In many regions across the world, hepatitis B virus (HBV) infection is still endemic and the transmission rate is much greater than majority of the known epidemic diseases. Numerous mathematical models (utilizing various differential operators) have been put forth over the past 20 years to understand the transmission mechanism of HBV in various nations and geographical areas. In this manuscript, an epidemic model with various novelties for capturing the dynamic of HBV while utilizing Caputo–Fabrizio fractional differential operator with asymptomatic carriers and vaccination effects is introduced. Initially, the model is formulated by using the ordinary derivative, and afterward, the fractional differential operator is applied to transform the model into arbitrary-ordered derivative. A few basic mathematical properties for the proposed integer-ordered model is presented. The existence of solution to the problem and its uniqueness of the fractional order model are established by transforming the problem into integral equations and then applying the standard results of fixed point theory. For boundedness and positivity of model’ solution is elaborated utilizing the techniques of fractional calculus. It is too much important to validate the theoretical findings through simulations; therefore, the solution curves of the model under consideration are displayed by using the well-known approach called the Mittag-Leffler. To show the behavior of the order of the operator on the dynamics of the disease, various graphical illustrations are presented at the end of the manuscript. By comparing the findings of the present study with the available literature, it is observed that fractional derivative is better to use than integer-order operator for capturing the realistic scenario of the disease.

Aggravation of the chronic obstructive pulmonary disease (COPD) often leads to a slew of complications, but the correlation between COPD aggravation and the complications on the basis of molecular level remains unclear. In this study, gene expression profiles of COPD in patients at early and aggravation stages were collected and differentially-expressed genes were selected. Meanwhile, gene expression data implicated in COPD complications were analyzed to establish a regulatory network of COPD aggravation and COPD related complications. In addition, the gene enrichment function of DAVID6.7 was utilized to evaluate the similarities between COPD aggravation and COPD complications in term of biological process. By analyzing the genes of COPD aggravation and the COPD complications, we found 18 genes highly related to COPD aggravation, among which haptoglobin (HP) was correlated with 14 complications, followed by ADRB2, LCK and CA1, which were related to 13, 11 and 11 complications, respectively. As far as the complications concerned, obesity was regulated by 17 of the 18 genes, which indicated that there was a close correlation between COPD aggravation and obesity. Meanwhile, lung cancer, diabetes and heart failure were regulated by 15, 15 and 14 genes, respectively, among the 18 selected genes. This study suggested the driver genes of COPD aggravation were capable of extensively regulating COPD complications, which would provide a theoretical basis for development of cures for COPD and its complications. Graphical Abstract