In this paper, a delayed virus model with two different transmission methods and treatments is investigated. This model is a time-delayed version of the model in (Zhang et al. in Comput. Math. Methods Med. 2015:758362, 2015). We show that the virus-free equilibrium is locally asymptotically stable if the basic reproduction number is smaller than one, and by regarding the time delay as a bifurcation parameter, the existence of local Hopf bifurcation is investigated. The results show that time delay can change the stability of the endemic equilibrium. Finally, we give some numerical simulations to illustrate the theoretical findings.

The nonlinear Schrödinger (NLS) equation is an ideal model for describing optical soliton transmission. This paper first introduces an integer-order generalized coupled NLS equation describing optical solitons in birefringence fibers. Secondly, the semi-inverse and fractional variational method is used to extend the integer‐order model to the space–time fractional order. Moreover, various nonlinear forms of fractional NLS equations are discussed, including the Kerr, power, parabolic, dual-power, and log law. The exact soliton solutions, such as bright, dark, and singular solitons, are given. Finally, the behavior of the solution is shown by three-dimensional figures with different fractional orders, which reveals the propagation characteristics of optical solitons in birefringence fibers described by the generalized coupled space–time fractional NLS equation.

In this study, we assume that blood is assumed to be a viscoelastic and incompressible homogeneous media in which several uniform sized oxygen bubbles are uniformly distributed. Based on this, we establish a (3 + 1)-dimensional modified Kadomtsev–Petviashvili (mKP) equation to describe the long nonlinear pressure waves in the gas bubbles–liquid mixtures. Using bell polynomials, a new bilinear form of the mKP equation is constructed, and then the one- and two-soliton solutions of the equation are obtained by the Hirota method. Via the one-soliton solutions, parametric conditions of the existence of shock wave, elevation and depression solitons, and the Mach reflection characters in the mixtures are discussed. Soliton interactions have been discussed on the basis of the two-soliton solutions. We find that the (i) parallel elastic interactions can exist between the shock and elevation solitons; (ii) oblique elastic interactions can exist between the (a) shock and depression solitons and (b) the elevation and depression solitons; and (iii) oblique inelastic interactions can exist between the two depression solitons.

The rapid development of artificial intelligence, “Internet +,” big data, 5G, and other technologies in the twenty-first century has not only brought great changes to the field of education but also brought unprecedented opportunities and challenges to the reform and innovation of mathematics teaching. The use of intelligent technology to carry out the reform of college mathematics teaching in depth and effectively improve the quality of teaching has become a hot topic discussed by the majority of mathematics teachers. This paper carefully analyzes and sorts out the problems existing in the current intelligent college mathematics teaching and systematically studies the related theories and design principles of the blended teaching mode. The ideas and approaches of the reform of college mathematics blended teaching supported by intelligent technology are deeply discussed from the aspects of improving the level of teachers, setting up a blended teaching team supported by intelligent technology, optimizing the informatization construction of teaching environment and establishing rich teaching resources, building an online and offline classroom teaching system, personalized learning under the background of microclass and cloud class, cross-university blended learning in the network environment, and mathematics precision teaching evaluation and optimization using big data.

Based on the generalized Darboux transformation, we construct high-order solitons for the Lakshmanan–Porsezian–Danie equation under the zero background. Then, we derive a Riemann–Hilbert problem for these high-order solitons. Using this Riemann–Hilbert problem, we find that when the variables (x, t) are proportional to the order n, we can analyze the large-order asymptotics. As the order increases, the (x, t) plane can be divided into four different asymptotic regions, which we refer to as the exponential-decay region, the algebraic-decay region, the genus-zero region, and the genus-one region. For each asymptotic region, we verify the consistency between the exact solution and the asymptotic solution.