# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $\germ R_0$. It is shown that if $\germ R_0$ is less than one, the disease-free equilibrium is globally asymptotically stable and in such a case the endemic equilibrium does not exist; if $\germ R_0$ 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 models (ODE) 92D30 Epidemiology 34D23 Global stability of ODE
Full Text:
##### References:
 [1] World Health Organization. Dengue haemorrhagic fever: diagnosis treatment and control, Geneva; 1986. [2] Gubler, D. J.: Dengue, The arbovirus: epidemiology and ecology 2, 213 (1986) [3] Anderson, R. M.; May, R. M.: Infectious diseases of humans, (1991) [4] Levin, S. A.; Hallam, T. G.; Gross, L. J.: Applied mathematical ecology, (1989) · Zbl 0688.92015 [5] Ma, Z.; Zhou, Y.; Wang, W.; Jin, Z.: Mathematical models and dynamics of infectious diseases, (2004) [6] Capasso, V.; Serio, G.: A generalization of the kermack -- mckendrick deterministic epidemic model, Math biosci 42, 43-61 (1978) · Zbl 0398.92026 · doi:10.1016/0025-5564(78)90006-8 [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 · doi:10.1007/BF00277162 [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 · doi:10.1007/BF00276956 [9] 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 [10] Esteva, L.; Vargas, C.: A model for dengue disease with variable human population, J math biol 38, 220-240 (1999) · Zbl 0981.92016 · doi:10.1007/s002850050147 [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 · doi:10.1016/S0895-7177(03)00111-0 [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 · doi:10.1016/j.jmaa.2007.02.061 [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 · doi:10.1007/s002850050064 [14] Jiang, Z.; Wei, J.: Stability and bifurcation analysis in a delayed SIR model, Chaos, solitons and fractals 35, No. 3, 609-619 (2008) · Zbl 1131.92055 · doi:10.1016/j.chaos.2006.05.045 [15] Tewa, J.; Dimi, J.; Bowong, S.: Lyapunov functions for a dengue disease transmission model, Chaos, solitons and fractals 39, No. 2, 936-941 (2009) · Zbl 1197.34099 · doi:10.1016/j.chaos.2007.01.069 [16] Den Driessche, P. Van; Watmough, J.: Reproduction numbers 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 [17] Cai, L.; Wu, J.: Analysis of an HIV/AIDS treatment model with a nonlinear incidence, Chaos, solitons and fractals 41, No. 1, 175-182 (2009) · Zbl 1198.34076 · doi:10.1016/j.chaos.2007.11.023 [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 · doi:10.1137/0522070 [20] Smith, H. L.: Systems of ordinary differential equations which generate an order preserving flow, SIAM rev 30, 87-98 (1988) · Zbl 0674.34012 · doi:10.1137/1030003 [21] Muldowney, J. S.: Compound matrices and ordinary differential equations, Rocky mountain J math 20, 857-872 (1990) · Zbl 0725.34049 · doi:10.1216/rmjm/1181073047 [22] Li, Y.; Muldowney, J. S.: Global stability for the SEIR model in epidemiology, Math biosci 125, 155-164 (1995) · Zbl 0821.92022 · doi:10.1016/0025-5564(95)92756-5