The infection of human immunodeficiency virus (HIV) is a serious and potentially incurable infection. There is no cure for HIV and is a public health issue around the world. That is why, it is valuable to investigate the intricate phenomena of HIV infection and provide some control interventions to lessen its economic burden. In this research work, the dynamics of HIV via fractional calculus to conceptualize the intricate phenomena of this viral infection has been formulated and conceptualized. We have shown the rudimentary concept of fractional calculus in Atangana–Baleanu framework. A novel numerical technique is presented for the chaotic and dynamic behaviour of the proposed model. The oscillatory and chaotic phenomena of the system have been shown with the fluctuation of different input factors of the system. Furthermore, we have shown the affect of fractional order on the proposed system of HIV infection. Most critical input parameters are highlighted through numerical simulations and suggested control intervention to the policy makers. Finally, we have shown the stability result and the convergence condition for the proposed numerical scheme.

It is renowned that the immune reaction in the tumour micro environment is a complex cellular process that requires additional research. Therefore, it is important to interrogate the tracking path behaviour of tumor-immune dynamics to alert policy makers about critical factors of the system. Here, we use fractional derivative to structure tumor-immune interactions. Furthermore, in our research, we concentrated on the qualitative investigation and time series analysis of tumor-immune cell interactions. The solution routes are examined using a new numerical technique to emphasis the impact of the factors on tumor-immune system. We focused on the behaviour of the system with fluctuation of different values. The most crucial components of the proposed system are identified and policymakers are advised. The outcomes of the present study are the strong predictor of clinical success and the in-out of immune cells in a tumour is also critical to treatment efficacy. As a result, studying the behaviour of tumor-immune cell interactions is important to predict crucial factors for the prevention and management to the health officials.

In this work, we study a plate equation with time delay in the velocity, frictional damping, and logarithmic source term. Firstly, we obtain the local and global existence of solutions by the logarithmic Sobolev inequality and the Faedo-Galerkin method. Moreover, we prove the stability and nonexistence results by the perturbed energy and potential well methods.

Passive control methods reduced the vulnerability of structures to earthquakes by decreasing the seismic demand and improving structural plasticity. One of the passive control systems is the eccentrically braced frame with a vertical shear link (V-EBF). The present study aims to direct the damage to the absorbing plates of the vertical link beam to allow the structure’s appropriate seismic performance and reparability. Yielding dampers are one of the most widely used types in systems and can provide perfect vibration control if used optimally. Different types of dampers were introduced and used; how to use them depends on the shape and the way they connect to the structure. This research investigates a new type of damper called box damper, an improved type of shear panel damper. The improvement in the way of connecting to the braced frame and the ease of using this damper in different situations are the features of this new damper. This research investigated the mechanism of these yielding dampers in structures and their strengths and weaknesses. In the next step in this study, a V-EBF with plates of thickness 4, 6, and 8 mm was analysed in the finite element software ABAQUS using the nonlinear static analysis and cyclic loading conditions. Some examples of this damper were attached to the braced frames to investigate the effect of using this damper on the seismic behaviour of the braced structures. The results show that the shear link performs like an electrical fuse absorbing all damage and plastic hinges so that other elements of the braced frame remain in their nonlinear elastic region. By increasing the thickness of the damper from 2 to 8 mm, the resistance increased by two times, and the flexibility of the structure had a noticeable change with the rise in thickness from 2 mm to 8 mm. Ductility increased from 38 to 75 mm.

In this paper, we investigate a BVP for a class impulsive fractional partial differential equations. We propose a new topological approach to prove the existence of at least one classical solution and at least two nonnegative classical solutions. The arguments are based upon recent theoretical results.