The Qualitative Research Of Several Kinds Of SEIRS Models Of Influenza H1N1

On the basis of the classical SIR compartment model, several SEIRS models areformulated to describe the transmission of H1N1influenza in differential patches. By thespectrum theory the matrix, the basic reproductive numberR0is defined and the concreteexpression is derived. Then, the dynamics properties such as existence and stability ofequilibrium of the models determine by the reproductive number are discussed. The resultswe derived can provide theoretical and quantitive basis to the prevention and control of H1N1influenza.Firstly, an SEIRS epidemic model without travelling between the two patches isformulated, and the basic reproductive numberR0defined by spectral radius of a matrix isobtained. The disease free equilibrium is locally asymptotically stable if R01and unstableif R01. By analyzing the expression of the basic reproductive number, it can be obtainedthat the greater cleads to smaller the basic reproduction number, and the disease can becontrol easier. In the sequel, the qualitative results are verified by numerical simulation.Secondly, we establish an SEIRS model with two patches and only one patch hasmigration. The basic reproduction numberR0is computed which determines the disease dieor not. The disease free equilibrium is locally asymptotically stable if R01and unstable ifR01, and numerical simulation results show that the number of infected individuals in thesecond patch quickly converges to zero, the one in the first patch gradually converges to zeroafter reaching a peak. Subsequently, we consider two SEIRS models with two patches, and inwhich different migration rates between the two patches are considered. Numerical simulationresults show that the number of infected individuals approaches to a positive constantwhen R01and approaches zero when R01, which means the disease will be popular inthe area when R01and extinct when R01. Finally, we extend the SEIRS model in two patches to a model in n patches. By usingspectral radius theory of matrix, we obtain the basic reproduction number of the n patchesmodel. The existence and stability of disease free equilibrium for this model is discussed.