camwa.2008.09.002 [10] Wiwanitkit, V.: Unusual mode of transmission of dengue, J. infect dev ctries. 30, 51-54 (2009) [11] Dushoff, J.; Huang, W.; Chavez, C. C.: Backwards bifurcations and catastrophe in simple models of fatal diseases, J. math. Biol. 36, 227-248 (1998) · Zbl 0917.92022 · doi:10.1007/s002850050099 [12] Garba, S. M.; Gumel, A. B.; Bakar, M. R. A.: Backward bifurcations in dengue transmission dynamics, Math. biosci. 215, 11-25 (2008) · Zbl 1156.92036 · doi:10.1016/j.mbs.2008.05.002 [13] Hui, J.; Zhu, D.: Global stability and periodicity on SIS epidemic models with backward bifurcation, Comput. math. Appl. 50, 1271-1290 (2005) · Zbl 1078.92058 · doi:10.1016/j.camwa.2005.06.003 [14] Cai, L.; Li, X.: Analysis of a simple vector-host epidemic model with direct transmission, Discrete dyn. Nat. soc. (2010) · Zbl 1190.92029 · doi:10.1155/2010/679613 [15] Lakshmikantham, V.; Leela, S.; Martynyuk, A. A.: Stability analysis of nonlinear systems, (1989) [16] Anderson, R. M.; May, R. M.: Infectious diseases of humans: dynamics and control, (1991) [17] Den Driessche, P. Van; Watmough, J.: Reproduction number and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. biosci. 180, 29-48 (2002) · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6 [18] Allen, L. J. S.: An introduction to mathematical biology, (2007) [19] Hadeler, K. P.; Den Driessche, P. Van: Backward bifurcation in epidemic control, Math. biosci. 146, 15-35 (1997) · Zbl 0904.92031 · doi:10.1016/S0025-5564(97)00027-8 [20] Sharomi, O.; Podder, C. N.; Gumel, A. B.; Elbasha, E. H.; Watmough, J.: Role of incidence function in vaccine-induced backward bifurcation in some HIV models, Math. biosci. 210, 436-463 (2007) · Zbl 1134.92026 · doi:10.1016/j.mbs.2007.05.012 [21] Lasalle, J. P.: The stability of dynamical systems, (1976) [22] Makinde, O. D.: Adomian decomposition approach to a SIR epidemic model with constant vaccination strategy, Appl. math. Comput. 184, 842-848 (2007) · Zbl 1109.92041 · doi:10.1016/j.amc.2006.06.074 [23] Khan, Y.; Wu, Q.: Homotopy perturbation transform method for nonlinear equations using he’s polynomials, Comput. math. Appl. (2010) [24] Khan, Y.: An effective modification of the Laplace decomposition method for nonlinear equations, Int. J. Nonlinear sci. Numer. simul. 10, 1373-1376 (2009) [25] Coutinho, F. A. B.; Burattini, M. N.; Lopez, L. F.; Massad, E.: Threshold conditions for a non-autonomous epidemic system describing the population dynamics of dengue, Bull. math. Biology 68, 2263-2282 (2006) [26] Patanarapelert, K.; Tang, I. M.: Effect of time delay on the transmission of dengue fever, I. J. Biol. life sci. 3, 238-246 (2007) [27] Esteva, L.; Vargas, C.: Analysis of a dengue disease transmission model, Math. biosci. 150, 131-151 (1998) · Zbl 0930.92020 · doi:10.1016/S0025-5564(98)10003-2