As of January 19, 2021, the cumulative number of people infected with coronavirus disease-2019 (COVID-19) in the United States has reached 24,433,486, and the number is still rising. The outbreak of the COVID-19 epidemic has not only affected the development of the global economy but also seriously threatened the lives and health of human beings around the world. According to the transmission characteristics of COVID-19 in the population, this study established a theoretical differential equation mathematical model, estimated model parameters through epidemiological data, obtained accurate mathematical models, and adopted global sensitivity analysis methods to screen sensitive parameters that significantly affect the development of the epidemic. Based on the established precise mathematical model, we calculate the basic reproductive number of the epidemic, evaluate the transmission capacity of the COVID-19 epidemic, and predict the development trend of the epidemic. By analyzing the sensitivity of parameters and finding sensitive parameters, we can provide effective control strategies for epidemic prevention and control. After appropriate modifications, the model can also be used for mathematical modeling of epidemics in other countries or other infectious diseases.

In this article, a fractional-order differential equation model of HBV infection was proposed with a Caputo derivative, delayed immune response, and logistic proliferation. Initially, infection-free and infection equilibriums and the basic reproduction number were computed. Thereafter, the stability of the two equilibriums was analyzed based on the fractional Routh–Hurwitz stability criterion, and the results indicated that the stability will change if the time delay or fractional order changes. In addition, the sensitivity of the basic reproduction number was analyzed to find out the most sensitive parameter. Lastly, the theoretical analysis was verified by numerical simulations. The results showed that the time delay of immune response and fractional order can significantly affect the dynamic behavior in the HBV infection process. Therefore, it is necessary to consider time delay and fractional order in modeling HBV infection and studying its dynamics.

Introduction: Mathematical model can be used to model complex biological processes, and have shown potential in describing apoptosis in chondrocytes.Method: In order to investigate the regulatory mechanisms of TNF signaling pathway in regulating chondrocyte apoptosis, a fractional-order differential equation model is proposed to describe the dynamic behavior and mutual interaction of apoptosis-related genes under the activation of TNF signaling pathway. Compared with the traditional molecular biology techniques, the proposed mathematical modeling has advantages to providing a more comprehensive understanding of the regulatory mechanisms of TNF signaling pathway in chondrocyte apoptosis.Result: In this paper, differentially expressed genes induced by IL-1β in human chondrocyte apoptosis are screened using high-throughput sequencing. It is found that they were significantly enriched in the TNF signaling pathway. Therefore, a mathematical model of the TNF signaling pathway is built. Using real-time PCR experiments, mRNA data is measured and used to identify the model parameters, as well as the correlation coefficient. Finally, the sensitivity of the model parameters is discussed by using numerical simulation methods, which can be used to predict the effects of different interventions and explore the optimal intervention strategies for regulating chondrocyte apoptosis.Discussion: Therefore, fractional-order differential equation modeling plays an important role in understanding the regulatory mechanisms of TNF signaling pathway in chondrocyte apoptosis and its potential clinical applications.

The boundary formation in the Drosophila large intestine is widely studied as an important biological problem. It has been shown that the Delta-Notch signaling pathway plays an essential role in the formation of boundary cells. In this paper, we propose a mathematical model for the Delta-Notch dependent boundary formation in the Drosophila large intestine in order to better interpret related experimental findings of this biological phenomenon. To achieve this, we not only perform stability analysis on the model from a theoretical point of view, but also perform numerical simulations to analyze the model with and without noises, the phenotype change with the change of Delta or Notch expression, and the perturbation influences of binding and inhibition parameters on the boundary formation. By doing all these work, we can assure that our model can better interpret the biological findings related to the boundary formation in the Drosophila large intestine.

We propose a comprehensive delayed HBV model, which not only considers the immune response to both infected cells and viruses and a time delay for the immune system to clear viruses but also incorporates an exposed state and the proliferation of hepatocytes. We prove the positivity and boundedness of solutions and analyze the global stability of two boundary equilibria and then study the local asymptotic stability and Hopf bifurcation of the positive (infection) equilibrium and also the stability of the bifurcating periodic solutions. Moreover, we illustrate how the factors such as the time delay, the immune response to infected cells and viruses, and the proliferation of hepatocytes affect the dynamics of the model by numerical simulation.

To develop and validate a mathematical model for predicting intracranial pressure (ICP) noninvasively using phase-contrast cine MRI (PC-MRI). We performed a retrospective analysis of PC-MRI from patients with communicating hydrocephalus (n = 138). The patients were recruited from Shenzhen Second People’s Hospital between November 2017 and April 2020, and randomly allocated into training (n = 97) and independent validation (n = 41) groups. All participants underwent lumbar puncture and PC-MRI in order to evaluate ICP and cerebrospinal fluid (CSF) parameters (i.e., aqueduct diameter and flow velocity), respectively. A novel ICP-predicting model was then developed based on the nonlinear relationships between the CSF parameters, using the Levenberg–Marquardt and general global optimisation methods. There was no significant difference in baseline demographic characteristics between the training and independent validation groups. The accuracy of the model for predicting ICP was 0.899 in the training cohort (n = 97) and 0.861 in the independent validation cohort (n = 41). We obtained an ICP-predicting model that showed excellent performance in the noninvasive diagnosis of clinically significant communicating hydrocephalus.