Dynamic Analysis Of Vector-Borne Disease Models With Delays

Vector-borne diseases,such as malaria,dengue fever,yellow fever,plague and so on,threaten the public health of human,and restrict the socioeconomic development in the world.By the mathematical model,the transmission dynamics of mosquito-borne diseases are investi-gated,which shows important practical significance.We mainly study mosquito-borne disease models with delays.Firstly,we incorporate nonlinear incidence rate and delay into biological mathematical model,and build a mosquito-borne disease model with two delays and nonlinear transmission rate,one is the incubation period in the host and the other is the incubation period in the vector,the effects of delay and nonlinear incidence on mosquito-borne disease transmission are studied.We prove that the global dynamics are completely determined by the basic reproduction number R0.Then when R0 ? 1,there exists a unique disease free equilibrium which is globally asymptotically stable;when R0>1,the mosquito-borne disease system is uniformly persistent,and the endemic equilibrium is globally asymptotically stable.Numerical simulations are carried out to illustrate the theoretical results.Secondly,we consider the dengue fever which is a rapidly spreading mosquito-borne dis-ease.Since transmission of dengue virus is greatly influenced by climatic factors and season-ality,then we formulate a dengue model with seasonality which includes maturation delay of mosquitoes,and the effects of delay and seasonality on dengue fever transmission are analyzed.We compute the basic reproduction number R0 by the theory of periodic system.Then we prove that,when R0<1,there exists a unique disease-free steady state which is globally asymptotically stable,and dengue fever will die out;when R0>1,there exists at least one positive periodic solution,and the system is uniformly persistent,and then the disease will persist.We further use these analytic results to study the dengue fever cases in Guangdong Province,China.The simulation results are good accordance with the reported cases of dengue fever in Guangdong Province.Finally,we can conclude the main results of this paper and investigate some ideas for future research.