×

zbMATH — the first resource for mathematics

Global dynamics of a dengue epidemic mathematical model. (English) Zbl 1198.34075
Summary: The paper investigates the global stability of a dengue epidemic model with saturation and bilinear incidence. The constant human recruitment rate and exponential natural death, as well as vector population with asymptotically constant population, are incorporated into the model. The model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. The stability of these two equilibria is controlled by the threshold number \(\mathfrak R_0\). It is shown that if \(\mathfrak R_0\) is less than one, the disease-free equilibrium is globally asymptotically stable and in such a case the endemic equilibrium does not exist; if \(\mathfrak R_0\) is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
92D30 Epidemiology
34D23 Global stability of solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] World Health Organization. Dengue haemorrhagic fever: diagnosis treatment and control, Geneva; 1986.
[2] Gubler, D.J., Dengue, (), 213
[3] Anderson, R.M.; May, R.M., Infectious diseases of humans, (1991), Oxford University Press London
[4] Levin, S.A.; Hallam, T.G.; Gross, L.J., Applied mathematical ecology, (1989), Springer New York
[5] Ma, Z.; Zhou, Y.; Wang, W.; Jin, Z., Mathematical models and dynamics of infectious diseases, (2004), China Science Press Beijing
[6] Capasso, V.; Serio, G., A generalization of the kermack – mckendrick deterministic epidemic model, Math biosci, 42, 43-61, (1978) · Zbl 0398.92026
[7] Liu, W.M.; Hethcote, H.W.; Levin, S.A., Dynamical behavior of epidemiological models with nonlinear incidence rates, J math biol, 25, 359-380, (1987) · Zbl 0621.92014
[8] Liu, W.M.; Levin, S.A.; Iwasa, Y., Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J math biol, 23, 187-204, (1986) · Zbl 0582.92023
[9] Esteva, L.; Vargas, C., Analysis of a dengue disease transmission model, Math biosci, 150, 131-151, (1998) · Zbl 0930.92020
[10] Esteva, L.; Vargas, C., A model for dengue disease with variable human population, J math biol, 38, 220-240, (1999) · Zbl 0981.92016
[11] Pongsumpun, P.; Tang, I.M., Transmission of dengue hemorrhagic fever in an age structured population, Math comput model, 37, 949-961, (2003) · Zbl 1045.92040
[12] Nucci, M.C.; Leach, P.G.L., Lie integrable cases of the simplified multistrain/two stream model for tuberculosis and dengue fever, J math anal appl, 333, 430-449, (2007) · Zbl 1118.92054
[13] Feng, Z.; Velasco-Hernández, J.X., Competitive exclusion in a vector – host model for dengue fever, J math biol, 35, 523-544, (1997) · Zbl 0878.92025
[14] Jiang, Z.; Wei, J., Stability and bifurcation analysis in a delayed SIR model, Chaos, solitons and fractals, 35, 3, 609-619, (2008) · Zbl 1131.92055
[15] Tewa, J.; Dimi, J.; Bowong, S., Lyapunov functions for a dengue disease transmission model, Chaos, solitons and fractals, 39, 2, 936-941, (2009) · Zbl 1197.34099
[16] Van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math biosci, 180, 29-48, (2002) · Zbl 1015.92036
[17] Cai, L.; Wu, J., Analysis of an HIV/AIDS treatment model with a nonlinear incidence, Chaos, solitons and fractals, 41, 1, 175-182, (2009) · Zbl 1198.34076
[18] Hirsch, M.W., System of differential equations which are competitive or cooperative, IV, SIAM J math anal, 21, 1225-1234, (1990) · Zbl 0734.34042
[19] Smith, H.L.; Thieme, H., Convergence for strongly ordered preserving semiflows, SIAM J math anal, 22, 1081-1101, (1991) · Zbl 0739.34040
[20] Smith, H.L., Systems of ordinary differential equations which generate an order preserving flow, SIAM rev, 30, 87-98, (1988) · Zbl 0674.34012
[21] Muldowney, J.S., Compound matrices and ordinary differential equations, Rocky mountain J math, 20, 857-872, (1990) · Zbl 0725.34049
[22] Li, Y.; Muldowney, J.S., Global stability for the SEIR model in epidemiology, Math biosci, 125, 155-164, (1995) · Zbl 0821.92022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.