With Nonlinear Incidence Rate And The Unsteady Infectious Disease Transmission Model Of The Population

It is well known that infectious diseases have always been the enemy of mankind. Some results have been achieved on the construction of epidemic models, the quatitative and quantitive analysis to these models and the prediction of disease trends. This paper studies epidemic spread networks with nonlinear incidence and varying population. Due to the varying population, the considered models emerge as large-scale networks; thanks to the nonlinear incidence, the usual Lyapunov anlysis seems to be complicated. In view of this, we proposed a method in which the stability of model and the type of equilibrium are dealt with separately. The main works are as follows:1.Combining cooperative system theory and the criterion based on approximate estimation of system states, we first use cooperative system theory to prove the global obtained global asymptotic stability of epidemic network; second, we use the criterion based on approximate estimation of system states to determin the type of equilibrium, that is, to determine when the origin is an equilibrium and when the points different from the origin are equilibria. The criterion based on approximate estimation of system states is clearly expressed by system parameters, which is easy to verify, and is also easy to obtain some quantitive estimations, for instance, to estimate the effect of recover rate when the disease exists in the population.2. Combining cooperative system theory and Driessche threshold theory,we first use cooperative system theory to prove the global obtained global asymptotic stability of epidemic network;second,we use Driessche threshold theory to determin the type of equilibrium, and obtained global threshold conditions of epidemic network.Different from Driessche threshold theory only gives criterions on local asymptotic stability of0equilibrium,we obtained global threshold conditions.3. Apllying our conclusion to prove the global asymptotic stability of SIS models with transport-related infection, we obtain the global threshold conditions to determine the type of equilibrium and estimate the quantitative effect of various factors on disease prevailing.The conclusions in this paper are of significance in the sense that they can be applied to determine when diseases disappear or prevail, and that in some cases they can be used to guide the disease control.