Fractional-Order Modeling of Viral Load Dynamics: Analyzing the Long-term Memory Effects in Chronic Infections
Keywords:
Fractional-order differential equations, Viral load dynamics, Chronic infections, Memory effects, Caputo derivative, Mathematical biology, Stability analysis, Numerical simulation, Immune response modelingAbstract
Classical integer-order differential equations have long been used to provide mathematical models of viral infections, and assume that the dynamics of the system only depend on its current state. This assumption, however, might not be adequate in the case of chronic infections, where the dynamics between the virus and the host immune response tends to be delayed and history dependent. Under these circumstances, history may affect the present dynamics, which results in patterns that are not readily modeled using conventional models.
In this paper, we suggest the use of a fractional-order model to model the dynamics of viral load in chronic infection with specific focus on the inclusion of the long-term memory effects. Caputo fractions are used to formulate the model, so that the cumulative effect of the previous states can be considered by the system. We look at the qualitative aspects of the model, such as existence and stability of equilibrium points, and investigate the changes in system behavior under variation of the fractional order.
The fractional-order model and classical one are compared by performing numerical simulations. It is noted that the fractional framework offers a more adaptable and, in certain instances, more realistic account of viral persistence and gradual immune adjustment. In particular, the model can recreate extended transient behavior and more regular decay behavior, commonly observed in clinical experiments, but challenging to model with integer-order methods.
Despite the fact that the model is a simplistic representation of complicated biological phenomena, the findings indicate that fractional-order dynamics may provide valuable insights into the behavior of chronic infections. The article adds to the current interest in modeling approaches that rely on memory and emphasize on their possible applicability to enhance our knowledge of disease progression and therapeutic response over time.
