The Dynamic Behavior Analysis Of Dengue Fever Model With Predator And Immunization

Dengue is an acute infectious diseases which caused by dengue fever virus and trans-mitted by the Aedes aegypti mosquito. A person would be infected if he bitted by Aedesaegypti mosquito carried with dengue fever virus which through mosquito's saliva intohis blood. If a person bitted by mosquito from having a temperature to bring down thefever (about six or seven days), the dengue fever virus may pass on to the mosquito andtransmitted. The disease can't transmitted through person to person, therefore, we can'tinfect by contacting with other infectious person. In1779, the disease was discoveredin Cairo (Egypt), Jakarta (Indonesia) and Philadelphia, and named aden fever or break-bone fever by its features. London, England in1869by the Royal Society of Medicinenamed as dengue fever. In1970, only DEN-2virus was present in the Americas, althoughDEN-3may have had a focal distribution in Colombia and Puerto Rico. In1977, DEN-1was introduced and caused major epidemics throughout the region over a16-year pe-riod. DEN-4was introduced in1981and caused similar widespread epidemics. Also in1981, a new strain of DEN-2from Southeast Asia caused the frst major DHF epidemicin the Americas (Cuba). so the studying of dengue fever model has important zoologymeaning. In this article, we would like to investigate the global asymptotic stability andglobal asymptotic attractivity of the system we formulate. The models are: Modelingtwo diferent serotype dengue virus control by introduction of Larvivorous fsh and thetransmission of dengue fever with age structured and immunized population. The ma-jor methods adopted are internally chain transmission theory, the LaSalle's invarianceprinciple, LaSalle-Lyapunov function, H-matrix, companion matrix, diferential equationcomparison principle and so on.The main contents in this paper can be summarized as follows:The frst section is introduction, in which we present research background, purposeand signifcance of the dengue fever model, and then the research status and results of the dengue fever model are given. Finally the organization of this paper is also presented.In Section2, Modeling two diferent serotype dengue virus control by introduction ofLarvivorous fsh is studied. At frst the preliminary information is given, at frst, we getthe basic reproduction number by the basic reproduction matrix, by using internallychain transmission theory, the LaSalle's invariance principle and diferential equationcomparison principle, we get the asymptotic stability and global asymptotic attractivityof the system. Finally, the numerical simulation is given. By the numerical simulation,the conclusions are valid.In Section3, the dengue fever model; the transmission of dengue fever with agestructured and immunized population is studied. At frst, we get the basic reproductionnumber by the basic reproduction matrix, Furthermore, by using Routh Hurwitz con-dition, LaSalle Lyapunov function, LaSalle invariant principle and matrix methods,we obtain the sufcient of disease free equilibrium's global asymptotic stability and theendemic equilibrium's local asymptotic stability, then the conjecture of endemic equilib-rium's global asymptotic stability is given, at last, we give two samples about the model,furthermore, the numerical simulation is given which demonstrate that the conjecture isvalid.