In the current study, a generalized SEIR epidemic model is studied. The generalized fractional-order SEIR model (susceptible-infected-recovered (SIR) epidemic) model differentiated the population into susceptible population, exposure population, infected population, and rehabilitation population and has fundamental mentoring importance for the forecast of the probable outburst of infectious ailments. The fundamental duplicated quantity R 0 is inferred. When R 0 < 1 , the disease-free equilibrium (DFE) is particular and tending towards stability. When R 0 > 1 , the endemic equilibrium is sole. In addition, certain circumstances are set up to make sure the local progressive stability of disease-free and endemic equilibrium. Considering the influence of the individual behavior, a broader SEIR epidemic model is raised, which classified the population into susceptible, exposure, infected, and rehabilitation. What is more, the basic reproduction number, that regulates whether the infection will die out or not, is obtained by the spectral radius of the next-generation matrix; moreover, the global stability of DFE and endemic equilibrium are analyzed by a geometry method.

A system of nonlinear diffusion equations with three components is studied via the potential symmetry method. It is shown that the system admits the potential symmetries for certain diffusion terms. The invariant solutions associated with the potential symmetries are obtained.

By means of asymptotically stable theory and infection model theory of ordinary differential equation, we do research on SIQS model with nonlinear and isolation. Firstly, we obtain the existence of threshold value R0 of disease-free equilibration point and local disease equilibration point. Secondly, we prove disease-free equilibration point is locally asymptotically stable when R0 < 1, and local disease equilibration point is locally asymptotically stable when R0 > 1. Furthermore, we have disease-free equilibration point and local disease equilibration point are globally asymptotically stable with the help of Liapunov function. Lastly, we explain at the point of biology.

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