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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