In this article, we’ll show how to solve the time-fractional seventh-order Lax’s Korteweg–de Vries and Kaup–Kupershmidt equations analytically using the homotopy perturbation approach, the Adomian decomposition method, and the Elzaki transformation. The KdV equation is a general integrable equation with an inverse scattering transform-based solution that arises in a variety of physical applications, including surface water waves, internal waves in a density stratified fluid, plasma waves, Rossby waves, and magma flow. Fractional derivative is described in the Caputo sense. The solutions to fractional partial differential equation is computed using convergent series. The numerical computations and graphical representations of the analytical results obtained using the homotopy perturbation and decomposition techniques. Moreover, plots that are simple to grasp are used to compare the integer order and fractional-order solutions. After only a few iterations, we may easily obtain numerical results that provide us better approximations. The exact solutions and the derived solutions were observed to be very similar. The suggested methods have also acquired the highest level of accuracy. The most prevalent and convergent techniques for resolving nonlinear fractional-order partial differential issues are the applied techniques.

Due to the increase in bacterial resistance, improving the anti-infectious immunity of the host is rapidly becoming a new strategy for the prevention and treatment of bacterial pneumonia. However, the specific lung immune responses and key immune cell subsets involved in bacterial infection are obscure. Actinobacillus pleuropneumoniae (APP) can cause porcine pleuropneumonia, a highly contagious respiratory disease that has caused severe economic losses in the swine industry. Here, using high-dimensional mass cytometry, the major immune cell repertoire in the lungs of mice with APP infection was profiled. Various phenotypically distinct neutrophil subsets and Ly-6C+ inflammatory monocytes/macrophages accumulated post-infection. Moreover, a linear differentiation trajectory from inactivated to activated to apoptotic neutrophils corresponded with the stages of uninfected, onset, and recovery of APP infection. CD14+ neutrophils, which mainly increased in number during the recovery stage of infection, were revealed to have a stronger ability to produce cytokines, especially IL-10 and IL-21, than their CD14- counterparts. Importantly, MHC-II+ neutrophils with antigen-presenting cell features were identified, and their numbers increased in the lung after APP infection. Similar results were further confirmed in the lungs of piglets infected with APP and Klebsiella pneumoniae infection by using a single-cell RNA-seq technique. Additionally, a correlation analysis between cluster composition and the infection process yielded a dynamic and temporally associated immune landscape where key immune clusters, including previously unrecognized ones, marked various stages of infection. Thus, these results reveal the characteristics of key neutrophil clusters and provide a detailed understanding of the immune response to bacterial pneumonia.

Chronic intestinal inflammation underlies inflammatory bowel disease (IBD). Previous studies indicated alterations in the cellular immune system; however, it has been challenging to interrogate the role of all immune cell subsets simultaneously. Therefore, we aimed to identify immune cell types associated with inflammation in IBD using high-dimensional mass cytometry. We analyzed 188 intestinal biopsies and paired blood samples of newly-diagnosed, treatment-naive patients (n=42) and controls (n=26) in two independent cohorts. We applied mass cytometry (36-antibody panel) to resolve single cells and analyzed the data with unbiased Hierarchical-SNE. In addition, imaging-mass cytometry (IMC) was performed to reveal the spatial distribution of the immune subsets in the tissue. We identified 44 distinct immune subsets. Correlation network analysis identified a network of inflammation-associated subsets, including HLA-DR⁺CD38⁺ EM CD4⁺ T cells, T regulatory-like cells, PD1⁺ EM CD8⁺ T cells, neutrophils, CD27⁺ TCRγδ cells and NK cells. All disease-associated subsets were validated in a second cohort. This network was abundant in a subset of patients, independent of IBD subtype, severity or intestinal location. Putative disease-associated CD4⁺ T cells were detectable in blood. Finally, imaging-mass cytometry revealed the spatial colocalization of neutrophils, memory CD4⁺ T cells and myeloid cells in the inflamed intestine. Our study indicates that a cellular network of both innate and adaptive immune cells colocalizes in inflamed biopsies from a subset of patients. These results contribute to dissecting disease heterogeneity and may guide the development of targeted therapeutics in IBD.