Abstract: Based on the mean field theory of complex networks, we  established a SEIR (susceptible, exposed, infectious, removed) model with periodic contagion  rates. Firstly, we determined the positive invariant set of the system and used  the spectral radius method of the linear integral operator to give the expression of the basic reproduction number R₀. Secondly, by using  the principle of comparison, we prove that the disease-free equilibrium point of the system is globally asymptotically stable when R₀<1, and there exists at least one positive periodic solution in the system and the system is uniformly persistent when R₀>1. Finally, the correctness of the theoretical analysis  is  verified by using numerical simulations, and  the results show  that the greater the maximum degree of nodes in the network, the greater the absolute density of infected people, indicating that the network structure has a significant impact on the spread of infectious diseases. In addition, through numerical simulations, we obtain that when R₀>1,  the system has a unique globally asymptotically stable  positive periodic solution, which enriches the theoretical analysis results.

Key words: periodicity, basic reproduction number, positive periodic solution, stability