Stability of a mathematical model of tumour-induced angiogenesis

A model consisting of three differential equations to simulate the interactions between cancer cells, the angiogenic factors and endothelial progenitor cells in tumor growth is developed. Firstly, the global existence, nonnegativity and boundedness of the solutions are discussed. Secondly, by analyzing the corresponding characteristic equations, the local stability of three boundary equilibria and the angiogenesis equilibrium of the model is discussed, respectively. We further consider global asymptotic stability of the boundary equilibria and the angiogenesis equilibrium by using the well-known Liapunov–LaSalle invariance principal. Finally, some numerical simulations are given to support the theoretical results

Dynamics of COVID-19 models with asymptomatic infections and quarantine measures

Considering the propagation characteristics of COVID-19 in different regions, the dynamics analysis and numerical demonstration of long-term and short-term models of COVID-19 are carried out, respectively. The long-term model is devoted to investigate the global stability of COVID-19 model with asymptomatic infections and quarantine measures. By using the limit system of the model and Lyapunov function method, it is shown that the COVID-19-free equilibrium $V^0$ is globally asymptotically stable if the control reproduction number $\mathcal{R}_{c}<1$ and globally attractive if $\mathcal{R}_{c}=1$, which means that COVID-19 will die out; the COVID-19 equilibrium $V^{\ast}$ is globally asymptotically stable if $\mathcal{R}_{c}>1$, which means that COVID-19 will be persistent. In particular, to obtain the local stability of $V^{\ast}$, we use proof by contradiction and the properties of complex modulus with some novel details, and we prove the weak persistence of the system to obtain the global attractivity of $V^{\ast}$. Moreover, the final size of the corresponding short-term model is calculated and the stability of its multiple equilibria is analyzed. Numerical simulations of COVID-19 cases show that quarantine measures and asymptomatic infections have a non-negligible impact on the transmission of COVID-19

Necessary and Sufficient Conditions of Oscillation in First Order Neutral Delay Differential Equations

We are concerned with oscillation of the first order neutral delay differential equation [x(t)−px(t−τ)]′+qx(t−σ)=0 with constant coefficients, and we obtain some necessary and sufficient conditions of oscillation for all the solutions in respective cases 01

Analysis of the current status of tuberculosis transmission in China based on a heterogeneity model

Tuberculosis (TB) is an infectious disease transmitted through the respiratory system. China is one of the countries with a high burden of TB. Since 2004, an average of more than 800,000 cases of active TB have been reported each year in China. Analyzing the case data from 2004-2018, we find significant differences in TB incidence by age group. Therefore, the effect of age heterogeneous structure on TB transmission needs further study. We develop a model of TB to explore the role of age heterogeneity as a factor in TB transmission. The model is fitted numerically using the nonlinear least squares method to obtain the key parameters in the model, and the basic reproduction number Rv 0.8017 is calculated and the sensitivity anal-ysis of Rv to the parameters is given. The simulation results show that reducing the number of new infections in the elderly population and increasing the recovery rate of elderly patients with the disease could significantly reduce the transmission of tuberculosis. Furthermore the feasibility of achieving the goals of the WHO End TB Strategy in China is assessed, and we obtain that with existing TB control measures it will take another 30 years for China to reach the WHO goal to reduce 90% of the number of new cases by year 2049. However, in theoretical it is feasible to reach the WHO strategic goal of ending tuberculosis by 2035 if the group contact rate in the elderly population can be reduced though it is difficulty to reduce the contact rate.Comment: We think this is a very interesting work that gives a good understanding of the current TB transmission in China and assesses the possibility of China achieving the 2035 TB control target and also explores possible ways for how to prevent and control the TB in Chin

Blank peak current-suppressed electrochemical aptameric sensing platform for highly sensitive signal-on detection of small molecule

In this contribution, an electrochemical aptameric sensing scheme for the sensitive detection of small molecules is proposed using adenosine as a target model. A ferrocene (Fc)-functionalized thiolated aptamer probe is adapted and immobilized onto an electrode surface. Introducing a recognition site for EcoRI into the aptamer sequence not only suppresses the peak current corresponding to blank sample but also provides a signal-on response mechanism. In the absence of adenosine, the aptamer can fold into a hairpin structure and form a cleavable double-stranded region. Fc is capable of being removed from electrode surface by treatment with endonuclease, and almost no peak current is observed. The adenosine/aptamer binding induces the conformational transition of designed aptamer, dissociating the cleavable double-stranded segment. Therefore, the integrated aptamer sequence is maintained when exposing to endonuclease, generating a peak current of Fc. Utilizing the present sensing scheme, adenosine even at a low concentration can give a detectable current signal. Thus, a detection limit of 10−10 M and a linear response range from 3.74 × 10−9 to 3.74 × 10−5 M are achieved. The proposed proof-of-principle of a novel electrochemical sensing is expected to extend to establish various aptameric platforms for the analysis of a broad range of target molecules of interest

Dark energy model with higher derivative of Hubble parameter

In this letter we consider a dark energy model in which the energy density is a function of the Hubble parameter $H$ and its derivative with respect to time $\rho_{de}=3\alpha \ddot{H}H^{-1}+3\beta\dot{H}+3\gamma H^2$. The behavior of the dark energy and the expansion history of the Universe depend heavily on the parameters of the model $\alpha$, $\beta$ and $\gamma$. It is very interesting that the age problem of the well-known three old objects can be alleviated in this models.Comment: 11 pages, 6 figures, the correct version accepted for publication in PL

A novel analysis approach of uniform persistence for a COVID-19 model with quarantine and standard incidence rate

A coronavirus disease 2019 (COVID-19) model with quarantine and standard incidence rate is first developed, then a novel analysis approach for finding the ultimate lower bound of COVID-19 infectious individuals is proposed, which means that the COVID-19 pandemic is uniformly persistent if the control reproduction number $\mathcal{R}_{c}>1$. This approach can be applied to other related biomathematical models, and some existing works can be improved by using it. In addition, the COVID-19-free equilibrium $V^0$ is locally asymptotically stable (LAS) if $\mathcal{R}_{c}<1$ and linearly stable if $\mathcal{R}_{c}=1$, respectively; while $V^0$ is unstable if $\mathcal{R}_{c}>1$.Comment: 13 pages, 1 figur

The surface geometry and shadow of a Schwarzschild black hole with halo

We have studied the surface geometry and shadows of Schwarzschild black hole with a halo containing quadrupolar and octopolar terms. We found the event horizon is prolate for the quadrupole strength $\mathcal{Q}<0$, and it becomes oblate for $\mathcal{Q}>0$. The event horizon stretches downward for $\mathcal{O}>0$. But for the case of $\mathcal{O}<0$, the event horizon stretches upward. We found the radius of light rings $r_{LR}$ in the space-time of Schwarzschild black hole with halo only depends on the quadrupole strength $\mathcal{Q}$. The black hole shadow is oblate when the quadrupole strength $\mathcal{Q}$ is larger than zero, and it is prolate when $\mathcal{Q}$ is less than zero. Black hole shadow shifts upward when the octopolar strength $\mathcal{O}$ is less than zero, and shifts downward when $\mathcal{O}$ is larger than zero. From the observable width $W$, height $H$, oblateness $K$ and distortion parameter $\delta$ of black hole shadow, one can determine the quadrupole strength $\mathcal{Q}$ and the octopolar strength $\mathcal{O}$ of Schwarzschild black hole with halo. Black hole shadow is always a circle for the observers with the inclination angle $\theta_{obs}=0$, and becomes bigger with the increase of $\mathcal{Q}$ or $\mathcal{O}$. Our results show that the quadrupolar and octopolar terms yield a series of interesting patterns for the shadow of a Schwarzschild black hole with halo.Comment: 18 pages,11 figure

Necessary and sufficient conditions for oscillation of neutral delay differential equations

In this article, we concerned with oscillation of the neutral delay differential equation $[x(t)-px(t-\tau)]'+qx(t-\sigma)=0$ with constant coefficients. By constructing several suitable auxiliary functions, we obtained some necessary and sufficient conditions for oscillation of all the solutions of the aforementioned equation for the cases $0<p<1$ and $p>1$