In this paper, we have mathematically analyzed a within-host model of SARS-CoV-2 which is used by Li et al. in the paper “The within-host viral kinetics of SARS-CoV-2” published in (Math. Biosci. Eng. 17(4):2853–2861, 2020). Important properties of the model, like nonnegativity of solutions and their boundedness, are established. Also, we have calculated the basic reproduction number which is an important parameter in the infection models. From stability analysis of the model, it is found that stability of the biologically feasible steady states are determined by the basic reproduction number $(\chi _{0})$( χ 0 ) . Numerical simulations are done in order to substantiate analytical results. A biological implication from this study is that a COVID-19 patient with less than one basic reproduction ratio can automatically recover from the infection.

The prime objective of the current study is to propose a novel mathematical framework under the fractional-order derivative, which describes the complex within-host behavior of SARS-CoV-2 by taking into account the effects of memory and carrier. To do this, we formulate a mathematical model of SARS-CoV-2 under the Caputo fractional-order derivative. We derived the conditions for the existence of equilibria of the model and computed the basic reproduction number ${\mathcal{R}}_0$. We used mathematical analysis to establish the proposed model’s local and global stability results. Some numerical resolutions of our theoretical results are presented. The main result of this study is that as the fractional derivative order increases, the approach of the solution to the equilibrium points becomes faster. It is also observed that the value of ${\mathcal{R}}_0$ increases as the value of $\beta$ and ${\pi }_v$ increases.

In this study, we discuss a cancer model considering discrete time-delay in tumor-immune interaction and stimulation processes. This study aims to analyze and observe the dynamics of the model along with variation of vital parameters and the delay effect on anti-tumor immune responses. We obtain sufficient conditions for the existence of equilibrium points and their stability. Existence of Hopf bifurcation at co-axial equilibrium is investigated. The stability of bifurcating periodic solutions is discussed, and the time length for which the solutions preserve the stability is estimated. Furthermore, we have derived the conditions for the direction of bifurcating periodic solutions. Theoretically, it was observed that the system undergoes different states if we vary the system’s parameters. Some numerical simulations are presented to verify the obtained mathematical results.

The interaction between the immune system and tumors is complex, and macrophages play a dual role in promoting and suppressing tumor growth. This research proposes a mathematical model that incorporates a single feedback control strategy in a tumor-macrophage interaction system. The model examines how feedback regulation affects tumor dynamics using Caputo fractional-order derivatives. Existence and uniqueness of solutions are demonstrated using Banach’s fixed-point theorem. Stability analysis of the model identified four equilibrium points. According to stability analysis, the tumor-free equilibrium is always unstable. The tumor dominant equilibrium in the presence of a feedback control function is conditionally asymptotically stable. The analysis identified two interior equilibrium points that are stable under the particular parameter conditions. Numerical simulation of the proposed model is performed using a predictor-corrector scheme. Parametric investigation is conducted to examine how different feedback strengths affect system dynamics. Through numerical simulation, several steady-state behaviors are identified, including uncontrolled growth, stable oscillations, and tumor elimination. These outcomes vary based on the fractional order parameter and the intensity of feedback control. The results reveal essential threshold values for tumor stability, highlighting conditions that may lead to either tumor elimination or persistence.