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Since the outbreak of coronavirus disease in 2019 (COVID-19), the disease has rapidly spread to the world, and the cumulative number of cases is now more than 2.3 million. We aim to study the spread mechanism of rumors on social network platform during the spread of COVID-19 and consider education as a control measure of the spread of rumors. Firstly, a novel epidemic-like model is established to characterize the spread of rumor, which depends on the nonautonomous partial differential equation. Furthermore, the registration time of network users is abstracted as 'age,' and the spreading principle of rumors is described from two dimensions of age and time. Specifically, the susceptible users are divided into higher-educators class and lower-educators class, in which the higher-educators class will be immune to rumors with a higher probability and the lower-educators class is more likely to accept and spread the rumors. Secondly, the existence and uniqueness of the solution is discussed and the stability of steady-state solution of the model is obtained.

Mobile-Edge Computing (MEC) is a new computing paradigm that provides a capillary distribution of cloud computing capabilities to the network edge. In this paper, we studied the security defense problem in MEC network environment. One big challenge is how to efficiently allocate resources to deploy Mobile-Edge Computing-Intrusion Detection Systems (MEC-IDS) in this system, since all the MEC hosts are composed of resource-constrained network devices. To tackle this challenge, a new resource allocation mechanism based on deterministic differential equation model is proposed and investigated. Existence, uniqueness and stability of the positive solution of this model are obtained by using Lyapuonv stability theory. Furthermore, we extended our study to MEC network environment with stochastic perturbation and established a new stochastic differential equation model. We proved the existence, uniqueness, persistence and oscillatory of the positive solution of this model and quantitatively analyzed the relationship between oscillation and intensity of stochastic perturbation. Numerical simulations are carried out to illustrate the effectiveness of the main results.

Considering that environmental factors, diet, subconscious mind and other uncertainties play an important role in the process of delaying and treating diseases, we propose, in this paper, an amended Hepatitis B virus (HBV) model with stochastic perturbation, and investigate the longtime dynamics of this stochastic model. First, if the basic reproductive number of the corresponding deterministic model is less than 1, some sufficient conditions for almost surely exponentially stable in the sense of the infected cells and free virus are established, and the stationary probability density function of the uninfected sell is also obtained. Further, some sufficient conditions for the existence of the stationary distribution are obtained for the basic reproductive number more than 1. In addition, oscillatory behaviors of this model about the equilibrium of the corresponding deterministic model are discussed. Finally, numerical simulations demonstrate the main theoretical results and show stochastic virus model has more dynamic behaviors relative to its corresponding deterministic model. Theoretical results and numerical simulations imply that the intensity and “type (divided into positive and negative)” of white noise play very important roles in the treatment of infectious disease, which can make the disease more and more repetitive and unpredictable. Of course, comfortable environment, reasonable diet, optimistic mood and other positive uncertainty factors have active effects on the treatment and delaying of diseases, but not the converse.

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