On Several Classes Of Epidemic Dynamical Models

Epidemic dynamical models is an important part of mathematical models in biology.Recently,they have received much attention by a lot of scholars.The aim of this work is to construct several epidemic mathematical models by the method of compartment modelling and study their dynamic behaviors.In the first part,two epidemic models without latent period are discussed.Firstly,an autonomous SIRS epidemic model with time delay is studied.The basic reproductive number R₀ is obtained which determines whether the disease is extinct or not.When the basic reproductive number is greater than 1,it is proved that the disease is permanent in the population,and explicit formula are obtained by which the eventual lower bound of the fraction of infectious individuals can be computed.We mainly use the technique of Liapunov functional to establish the global stability of the infection-free equilibrium and the local stability of the endemic equilibrium but need another sufficient condition.Secondly,an SIRVS epidemic model with pulse vaccination strategy is analyzed.We are interested in finding the basic reproductive number of the model which determine whether or not the disease dies out.The global attractivity of the disease-free periodic solution is obtained when the basic reproductive number is less than unity.The disease is permanent when the basic reproductive number is greater than unity,i.e.,the epidemic will turn out to endemic.Our results indicate that the disease will go to extinction when the vaccination rate reaches some critical value.In the second part,we discuss two epidemic models with latent period.Firstly, the asymptotic behavior of solutions of an autonomous SEIRS epidemic model with the saturation incidence is studied.Using the method of Liapunov functional,we obtain the disease-free equilibrium is globally stable if the basic reproduction number is not greater than one.Moreover,we show that the disease is permanent if the basic reproduction number is greater than one.Furthermore,the sufficient conditions of locally and globally asymptotically stable convergence to an endemic equilibrium are obtained base on the permanence.Secondly,a delayed SEIRS epidemic model with pulse vaccination and saturation incidence rate is investigated.Using Krasnoselskii's fixed-point theorem,we obtain the existence of infection-free periodic solution of the impulsive delayed epidemic system.We define some new threshold values R₁,R₂ and R₃.Further,using the comparison theorem,we obtain the explicit formulae of R₁ and R₂.Under the condition R₁＜1,the infection-free periodic solution is globally attractive,and R₂＞1 implies that the disease is permanent.Theoretical results show that the disease will be extinct if the vaccination rate is larger thanθ^* and the disease is uniformly persistent if the vaccination rate is less thanθ_*.These results indicate that a large pulse vaccination rate will lead to eradication of the disease.Moreover,we prove that the disease will be permanent as R₃＞1.In the last part,two epidemic models with time-dependent coefficients are considered. Firstly,We wish to investigate the dynamical behavior of an SIRVS epidemic model with time-dependent coefficients.Under the quite weak assumptions ,we give some new threshold conditions which determine whether or not the disease will go to extinction.The permanence and extinction of the infectious disease is studied.When the system degenerates into periodic or almost periodic system, the corresponding sharp threshold results are obtained for permanent endemicity versus extinction in terms of asymptotic time.Secondly,we derive some threshold conditions for permanence and extinction of diseases that can be described by a nonautonomous SEIRS epidemic model.Under the quite weak assumptions,we establish some sufficient conditions to prove the permanence and extinction of disease through analysis and auxiliary function.