GLOBAL CLASSICAL SOLUTIONS AND THE CLASSICAL LIMIT OF THE NON-RELATIVISTIC VLASOV-DARWIN SYSTEM WITH SMALL INITIAL DATA

We investigate the global classical solutions of the non-relativistic Vlasov-D arwin system with generalized variables(VDG)in three dimensions.We first prove the global existence and uniqueness for small initial data and derive the decay estimates of the Darwin potentials.Then,we show in this framework that the solutions converge in a pointwise sense to solutions of the classical Vlasov-Poisson system(VP)at the asymptotic rate of 1/c2 as the speed of light c tends to infinity for all time.Moreover,we obtain rigorously an asymptotic estimate of the difference between the two systems.

GLOBAL STABILITY OF AN SIRS EPIDEMIC MODEL WITH DELAYS

In this article, an SIRS epidemic model spread by vectors （mosquitoes） which have an incubation time to become infectious is formulated. It is shown that a disease-free equilibrium point is globally stable if no endemic equilibrium point exists. Further, the endemic equilibrium point （if it exists） is globally stable with a respect ＂weak delay＂. Some known results are generalized.

Global Stability Analysis of a Delayed SEIQR Epidemic Model with Quarantine and Latent

In this paper, we study a kind of the delayed SEIQR infectious disease model with the quarantine and latent, and get the threshold value which determines the global dynamics and the outcome of the disease. The model has a disease-free equilibrium which is unstable when the basic reproduction number is greater than unity. At the same time, it has a unique endemic equilibrium when the basic reproduction number is greater than unity. According to the mathematical dynamics analysis, we show that disease-free equilibrium and endemic equilibrium are locally asymptotically stable by using Hurwitz criterion and they are globally asymptotically stable by using suitable Lyapunov functions for any Besides, the SEIQR model with nonlinear incidence rate is studied, and the that the basic reproduction number is a unity can be found out. Finally, numerical simulations are performed to illustrate and verify the conclusions that will be useful for us to control the spread of infectious diseases. Meanwhile, the will effect changing trends of in system (1), which is obvious in simulations. Here, we take as an example to explain that.

Dynamic analysis of a sexually transmitted disease model on complex networks

In this paper,a sexually transmitted disease model is proposed on complex networks,where contacts between humans are treated as a scale-free social network.There are three groups in our model,which are dangerous male,non-dangerous male,and female.By mathematical analysis,we obtain the basic reproduction number for the existence of endemic equilibrium and study the effects of various immunization schemes about different groups.Furthermore,numerical simulations are undertaken to verify more conclusions.

Analysis of a Delayed SIR Model with Exponential Birth and Saturated Incidence Rate

In this paper, a delayed SIR model with exponential demographic structure and the saturated incidence rate is formulated. The stability of the equilibria is analyzed with delay: the endemic equilibrium is locally stable without delay;and the endemic equilibrium is stable if the delay is under some condition. Moreover the dynamical behaviors from stability to instability will change with an appropriate?critical value. At last, some numerical simulations of the model are given to illustrate the main theoretical results.

The Stability of Highly Pathogenic Avian Influenza Epidemic Model with Saturated Contact Rate

In this paper we present a highly pathogenic Avian influenza epidemic model with saturated contact rate. According to study of the dynamics, we calculated the basic reproduction number of the model. Through the analysis of this model, we have the following conclusion: if R0 ≤ 1, there is only one disease-free equilibrium which is globally stable, the disease will die;if R0 > 1, there is only one endemic equilibrium which is globally stable, disease will be popular.

2-Competition Index of Primitive Digraphs Using Graph Theory

Recently,scrambling index and competition index are widely applied to stochastic matrices and food webs. By analyzing the relationship of scrambling index and 2-competition index,n-「d/2」+ 1 was proved to be an upper bound of the 2-competition2 index of a primitive digraph with exact d loops in this article.Moreover,the maximum index problem and the index set problem for the 2-competition index of primitive digraphs with minimally strong digraphs were settled.

Spatial Pattern of an Epidemic Model with Cross-diffusion

Pattern formation of a spatial epidemic model with both self- and cross-diffusion is investigated. From the Turing theory, it is well known that Turing pattern formation cannot occur for the equal self-diffusion coefficients. However, combined with cross-diffusion, the system will show emergence of isolated groups, i.e., stripe-like or spotted or coexistence of both, which we show by both mathematical analysis and numerical simulations. Our study shows that the interaction of self- and cross-diffusion can be considered as an important mechanism for the appearance of complex spatiotemporal dynamics in epidemic models.

Modelling the spread of sexually transmitted diseases on scale-free networks

In this paper a new model for the spread of sexually transmitted diseases （STDs） is presented. The dynamic behaviors of the model on a heterogenons scale-free （SF） network are considered, where the absence of a threshold on the SF network is demonstrated, and the stability of the disease-free equilibrium is obtained. Three immunization strategies, uniform immunization, proportional immunization and targeted immunization, are applied in this model. Analytical and simulated results are given to show that the proportional immunization strategy in the model is effective on SF networks.

Emergence of spatiotemporal chaos arising from far-field breakup of spiral waves in the plankton ecological systems

It has been reported that the minimal spatially extended phytoplankton-zooplankton system exhibits both temporal regular/chaotic behaviour, and spatiotemporal chaos in a patchy environment. As a further investigation by means of computer simulations and theoretical analysis, in this paper we observe that the spiral waves may exist and the spatiotemporal chaos emerge when the parameters are within the mixed Turing-Hopf bifurcation region, which arises from the far-field breakup of the spiral waves over a large range of diffusion coefficients of phytoplankton and zooplankton. Moreover, the spatiotemporal chaos arising from the far-field breakup of spiral waves does not gradually invade the whole space of that region. Our results are confirmed by nonlinear bifurcation of wave trains, We also discuss ecological implications of these spatially structured patterns.

Study of TSP based on self-organizing map

Self-organizing map(SOM) proposed by Kohonen has obtained certain achievements in solving the traveling salesman problem(TSP).To improve Kohonen SOM,an effective initialization and parameter modification method is discussed to obtain a faster convergence rate and better solution.Therefore,a new improved self-organizing map(ISOM)algorithm is introduced and applied to four traveling salesman problem instances for experimental simulation,and then the result of ISOM is compared with those of four SOM algorithms:AVL,KL,KG and MSTSP.Using ISOM,the average error of four travelingsalesman problem instances is only 2.895 0%,which is greatly better than the other four algorithms:8.51%(AVL),6.147 5%(KL),6.555%(KG) and 3.420 9%(MSTSP).Finally,ISOM is applied to two practical problems:the Chinese 100 cities-TSP and102 counties-TSP in Shanxi Province,and the two optimal touring routes are provided to the tourists.

Emergence of Strange Spatial Pattern in a Spatial Epidemic Model

Pattern formation of a spatial epidemic model with nonlinear incidence rate hI^2 S/（1 ＋ αI^2） is investigated. Our results show that strange spatial dynamics, i.e., filament-like pattern, can be obtained by both mathematical analysis and numerical simulation, which are different from the previous results in the spatial epidemic model such as stripe-like or spotted or coexistence of both pattern and so on. The obtained results well extend the finding of pattern formation in the epidemic model and may well explain the distribution of the infected of some epidemic.

Nonpharmaceutical interventions contribute to the control of COVID-19 in China based on a pairwise model

Nonpharmaceutical interventions(NPIs),particularly contact tracing isolation and household quarantine,play a vital role in effectively bringing the Coronavirus Disease 2019(COVID-19)under control in China.The pairwise model,has an inherent advantage in characterizing those two NPIs than the classical well-mixed models.Therefore,in this paper,we devised a pairwise epidemic model with NPIs to analyze COVID-19 outbreak in China by using confirmed cases during February 3rde22nd,2020.By explicitly incorporating contact tracing isolation and family clusters caused by household quarantine,our model provided a good fit to the trajectory of COVID-19 infections.We calculated the reproduction number R=1.345(95%CI:1.230-1.460)for Hubei province and R=1.217(95%CI:1.207-1.227)for China(except Hubei).We also estimated the peak time of infections,the epidemic duration and the final size,which are basically consistent with real observation.We indicated by simulation that the traced high-risk contacts from incubated to susceptible decrease under NPIs,regardless of infected cases.The sensitivity analysis showed that reducing the exposure of the susceptible and increasing the clustering coefficient bolster COVID-19 control.With the enforcement of household quarantine,the reproduction number R and the epidemic prevalence declined effectively.Furthermore,we obtained the resumption time of work and production in China(except Hubei)on 10th March and in Hubei at the end of April 2020,respectively,which is broadly in line with the actual time.Our results may provide some potential lessons from China on the control of COVID-19 for other parts of the world.

Blow Up of Solutions to One Dimensional Initial-Boundary Value Problems for Semilinear Wave Equations with Variable Coefficients

This paper is devoted to studying the following initial-boundary value prob- lem for one-dimensional semilinear wave equations with variable coefficients and with subcritical exponent：We wili establish a blowup result for the above initial-boundary value problem, it is proved that there can be no global solutions no matter how small the initial data are, and also we give the lifespan estimate of solutions for above problem.

Pattern formation induced by cross-diffusion in a predator-prey system

This paper considers the Holling-Tanner model for predator-prey with self and cross-diffusion. From the Turing theory, it is believed that there is no Turing pattern formation for the equal self-diffusion coefficients. However, combined with cross-diffusion, it shows that the system will exhibit spotted pattern by both mathematical analysis and numerical simulations. Furthermore, asynchrony of the predator and the prey in the space. The obtained results show that cross-diffusion plays an important role on the pattern formation of the predator-prey system.

SPECTRAL ANALYSIS AND STABILIZATION OF A COUPLED WAVE-ODE SYSTEM

In this paper, an interconnected wave-ODE system with K-V damping in the wave equation and unknown parameters in the ODE is considered. It is found that the spectrum of the system operator is composed of two parts： Point spectrum and continuous spectrum. The continuous spectrum consists of an isolated point 1 1/d, and there are two branches of the asymptotic eigenvalues： The first branch is accumulating towards 1 -2, and the other branch tends to -∞. It is shown that there is a sequence of generalized eigenfunctions, which forms a Riesz basis for the Hilbert state space. As a consequence, the spectrum-determined growth condition and exponential stability of the system are concluded.

Global Dynamics of a Multi-group SEIR Epidemic Model with Infection Age

Consider the heterogeneity(e.g.,heterogeneous social behaviour,heterogeneity due to different geography,contrasting contact patterns and different numbers of sexual partners etc.)of host population,in this paper,the authors propose an infection age multigroup SEIR epidemic model.The model system also incorporates the feedback variables,where the infectivity of infected individuals may depend on the infection age.In the direction of mathematical analysis of model,the basic reproduction number R_0 has been computed.The global stability of disease-free equilibrium and endemic equilibrium have been established in the term of R_(0).More precisely,for R_(0)≤1,the disease-free equilibrium is globally asymptotically stable and for R_(0)>1,they establish global stability of endemic equilibrium using some graph theoretic techniques to Lyapunov function method.By considering a numerical example,they investigate the effects of infection age and feedback on the prevalence of the disease.Their result shows that feedback parameters have different and even opposite effects on different groups.However,by choosing an appropriate value of feedback parameters,the disease could be eradicated or maintained at endemic level.Besides,the infection age of infected individuals may also change the behaviour of the disease,global stable to damped oscillations or damped oscillations to global stable.

Transmission analysis of COVID-19 with discrete time imported cases: Tianjin and Chongqing as cases

In 2020,an unexpectedly large outbreak of the coronavirus disease 2019(COVID-19)epidemic was reported in China's Mainland.As we known,the epidemic was caused by imported cases in other provinces of China except for Hubei in 2020.In this paper,we developed a differential equation model with tracing isolation strategy with close contacts of newly confirmed cases and discrete time imported cases,to perform assessment and risk analysis for COVID-19 outbreaks in Tianjin and Chongqing city.Firstly,the model behavior without imported cases was given.Then,the real-time regeneration number in Tianjin and Chongqing city revealed a trend of rapidly rising,and then falling fast.Finally,sensitivity analysis demonstrates that the earlier with Wuhan lock-down,the fewer cases in these two cities.One can obtain that the tracing isolation of close contacts of newly confirmed cases could effectively control the spread of the disease.But it is not sensitive for the more contact tracing isolation days on confirmed cases,the fewer cases.Our investigation model could be potentially helpful to provide model building technology for the transmission of COVID-19.

The impact of media coverage with a piecewise transmission rate in the spread of diseases

A three-dimensional compartmental model with media coverage is proposed to describe the real characteristics of its impact in the spread of infectious diseases in a given region. A piecewise continuous transmission rate is introduced to describe that media coverage exhibits its effect only when the number of the infected exceeds a certain critical level. Further, it is assumed that the impact of media coverage on the contact transmission is described by an exponential decreasing factor. Stability analysis of the model shows that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than unity. On the other hand, when the basic reproduction number is greater than unity and media coverage impact is sufficiently small, a unique endemic equilibrium exists, which is globally asymptotically stable.

THE ■■-LEMMA UNDER SURJECTIVE MAPS

We consider the ■■-lemma for complex manifolds under surjective holomorphic maps.Furthermore,using Deligne-Griffiths-Morgan-Sullivan’s theorem,we prove that a product compact complex manifold satisfies the ■■-lemma if and only if all of its components do as well.