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Fundamental Properties, Comparative Analysis, and Application of Caputo and Riemann-Liouville Fractional Derivatives to a Cancer Cell Epidemic Model

This paper establishes the theoretical distinctions between Caputo and Riemann-Liouville fractional derivatives, demonstrating the former's superiority for biological initial-value problems, and applies this framework to a cancer cell epidemic model to analyze how fractional orders influence memory effects, disease progression, and immunotherapy response.

Original authors: Jenitha Borges M, John Borg S, Britto Antony Xavier G, Avinash N

Published 2026-08-10

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Jenitha Borges M, John Borg S, Britto Antony Xavier G, Avinash N

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the world of science as a giant, bustling city where everything is constantly moving and changing. For a long time, scientists used a very specific set of tools to predict how things move, grow, or spread. These tools, called "calculus," are like a high-speed camera that only takes snapshots of the present moment. They are amazing at describing how a ball rolls down a hill right now, based on how it was moving a split second ago. But what if the ball remembers how it felt yesterday? What if a virus doesn't just react to the person it's in right now, but carries the "memory" of every person it infected before?

This is where a special branch of math called "fractional calculus" comes in. Think of it as a camera that doesn't just take a snapshot, but records a whole video of the past, blending history into the present. In this video, the speed of change depends not just on the current moment, but on everything that happened before. Two main "lenses" exist for this kind of math: the Caputo lens and the Riemann-Liouville lens. Both can record the past, but they handle the very beginning of the story (the "initial conditions") in completely different ways. One lens treats the start of the story as a clean slate, while the other adds a strange, wobbly glitch right at the start. Scientists care about this because many real-world things—like how diseases spread or how cells grow—have deep memories. If you use the wrong lens, your prediction might look like a glitchy video game instead of real life.

This paper is a deep dive into comparing these two lenses and figuring out which one is the best tool for studying a very serious problem: the spread of cancer cells. The authors, a team of mathematicians from Sacred Heart College, first act like detectives, breaking down the fundamental rules of both the Caputo and Riemann-Liouville methods. They prove mathematically that if you try to measure a constant, unchanging value (like a steady population of healthy cells) using the Riemann-Liouville lens, it creates a weird, infinite spike right at the beginning that doesn't make sense in the real world. However, the Caputo lens handles these steady states perfectly, giving a clean zero. This is a huge deal because, in biology, we need to be able to say, "At the start, we had 100 healthy cells," without the math screaming that the number is actually infinite.

The team then takes these findings and builds a "fractional-order cancer cell epidemic model." Imagine a digital simulation of a battlefield where healthy cells and cancer cells are fighting. In a standard model, the battle happens instantly based on who is there right now. But in this new model, the cells have "memory." The cancer cells remember how fast they grew last week, and the healthy cells remember how many were lost in the past. The researchers ran computer simulations to see what happens when they turn the "memory dial" (called the fractional order, $\mu$ ) down from 1 (no memory) to 0.75 (strong memory).

The results were fascinating. When the memory was strong (lower $\mu$ values), the cancer didn't take over as quickly. The system seemed to "hesitate," remembering its initial healthy state for longer. It was as if the cancer cells were moving through thick syrup instead of water; the past held them back. The simulations showed that with a memory factor of 0.75, the cancer population grew much slower than in the standard model, and the healthy cells survived longer. The paper also tested a "treatment" scenario, adding a term to represent immunotherapy (a treatment that helps the immune system fight cancer). The math showed that this treatment worked in a straightforward, additive way: the more treatment you add, the more the cancer is suppressed, regardless of the memory settings.

Crucially, the authors demonstrated that trying to use the Riemann-Liouville lens for this biological model would have been a disaster. Because that method creates a mathematical "glitch" (a singularity) at the very start for any constant value, it would have made the model impossible to set up with real-world starting numbers. The Caputo lens, by contrast, allowed them to set the initial number of cells exactly as they are in reality and watch the story unfold naturally. The paper concludes that for modeling things like cancer growth, where history matters and starting numbers are real and fixed, the Caputo derivative is the superior tool. It provides a mathematically consistent way to include the "memory" of the disease without breaking the rules of physics or biology.

Technical Summary: Fundamental Properties, Comparative Analysis, and Application of Caputo and Riemann-Liouville Fractional Derivatives to a Cancer Cell Epidemic Model

Problem Statement
The paper addresses the theoretical and practical distinctions between the Caputo and Riemann-Liouville (RL) fractional derivatives, specifically within the context of modeling biological systems with memory effects. While fractional calculus is established as a tool for modeling hereditary characteristics in complex systems, the choice of operator is critical for initial-value problems. The authors identify a fundamental discrepancy: the RL derivative of a constant is non-zero (producing a singularity at the origin), whereas the Caputo derivative of a constant is zero. This distinction poses challenges for biological modeling, where initial conditions (e.g., initial cell populations) are physically meaningful constants, and singularities at $t=0$ lack biological interpretation. The study aims to rigorously establish the properties of both operators and determine which is more suitable for a cancer cell epidemic model.

Methodology
The research employs a combination of analytical derivation, comparative theorem proving, and numerical simulation:

Theoretical Framework: The authors define the Gamma and Beta functions, followed by the formal definitions of RL and Caputo fractional integrals and derivatives. They systematically derive fundamental properties, including:
- The fractional derivatives and integrals of power functions ( $z^m$ ).
- The linearity of both operators.
- The relationship between the two operators via a correction-term theorem, which expresses the Caputo derivative as the RL derivative minus a sum of initial value terms.
- Inverse properties, demonstrating how fractional integrals recover original functions up to polynomials determined by initial conditions.
Comparative Analysis: The paper utilizes graphical illustrations and specific examples (e.g., constant functions, $f(z)=1$ , and quadratic functions, $f(z)=z^2$ ) to visualize the behavior of both derivatives. This highlights the singular behavior of the RL derivative near the origin for constants versus the vanishing Caputo derivative.
Model Application: The theoretical results are applied to a coupled fractional-order differential system modeling healthy ( $H$ ) and cancer ( $C$ ) cell populations. The model incorporates logistic growth, inter-cellular competition, and immunotherapy control. The Caputo operator is selected for the model formulation to ensure compatibility with classical initial conditions ( $H(0)=H_0, C(0)=C_0$ ).
Numerical Simulation: The system is solved numerically using the Grunwald-Letnikov approximation scheme. Simulations are conducted over the time interval $z \in [0, 12]$ with varying fractional orders ( $\mu \in \{1.0, 0.9, 0.75, 0.7\}$ ) and treatment intensities to observe memory effects and disease progression.

Key Contributions

Analytical Clarification: The paper provides a rigorous proof that the Caputo derivative of a constant vanishes, while the RL derivative yields a singular term proportional to $z^{-\mu}$ . This is formalized through Theorem 3.4 and Lemma 3.3.
Correction-Term Relation: The authors establish Theorem 5.3, which explicitly quantifies the difference between the two operators as a sum of initial values weighted by singular kernels. This mathematically justifies why the RL formulation introduces "spurious" singular terms in biological contexts.
Operator Selection for Biology: The study demonstrates that the Caputo formulation is the natural choice for biological initial-value problems because it allows for the direct prescription of physical initial data without introducing non-physical singularities at $t=0$ .
Memory Effect Quantification: Through numerical simulations, the paper illustrates that lower fractional orders ( $\mu < 1$ ) induce sub-diffusive behavior, effectively slowing the rate of cancer progression and prolonging the system's "memory" of its initial state compared to the classical integer-order model ( $\mu = 1$ ).

Results

Theoretical: The analysis confirms that for functions where initial values and lower-order derivatives vanish at the origin (e.g., $f(z)=z^2$ ), the Caputo and RL derivatives coincide. However, for constant functions or functions with non-zero initial values, the operators diverge significantly.
Numerical:
- Fractional Order Impact: Decreasing $\mu$ from 1.0 to 0.75 results in a slower convergence of the cancer cell population to the equilibrium and a delayed takeover of healthy cells. The total cell population approaches the carrying capacity ( $K$ ) more slowly, reflecting the retention of initial low-density history.
- Treatment Response: The inclusion of an immunotherapy term ( $-\sigma C$ ) shows an additive reduction in cancer cell density, consistent with the linearity of the Caputo operator. Higher treatment intensity ( $\sigma$ ) leads to faster recovery of healthy cells.
- Phase Portraits: Trajectories for $\mu < 1$ traverse a wider arc in the phase plane before converging to the cancer-dominant equilibrium, indicating increased memory and slower information propagation.

Significance
The paper claims that its primary significance lies in providing a mathematically consistent and biologically interpretable foundation for using fractional differential equations in applied mathematical biology. By rigorously proving that the Caputo derivative avoids the singularities inherent in the RL formulation for constant initial states, the study validates the Caputo operator as the appropriate tool for modeling cancer cell dynamics. The work bridges the gap between abstract fractional calculus theory and practical epidemiological modeling, demonstrating how fractional orders can be used to tune the "memory" of a biological system. The authors conclude that their analysis offers a necessary theoretical basis for future investigations into the stability, existence, and uniqueness of equilibria in fractional biological models, as well as the development of more accurate numerical schemes for such systems.

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