Novel codynamics of the HIV-1/HTLV-I model involving humoral immune response and cellular outbreak: A new approach to probability density functions and fractional operators

Both human immunodeficiency virus type 1 (HIV-1) and human T-lymphotropic virus type I (HTLV-I) are retroviruses that afflict CD4+ T cells. In this article, the codynamics of within-host HIV-1 and HTLV-I are presented via piecewise fractional differential equations by employing a stochastic system with an influential strategy for biological research. It is demonstrated that the scheme is mathematically and biologically feasible by illustrating that the framework has positive and bounded global findings. The necessary requirements are deduced, ensuring the virus's extinction. In addition, the structure is evaluated for the occurrence of an ergodic stationary distribution and sufficient requirements are developed. A deterministic-stochastic mechanism for simulation studies is constructed and executed in MATLAB to reveal the model's long-term behavior. Utilizing rigorous analysis, we predict that the aforesaid model is an improvement of the existing virus-to-cell and cell-to-cell interactions by investigating an assortment of behaviour patterns that include cross-over to unpredictability processes. Besides that, the piecewise differential formulations, which can be consolidated with integer-order, Caputo, Caputo-Fabrizio, Atangana-Baleanu and stochastic processes, have been declared to be exciting opportunities for researchers in a spectrum of disciplines by enabling them to incorporate distinctive features in various temporal intervals. As a result, by applying these formulations to difficult problems, researchers can achieve improved consequences in reporting realities with white noise. White noise in fractional HIV-1/HTLV-I codynamics plays an extremely important function in preventing the proliferation of an outbreak when the proposed flow is constant and disease extermination is directly proportional to the magnitude of the white noise.