Global stability of a delayed SEIRS epidemic model with saturation incidence rate

In this paper, an SEIRS epidemic model with a saturation incidence rate and a time delay describing a latent period is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is established. When the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. Numerical simulations are carried out to illustrate the main theoretical results.     

The pulse treatment of computer viruses: a modeling study

Unlike new medical procedures, new antivirus software can be disseminated rapidly through the Internet and takes effect immediately after it is run. As a result, a considerable number of infected computers can be cured almost simultaneously. Consequently, it is of practical importance to understand how pulse treatment affects the spread of computer viruses. For this purpose, an impulsive malware propagation model is proposed. To the best of our knowledge, this is the first computer virus model that takes into account the effect of pulse treatment. The dynamic properties of this model are investigated comprehensively. Specifically, it is found that (a) the virus-free periodic solution is globally asymptotically stable when the basic reproduction ratio (BRR) is less than unity, (b) infections are permanent when the BRR exceeds unity, and (c) a locally asymptotically stable viral periodic solution bifurcates from the virus-free periodic solution when the BRR goes through unity. A close inspection of the influence of different model parameters on the BRR allows us to suggest some feasible measures of eradicating electronic infections.     

Basic Reproduction Ratios for Almost Periodic Compartmental Epidemic Models

The theory of the basic reproduction ratio $R_{0}$ R 0 and its computation formulae for almost periodic compartmental epidemic models are established. It is shown that the disease-free almost periodic solution is stable if $R_{0}<1$ R 0 < 1 , and unstable if $R_{0}>1$ R 0 > 1 . We also apply the developed theory to a patchy model with almost periodic population dispersal and disease transmission coefficients to obtain a threshold type result for uniform persistence and global extinction of the disease.     

Global properties of a class of virus infection models with multitarget cells

In this paper, we propose a class of virus infection models with multitarget cells and study their global properties. We first study three models with specific forms of incidence rate function, then study a model with a more general nonlinear incidence rate. The basic model is a (2n+1)-dimensional nonlinear ODEs that describes the population dynamics of the virus, n classes of uninfected target cells, and n classes of infected target cells. Model with exposed state and model with saturated infection rate are also studied. For these models, Lyapunov functions are constructed to establish the global asymptotic stability of the uninfected and infected steady states of these models. We have proven that if the basic reproduction number is less than unity then the uninfected steady state is globally asymptotically stable, and if the basic reproduction number is greater than unity then the infected steady state is globally asymptotically stable. For the model with general nonlinear incidence rate, we construct suitable Lyapunov functions and establish the sufficient conditions for the global stability of the uninfected and infected steady states of this model.     

Threshold dynamics for a HFMD epidemic model with?periodic transmission rate

In this paper, a periodic epidemic model is proposed in order to simulate the dynamics of HFMD transmission. We consider the effects of quarantine in the children population. We obtain a threshold value which determines the extinction and uniform persistence of the disease. Our results show that the disease-free equilibrium is globally asymptotically stable if the threshold value is less than unity. Otherwise, the system has a positive periodic solution and the disease persists. Numerical simulations show that quarantine has a positive impact on the spread of disease, i.e., quarantine is beneficial to the intervention and control of the disease outbreak in the children population.     

Impacts of the Cell-Free and Cell-to-Cell Infection Modes on Viral Dynamics

Virus can disseminate between uninfected target cells via two modes, namely, the diffusion-limited cell-free viral spread and the direct cell-to-cell transfer using virological synapses. To examine how these two viral infection modes impact the viral dynamics, in this paper, we propose and analyze a general viral infection model that incorporates these two viral infection modes. The model also includes nonlinear target-cell dynamics, infinitely distributed intracellular delays, nonlinear incidences, and concentration-dependent clearance rates. It is shown that the numbers of secondly infected cells through the cell-free infection mode and the cell-to-cell infection mode both contribute to the basic reproduction number. Under some reasonable assumptions, the model exhibits a global threshold dynamics: the infection is cleared out if the basic reproduction number is less than one and the infection persists if the basic reproduction number is larger than one. Two specific examples are provided to illustrate that our theoretical results cover and improve some existing ones. When the underlying assumptions are not satisfied, oscillations via global Hopf bifurcation can be observed. A brief simulation of two-parameter bifurcation analysis is given to explore the joint impacts on viral dynamics for the interplay between nonlinear target-cell dynamics and intracellular delays.     

Modelling the effects of heavy alcohol consumption on the transmission dynamics of gonorrhea

Prior studies have indicated that heavy alcohol drinkers are likely to engage in risky sexual behaviours and thus, more likely to get sexually transmitted infections (STIs) than social drinkers. Here, we formulate a deterministic model for evaluating the impact of heavy alcohol drinking on the reemerging gonorrhea epidemic. The model is rigorously analysed, showing the existence of a globally asymptotically stable disease-free equilibrium whenever the reproductive number is less than unity. If the disease threshold number is greater than unity, a unique endemic equilibrium exists and is globally asymptotically stable in the interior of the feasible region and the disease persists at endemic proportions if it is initially present. Both analytical and numerical results are provided to ascertain whether heavy alcohol drinking has an impact on the transmission dynamics of gonorrhea.     

Modeling and analysis of the transmission dynamics of tuberculosis without and with seasonality

A deterministic model of tuberculosis without and with seasonality is designed and analyzed into its transmission dynamics. We first present and analyze a tuberculosis model without seasonality, which incorporates the essential biological and epidemiological features of the disease. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with one or more stable endemic equilibria when the associated basic reproduction number is less than unity. The statistical data of tuberculosis (TB) cases show seasonal fluctuations in many countries. Then, the extension of our TB model by incorporating seasonality is developed and the basic reproduction ratio is defined. Parameter values of the model are estimated according to demographic and epidemiological data in Cameroon. The simulation results are in good accordance with the seasonal variation of the reported cases of active TB in Cameroon.     

Periodic solutions of an epidemic model with saturated treatment

Based on the fact that many infectious diseases exhibit periodic fluctuations and there is a saturated phenomenon during disease treatment, we study an SIR model with periodic incidence rate and saturated treatment function. Firstly, we find that the basic reproduction number less than 1 cannot insure the global stability of disease-free equilibrium and it needs to add other conditions. Moreover, we establish sufficient conditions for the multiplicity of positive periodic solutions. We also apply the numerical method to confirm theoretical results and show the stability of the periodic solutions. We observe that there are two periodic solutions in the system where one is stable and the other one is unstable. These results will provide some guidance for control measures of disease.     

QUALITATIVE ANALYSIS OF AN SEIS EPIDEMIC MODEL WITH NONLINEAR INCIDENCE RATE

By means of limit theory and Fonda's theorem, an SEIS epidemic model with constant recruitment and the disease-related rate is considered. The incidence term is of the nonlinear form, and the basic reproduction number is found. If the basic reproduction number is less than one, there exists only the disease-free equilibrium, which is globally asymptotically stable, and the disease dies out eventually. If the basic reproduction number is greater than one, besides the unstable disease-free equilibrium, there exists also a unique endemic equilibrium, which is locally asymptotically stable, and the disease is uniformly persistent.     

Longwave Approximation in Film Flow Theory

An asymptotic longwave model which takes dispersive terms into account is constructed for describing the motion of thin films with finite deviations from the middle surface. An exact periodic solution describing a nonlinear capillary wave is constructed within the framework of the model. Small deviations from the nonlinear capillary wave are described by a linear system with periodic coefficients. It is shown that for wave perturbation periods greater than a certain critical value the monodromy matrix of this system has eigenvalues whose absolute values are equal to unity. For perturbation periods less than the critical period the absolute value of one of the eigenvalues becomes greater than unity.     

Dynamic and wear characteristics of self-lubricating bearing cage: effects of cage pocket shape

34,354,966 active cases and 460,787 deaths because of COVID-19 pandemic were recorded on November 06, 2021, in India. To end this ongoing global COVID-19 pandemic, there is an urgent need to implement multiple population-wide policies like social distancing, testing more people and contact tracing. To predict the course of the pandemic and come up with a strategy to control it effectively, a compartmental model has been established. The following six stages of infection are taken into consideration: susceptible (S), asymptomatic infected (A), clinically ill or symptomatic infected (I), quarantine (Q), isolation (J) and recovered (R), collectively termed as SAIQJR. The qualitative behavior of the model and the stability of biologically realistic equilibrium points are investigated in terms of the basic reproduction number. We performed sensitivity analysis with respect to the basic reproduction number and obtained that the disease transmission rate has an impact in mitigating the spread of diseases. Moreover, considering the non-pharmaceutical and pharmaceutical intervention strategies as control functions, an optimal control problem is implemented to mitigate the disease fatality. To reduce the infected individuals and to minimize the cost of the controls, an objective functional has been constructed and solved with the aid of Pontryagin’s maximum principle. The implementation of optimal control strategy at the start of a pandemic tends to decrease the intensity of epidemic peaks, spreading the maximal impact of an epidemic over an extended time period. Extensive numerical simulations show that the implementation of intervention strategy has an impact in controlling the transmission dynamics of COVID-19 epidemic. Further, our numerical solutions exhibit that the combination of three controls are more influential when compared with the combination of two controls as well as single control. Therefore, the implementation of all the three control strategies may help to mitigate novel coronavirus disease transmission at this present epidemic scenario.

     

动力学平衡方程的Euler中点辛差分求解格式

给出了动力学方程${\pmb M}\ddot {\pmb x} + {\pmb C}\dot {\pmb x} + {\pmb K \pmb x} = {\pmb R}$的二阶Euler中点隐式差分求解格式，分保守系统、无 阻尼受迫振动系统和阻尼系统3种情况, 讨论了算法中Jacobi矩阵${\pmb A}$的性质，譬 如${\pmb A}$是否为辛矩阵以及谱半径等. 对于无阻尼系统，证明了无论是否存在外 载荷，Jacobi 矩阵都是辛矩阵. 证明了辛矩阵的所有本征值的模为1，其谱半径永远 为1, 以及$\delta = 0.5$和$\alpha = 0.25$的Newmark算法就是Euler中点隐式差 分格式，对保守系统它们都是辛算法. 严格证 明了Euler中点辛格式是严格保持系统能量的. 通过算例详细讨论了保辛算法用于求解非保 守系统动态特性的优越性，如广义保结构特性等；分析了保辛算法的相位误差以及由其引起 的系统的附加能量特性；分析了保辛算法和$\delta \ne 0.5$的Newmark算法的精度随着激励频率与系统固有频率比的变化情况等     

Eigenanalysis and continuum modelling of pre-twisted repetitive beam-like structures

A repetitive pin-jointed, pre-twisted structure is analysed using a state variable transfer matrix technique. Within a global coordinate system the transfer matrix is periodic, but introduction of a local coordinate system rotating with nodal cross-sections results in an autonomous transfer matrix for this Floquet system. Eigenanalysis reveals four real unity eigenvalues, indicating tension–torsion coupling, and equivalent continuum properties such as Poisson’s ratio, cross-sectional area, torsion constant and the tension–torsion coupling coefficient are determined. A variety of real and complex near diagonal Jordan decompositions are possible for the multiple (eight) complex unity eigenvalues and these are discussed in some detail. Analysis of the associated principal vectors shows that a bending moment produces curvature in the plane of the moment, together with shear deformation in the perpendicular plane, but no bending–bending coupling; the choice of a structure having an equilateral triangular cross-section is thought responsible for this unexpected behaviour, as the equivalent continuum second moments of area are equal about all cross-sectional axes. In addition, an asymmetric stiffness matrix is obtained for bending moment and shearing force coupling, and possible causes are discussed.     

Basic Reproduction Ratios for Periodic Compartmental Models with Time Delay

In this paper, we establish the theory of basic reproduction ratio \(R_0\) for a large class of time-delayed compartmental population models in a periodic environment. It is proved that \(R_0\) serves as a threshold value for the stability of the zero solution of the associated periodic linear systems. As an illustrative example, we also apply the developed theory to a periodic SEIR model with an incubation period and obtain a threshold result on its global dynamics in terms of \(R_0\).     

Global stability of a delayed HIV infection model with nonlinear incidence rate

Under the assumption that the incidence rate of the infection and the removal rate of the infective by cytotoxic T lymphocytes are nonlinear, we study the global dynamics of a HIV infection model with the response of the immune system using characteristic equation, the Fluctuation lemma, and the direct Lyapunov method. The existence of a threshold parameter, i.e., the basic reproduction number or basic reproductive ratio is established and the global stability of the equilibria is discussed.     

Analysis of non-pharmaceutical interventions impacts on COVID-19 pandemic in Iran

The COVID-19 pandemic shows to have a huge impact on people's health and countries' infrastructures around the globe. Iran was one of the first countries that experienced the vast prevalence of the coronavirus outbreak. The Iranian authorities applied various non-pharmaceutical interventions to eradicate the epidemic in different periods. This study aims to investigate the effectiveness of non-pharmaceutical interventions in managing the current Coronavirus pandemic and to predict the next wave of infection in Iran. To achieve the research objective, the number of cases and deaths before and after the interventions was studied and the effective reproduction number of the infection was analyzed under various scenarios. The SEIR generic model was applied to capture the dynamic of the pandemic in Iran. To capture the effects of different interventions, the corresponding reproduction number was considered. Depending on how people are responsive to interventions, the effectiveness of each intervention has been investigated. Results show that the maximum number of the total of infected individuals will occur around the end of May and the start of June 2021. It is concluded that the outbreak could be smoothed if full lockdown and strict quarantine continue. The proposed modeling could be used as an assessment tool to evaluate the effects of different interventions in new outbreaks.

     

The importance of immune responses in a model of hepatitis B virus

In this paper, the dynamical behavior of a hepatitis B virus model with CTL immune responses is studied. Analyzing the model, we show that the virus-free equilibrium is globally asymptotically stable if the basic reproductive ratio of virus is less than one and the endemic equilibrium is locally asymptotically stable if the basic reproductive ratio is greater than one. When the basic reproductive ratio is greater than one, the system is uniformly persistent, which means the virus is endemic. Mathematical analysis and numerical simulations show that the CTL immune responses play a significant and decisive role in eradication of disease. The study and information derived from this model may have an important impact on treatment protocols of hepatitis B virus in the future.     

Dynamics of a novel nonlinear SIR model with double epidemic hypothesis and impulsive effects

In this paper, the propagation of a nonlinear delay SIR epidemic using the double epidemic hypothesis is modeled. In the model, a system of impulsive functional differential equations is studied and the sufficient conditions for the global attractivity of the semi-trivial periodic solution are drawn. By use of new computational techniques for impulsive differential equations with delay, we prove that the system is permanent under appropriate conditions. The results show that time delay, pulse vaccination, and nonlinear incidence have significant effects on the dynamics behaviors of the model. The conditions for the control of the infection caused by viruses A and B are given.     

Effects of aspect ratio on unsteady solutions through curved duct flow

The effects of the aspect ratio on unsteady solutions through the curved duct flow are studied numerically by a spectral based computational procedure with a temperature gradient between the vertical sidewalls for the Grashof number 100 ≤ Gr ≤ 2 000. The outer wall of the duct is heated while the inner wall is cooled and the top and bottom walls are adiabatic. In this paper, unsteady solutions are calculated by the time history analysis of the Nusselt number for the Dean numbers Dn = 100 and Dn = 500 and the aspect ratios 1≤γ≤ 3. Water is taken as a working fluid （Pr =7.0）. It is found that at Dn = 100, there appears a steady-state solution for small or large Gr. For moderate Gr, however, the steady-state solution turns into the periodic solution if γ is increased. For Dn = 500, on the other hand, it is analyzed that the steady-state solution turns into the chaotic solution for small and large Gr for any γ lying in the range. For moderate Gr at Dn = 500, however, the steady-state flow turns into the chaotic flow through the periodic oscillating flow if the aspect ratio is increased.