# 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 analysis of a vector-host epidemic model with nonlinear incidences. (English) Zbl 1202.92075
Summary: An epidemic model with nonlinear incidences is proposed to describe the dynamics of diseases spread by vectors (mosquitoes), such as malaria, yellow fever, dengue and so on. The constant human recruitment rate and exponential natural death, as well as a vector population with asymptotically constant population, are incorporated into the model. The stability of the system is analyzed for the disease-free and endemic equilibria. The stability of the system can be controlled by the threshold number ${\Re }_{0}$. It is shown that if ${\Re }_{0}$ is less than one, the disease free equilibrium is globally asymptotically stable and in such a case the endemic equilibrium does not exist; if ${\Re }_{0}$ is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Our results imply that the threshold condition of the system provides important guidelines for accessing control of the vector diseases, and the spread of vector epidemic in an efficient way can be prevented. The contribution of the nonlinear saturating incidence to the basic reproduction number and the level of the endemic equilibrium are also analyzed, respectively.
##### MSC:
 92D30 Epidemiology 34D23 Global stability of ODE 37N25 Dynamical systems in biology 92C60 Medical epidemiology 93C95 Applications of control theory
##### References:
 [1] Anderson, R. M.; May, R. M.: Infectious diseases of humans: dynamics and control, (1991) [2] Zhou, Z. Ma Y.; Wu, J.: Modeling and dynamics of infectious diseases, (2009) [3] Thieme, H. R.: Mathematics in population biology, (2003) [4] lt;http://www.who.int/whosis/whostat/2008/en/index.htmlgt;. [5] Ross, R.: The prevention of malaria, (1911) [6] Feng, Z.; Velasco-Hernandez, J. X.: 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 [7] Esteva, L.; Yang, H. Mo: Mathematical model to assess the control of aedes aegypti mosquitoes by the sterile insect technique, Math. biosci. 198, 132-147 (2005) · Zbl 1090.92048 · doi:10.1016/j.mbs.2005.06.004 [8] Bowman, C.; Gumel, A. B.; Den Driessche, P. Van; Wu, J.; Zhu, H.: A mathematical model for assessing control strategies against west nile virus, Bull. math. Biol. 67, 1107-1133 (2005) [9] Mackinnon, M. J.: Drug resistance models for malaria, Acta tropica 94, No. 3, 207-217 (2005) [10] Qiu, Z.: Dynamical behavior of a vector-host epidemic model with demographic structure, Comput. math. Appl. 56, No. 12, 3118-3129 (2008) · Zbl 1165.34382 · doi:10.1016/j.camwa.2008.09.002 [11] Koella, J. C.: On the use of mathematical models of malaria transmission, Acta tropica 49, 1-25 (1991) [12] Sutherst, R. W.: Global change and human vulnerability to vector-borne diseases, Clin. microbiol. Rev. 17, No. 1, 136-173 (2004) [13] Khasnis, A. A.; Nettleman, M. D.: Global warming and infectious disease, Arch. med. Res. 36, No. 6, 689-696 (2005) [14] Ma, Z.; Zhou, Y.; Wang, W.; Jin, Z.: Mathematical models and dynamics of infectious diseases, (2004) [15] Colwell, R. R.; Bryaton, P.; Herrington, D.; Tall, B.; Huq, A.; Levine, M. M.: Viable but nonculturable vibrio cholerae revert to a cultivable state in the human intestine, World J. Microbiol. biotechnol. 12, 28-31 (1996) [16] Aron, J. L.; May, R. M.: The population dynamics of malaria, The population dynamics of infectious disease: theory, and application, 139-179 (1982) [17] Wang, W.: Epidemic models with nonlinear infection forces, Math. biosci. Eng. 3, 267-279 (2006) · Zbl 1089.92052 · doi:10.3934/mbe.2006.3.267 [18] Liu, W.; 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 [19] Cai, L.; Li, X.; Gosh, M.: Global stability of a stage-structured epidemic model with a nonlinear incidence, Appl. math. Comput. 214, No. 1, 73-82 (2009) · Zbl 1172.92027 · doi:10.1016/j.amc.2009.03.088 [20] Zhang, J.; Ma, Z.: Global dynamics of an SEIR epidemic model with saturating contact rate, Math. biosci. 185, 15-32 (2003) · Zbl 1021.92040 · doi:10.1016/S0025-5564(03)00087-7 [21] Cai, L.; Li, X.: Analysis of a SEIV epidemic model with a nonlinear incidence rate, Appl. math. Model. 33, No. 7, 2919-2926 (2009) · Zbl 1205.34049 · doi:10.1016/j.apm.2008.01.005 [22] Li, J.; Zhou, Y.; Wu, J.; Ma, Z.: Complex dynamics of a simple epidemic model with a nonlinear incidence, Discrete continuous dyn. Syst.-ser. B 8, No. 1, 161-173 (2007) · Zbl 1128.92039 · doi:10.3934/dcdsb.2007.8.161 [23] Anderson, R. M.: The persistence of direct life cycle infectious diseases within populations of hosts, Lectures on mathematics in the life sciences, 1-67 (1979) · Zbl 0422.92022 [24] Capasso, V.; Serio, G.: A generalisation of the kermack-mckendrick deterministic epidemic model, Math. biosci. 42, 43-61 (1978) · Zbl 0398.92026 · doi:10.1016/0025-5564(78)90006-8 [25] Heesterbeek, J. A. P.; Metz, J. A. J.: The saturating contact rate in marriage and epidemic models, J. math. Biol. 31, 529-539 (1993) · Zbl 0770.92021 · doi:10.1007/BF00173891 [26] Xu, R.; Ma, Z.: Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear anal. RWA 10, 3175-3189 (2009) · Zbl 1183.34131 · doi:10.1016/j.nonrwa.2008.10.013 [27] Mccluskey, C. Connell: Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear anal. RWA 11, 3106-3109 (2010) · Zbl 1197.34166 · doi:10.1016/j.nonrwa.2009.11.005 [28] Aron, J. L.: Mathematical modeling of immunity to malaria, Math. biosci. 90, 385-396 (1988) · Zbl 0651.92018 · doi:10.1016/0025-5564(88)90076-4 [29] Smith, D. L.; Mckenzie, F. E.: Statics and dynamics of malaria infection in anopheles mosquitoes, Malaria J. 3, 13-27 (2004) [30] 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 [31] Arino, J.; Mccluskey, C. C.; Den Driessche, P. Van: Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. math. 64, 260-276 (2003) · Zbl 1034.92025 · doi:10.1137/S0036139902413829 [32] J.P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM Philadelphia, PA, 1976. [33] Fonda, A.: Uniformly persistent semidynamical systems, Proc. amer. Math. soc. 104, 111-116 (1988) · Zbl 0667.34065 · doi:10.2307/2047471 [34] Hirsch, M. W.: System of differential equations which are competitive or cooperative, IV, SIAM J. Math. anal. 21, 1225-1234 (1990) [35] 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 [36] 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 [37] 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 [38] 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 [39] Zhang, J.; Li, J.; Ma, Z.: Global dynamics of an SEIR epidemic model with immigration of different compartments, Acta Mathematica scientia 26, 551-567 (2006) · Zbl 1096.92039 · doi:10.1016/S0252-9602(06)60081-7