In this paper, a discrete-time SIR epidemic model with nonlinear incidence and recovery rates is obtained by using the forward Euler’s method. The existence and stability of fixed points in this model are well studied. The center manifold theorem and bifurcation theory are applied to analyze the bifurcation properties by using the discrete time step and the intervention level as control parameters. We discuss in detail some codimension-one bifurcations such as transcritical, period-doubling and Neimark–Sacker bifurcations, and a codimension-two bifurcation with 1:2 resonance. In addition, the phase portraits, bifurcation diagrams and maximum Lyapunov exponent diagrams are drawn to verify the correctness of our theoretical analysis. It is found that the numerical results are consistent with the theoretical analysis. More interestingly, we also found other bifurcations in the model during the numerical simulation, such as codimension-two bifurcations with 1:1 resonance, 1:3 resonance and 1:4 resonance, generalized period-doubling and fold-flip bifurcations. The results show that the dynamics of the discrete-time model are richer than that of the continuous-time SIR epidemic model. Such a discrete-time model may not only be widely used to detect the pathogenesis of infectious diseases, but also make a great contribution to the prevention and control of infectious diseases.

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The nonlocal boundary value problem with eigenparameter dependent boundary conditions is studied in this paper. Firstly, we give the asymptotic expressions of the general solution for the equation corresponding to the initial conditions with eigenparameters, then we prove the multiplicity of eigenvalues some properties of the eigenvalues and eigenfunctions. Finally, the asymptotic formulas of eigenvalues and eigenfunctions are obtained under certain mild conditions. Our method is to incorporate the perturbation theory and asymptotic analysis in the framework of classical Sturm-Liouville problems, which provides a new sight for the investigating of the Sturm-Liouville problems with eigenparameter in boundary conditions.

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