Mathematical Investigation of Cancer-Immune-Angiogenesis Model Using Fuzzy Piecewise Fractional Derivatives

Abstract

This work develops a fuzzy piecewise fractional derivative (FPFD) model for cancer-immune-angiogenesis dynamics under uncertainty. Five fuzzy state variables track tumor cells, immune effectors, vessel density, oxygen, and drug concentration. We employ fuzzy triangular numbers with $\alpha$-cut interval arithmetic using constrained fuzzy arithmetic model parametric uncertainty, with numerical values. Oxygen-dependent carrying capacity follows a Hill-type function; hypoxia-induced angiogenesis follows a decreasing Michaelis–Menten function. The model transitions at $t_1=50$ days from memoryless fuzzy classical derivative to fuzzy ABC fractional derivative of order $\psi$. The transition time $t_1=50$ days is biologically justified based on experimental observations of the angiogenic switch in solid tumors, which typically occurs within 4–8 weeks post-inoculation. Positivity, boundedness, Lipschitz continuity, existence, and uniqueness of fuzzy solutions are proved via Banach fixed-point theorem in a weighted norm. A basic reproduction number interval ${\mathcal{R}}_0=[{\underline{\mathcal{R}}}_0,{\overline{\mathcal{R}}}_0]$ is derived; local and global stability conditions are established for disease-free and endemic equilibria using fuzzy differential inclusions. Global sensitivity analysis using latin hypercube sampling with $N=500$ samples explores the range of possible outcomes across the fuzzy parameter support. In the numerical implementation, we use a fourth-order fuzzy Runge–Kutta method (Phase I), and a fractional Adams–Bashforth–Moulton predictor-corrector method (Phase II), ensuring preservation of fuzzy number characteristics.

1. Introduction and Motivations

The non-local nature of tumor proliferation and the inherent uncertainty in biological parameters constitute two significant challenges to the mathematical modeling of cancer systems. Genetic factors play a role in the onset of cancer, and historical events can influence present and future outcomes [1,2]. This includes complex pathophysiological processes such as immune evasion, metastasis, and proliferation [3,4]. Standard integer-order differential equations [5,6] do not adequately describe these non-local behaviors.

Fractional calculus employs derivatives of non-integer orders to model memory-dependent phenomena. The Atangana–Baleanu–Caputo (ABC) derivative [7] represents biological relaxation processes owing to its non-singular Mittag–Leffler kernel [8,9]. The Caputo derivative has a singular kernel, but the ABC derivative has a non-singular Mittag–Leffler kernel that shows how memory effects in biological systems, like immune exhaustion and vascular remodeling, fade away smoothly. This makes it suitable for modeling tumor-immune interactions where the influence of past states decays gradually rather than exhibiting an initial singularity [10,11]. It has been utilized to model lung cancer stem cells [10] and address colorectal cancer [11] and treatment-resistant breast cancer [12]. Advanced computational methods have enabled the examination of tumor system stability [13,14].

To address the uncertainty in clinical standards, we need frameworks that use interval values rather than probability distributions to represent probabilities. Fuzzy set theory produces probability-based predictions using fuzzy numbers to represent uncertain variables [15,16]. Recent developments include numerical simulations using implicit finite-difference schemes [17], hybrid frameworks for cancer dissemination [18], fuzzy ABC liver dynamics [19], and fuzzy ABC infectious diseases [20].

Cancer progression involves distinct physiological stages, prompting the use of categorical operators [21]. Fractional categorical frameworks have been used to model temporal variations in viral responses [22] and estrogen-mediated tumor proliferation [23].

Despite these developments, current approaches treat fractional calculus, fuzzy uncertainty, and categorical dynamics as separate methodologies [24,25,26]. Fractional cancer models typically use different fractional derivatives without quantifying uncertainty. Fuzzy models use simple interval calculus, which deals with the frequency of variables independently, leading to an overestimation of uncertainty. Hyperbolic frameworks lack integration with both fuzzy and fractional approaches. No unified framework addresses all three of these features simultaneously.

This work addresses this gap by developing a mathematical framework that integrates fuzzy uncertainty quantification, fractional ABC memory, and hyperbolic phase transitions. The main contributions are: (i) the first integration of fractional ABC derivatives with hyperbolic switching in a cancer-immunity-angiogenesis model; (ii) the application of constrained fuzzy calculus to solve the dependency problem inherent in simple interval methods; (iii) a rigorous clarity analysis that establishes positivity, constraint, Lipschitz continuity, and existence-singularity via Banach’s fixed-point theory; (iv) stability analysis using fuzzy differential embeddings leading to threshold conditions based on the fuzzy reproduction number ${\mathcal{R}}_0=[{\underline{\mathcal{R}}}_0,{\overline{\mathcal{R}}}_0]$; and (v) range-based sensitivity analysis taking into account the probabilistic nature of fuzzy coefficients.

2. Preliminaries: Fuzzy Numbers and Constrained Arithmetic

Let ${\mathbb{R}}_{\mathcal{G}}$ denote the space of compact, convex fuzzy numbers on $\mathbb{R}$. For more information about the following definitions, see [27,28,29,30].

Definition 1.

$\tilde{y}\in {\mathbb{R}}_{\mathcal{G}}$ iff:

(i)

$\tilde{y}:\mathbb{R}\to [0,1]$ is upper semicontinuous;

(ii)

$\tilde{y}(\lambda x+(1-\lambda )y)\ge min\{\tilde{y}\left(x\right),\tilde{y}\left(y\right)\}$, $\forall x,y\in \mathbb{R}$, $\lambda \in [0,1]$;

(iii)

$\exists x_0\in \mathbb{R}$: $\tilde{y}\left(x_0\right)=1$;

(iv)

$supp\left(\tilde{y}\right)=\overline{\{x\in \mathbb{R}:\tilde{y}\left(x\right)>0\}}$ is compact.

Definition 2.

For $\alpha \in [0,1]$, the α-cut of $\tilde{y}$ is

$${\left[\tilde{y}\right]}_{\alpha }=\{x\in \mathbb{R}:\tilde{y}\left(x\right)\ge \alpha \}=[{\underline{y}}_{\alpha },{\overline{y}}_{\alpha }],$$

where ${\underline{y}}_{\alpha }=inf{\left[\tilde{y}\right]}_{\alpha }$, ${\overline{y}}_{\alpha }=sup{\left[\tilde{y}\right]}_{\alpha }$. The functions $\alpha \mapsto {\underline{y}}_{\alpha }$, $\alpha \mapsto {\overline{y}}_{\alpha }$ are left-continuous, non-decreasing, and non-increasing, respectively, with ${\underline{y}}_{\alpha }\le {\overline{y}}_{\alpha }$.

Definition 3 (Constrained Fuzzy Arithmetic).

For fuzzy numbers $\tilde{x},\tilde{y}$ with joint constraint set $\mathcal{C}\subseteq {\mathbb{R}}^2$ representing known dependencies, the constrained arithmetic operation $★\in \{+,-,\times ,÷\}$ is defined via:

$${[\tilde{x}★\tilde{y}]}_{\alpha }=\{x★y:x\in {\left[\tilde{x}\right]}_{\alpha },y\in {\left[\tilde{y}\right]}_{\alpha },(x,y)\in \mathcal{C}\}.$$

For expressions with repeated variables, the consistency constraint that each occurrence of the same variable takes the same value is enforced. For a function $g:{\mathbb{R}}^n\to \mathbb{R}$ applied to fuzzy arguments ${\tilde{x}}_1,\dots ,{\tilde{x}}_n$ with dependency constraints, the fuzzy extension is:

$${\left[\tilde{g}({\tilde{x}}_1,\dots ,{\tilde{x}}_n)\right]}_{\alpha }=\{g(u_1,\dots ,u_n):u_i\in {\left[{\tilde{x}}_i\right]}_{\alpha },(u_1,\dots ,u_n)\in \mathcal{C}\},$$

where $\mathcal{C}$ encodes all known dependencies (e.g., $u_i=u_j$ when the same variable appears multiple times).

Definition 4 (Hausdorff Metric).

For $\tilde{A},\tilde{B}\in {\mathbb{R}}_{\mathcal{G}}$,

$$d_H(\tilde{A},\tilde{B})=\underset{\alpha \in [0,1]}{sup}max\left\{\right|{\underline{A}}_{\alpha }-{\underline{B}}_{\alpha }|,|{\overline{A}}_{\alpha }-{\overline{B}}_{\alpha }\left|\right\}.$$

Note that $({\mathbb{R}}_{\mathcal{G}},d_H)$ is a complete metric space.

Definition 5 (gH-Difference and gH-Derivative).

For $\tilde{y},\tilde{z}\in {\mathbb{R}}_{\mathcal{G}}$, the generalized Hukuhara difference $\tilde{y}{\ominus }_{gH}\tilde{z}=\tilde{w}$ exists iff

$$\tilde{y}=\tilde{z}+\tilde{w}\mathit{or}\tilde{z}=\tilde{y}+(-1)\tilde{w}.$$

When it exists, the α-cut is:

$${\left[\tilde{y}{\ominus }_{gH}\tilde{z}\right]}_{\alpha }=[min\{{\underline{y}}_{\alpha }-{\underline{z}}_{\alpha },{\overline{y}}_{\alpha }-{\overline{z}}_{\alpha }\},max\{{\underline{y}}_{\alpha }-{\underline{z}}_{\alpha },{\overline{y}}_{\alpha }-{\overline{z}}_{\alpha }\}].$$

A function $f:[a,b]\to {\mathbb{R}}_{\mathcal{G}}$ is gH-differentiable at $t_0$ if $\exists D_{gH}^{1}f\left(t_0\right)\in {\mathbb{R}}_{\mathcal{G}}$:

$$D_{gH}^{1}f\left(t_0\right)=\underset{h\to 0^{+}}{lim}{\displaystyle \frac{f(t_0+h){\ominus }_{gH}f\left(t_0\right)}{h}}=\underset{h\to 0^{+}}{lim}{\displaystyle \frac{f\left(t_0\right){\ominus }_{gH}f(t_0-h)}{h}},$$

with the limits taken in $({\mathbb{R}}_{\mathcal{G}},d_H)$.

Definition 6

(Fuzzy ABC Derivative [31]). For $\psi \in (0,1)$, let $f\in C^1([0,T],{\mathbb{R}}_{\mathcal{G}})$ be gH-differentiable. The fuzzy ABC fractional derivative is defined α-cutwise:

$$\begin{array}{cc}\hfill {{[}^{ABC}{\mathcal{D}}_t^{\psi }f\left(t\right)]}_{\alpha }& ={\displaystyle \frac{B\left(\psi \right)}{1-\psi }}{\int }_0^t{E}_{\psi }\left(-{\displaystyle \frac{\psi }{1-\psi }}{(t-\tau )}^{\psi }\right){\left[f^{\prime }\left(\tau \right)\right]}_{\alpha }d\tau ,\hfill \\ \hfill B\left(\psi \right)& =1-\psi +{\displaystyle \frac{\psi }{\mathrm{\Gamma }\left(\psi \right)}},E_{\psi }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathrm{\Gamma }(\psi k+1)}},\hfill \end{array}$$

where the integral is interpreted as the Aumann integral for interval-valued functions:

$${\int }_0^t{\left[g\left(\tau \right)\right]}_{\alpha }d\tau =\left[{\int }_0^t\underline{g}(\tau ;\alpha )d\tau ,{\int }_0^t\overline{g}(\tau ;\alpha )d\tau \right].$$

Definition 7 

(Piecewise Operator [32]). For $0<t_1<T$, define

$${}^{\mathcal{P}}\mathcal{D}_t^{\psi }\tilde{\mathbb{X}}\left(t\right)=\left\{\begin{array}{cc}{D}_{gH}^1\tilde{\mathbb{X}}\left(t\right),\hfill & t\in [0,t_1],\hfill \\ {}^{\mathit{ABC}}\mathcal{D}_{t1,t}^{\psi }\tilde{\mathbb{X}}\left(t\right),\hfill & t\in (t_1,T],\hfill \end{array}\right.$$

where ${}^{\mathit{ABC}}\mathcal{D}_{t1,t}^{\psi }$ denotes the ABC derivative with lower limit $t_1$. To maintain dimensional consistency, the transition to fractional order involves a scaling parameter σ with the dimension of time (days), as applied in the system definitions.

Lemma 1 (Integral Representation).

For $\tilde{\mathbb{X}}\in C([0,T],{\mathbb{R}}_{\mathcal{G}}^5)$ satisfying the piecewise fractional system, the following integral equation holds:

$$\tilde{\mathbb{X}}\left(t\right)=\left\{\begin{array}{cc}{\displaystyle {\tilde{\mathbb{X}}}_0+{\int }_0^t\overrightarrow{\mathbb{G}}(s,\tilde{\mathbb{X}}\left(s\right))ds,}\hfill & t\in [0,t_1],\hfill \\ \tilde{\mathbb{X}}\left(t_1\right)+{\sigma }^{1-\psi }\left[{\displaystyle {\displaystyle \frac{1-\psi }{B\left(\psi \right)}}\overrightarrow{\mathbb{G}}(t,\tilde{\mathbb{X}}\left(t\right))+{\displaystyle \frac{\psi }{B\left(\psi \right)\mathrm{\Gamma }\left(\psi \right)}}{\int }_{t_1}^t{(t-s)}^{\psi -1}\overrightarrow{\mathbb{G}}(s,\tilde{\mathbb{X}}\left(s\right))ds}\right],\hfill & t\in (t_1,T].\hfill \end{array}\right.$$

Proof. 

For $t\in [0,t_1]$, the result follows from the fundamental theorem of calculus for gH-differentiable functions. For $t\in (t_1,T]$, apply the AB fractional integral operator to both sides of the ABC derivative definition and use continuity at $t_1$. □

3. Model Formulation with Constrained Fuzzy Arithmetic

Biological Background and Model Rationale: Mathematical models of tumor-immune dynamics, which originate from predator–prey formulations [3,4], encounter three fundamental limitations that hinder their clinical applicability:

• Parameter uncertainty: Conventional models assume fixed parameters, whereas clinical measurements are inherently interval-valued. Stochastic approaches require assumed probability distributions, while fuzzy set theory requires only the interval estimates that clinicians actually possess.
• Memory dependence: Tumor growth exhibits memory dependency due to immune exhaustion and epigenetic changes [5]. The angiogenic switch necessitates continuous activation of hypoxia-inducible factors, requiring fractional calculus to integrate the preceding effects.
• Structural changes: Cancer progresses through specific stages, commencing with avascular growth constrained by oxygen diffusion, followed by angiogenesis and vascularized growth. For accurate prediction, models that capture this change are necessary.

The current framework addresses these limitations by: (i) employing non-singular fractional derivatives to incorporate memory effects, (ii) using fuzzy numbers to depict parametric uncertainty, and (iii) introducing a piecewise approach that shifts between memoryless and memory-dominated dynamics at the crossover point angiogenic switch $t_1$.

The angiogenic switch time ($t_1=50$ days) is the most important time for a tumor to change from growing without blood vessels to growing with blood vessels. The selection of angiogenic switch time is predicated on experimental findings: research [33,34] indicates that the transition from avascular to vascular growth generally transpires within 4–8 weeks following inoculation.

The piecewise framework lets the model show Phase I ($t\le t_1$), the memoryless avascular phase, in which growth is limited by how well oxygen diffuses. In Phase II ($t>t_1$), the vascular phase, dominated by memory, angiogenesis speeds up tumor growth.

The value of $t_1$ (50 days) can be changed to fit different types of tumors or the needs of individual patients. Section 6.2 shows the sensitivity analysis of $t_1$.

3.1. State Variables

The proposed model comprises five fuzzy-number-valued functions $\tilde{X}:[0,T]\to {\mathbb{R}}_{\mathcal{G}}$, each defined by its $\alpha$-cut intervals $[{\underline{X}}_{\alpha }\left(t\right),{\overline{X}}_{\alpha }\left(t\right)]$:

• Tumor cells $\tilde{C}\left(t\right)$: Represent the malignant population, progressing from exponential growth to immune-constrained and nutrient-limited dynamics.
• Immune effectors $\tilde{I}\left(t\right)$: Cytotoxic lymphocytes recruited at rate ${\tilde{r}}_I$, expanding upon antigen encounter at rate ${\tilde{\beta }}_{IC}$, and undergoing tumor-induced immunosuppression at rate ${\tilde{\gamma }}_{IS}$.
• Blood vessel density $\tilde{V}\left(t\right)$: Measures vascularization. Avascular tumors ($\tilde{V}\approx 0$) are constrained by oxygen diffusion until the angiogenic switch initiates endothelial proliferation at rate ${\tilde{r}}_V$, offset by vessel regression at rate ${\tilde{d}}_V$.
• Oxygen concentration $\tilde{O}\left(t\right)$: Supplied by functional vasculature (${\tilde{D}}_O\tilde{V}$), consumed by tumor and immune cells (${\tilde{\rho }}_C,{\tilde{\rho }}_I$), and lost to diffusion (${\tilde{\lambda }}_O$). The Warburg effect is reflected in higher tumor consumption (${\tilde{\rho }}_C\gg {\tilde{\rho }}_I$).
• Drug concentration $\tilde{T}\left(t\right)$: In this control study, $\tilde{T}\equiv 0$, laying the groundwork for future therapeutic extensions.

Inter-patient variability and inherent uncertainties in clinical measurements are captured through fuzzy representation. Before introducing uncertainty and memory effects, the underlying deterministic integer-order model is given by:

$$\begin{array}{cc}\hfill {\displaystyle \frac{dC}{dt}}& =r_{C}C\left(1-{\displaystyle \frac{C}{k_C\left(O\right)}}\right)-d_{C}C-{\alpha }_{CI}CI,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\displaystyle \frac{dI}{dt}}& =r_{I}I\left(1-{\displaystyle \frac{I}{K_I}}\right)+{\beta }_{IC}CI-d_{I}I-{\gamma }_{IS}CI,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\displaystyle \frac{dV}{dt}}& =r_{V}V\left(1-{\displaystyle \frac{V}{K_V}}\right)+{\sigma }_{VC}f\left(O\right)C-d_{V}V,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\displaystyle \frac{dO}{dt}}& =D_{O}V-{\rho }_{C}C-{\rho }_{I}I-{\lambda }_{O}O,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\displaystyle \frac{dT}{dt}}& =-{\lambda }_{T}T,\hfill \end{array}$$

(5)

with initial conditions $C\left(0\right)=C_0$, $I\left(0\right)=I_0$, $V\left(0\right)=V_0$, $O\left(0\right)=O_0$, $T\left(0\right)=0$.

This system serves as the crisp prototype from which the fuzzy fractional model is derived by: (i) replacing integer-order derivatives with the piecewise fractional operator, (ii) treating all parameters as fuzzy numbers, and (iii) applying constrained fuzzy arithmetic to handle variable dependencies.

3.2. Constrained Fuzzy Arithmetic

3.2.1. The Dependency Problem in Naive Interval Arithmetic

Naive interval arithmetic treats each occurrence of a variable in an expression independently. Consider the term ${\tilde{r}}_C\tilde{C}\left(1-{\displaystyle \frac{\tilde{C}}{{\tilde{k}}_C\left(\tilde{O}\right)}}\right)$, where $\tilde{C}$ appears three times. Standard interval arithmetic computes:

$$\begin{array}{cc}\hfill {\left[\tilde{C}\right]}_{\alpha }& =[{\underline{C}}_{\alpha },{\overline{C}}_{\alpha }],\hfill \\ \hfill \tilde{C}\oslash {\tilde{k}}_C\left(\tilde{O}\right){]}_{\alpha }& =\left[{\displaystyle \frac{{\underline{C}}_{\alpha }}{{\overline{k}}_{C,\alpha }}},{\displaystyle \frac{{\overline{C}}_{\alpha }}{{\underline{k}}_{C,\alpha }}}\right],\hfill \\ \hfill [1\ominus (\tilde{C}\oslash {\tilde{k}}_C\left(\tilde{O}\right)){]}_{\alpha }& =\left[1-{\displaystyle \frac{{\overline{C}}_{\alpha }}{{\underline{k}}_{C,\alpha }}},1-{\displaystyle \frac{{\underline{C}}_{\alpha }}{{\overline{k}}_{C,\alpha }}}\right],\hfill \end{array}$$

and then multiplies this result by ${\left[\tilde{C}\right]}_{\alpha }$. This permits distinct values of c in each occurrence, producing intervals that contain impossible combinations and artificially wide uncertainty bands.

3.2.2. Constrained Fuzzy Arithmetic Solution

Constrained fuzzy arithmetic enforces that each occurrence of the same variable takes the identical value:

$${[\tilde{C}\odot (1\ominus \tilde{C}\oslash {\tilde{k}}_C\left(\tilde{O}\right))]}_{\alpha }=\left\{c\left(1-{\displaystyle \frac{c}{k}}\right):c\in {\left[\tilde{C}\right]}_{\alpha },k\in {\left[{\tilde{k}}_C\left(\tilde{O}\right)\right]}_{\alpha }\right\},$$

with the constraint that the same c is used in both linear and quadratic terms.

This approach:

• Yields significantly tighter bounds;
• Ensures the fuzzy solution corresponds to actual possible trajectories;
• Addresses the dependency problem ignored by existing fuzzy cancer models [17,18];
• Produces uncertainty bands approximately 40% narrower than naive arithmetic for the same parameter ranges.

3.3. Functional Relationships

3.3.1. Oxygen-Dependent Carrying Capacity (Hill Function)

$${\tilde{k}}_C\left(\tilde{O}\right)={\tilde{K}}_C^0\left(\tilde{1}\oplus {\tilde{\delta }}_K\odot {\displaystyle \frac{{\tilde{O}}^{\odot 2}}{{\tilde{\theta }}_H^{\odot 2}\oplus {\tilde{O}}^{\odot 2}}}\right).$$

This function models tumor acclimatization to hypoxia:

• Normoxia ($\tilde{O}\gg {\tilde{\theta }}_H$): Fraction approaches unity → ${\tilde{k}}_C\approx {\tilde{K}}_C^0(\tilde{1}\oplus {\tilde{\delta }}_K)$;
• Severe hypoxia ($\tilde{O}\ll {\tilde{\theta }}_H$): Fraction approaches zero → ${\tilde{k}}_C\approx {\tilde{K}}_C^0$.

Parameter ${\tilde{\theta }}_H$ is the oxygen tension at half-maximal carrying capacity.

3.3.2. Hypoxia-Induced Angiogenesis (Michaelis–Menten Function)

$$f\left(\tilde{O}\right)={\displaystyle \frac{{\tilde{\sigma }}_{max}\odot {\tilde{\theta }}_V}{{\tilde{\theta }}_V\oplus \tilde{O}}}.$$

This function models VEGF secretion in response to hypoxia:

• Normoxia ($\tilde{O}\gg {\tilde{\theta }}_V$): $f\left(\tilde{O}\right)\approx 0$;
• Hypoxia: VEGF secretion rises as oxygen levels drop;
• Half-maximal effect occurs at $\tilde{O}={\tilde{\theta }}_V$;
• Maximum effect ${\tilde{\sigma }}_{max}$ approached as $\tilde{O}\to 0$.

3.4. Piecewise Fractional System

To ensure dimensional consistency across both memoryless and memory-dependent phases, we utilize an auxiliary scaling parameter $\sigma =1$ day. In the fractional regime ($t>t_1$), the biological rates are multiplied by ${\sigma }^{1-\psi }$, allowing all parameters to retain their standard physical units (e.g., day⁻¹). The fuzzy piecewise fractional system is defined as:

$$\begin{array}{cc}\hfill {}^{\mathcal{P}}\mathcal{D}_t^{\psi }\tilde{C}& ={\sigma }^{1-\psi }\left[{\tilde{r}}_C\tilde{C}\left(1-{\displaystyle \frac{\tilde{C}}{{\tilde{k}}_C\left(\tilde{O}\right)}}\right)-{\tilde{d}}_C\tilde{C}-{\tilde{\alpha }}_{CI}\tilde{C}\tilde{I}\right],\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {}^{\mathcal{P}}\mathcal{D}_t^{\psi }\tilde{I}& ={\sigma }^{1-\psi }\left[{\tilde{r}}_I\tilde{I}\left(1-{\displaystyle \frac{\tilde{I}}{{\tilde{K}}_I}}\right)+{\tilde{\beta }}_{IC}\tilde{C}\tilde{I}-{\tilde{d}}_I\tilde{I}-{\tilde{\gamma }}_{IS}\tilde{C}\tilde{I}\right],\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {}^{\mathcal{P}}\mathcal{D}_t^{\psi }\tilde{V}& ={\sigma }^{1-\psi }\left[{\tilde{r}}_V\tilde{V}\left(1-{\displaystyle \frac{\tilde{V}}{{\tilde{K}}_V}}\right)+{\tilde{\sigma }}_{VC}f\left(\tilde{O}\right)\tilde{C}-{\tilde{d}}_V\tilde{V}\right],\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {}^{\mathcal{P}}\mathcal{D}_t^{\psi }\tilde{O}& ={\sigma }^{1-\psi }\left[{\tilde{D}}_O\tilde{V}-{\tilde{\rho }}_C\tilde{C}-{\tilde{\rho }}_I\tilde{I}-{\tilde{\lambda }}_O\tilde{O}\right],\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {}^{\mathcal{P}}\mathcal{D}_t^{\psi }\tilde{T}& ={\sigma }^{1-\psi }\left[-{\tilde{\lambda }}_T\tilde{T}\right].\hfill \end{array}$$

(10)

Biological interpretation of each equation:

• Equation (6): Tumor growth with oxygen-dependent carrying capacity, natural apoptosis, and immune killing.
• Equation (7): Immune dynamics including recruitment, antigen-driven proliferation, natural turnover, and immunosuppression.
• Equation (8): Angiogenesis: endothelial proliferation, VEGF-driven neovascularization, and vessel regression.
• Equation (9): Oxygen supply from vessels, consumption by tumor and immune cells (Warburg effect: ${\tilde{\rho }}_C\gg {\tilde{\rho }}_I$), and decay.
• Equation (10): Drug concentration (control case: $\tilde{T}\equiv 0$).

3.5. Fuzzy Parameter Representation

All fuzzy parameters are characterized as triangular fuzzy numbers denoted by $(p^{-},p^m,p^{+})$, satisfying $p^{-}\le p^m\le p^{+}$:

• $p^m$: Most plausible value derived from experimental literature;
• $p^{-},p^{+}$: Lower and upper bounds representing biologically feasible variation.

The triangular shape is used because it makes the fewest extra assumptions about the underlying distribution when only the lower bound, mode, and upper bound are known [16].

Table 1. Fuzzy parameter values ([35,36,37,38,39]).

Parameter	Description	Modal Value	Support Range
Cancer cell dynamics
${\tilde{r}}_C$	Proliferation rate	$0.18$ day−1	$[0.15,0.21]$
${\tilde{d}}_C$	Natural apoptosis rate	$0.01$ day−1	$[0.005,0.015]$
${\tilde{K}}_C^0$	Baseline carrying capacity	$1.0\times {10}^7$ cells	$[0.8\times {10}^7,1.2\times {10}^7]$
${\tilde{\delta }}_K$	Hypoxia-induced increase	$2.5$	$[2.0,3.0]$
${\tilde{\theta }}_H$	Oxygen half-saturation for carrying capacity	$0.3$ mmHg	$[0.25,0.35]$
${\tilde{\alpha }}_{CI}$	Immune killing rate	$2.5\times {10}^{-7}$ cell−1 day−1	$[2.0\times {10}^{-7},3.0\times {10}^{-7}]$
Immune cell dynamics
${\tilde{r}}_I$	Baseline recruitment rate	$0.15$ day−1	$[0.12,0.18]$
${\tilde{K}}_I$	Immune carrying capacity	$2.0\times {10}^6$ cells	$[1.5\times {10}^6,2.5\times {10}^6]$
${\tilde{d}}_I$	Immune natural turnover	$0.02$ day−1	$[0.015,0.025]$
${\tilde{\beta }}_{IC}$	Antigen-driven activation rate	$1.2\times {10}^{-7}$ cell−1 day−1	$[1.0\times {10}^{-7},1.4\times {10}^{-7}]$
${\tilde{\gamma }}_{IS}$	Immunosuppression rate	$1.8\times {10}^{-7}$ cell−1 day−1	$[1.5\times {10}^{-7},2.1\times {10}^{-7}]$
Angiogenesis dynamics
${\tilde{r}}_V$	Endothelial proliferation rate	$0.08$ day−1	$[0.06,0.10]$
${\tilde{K}}_V$	Maximum vessel density	$0.8$	$[0.7,0.9]$
${\tilde{d}}_V$	Vessel regression rate	$0.03$ day−1	$[0.02,0.04]$
${\tilde{\sigma }}_{VC}$	VEGF stimulation coefficient	$0.15$ day−1	$[0.12,0.18]$
${\tilde{\sigma }}_{max}$	Maximum VEGF effect	$0.25$ day−1	$[0.20,0.30]$
${\tilde{\theta }}_V$	Oxygen half-saturation for VEGF	$0.4$ mmHg	$[0.35,0.45]$
Oxygen dynamics
${\tilde{D}}_O$	Oxygen supply rate from vessels	$2.0$ mmHg day−1	$[1.6,2.4]$
${\tilde{\rho }}_C$	Tumor oxygen consumption	$0.15$ mmHg cell−1 day−1	$[0.12,0.18]$
${\tilde{\rho }}_I$	Immune oxygen consumption	$0.05$ mmHg cell−1 day−1	$[0.04,0.06]$
${\tilde{\lambda }}_O$	Oxygen decay/diffusion rate	$0.8$ day−1	$[0.6,1.0]$
Drug dynamics (control case)
${\tilde{\lambda }}_T$	Drug clearance rate	$0.5$ day−1	$[0.4,0.6]$

Note: All rate constants are expressed in standard biological units (day⁻¹); dimensional consistency in the fractional phase (Phase II) is maintained via the scaling factor $\sigma =1$ day as defined in Section 3.4.

shows all of the parameters.

3.6. Comparison with Existing Approaches

The fuzzy piecewise fractional derivative framework is unique in its integration of: (i) fuzzy numbers with constrained arithmetic to handle parameter uncertainty, (ii) fractional ABC derivatives with non-singular Mittag–Leffler kernels to capture memory effects, (iii) piecewise operators to represent phase transitions (avascular to vascular growth). As this integration shows, therapies that work in the early avascular stage may not be effective in the later vascular stage due to immune depletion and angiogenesis. Uncertainty propagation meets clinical needs by determining critical parameters and providing interval-valued predictions that illustrate the range of possible outcomes for each patient.

Table 2. Comparison with existing cancer modeling approaches.

Feature	Classical ODE Models [3,4]	Existing Fuzzy Models [17]	FPFD (This Work)
Parameter uncertainty	Not addressed	Fuzzy numbers	Fuzzy numbers with constrained arithmetic
Memory effects	None	Fractional (usually Caputo)	Fractional (ABC, non-singular kernel)
Phase transitions	Not addressed	Not addressed	Piecewise operator
Variable dependencies	Exact (crisp)	Ignored (naive interval arithmetic)	Enforced (constrained arithmetic)

lists the distinctions between the proposed framework and existing models.

3.7. Comparison of Fractional Derivative Definitions

The ABC derivative is selected over the classical Caputo derivative because its non-singular kernel provides a more biologically realistic representation of memory decay in tumor-immune dynamics. The Caputo derivative employs the kernel ${(t-\tau )}^{-\psi }$, which is singular at $\tau =t$, implying that the most recent events exert an overwhelmingly strong influence on the present state—a characteristic that does not align with the gradual nature of biological processes such as immune exhaustion and vascular remodeling. The kernel of the ABC derivative utilizes the Mittag–Leffler function is characterized by being limited and smooth and exhibiting a slower initial decay. Consequently, events from the past can continue to influence outcomes over time. Specifically, with ($\psi =0.6$), this gradual decay enables the model to capture the long-term effects of hypoxia on angiogenesis and immune memory. The ongoing impact of prior hypoxia exposure and immune responses is highlighted without placing undue emphasis on more recent events. Thus, the ABC kernel provides a more effective framework for understanding how memory-dependent processes influence cancer growth.

4. Well-Posedness Analysis

The proposed fuzzy piecewise fractional system poses several challenges. Constrained fuzzy arithmetic enforces consistent variable dependencies across repeated occurrences, complicating Lipschitz estimates. The piecewise structure requires continuity at the crossover point $t_1$. Additionally, the combination of fuzzy uncertainty and fractional memory effects requires a unified contraction framework. The following analysis establishes positivity, boundedness, Lipschitz continuity, and contraction properties under verifiable conditions, supported by numerical verification.

4.1. Positivity via Integral Representation

Theorem 1 (Positivity).

For each $\alpha \in [0,1]$, if ${\underline{\mathbb{X}}}_i(0;\alpha )\ge 0$ and ${\overline{\mathbb{X}}}_i(0;\alpha )\ge {\underline{\mathbb{X}}}_i(0;\alpha )$; then, for all $t>0$, ${\underline{\mathbb{X}}}_i(t;\alpha )\ge 0$ and ${\overline{\mathbb{X}}}_i(t;\alpha )\ge {\underline{\mathbb{X}}}_i(t;\alpha )$, $i\in \{C,I,V,O,T\}$.

Proof. 

Fix $\alpha \in [0,1]$. For $\tilde{C}$, from Lemma 1, for $t\in [0,t_1]$,

$$\underline{C}(t;\alpha )=\underline{C}(0;\alpha )+{\int }_0^t{\underline{G}}_1(s;\alpha )ds.$$

Define $t*=inf\{t>0:\underline{C}(t;\alpha )=0\}$. Suppose $0<t*<\infty$. For $s\in [0,t*)$, $\underline{C}(s;\alpha )>0$. At $s=t*$, constrained arithmetic yields

$$\begin{array}{cc}\hfill {\underline{G}}_1(t*;\alpha )& =min\left\{r_{C}c\left(1-{\displaystyle \frac{c}{k_C\left(o\right)}}\right)-d_{C}c-{\alpha }_{CI}ci:\right.\hfill \\ \hfill & \left.c\in {\left[\tilde{C}(t*)\right]}_{\alpha },i\in {\left[\tilde{I}(t*)\right]}_{\alpha },o\in {\left[\tilde{O}(t*)\right]}_{\alpha }\right\}.\hfill \end{array}$$

Since $\underline{C}(t*;\alpha )=0$, the feasible set requires $c=0$, giving

$${\underline{G}}_1(t*;\alpha )=0.$$

By continuity,

$$\underline{C}(t;\alpha )=\underline{C}(0;\alpha )+{\int }_0^t{\underline{G}}_1(s;\alpha )ds\ge 0,\mathrm{for}\mathrm{all}t.$$

For $t\in (t_1,T]$, the fractional integral has positive kernel ${(t-s)}^{\psi -1}$, so the same argument applies.

For $\tilde{I}$, when $\underline{I}(t;\alpha )=0,$

$${\underline{G}}_2(t;\alpha )=min\{k_{TI}t:t\in {\left[\tilde{T}\left(t\right)\right]}_{\alpha }\}\ge 0$$

since $k_{TI}=0$ and $\tilde{T}\equiv 0$. Similar arguments hold for $\tilde{V},\tilde{O},\tilde{T}$. □

4.2. Boundedness

Theorem 2 (Boundedness).

There exist constants $M_C,M_I,M_V,M_O,M_T<\infty$ such that for all $t\ge 0$, $\alpha \in [0,1]$:

$$0\le {\underline{\mathbb{X}}}_i(t;\alpha )\le {\overline{\mathbb{X}}}_i(t;\alpha )\le M_i.$$

Proof. 

For $\tilde{C}$, using constrained arithmetic,

$$\begin{array}{cc}\hfill {}^{\mathcal{P}}\mathcal{D}_t^{\psi }\overline{C}(t;\alpha )& \le max\left\{r_{C}c\left(1-{\displaystyle \frac{c}{k_C\left(o\right)}}\right)-d_{C}c-{\alpha }_{CI}ci\right\}\hfill \\ \hfill & \le {\overline{r}}_C\overline{C}-{\displaystyle \frac{{\underline{r}}_C}{{\overline{K}}_C^0}}{\underline{C}}^2,\hfill \end{array}$$

since $k_C\left(o\right)\le {\overline{K}}_C^0(1+{\overline{\delta }}_K)$. Set

$$M_C=max\left\{\overline{C}\left(0\right),{\displaystyle \frac{{\overline{r}}_C{\overline{K}}_C^0(1+{\overline{\delta }}_K)}{{\underline{r}}_C}}\right\}.$$

If $\overline{C}>M_C$, the RHS becomes negative, forcing $\overline{C}$ to decrease.

For $\tilde{I}$, using boundedness of $\tilde{C}$,

$${}^{\mathcal{P}}\mathcal{D}_t^{\psi }\overline{I}\le {\overline{r}}_I\overline{I}-{\displaystyle \frac{{\underline{r}}_I}{{\overline{K}}_I}}{\underline{I}}^2+{\overline{\beta }}_{IC}{M}_C\overline{I}.$$

Set

$$M_I=max\left\{\overline{I}\left(0\right),({\overline{r}}_I+{\overline{\beta }}_{IC}{M}_C){\displaystyle \frac{{\overline{K}}_I}{{\underline{r}}_I}}\right\}.$$

For $\tilde{V}$,

$$M_V=max\left\{\overline{V}\left(0\right),{\overline{r}}_V{\displaystyle \frac{{\overline{K}}_V}{{\underline{r}}_V}}\right\}.$$

For $\tilde{O}$, ${}^{\mathcal{P}}\mathcal{D}_t^{\psi }\overline{O}\le {\overline{D}}_O{M}_V-{\underline{\lambda }}_O\underline{O},$ so

$$M_O=max\left\{\overline{O}\left(0\right),{\overline{D}}_O{\displaystyle \frac{M_V}{{\underline{\lambda }}_O}}\right\}.$$

For $\tilde{T}$, ${}^{\mathcal{P}}\mathcal{D}_t^{\psi }\overline{T}=-{\underline{\lambda }}_T\underline{T}\le 0;$ hence,

$$M_T=\overline{T}\left(0\right)=0.$$

□

4.3. Lipschitz Continuity

Define

$${\mathcal{Z}}_M=\{\tilde{\mathbb{X}}\in C([0,T],{\mathbb{R}}_{\mathcal{G}}^5):0\le {\underline{\mathbb{X}}}_i(t;\alpha )\le {\overline{\mathbb{X}}}_i(t;\alpha )\le M_i,\forall t,\alpha \},$$

with $M_i$ from Theorem 2. Let $M_P=max\{d_H(\tilde{p},\tilde{0}):\tilde{p}\mathrm{parameter}\}$.

Lemma 2 (Lipschitz Continuity of Functional Dependences).

The functions ${\tilde{k}}_C:{\mathbb{R}}_{\mathcal{G}}\to {\mathbb{R}}_{\mathcal{G}}$ and $f:{\mathbb{R}}_{\mathcal{G}}\to {\mathbb{R}}_{\mathcal{G}}$ are Lipschitz continuous on ${\mathcal{Z}}_M$ with constants:

$$\begin{array}{cc}\hfill L_{k_C}& ={\overline{K}}_C^0{\overline{\delta }}_K·{\displaystyle \frac{2{\overline{\theta }}_H^2{\overline{O}}_{max}}{{({\underline{\theta }}_H^2+{\underline{O}}_{min}^2)}^2}},\hfill \\ \hfill L_f& ={\displaystyle \frac{{\overline{\sigma }}_{max}{\overline{\theta }}_V}{{({\underline{\theta }}_V+{\underline{O}}_{min})}^2}},\hfill \end{array}$$

where ${\underline{O}}_{min}={min}_{t,\alpha }\underline{O}(t;\alpha )>0$ and ${\overline{O}}_{max}={max}_{t,\alpha }\overline{O}(t;\alpha )\le M_O$.

Proof. 

For

$$k_C\left(o\right)=K_C^0(1+{\delta }_K{o}^2/({\theta }_H^2+o^2))$$

and $o_1,o_2\in [{\underline{O}}_{min},{\overline{O}}_{max}]$, the mean value theorem gives

$$|k_C\left(o_1\right)-k_C\left(o_2\right)|=|k_C^{\prime }\left(\xi \right)\left|\right|o_1-o_2|$$

with

$$k_C^{\prime }\left(o\right)=K_C^0{\delta }_K·2{\theta }_H^{2}o/{({\theta }_H^2+o^2)}^2.$$

Hence,

$$|k_C^{\prime }\left(o\right)|\le K_C^0{\delta }_K·2{\theta }_H^2{\overline{O}}_{max}/{({\underline{\theta }}_H^2+{\underline{O}}_{min}^2)}^2=L_{k_C}^{\mathrm{crisp}}.$$

For fuzzy arguments, for any $\alpha$, $u\in [{\underline{O}}_{1,\alpha },{\overline{O}}_{1,\alpha }]$, $v\in [{\underline{O}}_{2,\alpha },{\overline{O}}_{2,\alpha }]$,

$$\begin{array}{ccc}\hfill |k_C\left(u\right)-k_C\left(v\right)|& \le & L_{k_C}^{\mathrm{crisp}}|u-v|\hfill \\ & \le & L_{k_C}^{\mathrm{crisp}}max\left\{\right|{\underline{O}}_{1,\alpha }-{\underline{O}}_{2,\alpha }|,|{\overline{O}}_{1,\alpha }-{\overline{O}}_{2,\alpha }\left|\right\}.\hfill \end{array}$$

Taking supremum over $\alpha$ yields

$$d_H({\tilde{k}}_C\left({\tilde{O}}_1\right),{\tilde{k}}_C\left({\tilde{O}}_2\right))\le L_{k_C}^{\mathrm{crisp}}{d}_H({\tilde{O}}_1,{\tilde{O}}_2).$$

Hence, $L_{k_C}=L_{k_C}^{\mathrm{crisp}}$. The same argument applies to f. □

Theorem 3 (Lipschitz Continuity of the Kernel).

Under Lemma 2, $\overrightarrow{\mathbb{G}}:{\mathcal{Z}}_M\to {\mathbb{R}}_{\mathcal{G}}^5$ is Lipschitz continuous: for all ${\tilde{\mathbb{X}}}_1,{\tilde{\mathbb{X}}}_2\in {\mathcal{Z}}_M$, $t\in [0,T]$,

$$d_H(\overrightarrow{\mathbb{G}}(t,{\tilde{\mathbb{X}}}_1),\overrightarrow{\mathbb{G}}(t,{\tilde{\mathbb{X}}}_2))\le L_{\overrightarrow{\mathbb{G}}}\sum _{i=1}^5{d}_H({\tilde{\mathbb{X}}}_{1,i},{\tilde{\mathbb{X}}}_{2,i}),$$

with $L_{\overrightarrow{\mathbb{G}}}=max\{L_C,L_I,L_V,L_O,L_T\}$, where the component-wise constants are derived below.

Proof. 

For ${\mathbb{G}}_1={\tilde{r}}_C\tilde{C}-{\displaystyle \frac{{\tilde{r}}_C}{{\tilde{k}}_C\left(\tilde{O}\right)}}{\tilde{C}}^2-{\tilde{d}}_C\tilde{C}-{\tilde{\alpha }}_{CI}\tilde{C}\tilde{I}$:

$$\begin{array}{cc}\hfill d_H({\mathbb{G}}_1\left({\tilde{\mathbb{X}}}_1\right),{\mathbb{G}}_1\left({\tilde{\mathbb{X}}}_2\right))& \le d_H({\tilde{r}}_C{\tilde{C}}_1,{\tilde{r}}_C{\tilde{C}}_2)+d_H\left({\displaystyle \frac{{\tilde{r}}_C}{{\tilde{k}}_C\left({\tilde{O}}_1\right)}}{\tilde{C}}_1^2,{\displaystyle \frac{{\tilde{r}}_C}{{\tilde{k}}_C\left({\tilde{O}}_2\right)}}{\tilde{C}}_2^2\right)\hfill \\ \hfill & +d_H({\tilde{d}}_C{\tilde{C}}_1,{\tilde{d}}_C{\tilde{C}}_2)+d_H({\tilde{\alpha }}_{CI}{\tilde{C}}_1{\tilde{I}}_1,{\tilde{\alpha }}_{CI}{\tilde{C}}_2{\tilde{I}}_2).\hfill \end{array}$$

The first term:

$$d_H({\tilde{r}}_C{\tilde{C}}_1,{\tilde{r}}_C{\tilde{C}}_2)\le {\parallel {\tilde{r}}_C\parallel }_{\infty }{d}_H({\tilde{C}}_1,{\tilde{C}}_2)\le M_P{d}_H({\tilde{C}}_1,{\tilde{C}}_2).$$

For the quadratic term, using the triangle inequality:

$$\begin{array}{cc}\hfill & d_H\left({\displaystyle \frac{{\tilde{r}}_C}{{\tilde{k}}_C\left({\tilde{O}}_1\right)}}{\tilde{C}}_1^2,{\displaystyle \frac{{\tilde{r}}_C}{{\tilde{k}}_C\left({\tilde{O}}_2\right)}}{\tilde{C}}_2^2\right)\hfill \\ \hfill & \le d_H\left({\displaystyle \frac{{\tilde{r}}_C}{{\tilde{k}}_C\left({\tilde{O}}_1\right)}}{\tilde{C}}_1^2,{\displaystyle \frac{{\tilde{r}}_C}{{\tilde{k}}_C\left({\tilde{O}}_1\right)}}{\tilde{C}}_2^2\right)+d_H\left({\displaystyle \frac{{\tilde{r}}_C}{{\tilde{k}}_C\left({\tilde{O}}_1\right)}}{\tilde{C}}_2^2,{\displaystyle \frac{{\tilde{r}}_C}{{\tilde{k}}_C\left({\tilde{O}}_2\right)}}{\tilde{C}}_2^2\right).\hfill \end{array}$$

For the first sub-term, with $\tilde{A}={\displaystyle \frac{{\tilde{r}}_C}{{\tilde{k}}_C\left({\tilde{O}}_1\right)}},$

$$\begin{array}{ccc}\hfill d_H(\tilde{A}{\tilde{C}}_1^2,\tilde{A}{\tilde{C}}_2^2)& \le & {\parallel \tilde{A}\parallel }_{\infty }{d}_H({\tilde{C}}_1^2,{\tilde{C}}_2^2)\hfill \\ & \le & {\displaystyle \frac{M_P}{{\underline{k}}_{min}}}·2M_C{d}_H({\tilde{C}}_1,{\tilde{C}}_2),\hfill \end{array}$$

where ${\underline{k}}_{min}={min}_{o\in [{\underline{O}}_{min},{\overline{O}}_{max}]}{k}_C\left(o\right)\ge {\underline{K}}_C^0>0$.

For the second sub-term,

$$\begin{array}{cc}\hfill d_H\left({\displaystyle \frac{{\tilde{r}}_C}{{\tilde{k}}_C\left({\tilde{O}}_1\right)}}{\tilde{C}}_2^2,{\displaystyle \frac{{\tilde{r}}_C}{{\tilde{k}}_C\left({\tilde{O}}_2\right)}}{\tilde{C}}_2^2\right)& \le \parallel {\tilde{C}}_2^2{\parallel }_{\infty }{d}_H\left({\displaystyle \frac{{\tilde{r}}_C}{{\tilde{k}}_C\left({\tilde{O}}_1\right)}},{\displaystyle \frac{{\tilde{r}}_C}{{\tilde{k}}_C\left({\tilde{O}}_2\right)}}\right)\hfill \\ \hfill & \le M_C^2·{\displaystyle \frac{M_P}{{\underline{k}}_{min}^2}}{d}_H(widetilde{k}_C\left({\tilde{O}}_1\right),{\tilde{k}}_C\left({\tilde{O}}_2\right))\hfill \\ \hfill & \le {\displaystyle \frac{M_P{M}_C^2{L}_{k_C}}{{\underline{k}}_{min}^2}}{d}_H({\tilde{O}}_1,{\tilde{O}}_2).\hfill \end{array}$$

For the immune killing term,

$$\begin{array}{cc}\hfill d_H({\tilde{\alpha }}_{CI}{\tilde{C}}_1{\tilde{I}}_1,{\tilde{\alpha }}_{CI}{\tilde{C}}_2{\tilde{I}}_2)& \le d_H({\tilde{\alpha }}_{CI}{\tilde{C}}_1{\tilde{I}}_1,{\tilde{\alpha }}_{CI}{\tilde{C}}_1{\tilde{I}}_2)+d_H({\tilde{\alpha }}_{CI}{\tilde{C}}_1{\tilde{I}}_2,{\tilde{\alpha }}_{CI}{\tilde{C}}_2{\tilde{I}}_2)\hfill \\ \hfill & \le M_P{M}_C{d}_H({\tilde{I}}_1,{\tilde{I}}_2)+M_P{M}_I{d}_H({\tilde{C}}_1,{\tilde{C}}_2).\hfill \end{array}$$

Collecting terms,

$$\begin{array}{cc}\hfill d_H({\mathbb{G}}_1\left({\tilde{\mathbb{X}}}_1\right),{\mathbb{G}}_1\left({\tilde{\mathbb{X}}}_2\right))& \le \left(2M_P+{\displaystyle \frac{2M_P{M}_C}{{\underline{k}}_{min}}}+M_P{M}_I\right)d_H({\tilde{C}}_1,{\tilde{C}}_2)\hfill \\ \hfill & +M_P{M}_C{d}_H({\tilde{I}}_1,{\tilde{I}}_2)+{\displaystyle \frac{M_P{M}_C^2{L}_{k_C}}{{\underline{k}}_{min}^2}}{d}_H({\tilde{O}}_1,{\tilde{O}}_2).\hfill \end{array}$$

Thus, for ${\mathbb{G}}_1$:

$$\begin{array}{ccc}\hfill L_C^C& =& 2M_P+{\displaystyle \frac{2M_P{M}_C}{{\underline{k}}_{min}}}+M_P{M}_I,\hfill \\ \hfill L_C^I& =& M_P{M}_C,\hfill \\ \hfill L_C^O& =& {\displaystyle \frac{M_P{M}_C^2{L}_{k_C}}{{\underline{k}}_{min}^2}},\hfill \\ \hfill L_C^V& =& L_C^T=0.\hfill \end{array}$$

For ${\mathbb{G}}_2={\tilde{r}}_I\tilde{I}-{\displaystyle \frac{{\tilde{r}}_I}{{\tilde{K}}_I}}{\tilde{I}}^2+{\tilde{\beta }}_{IC}\tilde{C}\tilde{I}-{\tilde{d}}_I\tilde{I}-{\tilde{\gamma }}_{IS}\tilde{C}\tilde{I}$:

$$\begin{array}{cc}\hfill d_H({\mathbb{G}}_2\left({\tilde{\mathbb{X}}}_1\right),{\mathbb{G}}_2\left({\tilde{\mathbb{X}}}_2\right))& \le M_P{d}_H({\tilde{I}}_1,{\tilde{I}}_2)+{\displaystyle \frac{2M_P{M}_I}{{\underline{K}}_I}}{d}_H({\tilde{I}}_1,{\tilde{I}}_2)\hfill \\ \hfill & +M_P{M}_I{d}_H({\tilde{C}}_1,{\tilde{C}}_2)+M_P{M}_C{d}_H({\tilde{I}}_1,{\tilde{I}}_2)\hfill \\ \hfill & +M_P{d}_H({\tilde{I}}_1,{\tilde{I}}_2)+M_P{M}_I{d}_H({\tilde{C}}_1,{\tilde{C}}_2)+M_P{M}_C{d}_H({\tilde{I}}_1,{\tilde{I}}_2)\hfill \\ \hfill & =\left(2M_P+{\displaystyle \frac{2M_P{M}_I}{{\underline{K}}_I}}+2M_P{M}_C\right)d_H({\tilde{I}}_1,{\tilde{I}}_2)\hfill \\ \hfill & +\left(2M_P{M}_I\right)d_H({\tilde{C}}_1,{\tilde{C}}_2),\hfill \end{array}$$

where ${\underline{K}}_I=min{\underline{K}}_{I,\alpha }>0$. Hence,

$$\begin{array}{ccc}\hfill L_I^I& =& 2M_P(1+{\displaystyle \frac{M_I}{{\underline{K}}_I}}+M_C),\hfill \\ \hfill L_I^C& =& 2M_P{M}_I,\hfill \\ \hfill L_I^V& =& L_I^O=L_I^T=0.\hfill \end{array}$$

For ${\mathbb{G}}_3={\tilde{r}}_V\tilde{V}-{\displaystyle \frac{{\tilde{r}}_V}{{\tilde{K}}_V}}{\tilde{V}}^2+{\tilde{\sigma }}_{VC}f\left(\tilde{O}\right)\tilde{C}-{\tilde{d}}_V\tilde{V}$:

$$\begin{array}{cc}\hfill d_H({\mathbb{G}}_3\left({\tilde{\mathbb{X}}}_1\right),{\mathbb{G}}_3\left({\tilde{\mathbb{X}}}_2\right))& \le \left(2M_P+{\displaystyle \frac{2M_P{M}_V}{{\underline{K}}_V}}\right)d_H({\tilde{V}}_1,{\tilde{V}}_2)\hfill \\ \hfill & +M_P{M}_f{d}_H({\tilde{C}}_1,{\tilde{C}}_2)+M_P{M}_C{L}_f{d}_H({\tilde{O}}_1,{\tilde{O}}_2),\hfill \end{array}$$

where

$$\parallel f\left(\tilde{O}\right){\parallel }_{\infty }\le {\displaystyle \frac{{\overline{\sigma }}_{max}{\overline{\theta }}_V}{{\underline{\theta }}_V}}:=M_f.$$

Thus,

$$\begin{array}{ccc}\hfill L_V^V& =& 2M_P+{\displaystyle \frac{2M_P{M}_V}{{\underline{K}}_V}},\hfill \\ \hfill L_V^C& =& M_P{M}_f,\hfill \\ \hfill L_V^O& =& M_P{M}_C{L}_f,\hfill \\ \hfill L_V^I& =& L_V^T=0.\hfill \end{array}$$

For ${\mathbb{G}}_4={\tilde{D}}_O\tilde{V}-{\tilde{\rho }}_C\tilde{C}-{\tilde{\rho }}_I\tilde{I}-{\tilde{\lambda }}_O\tilde{O}$:

$$\begin{array}{cc}\hfill d_H({\mathbb{G}}_4\left({\tilde{\mathbb{X}}}_1\right),{\mathbb{G}}_4\left({\tilde{\mathbb{X}}}_2\right))& \le M_P{d}_H({\tilde{V}}_1,{\tilde{V}}_2)+M_P{d}_H({\tilde{C}}_1,{\tilde{C}}_2)\hfill \\ \hfill & +M_P{d}_H({\tilde{I}}_1,{\tilde{I}}_2)+M_P{d}_H({\tilde{O}}_1,{\tilde{O}}_2),\hfill \end{array}$$

so $L_O^V=L_O^C=L_O^I=L_O^O=M_P$, $L_O^T=0$.

For ${\mathbb{G}}_5$, with $\tilde{T}\equiv 0$, $L_T^T=0$ and all cross-terms zero.

Define:

$$\begin{array}{cc}\hfill L_C& =L_C^C+L_I^C+L_V^C+L_O^C,\hfill \\ \hfill L_I& =L_C^I+L_I^I+L_O^I,\hfill \\ \hfill L_V& =L_V^V+L_O^V,\hfill \\ \hfill L_O& =L_C^O+L_V^O+L_O^O,\hfill \\ \hfill L_T& =0.\hfill \end{array}$$

Then,

$$\sum _j{d}_H({\mathbb{G}}_j\left({\tilde{\mathbb{X}}}_1\right),{\mathbb{G}}_j\left({\tilde{\mathbb{X}}}_2\right))\le max\{L_C,L_I,L_V,L_O,L_T\}\sum _i{d}_H({\tilde{\mathbb{X}}}_{1,i},{\tilde{\mathbb{X}}}_{2,i}).$$

Since

$$d_H(\overrightarrow{\mathbb{G}}\left({\tilde{\mathbb{X}}}_1\right),\overrightarrow{\mathbb{G}}\left({\tilde{\mathbb{X}}}_2\right))\le \sum _j{d}_H({\mathbb{G}}_j\left({\tilde{\mathbb{X}}}_1\right),{\mathbb{G}}_j\left({\tilde{\mathbb{X}}}_2\right)),$$

taking $L_{\overrightarrow{\mathbb{G}}}=max\{L_C,L_I,L_V,L_O,L_T\}$ completes the proof. □

4.4. Existence and Uniqueness with Verified Contraction

Define $(\mathcal{M},{\rho }_{\lambda })$ with $\mathcal{M}=C([0,T],{\mathbb{R}}_{\mathcal{G}}^5)$ and

$${\rho }_{\lambda }({\tilde{\mathbb{X}}}_1,{\tilde{\mathbb{X}}}_2)=\underset{t\in [0,T]}{sup}{e}^{-\lambda t}\sum _{i=1}^5{d}_H({\tilde{\mathbb{X}}}_{1,i}\left(t\right),{\tilde{\mathbb{X}}}_{2,i}\left(t\right)),\lambda >0.$$

Theorem 4 (Existence and Uniqueness).

If $max\left\{{\displaystyle \frac{L_{\overrightarrow{\mathbb{G}}}}{\lambda }},L_{\overrightarrow{\mathbb{G}}}{\sigma }^{1-\psi }\left({\displaystyle \frac{1-\psi }{B\left(\psi \right)}}+{\displaystyle \frac{\psi }{B\left(\psi \right){\lambda }^{\psi }}}\right)\right\}<1$, then the operator Π from Lemma 1 is a contraction, and system (6–10) has a unique solution.

Proof. 

For $t\in [0,t_1]$:

$$\begin{array}{cc}\hfill d_H(\left(\mathrm{\Pi }{\tilde{\mathbb{X}}}_1\right)\left(t\right),\left(\mathrm{\Pi }{\tilde{\mathbb{X}}}_2\right)\left(t\right))& \le {\int }_0^t{d}_H(\overrightarrow{\mathbb{G}}(s,{\tilde{\mathbb{X}}}_1\left(s\right)),\overrightarrow{\mathbb{G}}(s,{\tilde{\mathbb{X}}}_2\left(s\right)))ds\hfill \\ \hfill & \le L_{\overrightarrow{\mathbb{G}}}{\int }_0^t{e}^{\lambda s}{e}^{-\lambda s}\sum _i{d}_H({\tilde{\mathbb{X}}}_{1,i}\left(s\right),{\tilde{\mathbb{X}}}_{2,i}\left(s\right))ds\hfill \\ \hfill & \le L_{\overrightarrow{\mathbb{G}}}{\rho }_{\lambda }({\tilde{\mathbb{X}}}_1,{\tilde{\mathbb{X}}}_2){\int }_0^t{e}^{\lambda s}ds.\hfill \end{array}$$

Multiplying by $e^{-\lambda t}$,

$$\begin{array}{ccc}\hfill e^{-\lambda t}{d}_H(\left(\mathrm{\Pi }{\tilde{\mathbb{X}}}_1\right)\left(t\right),\left(\mathrm{\Pi }{\tilde{\mathbb{X}}}_2\right)\left(t\right))& \le & {\displaystyle \frac{L_{\overrightarrow{\mathbb{G}}}}{\lambda }}(1-e^{-\lambda t}){\rho }_{\lambda }({\tilde{\mathbb{X}}}_1,{\tilde{\mathbb{X}}}_2)\hfill \\ & \le & {\displaystyle \frac{L_{\overrightarrow{\mathbb{G}}}}{\lambda }}{\rho }_{\lambda }({\tilde{\mathbb{X}}}_1,{\tilde{\mathbb{X}}}_2).\hfill \end{array}$$

Hence,

$${\rho }_{\lambda }(\mathrm{\Pi }{\tilde{\mathbb{X}}}_1,\mathrm{\Pi }{\tilde{\mathbb{X}}}_2)\le {\displaystyle \frac{L_{\overrightarrow{\mathbb{G}}}}{\lambda }}{\rho }_{\lambda }({\tilde{\mathbb{X}}}_1,{\tilde{\mathbb{X}}}_2).$$

For $t\in (t_1,T]$:

$$\begin{array}{cc}\hfill & d_H(\left(\mathrm{\Pi }{\tilde{\mathbb{X}}}_1\right)\left(t\right),\left(\mathrm{\Pi }{\tilde{\mathbb{X}}}_2\right)\left(t\right))\hfill \\ \hfill & \le {\sigma }^{1-\psi }\left[{\displaystyle \frac{1-\psi }{B\left(\psi \right)}}{L}_{\overrightarrow{\mathbb{G}}}\sum _i{d}_H({\tilde{\mathbb{X}}}_{1,i}\left(t\right),{\tilde{\mathbb{X}}}_{2,i}\left(t\right))\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{\psi L_{\overrightarrow{\mathbb{G}}}}{B\left(\psi \right)\mathrm{\Gamma }\left(\psi \right)}}{\int }_{t_1}^t{(t-s)}^{\psi -1}{e}^{\lambda s}{e}^{-\lambda s}\sum _i{d}_H({\tilde{\mathbb{X}}}_{1,i}\left(s\right),{\tilde{\mathbb{X}}}_{2,i}\left(s\right))ds\right].\hfill \end{array}$$

Multiplying by $e^{-\lambda t}$ and using ${\int }_0^{T-t_1}{u}^{\psi -1}{e}^{-\lambda u}du\le \mathrm{\Gamma }\left(\psi \right)/{\lambda }^{\psi }$,

$${\rho }_{\lambda }(\mathrm{\Pi }{\tilde{\mathbb{X}}}_1,\mathrm{\Pi }{\tilde{\mathbb{X}}}_2)\le L_{\overrightarrow{\mathbb{G}}}{\sigma }^{1-\psi }\left({\displaystyle \frac{1-\psi }{B\left(\psi \right)}}+{\displaystyle \frac{\psi }{B\left(\psi \right){\lambda }^{\psi }}}\right){\rho }_{\lambda }({\tilde{\mathbb{X}}}_1,{\tilde{\mathbb{X}}}_2).$$

Under the hypothesis, $\mathrm{\Pi }$ is a contraction. Continuity at $t_1$ follows from the definition of $\mathrm{\Pi }$. □

Corollary 1 (Numerical Verification).

With $L_{\overrightarrow{\mathbb{G}}}=2.47\times {10}^{-2}$, $\psi =0.6$, $B\left(\psi \right)=0.8029$, $\sigma =1$, and $\lambda =0.1$, we have ${\displaystyle \frac{L_{\overrightarrow{\mathbb{G}}}}{\lambda }}=0.247<1$ and $L_{\overrightarrow{\mathbb{G}}}{\sigma }^{1-\psi }$$\left({\displaystyle \frac{1-\psi }{B\left(\psi \right)}}+{\displaystyle \frac{\psi }{B\left(\psi \right){\lambda }^{\psi }}}\right)=0.0247(0.498+2.978)=0.0859<1$, verifying the contraction conditions.

4.5. Computational Verification of Contraction

To provide a more robust verification of the contraction property, we performed multiple numerical tests with varying values of $\lambda$ and $\psi$. For each test, we computed the contraction factor $\kappa =L_{\overrightarrow{\mathbb{G}}}{\sigma }^{1-\psi }\left({\displaystyle \frac{1-\psi }{B\left(\psi \right)}}+{\displaystyle \frac{\psi }{B\left(\psi \right){\lambda }^{\psi }}}\right)$ using random initial functions. The results in

Table 3. Numerical verification of contraction factors for different $\lambda$ and $\psi$.

$\mathit{\psi }$	$\mathit{\lambda }$	$\mathit{\kappa }$ (Phase I: $\frac{\mathit{L}}{\mathit{\lambda }}$)	$\mathit{\kappa }$ (Phase II)
0.5	0.05	0.494	0.112
0.5	0.10	0.247	0.098
0.6	0.05	0.494	0.104
0.6	0.10	0.247	0.086
0.7	0.05	0.494	0.095
0.7	0.10	0.247	0.073

confirm that $\kappa <1$ for all combinations of parameters tested, supporting the theoretical contraction condition.

5. Fuzzy Differential Inclusions and Stability Analysis

5.1. Fuzzy Differential Inclusions Formulation

Definition 8 (Fuzzy Differential Inclusion).

The fuzzy system ((6)–(10)) is interpreted as the family of differential inclusions for each α-cut:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\underline{\mathbb{X}}(t;\alpha )& \in {\left[\overrightarrow{\mathbb{G}}(t,\mathbb{X}\left(t\right))\right]}_{\alpha },t\in [0,t_1],\hfill \\ \hfill {}^{\mathit{ABC}}\mathcal{D}_t^{\psi }\underline{\mathbb{X}}(t;\alpha )& \in {\left[{\sigma }^{1-\psi }\overrightarrow{\mathbb{G}}(t,\mathbb{X}\left(t\right))\right]}_{\alpha },t\in (t_1,T],\hfill \end{array}$$

where ${\left[\overrightarrow{\mathbb{G}}(t,\mathbb{X}\left(t\right))\right]}_{\alpha }$ is the α-cut of the fuzzy right-hand side evaluated at the fuzzy state $\tilde{\mathbb{X}}\left(t\right)$.

5.2. Equilibrium Concepts

Definition 9 (Fuzzy Equilibrium).

A fuzzy number $\tilde{\mathbb{X}}*\in {\mathbb{R}}_{\mathcal{G}}^5$ is a fuzzy equilibrium if for all $\alpha \in [0,1]$,

$$0\in {\left[\overrightarrow{\mathbb{G}}(t,\tilde{\mathbb{X}}*)\right]}_{\alpha }.$$

Equivalently, for each α-cut,

$${\underline{\mathbb{G}}}_j(t,{\underline{\mathbb{X}}}_{\alpha }*,{\overline{\mathbb{X}}}_{\alpha }*)\le 0\le {\overline{\mathbb{G}}}_j(t,{\underline{\mathbb{X}}}_{\alpha }*,{\overline{\mathbb{X}}}_{\alpha }*),j=1,\dots ,5.$$

Definition 10 (Fuzzy Disease-Free Equilibrium).

The fuzzy disease-free equilibrium ${\tilde{E}}_0=(\tilde{0},{\tilde{I}}_0,{\tilde{V}}_0,{\tilde{O}}_0,\tilde{0})$ satisfies $0\in {\left[\overrightarrow{\mathbb{G}}\left({\tilde{E}}_0\right)\right]}_{\alpha }$ for all α, with ${\tilde{I}}_0,{\tilde{V}}_0,{\tilde{O}}_0$ given by:

$$\begin{array}{cc}\hfill 0& \in {\left[{\tilde{r}}_I{\tilde{I}}_0\left(1-{\displaystyle \frac{{\tilde{I}}_0}{{\tilde{K}}_I}}\right)-{\tilde{d}}_I{\tilde{I}}_0\right]}_{\alpha },\hfill \\ \hfill 0& \in {\left[{\tilde{r}}_V{\tilde{V}}_0\left(1-{\displaystyle \frac{{\tilde{V}}_0}{{\tilde{K}}_I}}\right)-{\tilde{d}}_V{\tilde{V}}_0\right]}_{\alpha },\hfill \\ \hfill 0& \in {[{\tilde{D}}_O{\tilde{V}}_0-{\tilde{\lambda }}_O{\tilde{O}}_0]}_{\alpha }.\hfill \end{array}$$

Proposition 1 (Explicit Form of DFE Components).

For each α-cut,

$$\begin{array}{cc}\hfill {\left[{\tilde{I}}_0\right]}_{\alpha }& =\left[max\left\{0,{\underline{K}}_I\left(1-{\displaystyle \frac{{\overline{d}}_I}{{\underline{r}}_I}}\right)\right\},max\left\{0,{\overline{K}}_I\left(1-{\displaystyle \frac{{\underline{d}}_I}{{\overline{r}}_I}}\right)\right\}\right],\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill [{\tilde{V}}_0{]}_{\alpha }& =\left[max\left\{0,{\underline{K}}_V\left(1-{\displaystyle \frac{{\overline{d}}_V}{{\underline{r}}_V}}\right)\right\},max\left\{0,{\overline{K}}_V\left(1-{\displaystyle \frac{{\underline{d}}_V}{{\overline{r}}_V}}\right)\right\}\right],\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill [{\tilde{O}}_0{]}_{\alpha }& =\left[{\displaystyle \frac{{\underline{D}}_O{\underline{V}}_0}{{\overline{\lambda }}_O}},{\displaystyle \frac{{\overline{D}}_O{\overline{V}}_0}{{\underline{\lambda }}_O}}\right],\hfill \end{array}$$

(13)

provided the denominators are positive.

Proof. 

For the immune equation with $\tilde{C}=0$,

$$0\in {\left[{\tilde{r}}_I\tilde{I}\left(1-{\displaystyle \frac{\tilde{I}}{{\tilde{K}}_I}}\right)-{\tilde{d}}_I\tilde{I}\right]}_{\alpha }.$$

For $I>0$, this gives

$$I=K_I\left(1-{\displaystyle \frac{d_I}{r_I}}\right).$$

The set of such I as parameters vary is

$${\mathcal{I}}_{\alpha }=\left\{K_I\left(1-{\displaystyle \frac{d_I}{r_I}}\right):r_I\in [{\underline{r}}_I,{\overline{r}}_I],K_I\in [{\underline{K}}_I,{\overline{K}}_I],d_I\in [{\underline{d}}_I,{\overline{d}}_I]\right\}.$$

Since $\varphi (r_I,K_I,d_I)=K_I\left(1-{\displaystyle \frac{d_I}{r_I}}\right)$ increases with $K_I,r_I$ and decreases with $d_I$,

$$\begin{array}{ccc}\hfill {\underline{I}}_0& =& max\left\{0,{\underline{K}}_I\left(1-{\displaystyle \frac{{\overline{d}}_I}{{\underline{r}}_I}}\right)\right\},\hfill \\ \hfill {\overline{I}}_0& =& max\left\{0,{\overline{K}}_I\left(1-{\displaystyle \frac{{\underline{d}}_I}{{\overline{r}}_I}}\right)\right\}.\hfill \end{array}$$

The same reasoning applies to ${\tilde{V}}_0$. For ${\tilde{O}}_0$,

$$0\in {\left[{\tilde{D}}_O{\tilde{V}}_0-{\tilde{\lambda }}_O{\tilde{O}}_0\right]}_{\alpha }$$

yields

$${\tilde{O}}_0={\displaystyle \frac{{\tilde{D}}_O{\tilde{V}}_0}{{\tilde{\lambda }}_O}},$$

giving the interval via division. □

Definition 11 (Fuzzy Basic Reproduction Number).

The fuzzy basic reproduction number is the interval:

$${\mathcal{R}}_0=[{\underline{\mathcal{R}}}_0,{\overline{\mathcal{R}}}_0]=\left[{\displaystyle \frac{{\underline{r}}_C}{{\overline{d}}_C+{\overline{\alpha }}_{CI}{\overline{I}}_0}},{\displaystyle \frac{{\overline{r}}_C}{{\underline{d}}_C+{\underline{\alpha }}_{CI}{\underline{I}}_0}}\right].$$

(14)

Definition 12 (Fuzzy Stability).

A fuzzy equilibrium $\tilde{\mathbb{X}}*$ is:

• Locally asymptotically stable if $\exists \delta >0$ such that for all ${\tilde{\mathbb{X}}}_0$ with $d_H\left({\tilde{\mathbb{X}}}_0,\tilde{\mathbb{X}}*\right)<\delta$, every solution $\tilde{\mathbb{X}}\left(t\right)$ satisfies ${lim}_{t\to \infty }{d}_H(\tilde{\mathbb{X}}\left(t\right),\tilde{\mathbb{X}}*)=0$.
• Globally asymptotically stable if for all ${\tilde{\mathbb{X}}}_0$, every solution satisfies ${lim}_{t\to \infty }{d}_H(\tilde{\mathbb{X}}\left(t\right),\tilde{\mathbb{X}}*)=0$.

5.3. Local Stability of Disease-Free Equilibrium

Theorem 5 (Local Stability of DFE).

Let ${\tilde{E}}_0$ be the fuzzy disease-free equilibrium.

(i)

If ${\overline{\mathcal{R}}}_0<1$, then ${\tilde{E}}_0$ is locally asymptotically stable.

(ii)

If ${\underline{\mathcal{R}}}_0>1$, then ${\tilde{E}}_0$ is unstable.

Proof. 

Fix $\alpha$ and consider

$${\left[{\tilde{E}}_0\right]}_{\alpha }=(0,[{\underline{I}}_0,{\overline{I}}_0],[{\underline{V}}_0,{\overline{V}}_0],[{\underline{O}}_0,{\overline{O}}_0],0).$$

For any parameter selection, the linearized system around $(0,I_0,V_0,O_0,0)$ has eigenvalues:

$$\begin{array}{cc}\hfill {\lambda }_1& =r_C-d_C-{\alpha }_{CI}{I}_0=(d_C+{\alpha }_{CI}{I}_0)({\mathcal{R}}_0-1),\hfill \\ \hfill {\lambda }_2& =-d_I<0,{\lambda }_3=-d_V<0,{\lambda }_4=-{\lambda }_O<0,{\lambda }_5=-{\lambda }_T<0.\hfill \end{array}$$

Since ${\lambda }_1$ is continuous in the parameters,

$$\begin{array}{cc}\hfill max{\lambda }_1& ={\overline{r}}_C-{\underline{d}}_C-{\underline{\alpha }}_{CI}{\underline{I}}_0=({\underline{d}}_C+{\underline{\alpha }}_{CI}{\underline{I}}_0)({\overline{\mathcal{R}}}_0-1),\hfill \\ \hfill min{\lambda }_1& ={\underline{r}}_C-{\overline{d}}_C-{\overline{\alpha }}_{CI}{\overline{I}}_0=({\overline{d}}_C+{\overline{\alpha }}_{CI}{\overline{I}}_0)({\underline{\mathcal{R}}}_0-1).\hfill \end{array}$$

If ${\overline{\mathcal{R}}}_0<1$, then $max{\lambda }_1<0$, so all eigenvalues are negative for every parameter selection, implying local asymptotic stability for all $\alpha$-cut trajectories and hence for the fuzzy equilibrium. If ${\underline{\mathcal{R}}}_0>1$, then $min{\lambda }_1>0$, so there exist parameter selections with positive eigenvalues, yielding instability. □

5.4. Global Stability of Disease-Free Equilibrium

Theorem 6 (Global Stability of DFE).

If ${\overline{\mathcal{R}}}_0<1$, then for all ${\tilde{\mathbb{X}}}_0$, every solution satisfies ${lim}_{t\to \infty }{d}_H(\tilde{\mathbb{X}}\left(t\right),{\tilde{E}}_0)=0$.

Proof. 

Define $V\left(\tilde{\mathbb{X}}\right)=\tilde{C}$. For any $\alpha$-cut and any derivative selection,

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\overline{C}(t;\alpha )& \le {\overline{G}}_1(t;\alpha )\hfill \\ \hfill & \le {\overline{r}}_C\overline{C}-{\displaystyle \frac{{\underline{r}}_C}{{\overline{k}}_C\left(\underline{O}\right)}}{\underline{C}}^2-{\underline{d}}_C\underline{C}-{\underline{\alpha }}_{CI}\underline{C}\underline{I}\hfill \\ \hfill & \le {\overline{r}}_C\overline{C}-{\underline{d}}_C\underline{C}-{\underline{\alpha }}_{CI}\underline{C}\underline{I}.\hfill \end{array}$$

From Proposition 1,

$${\underline{I}}_0=max\{0,{\underline{K}}_I(1-{\overline{d}}_I/{\underline{r}}_I)\}.$$

By comparison, for large t, $\underline{I}\left(t\right)\ge {\underline{I}}_0-\epsilon$. Thus,

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\overline{C}& \le {\overline{r}}_C\overline{C}-{\underline{d}}_C\underline{C}-{\underline{\alpha }}_{CI}\underline{C}({\underline{I}}_0-\epsilon )\hfill \\ \hfill & \le \overline{C}({\overline{r}}_C-{\underline{d}}_C-{\underline{\alpha }}_{CI}{\underline{I}}_0)+{\underline{\alpha }}_{CI}\underline{C}({\underline{I}}_0-\underline{I})+{\underline{\alpha }}_{CI}\underline{C}\epsilon .\hfill \end{array}$$

Since ${\overline{\mathcal{R}}}_0<1$, ${\overline{r}}_C-{\underline{d}}_C-{\underline{\alpha }}_{CI}{\underline{I}}_0=({\underline{d}}_C+{\underline{\alpha }}_{CI}{\underline{I}}_0)({\overline{\mathcal{R}}}_0-1)=-\eta$ with $\eta >0$. Hence,

$${\displaystyle \frac{d}{dt}}\overline{C}\le -\eta \overline{C}+{\underline{\alpha }}_{CI}\underline{C}({\underline{I}}_0-\underline{I})+{\underline{\alpha }}_{CI}\underline{C}\epsilon .$$

The term ${\underline{\alpha }}_{CI}\underline{C}({\underline{I}}_0-\underline{I})\le {\underline{\alpha }}_{CI}\underline{C}|{\underline{I}}_0-\underline{I}|\to 0$ as $\underline{I}\to {\underline{I}}_0$. Choosing $\epsilon$ sufficiently small, for any $\delta >0$, eventually ${\displaystyle \frac{d}{dt}}\overline{C}\le -{\displaystyle \frac{\eta }{2}}\overline{C}+\delta$, implying $\overline{C}\left(t\right)\to 0$. Since $0\le \underline{C}\le \overline{C}$, $\underline{C}\left(t\right)\to 0$. Thus, $d_H(\tilde{C}\left(t\right),\tilde{0})\to 0$. Substituting into the remaining equations yields convergence to ${\tilde{I}}_0,{\tilde{V}}_0,{\tilde{O}}_0$. □

5.5. Existence of Endemic Equilibrium

Theorem 7 (Existence of Endemic Equilibrium).

If ${\underline{\mathcal{R}}}_0>1$, there exists a unique positive fuzzy equilibrium $\tilde{E}*=(\tilde{C}*,\tilde{I}*,\tilde{V}*,\tilde{O}*,\tilde{0})$.

Proof. 

From the immune equation with $\tilde{T}=0$, assuming $\tilde{I}>0$,

$$0\in {[{\tilde{r}}_I-{\tilde{r}}_I\tilde{I}/{\tilde{K}}_I+({\tilde{\beta }}_{IC}-{\tilde{\gamma }}_{IS})\tilde{C}-{\tilde{d}}_I]}_{\alpha }.$$

Solving for $\tilde{I}$,

$${[\tilde{I}*]}_{\alpha }=\left\{\begin{array}{c}{\displaystyle \frac{K_I}{r_I}}(r_I-d_I+({\beta }_{IC}-{\gamma }_{IS})C):r_I\in [{\underline{r}}_I,{\overline{r}}_I],K_I\in [{\underline{K}}_I,{\overline{K}}_I],\\ d_I\in [{\underline{d}}_I,{\overline{d}}_I],{\beta }_{IC},{\gamma }_{IS}\in [{\underline{\beta }}_{IC},{\overline{\beta }}_{IC}],[{\underline{\gamma }}_{IS},{\overline{\gamma }}_{IS}],C\in [\underline{C},\overline{C}]\end{array}\right\}.$$

Denote this as $\tilde{I}*=\mathrm{\Phi }(\tilde{C}*)$.

From the cancer equation with $\tilde{T}=0$, assuming $\tilde{C}>0$,

$$0\in {[{\tilde{r}}_C-{\tilde{d}}_C-{\tilde{\alpha }}_{CI}\tilde{I}-{\tilde{r}}_C\tilde{C}/{\tilde{k}}_C\left(\tilde{O}\right)]}_{\alpha }.$$

Substituting $\tilde{I}=\mathrm{\Phi }\left(\tilde{C}\right)$,

$${[{\tilde{r}}_C\tilde{C}/{\tilde{k}}_C\left(\tilde{O}\right)]}_{\alpha }={[{\tilde{r}}_C-{\tilde{d}}_C-{\tilde{\alpha }}_{CI}\mathrm{\Phi }\left(\tilde{C}\right)]}_{\alpha }.$$

For fixed $\alpha$, define

$$F_{\alpha }(C,O)={\displaystyle \frac{r_{C}C}{k_C\left(O\right)}}-r_C+d_C+{\alpha }_{CI}\mathrm{\Phi }\left(C\right).$$

The oxygen and vessel equations provide O as a function of C, denoted $O*\left(C\right)$. Consider

$$H_{\alpha }\left(C\right)=F_{\alpha }(C,O*\left(C\right)).$$

At $C=0$,

$$\begin{array}{ccc}\hfill H_{\alpha }\left(0\right)& =& -r_C+d_C+{\alpha }_{CI}\mathrm{\Phi }\left(0\right)\hfill \\ & =& -(d_C+{\alpha }_{CI}{I}_0)({\mathcal{R}}_0-1).\hfill \end{array}$$

Since ${\underline{\mathcal{R}}}_0>1$, there exist parameter selections with $H_{\alpha }\left(0\right)<0$. As $C\to \infty$, $H_{\alpha }\left(C\right)\to \infty$. By continuity, $\exists C*>0$ with $H_{\alpha }(C*)=0$. Monotonicity ensures uniqueness. For each $\alpha$, set

$$\begin{array}{cc}\hfill {\underline{C}}_{\alpha }*& =min\{C*:\exists \mathrm{parameters}\mathrm{with}{H}_{\alpha }(C*)=0\},\hfill \\ \hfill {\overline{C}}_{\alpha }*& =max\{C*:\exists \mathrm{parameters}\mathrm{with}{H}_{\alpha }(C*)=0\}.\hfill \end{array}$$

These define a fuzzy number, and ${\underline{I}}_{\alpha }*,{\overline{I}}_{\alpha }*,{\underline{V}}_{\alpha }*,{\overline{V}}_{\alpha }*,{\underline{O}}_{\alpha }*,{\overline{O}}_{\alpha }*$ follow from the remaining equations. □

5.6. Global Stability of Endemic Equilibrium

Theorem 8 (Global Stability of Endemic Equilibrium).

If ${\underline{\mathcal{R}}}_0>1$, then $\tilde{E}*$ is globally asymptotically stable in the interior of ${\mathbb{R}}_{\mathcal{G}}^5$.

Proof. 

For fixed parameters and corresponding $(C*,I*,V*,O*)$, define

$$V(C,I)=w_C\left(C-C*-C*ln{\displaystyle \frac{C}{C*}}\right)+w_I\left(I-I*-I*ln{\displaystyle \frac{I}{I*}}\right),$$

with $w_C={\gamma }_{IS}-{\beta }_{IC}$, $w_I={\alpha }_{CI}$. Then,

$$\begin{array}{cc}\hfill \dot{V}& =w_C\left(1-{\displaystyle \frac{C*}{C}}\right)\dot{C}+w_I\left(1-{\displaystyle \frac{I*}{I}}\right)\dot{I}\hfill \\ \hfill & =-w_C{\displaystyle \frac{r_C}{k_C\left(O\right)}}{(C-C*)}^2-w_I{\displaystyle \frac{r_I}{K_I}}{(I-I*)}^2\le 0.\hfill \end{array}$$

For $t>t_1$, since the ABC kernel is positive,

$${}^{\mathit{ABC}}\mathcal{D}_t^{\psi }V\left(t\right)={\displaystyle \frac{B\left(\psi \right)}{1-\psi }}{\int }_{t_1}^t{E}_{\psi }\left(-{\displaystyle \frac{\psi }{1-\psi }}{(t-\tau )}^{\psi }\right)\dot{V}\left(\tau \right)d\tau \le 0.$$

By LaSalle’s principle, trajectories converge to the largest invariant set where $\dot{V}=0$, which is $\{C=C*,I=I*\}$, forcing $V=V*$, $O=O*$. For the fuzzy inclusion, at each $\alpha$-cut, the maximum possible derivative over selections is attained at extreme points, each corresponding to a crisp system with $\dot{V}\le 0$. Hence, $\dot{V}\le 0$ for all selections, and convergence in $d_H$ follows from convergence of all $\alpha$-cuts. □

5.7. Sensitivity Analysis: Range Exploration

We perform global sensitivity analysis to explore the range of possible outcomes across the fuzzy parameter support (

Table 4. Range of tumor sizes at $T=200$ days when varying each parameter across its support.

Parameter	Min Outcome (Cells)	Max Outcome (Cells)	Range Width (Cells)
${\tilde{r}}_C$	$2.34\times {10}^6$	$8.92\times {10}^6$	$6.58\times {10}^6$
${\tilde{d}}_C$	$3.12\times {10}^6$	$7.45\times {10}^6$	$4.33\times {10}^6$
${\tilde{\rho }}_C$	$3.89\times {10}^6$	$7.01\times {10}^6$	$3.12\times {10}^6$
${\tilde{\alpha }}_{CI}$	$4.12\times {10}^6$	$6.89\times {10}^6$	$2.77\times {10}^6$
${\tilde{\beta }}_{IC}$	$4.45\times {10}^6$	$6.67\times {10}^6$	$2.22\times {10}^6$
${\tilde{K}}_C^0$	$4.78\times {10}^6$	$6.34\times {10}^6$	$1.56\times {10}^6$
${\tilde{D}}_O$	$5.01\times {10}^6$	$6.12\times {10}^6$	$1.11\times {10}^6$
${\tilde{r}}_I$	$5.45\times {10}^6$	$5.67\times {10}^6$	$0.22\times {10}^6$

). For each fuzzy parameter $\tilde{p}$ with triangular support $[p^{-},p^{+},p^{+}]$, we sample $N=500$ points uniformly from $[p^{-},p^{+}]$, compute the corresponding tumor size at $t=200$ days, and record the minimum and maximum outcomes.

6. Numerical Simulations and Discussion

For the numerical implementation of the fractional dynamics in Phase II, we set the scaling parameter $\sigma =1$ day. This ensures that the numerical step size h (which has units of days) remains dimensionally consistent with the biological rates in Table 1. For numerical implementation, the fuzzy system is decomposed into interval-valued equations at each $\alpha$-cut. For ${\mathbb{G}}_1$, this yields:

$$\begin{array}{cc}\hfill {\underline{G}}_1(t;\alpha )& =min\left\{r_{C}c\left(1-{\displaystyle \frac{c}{k_C\left(o\right)}}\right)-d_{C}c-{\alpha }_{CI}CI\right\},\hfill \\ \hfill {\overline{G}}_1(t;\alpha )& =max\left\{r_{C}c\left(1-{\displaystyle \frac{c}{k_C\left(o\right)}}\right)-d_{C}c-{\alpha }_{CI}CI\right\}\hfill \end{array}$$

for

$$\begin{array}{cc}\hfill r_C& \in {\left[{\tilde{r}}_C\right]}_{\alpha },c\in {\left[\tilde{C}\left(t\right)\right]}_{\alpha },d_C\in {\left[{\tilde{d}}_C\right]}_{\alpha },\hfill \\ \hfill {\alpha }_{CI}& \in {\left[{\tilde{\alpha }}_{CI}\right]}_{\alpha },I\in {\left[\tilde{I}\left(t\right)\right]}_{\alpha },o\in {\left[\tilde{O}\left(t\right)\right]}_{\alpha },\hfill \\ \hfill k_C\left(o\right)& =K_C^0\left(1+{\delta }_K{\displaystyle \frac{o^2}{{\theta }_H^2+o^2}}\right),\hfill \\ \hfill K_C^0& \in {\left[{\tilde{K}}_C^0\right]}_{\alpha },{\delta }_K\in {\left[{\tilde{\delta }}_K\right]}_{\alpha },{\theta }_H\in {\left[{\tilde{\theta }}_H\right]}_{\alpha },\hfill \end{array}$$

with the consistency constraint that the same value c is used for all three occurrences of the tumor cell variable.

Analogous expressions hold for ${\mathbb{G}}_2,{\mathbb{G}}_3,{\mathbb{G}}_4$, with ${\mathbb{G}}_5\equiv 0$ for the control case. The minimization and maximization are performed over all parameter and state combinations within the $\alpha$-cut intervals, enforcing variable consistency across repeated occurrences.

For monotonic terms, the extrema occur at vertices:

$$\underset{(\mathbf{p},\mathbf{x})\in \mathcal{V}}{min}{G}_j(\mathbf{p},\mathbf{x}),\underset{(\mathbf{p},\mathbf{x})\in \mathcal{V}}{max}{G}_j(\mathbf{p},\mathbf{x}),$$

where $\mathcal{V}={\prod }_i\{{\underline{p}}_i^{\alpha },{\overline{p}}_i^{\alpha }\}\times {\prod }_k\{{\underline{X}}_k^{\alpha },{\overline{X}}_k^{\alpha }\}$. For non-monotonic components, the extrema are approximated via latin hypercube sampling ($N_s=50$) followed by local refinement.

Discretization. Let $h>0$, $t_m=mh$, $m=0,\dots ,N$, $N=\lfloor {\displaystyle \frac{T}{h}}\rfloor$, $N_1=\lfloor {\displaystyle \frac{t_1}{h}}\rfloor$. For each $\alpha$, denote ${\underline{\mathbb{X}}}_m=\underline{\mathbb{X}}(t_m;\alpha )$, ${\overline{\mathbb{X}}}_m=\overline{\mathbb{X}}(t_m;\alpha )$.

Phase I: Constrained Fuzzy Runge–Kutta 4

$$\begin{array}{cc}\hfill {\underline{S}}_1& =hmin\{{\mathbb{G}}_1(c,i,v,o,t):c\in [{\underline{C}}_m,{\overline{C}}_m],i\in [{\underline{I}}_m,{\overline{I}}_m],v\in [{\underline{V}}_m,{\overline{V}}_m],o\in [{\underline{O}}_m,{\overline{O}}_m],t\in [{\underline{T}}_m,{\overline{T}}_m]\},\hfill \\ \hfill {\overline{S}}_1& =hmax\{{\mathbb{G}}_1(c,i,v,o,t):c\in [{\underline{C}}_m,{\overline{C}}_m],i\in [{\underline{I}}_m,{\overline{I}}_m],v\in [{\underline{V}}_m,{\overline{V}}_m],o\in [{\underline{O}}_m,{\overline{O}}_m],t\in [{\underline{T}}_m,{\overline{T}}_m]\},\hfill \end{array}$$

with similar definitions for ${\underline{S}}_2,{\overline{S}}_2,{\underline{S}}_3,{\overline{S}}_3,{\underline{S}}_4,{\overline{S}}_4$ using intermediate points:

$$\begin{array}{cc}\hfill [{\underline{C}}_{\mathrm{mid}},{\overline{C}}_{\mathrm{mid}}]& =[{\underline{C}}_m,{\overline{C}}_m]+\frac{h}{2}[{\underline{k}}_1,{\overline{k}}_1],\hfill \\ \hfill [{\underline{I}}_{\mathrm{mid}},{\overline{I}}_{\mathrm{mid}}]& =[{\underline{I}}_m,{\overline{I}}_m]+\frac{h}{2}[{\underline{k}}_1,{\overline{k}}_1],\hfill \end{array}$$

where $[{\underline{k}}_1,{\overline{k}}_1]=[{\underline{S}}_1,{\overline{S}}_1]/h$, with analogous expressions for $V,O,T$. Updates preserve nesting:

$${\underline{\mathbb{X}}}_{m+1}={\underline{\mathbb{X}}}_m+{\displaystyle \frac{1}{6}}\left({\underline{S}}_1+2{\underline{S}}_2+2{\underline{S}}_3+{\underline{S}}_4\right),{\overline{\mathbb{X}}}_{m+1}={\overline{\mathbb{X}}}_m+{\displaystyle \frac{1}{6}}\left({\overline{S}}_1+2{\overline{S}}_2+2{\overline{S}}_3+{\overline{S}}_4\right).$$

Phase II: Constrained Fuzzy Adams–Bashforth–Moulton Predictor:

$$\begin{array}{cc}\hfill {\underline{\mathbb{X}}}_{m+1}^P& ={\underline{\mathbb{X}}}_{N_1}+{\sigma }^{1-\psi }{\displaystyle \frac{1-\psi }{B\left(\psi \right)}}{\underline{\mathbb{G}}}_m+{\sigma }^{1-\psi }{\displaystyle \frac{\psi h^{\psi }}{B\left(\psi \right)\mathrm{\Gamma }(\psi +1)}}\sum _{j=N_1}^m{w}_{j,m+1}{\underline{\mathbb{G}}}_j,\hfill \\ \hfill {\overline{\mathbb{X}}}_{m+1}^P& ={\overline{\mathbb{X}}}_{N_1}+{\sigma }^{1-\psi }{\displaystyle \frac{1-\psi }{B\left(\psi \right)}}{\overline{\mathbb{G}}}_m+{\sigma }^{1-\psi }{\displaystyle \frac{\psi h^{\psi }}{B\left(\psi \right)\mathrm{\Gamma }(\psi +1)}}\sum _{j=N_1}^m{w}_{j,m+1}{\overline{\mathbb{G}}}_j,\hfill \end{array}$$

where $w_{j,m+1}={(m-j+1)}^{\psi }-{(m-j)}^{\psi }$, and ${\underline{\mathbb{G}}}_j,{\overline{\mathbb{G}}}_j$ are stored intervals from previous steps.

Corrector:

$$\begin{array}{cc}\hfill {\underline{\mathbb{X}}}_{m+1}& ={\underline{\mathbb{X}}}_{N_1}+{\sigma }^{1-\psi }{\displaystyle \frac{1-\psi }{B\left(\psi \right)}}{\underline{\mathbb{G}}}_{m+1}^P+{\sigma }^{1-\psi }{\displaystyle \frac{\psi h^{\psi }}{B\left(\psi \right)\mathrm{\Gamma }(\psi +2)}}\sum _{j=N_1}^{m+1}{q}_{j,m+1}{\underline{\mathbb{G}}}_j,\hfill \\ \hfill {\overline{\mathbb{X}}}_{m+1}& ={\overline{\mathbb{X}}}_{N_1}+{\sigma }^{1-\psi }{\displaystyle \frac{1-\psi }{B\left(\psi \right)}}{\overline{\mathbb{G}}}_{m+1}^P+{\sigma }^{1-\psi }{\displaystyle \frac{\psi h^{\psi }}{B\left(\psi \right)\mathrm{\Gamma }(\psi +2)}}\sum _{j=N_1}^{m+1}{q}_{j,m+1}{\overline{\mathbb{G}}}_j,\hfill \end{array}$$

with

$$q_{j,m+1}=\left\{\begin{array}{cc}{(m-j+2)}^{\psi +1}-2{(m-j+1)}^{\psi +1}+{(m-j)}^{\psi +1},\hfill & j\le m,\hfill \\ 1,\hfill & j=m+1,\hfill \end{array}\right.$$

where ${\underline{\mathbb{G}}}_{m+1}^P,{\overline{\mathbb{G}}}_{m+1}^P$ are obtained via constrained optimization at the predictor state intervals.

Convergence. Phase I RK4 has local error $O\left(h^5\right)$; Phase II FABM has order $min\{2,1+\psi \}$. The hybrid scheme converges with rate $min\{4,1+\psi \}$. Constrained arithmetic preserves interval nesting,

$${\alpha }_1<{\alpha }_2⟹[{\underline{\mathbb{X}}}^{{\alpha }_1},{\overline{\mathbb{X}}}^{{\alpha }_1}]\supseteq [{\underline{\mathbb{X}}}^{{\alpha }_2},{\overline{\mathbb{X}}}^{{\alpha }_2}],$$

and does not affect the theoretical convergence rates.

Numerical simulations were conducted, and the figures were plotted using the parameters in Table 1 with a time step of $h=0.1$ days over a period of $T=200$ days, with a crossover at $t_1=50$ days. The resulting fuzzy trajectories for cancer cell population ($\tilde{C}\left(t\right)$), immune cell density ($\tilde{I}\left(t\right)$), blood vessel density ($\tilde{V}\left(t\right)$), and oxygen concentration ($\tilde{O}\left(t\right)$) are shown in Figure 1, Figure 2, Figure 3, and Figure 4, respectively.

6.1. Effect of Fractional Order $\psi$ on Model Dynamics

To investigate the impact of memory effects, we performed simulations for different fractional orders $\psi \in \{0.5,0.6,0.7,0.8,0.9\}$ while keeping all other parameters fixed at their modal values.

Table 5. Sensitivity analysis: impact of fractional order $\psi$ on tumor dynamics (incorporating fuzzy parameter uncertainty).

$\mathit{\psi }$	Tumor Size at $\mathit{T}=200$ Days (Cells)			Change from $\mathit{\psi }=0.6$
	Median	5th Perc.	95th Perc.	Median	5th Perc.	95th Perc.
0.5	$3.42\times {10}^6$	$2.15\times {10}^6$	$4.89\times {10}^6$	$-15.8\%$	$-18.9\%$	$-12.3\%$
0.6	$4.06\times {10}^6$	$2.65\times {10}^6$	$5.58\times {10}^6$	$0\%$	$0\%$	$0\%$
0.7	$4.58\times {10}^6$	$3.02\times {10}^6$	$6.21\times {10}^6$	$+12.8\%$	$+14.0\%$	$+11.3\%$
0.8	$5.01\times {10}^6$	$3.34\times {10}^6$	$6.75\times {10}^6$	$+23.4\%$	$+26.0\%$	$+21.0\%$
0.9	$5.34\times {10}^6$	$3.58\times {10}^6$	$7.18\times {10}^6$	$+31.5\%$	$+35.1\%$	$+28.7\%$

Note: Findings based on $n=50$ parameter samples at $\alpha =0$. $t_1=50$ days.

shows how $\psi$ affects the size of a tumor after 200 days when the parameters are not clear. The findings demonstrate that diminished values of $\psi$ (enhanced memory) inhibit tumor proliferation, with $\psi =0.5$ decreasing the median tumor size by 15.8% relative to $\psi =0.6$. On the other hand, higher values of $\psi$ get closer to the classical integer-order behavior. For example, $\psi =0.9$ makes the tumor size grow by 31.5%. The 5th and 95th percentiles show the uncertainty bands getting wider as $\psi$ goes down. This suggests that memory causes parameter uncertainty to have a bigger effect.

Table 6. A comprehensive sensitivity analysis regarding the impact of fractional order $\psi$ on all state variables at $T=200$ days.

$\mathit{\psi }$	Tumor Cells ($\times {10}^6$)		Immune Cells ($\times {10}^6$)		Vessel Density
	Mean	Std Dev	Mean	Std Dev	Mean	Std Dev
0.5	$3.42$	$0.68$	$1.89$	$0.32$	$0.67$	$0.09$
0.6	$4.06$	$0.71$	$1.76$	$0.28$	$0.72$	$0.08$
0.7	$4.58$	$0.78$	$1.68$	$0.25$	$0.76$	$0.07$
0.8	$5.01$	$0.83$	$1.61$	$0.23$	$0.79$	$0.06$
0.9	$5.34$	$0.88$	$1.55$	$0.21$	$0.81$	$0.05$

Note: Oxygen concentration at $T=200$ days: $\psi =0.5$: $1.24$ mmHg, $\psi =0.6$: $1.18$ mmHg, $\psi =0.7$: $1.12$ mmHg, $\psi =0.8$: $1.07$ mmHg, $\psi =0.9$: $1.03$ mmHg.

shows how $\psi$ changes all of the state variables. As $\psi$ increases, tumor cells multiply and immune cells decrease, which means that the immune system is less effective and memory effects are less strong. As $\psi$ goes up, the density of blood vessels goes up, which means that more blood is flowing through them. The body needs more energy, so the amount of oxygen in the air goes down. These findings underscore the imperative for accurate estimation of the fractional order in personalized patient applications.

6.2. Sensitivity Analysis for Transition Time $t_1$

We conducted a sensitivity analysis to evaluate the resilience of model predictions concerning the transition time ($t_1$). We found that changing $t_1$ from 30 to 70 days (in 10-day increments) while keeping $\psi =0.6$ the same caused earlier transitions to lead to earlier angiogenic onset and larger tumor sizes, while later transitions delayed vascularization.

Table 7. Sensitivity analysis: effect of crossover time $t_1$ on tumor dynamics (with fuzzy parameter uncertainty).

${\mathit{t}}_1$ (Days)	Tumor Size at $\mathit{T}=200$ Days (Cells)			Change from ${\mathit{t}}_1=50$ Days
	Median	5th Perc.	95th Perc.	Median	5th Perc.	95th Perc.
30	$5.12\times {10}^6$	$3.45\times {10}^6$	$6.89\times {10}^6$	$+26.1\%$	$+30.2\%$	$+23.5\%$
40	$4.52\times {10}^6$	$2.98\times {10}^6$	$6.12\times {10}^6$	$+11.3\%$	$+12.5\%$	$+9.7\%$
50	$4.06\times {10}^6$	$2.65\times {10}^6$	$5.58\times {10}^6$	$0\%$	$0\%$	$0\%$
60	$3.71\times {10}^6$	$2.41\times {10}^6$	$5.12\times {10}^6$	$-8.6\%$	$-9.1\%$	$-8.2\%$
70	$3.45\times {10}^6$	$2.23\times {10}^6$	$4.78\times {10}^6$	$-15.0\%$	$-15.8\%$	$-14.3\%$

Note: Results based on $n=50$ parameter samples at $\alpha =0$ (full uncertainty). $\psi =0.6$.

quantifies this effect, showing that $t_1=30$ days increases median tumor size by $26.1\%$ compared to the baseline $t_1=50$ days, while $t_1=70$ days reduces tumor size by $15.0\%$. The uncertainty bands become narrower with later transitions, which means that a delayed angiogenic switch lessens the effect of parameter uncertainty.

Table 8. Detailed sensitivity analysis: effect of crossover time $t_1$ on all state variables at $T=200$ days.

${\mathit{t}}_1$ (Days)	Tumor Cells ($\times {10}^6$)		Immune Cells ($\times {10}^6$)		Vessel Density
	Mean	Std Dev	Mean	Std Dev	Mean	Std Dev
30	$5.12$	$0.86$	$1.45$	$0.19$	$0.84$	$0.07$
40	$4.52$	$0.79$	$1.62$	$0.24$	$0.78$	$0.08$
50	$4.06$	$0.71$	$1.76$	$0.28$	$0.72$	$0.08$
60	$3.71$	$0.65$	$1.87$	$0.31$	$0.67$	$0.09$
70	$3.45$	$0.61$	$1.95$	$0.34$	$0.63$	$0.10$

Note: Oxygen concentration at $T=200$ days: $t_1=30$: $0.98$ mmHg, $t_1=40$: $1.08$ mmHg, $t_1=50$: $1.18$ mmHg, $t_1=60$: $1.27$ mmHg, $t_1=70$: $1.34$ mmHg.

shows how all state variables are affected in detail. Later transitions ($t_1=70$ days) lead to a lower tumor burden, a higher number of immune cells ($1.95\times {10}^6$ vs $1.45\times {10}^6$ at $t_1=30$), and a higher oxygen concentration ($1.34$ mmHg vs $0.98$ mmHg). This shows that the immune system is better able to control the situation and that the body’s metabolic needs are lower. These results validate the model’s conclusions as resilient to the particular selection of $t_1$ within a biologically plausible spectrum, while illustrating that postponing the angiogenic switch yields clinical advantages.

We looked at the relative difference between the ABC, Caputo, and CF derivatives to see how changing the definition of the derivative changed the final tumor size at ($T=200$) days. The Caputo derivative made tumors that were about 12–18% bigger than those made by the ABC derivative. This was because its unique kernel was more focused on how things were changing recently. The CF derivative, on the other hand, gave results that were within 5% of the ABC derivative. This means that this system acts in a similar way. These differences show how important it is to choose a kernel based on the biological memory needs of the application.

6.3. Summary of Sensitivity Findings

The sensitivity analyses provide significant insights into model behavior amid ambiguous parameter uncertainty.

Table 9. Summary: key findings from fuzzy sensitivity analysis.

Parameter	Key Findings
Fractional Order $\psi$	Lower $\psi$ (stronger memory) suppresses tumor growth $\psi =0.5$ reduces median tumor size by $15.8\%$ compared to $\psi =0.6$ $\psi =0.9$ increases median tumor size by $31.5\%$ compared to $\psi =0.6$ Uncertainty band width increases as $\psi$ decreases Immune cell population decreases with higher $\psi$ (from $1.89\times {10}^6$ at $\psi =0.5$ to $1.55\times {10}^6$ at $\psi =0.9$)
Crossover Time $t_1$	Earlier transition ($t_1=30$ days) leads to larger tumors ($+26.1\%$) Later transition ($t_1=70$ days) reduces tumor size ($-15.0\%$) Delay in angiogenic switch allows better immune control Oxygen concentration increases with later transition (from $0.98$ mmHg at $t_1=30$ to $1.34$ mmHg at $t_1=70$) Immune cell population increases with later transition (from $1.45\times {10}^6$ at $t_1=30$ to $1.95\times {10}^6$ at $t_1=70$)

shows that lower fractional orders (better memory) and longer crossover times (delayed angiogenic switch) are linked to a smaller tumor burden and better immune regulation.

Table 10. Uncertainty quantification: effect of $\psi$ and $t_1$ on prediction reliability.

Parameter	Uncertainty Band Width at $\mathit{T}=200$ Days (Cells)
	Value	Tumor Size Range	Relative Width
Fractional Order $\psi$ (at $t_1=50$)
$\psi =0.5$	–	$[2.15,4.89]\times {10}^6$	$2.74\times {10}^6$
$\psi =0.6$	–	$[2.65,5.58]\times {10}^6$	$2.93\times {10}^6$
$\psi =0.7$	–	$[3.02,6.21]\times {10}^6$	$3.19\times {10}^6$
$\psi =0.8$	–	$[3.34,6.75]\times {10}^6$	$3.41\times {10}^6$
$\psi =0.9$	–	$[3.58,7.18]\times {10}^6$	$3.60\times {10}^6$
Crossover Time $t_1$ (at $\psi =0.6$)
$t_1=30$ days	–	$[3.45,6.89]\times {10}^6$	$3.44\times {10}^6$
$t_1=40$ days	–	$[2.98,6.12]\times {10}^6$	$3.14\times {10}^6$
$t_1=50$ days	–	$[2.65,5.58]\times {10}^6$	$2.93\times {10}^6$
$t_1=60$ days	–	$[2.41,5.12]\times {10}^6$	$2.71\times {10}^6$
$t_1=70$ days	–	$[2.23,4.78]\times {10}^6$	$2.55\times {10}^6$

Note: Uncertainty bands represent 5th–95th percentile ranges from $n=50$ parameter samples.

shows how the reliability of predictions changes when there is uncertainty about the parameters. As $\psi$ goes down, the width of the uncertainty band grows. It goes from $2.74\times {10}^6$ cells at $\psi =0.5$ to $3.60\times {10}^6$ cells at $\psi =0.9$. This shows that memory effects make the spread of parameter uncertainty worse. The uncertainty bands get smaller as the crossover times get later, going from $3.44\times {10}^6$ cells at $t_1=30$ to $2.55\times {10}^6$ cells at $t_1=70$. This indicates that a postponed angiogenic switch mitigates the impact of parametric imprecision.

Table 11. Correlation analysis: impact of key parameters on final tumor size.

Parameter	Correlation with Final Tumor Size			Sensitivity Rank
	$\mathit{\psi }=\mathbf{0}.\mathbf{6}$, ${\mathit{t}}_{\mathbf{1}}=\mathbf{50}$	$\mathit{\psi }=\mathbf{0}.\mathbf{6}$, ${\mathit{t}}_{\mathbf{1}}=\mathbf{70}$	$\mathit{\psi }=\mathbf{0}.\mathbf{9}$, ${\mathit{t}}_{\mathbf{1}}=\mathbf{50}$
$r_C$ (Proliferation rate)	$+0.82$	$+0.79$	$+0.85$	1
$d_C$ (Apoptosis rate)	$-0.76$	$-0.73$	$-0.78$	2
${\alpha }_{CI}$ (Immune killing)	$-0.68$	$-0.65$	$-0.71$	3
${\beta }_{IC}$ (Activation rate)	$-0.52$	$-0.49$	$-0.55$	4
$K_C^0$ (Carrying capacity)	$+0.45$	$+0.42$	$+0.48$	5
$D_O$ (Oxygen supply)	$-0.38$	$-0.35$	$-0.41$	6
$r_I$ (Immune recruitment)	$-0.31$	$-0.28$	$-0.34$	7

Note: Correlation coefficients calculated from $n=500$ parameter samples. Positive correlation indicates parameter increases tumor size.

displays a correlation analysis identifying the parameters that most significantly influence the tumor’s final size. The proliferation rate $r_C$ exhibits the most significant positive correlation ($+0.82$), while the apoptosis rate $d_C$ demonstrates the most substantial negative correlation ($-0.76$). These correlations endure across different $\psi$ and $t_1$ values, confirming the robustness of these results.

Finally,

Table 12. Clinical implications: recommended parameter ranges for treatment planning.

Parameter	Optimal Range	Clinical Interpretation
Fractional Order $\psi$	0.5–0.6	Patients with stronger memory effects (lower $\psi$) show better response to immunotherapy. Memory strength may be influenced by tumor microenvironment.
Crossover Time $t_1$	60–70 days	Delaying the angiogenic switch (later transition) significantly reduces tumor burden. Anti-angiogenic therapies should be initiated early to maintain avascular state.
Uncertainty Band Width	Narrower with $\psi >0.7$ and $t_1<40$	Predictions are most reliable for high $\psi$ (weak memory) and early crossover. Clinical decisions should account for wider uncertainty in memory-dominated regimes.

turns these math results into clinical advice. The ideal fractional order range is determined to be $0.5–0.6$, indicating that patients exhibiting more pronounced memory effects may have a more favorable response to immunotherapy. The best time range for crossover is between 60 and 70 days. This means that early anti-angiogenic treatments that delay the angiogenic switch can greatly lower the amount of tumor.

The simulated dynamics qualitatively capture recognized biological phenomena such as the angiogenic switch and immune suppression. Stability analysis via fuzzy differential inclusions yields threshold conditions expressed through a fuzzy reproduction number: tumor elimination is indicated when ${\overline{\mathcal{R}}}_0<1$, and persistent disease when ${\underline{\mathcal{R}}}_0>1$. Sensitivity analysis identifies the proliferation rate ${\tilde{r}}_C$ and apoptosis rate ${\tilde{d}}_C$ as the dominant parameters influencing model output. The current framework has some limitations as follows: simulations are limited to a no-treatment control scenario, spatial heterogeneity is ignored, and parameters are derived from pooled estimates rather than individual patient data. The main areas of future research will focus on the calibration of Bayesian parameters, the development of spatial extensions utilizing fuzzy fractional partial differential equations, and the integration of adaptive therapy optimization algorithms.

7. Conclusions

This study established a mathematical fuzzy piecewise fractional derivative framework for the dynamics of cancer-immune-angiogenesis under conditions of uncertainty. The framework includes five components: (i) constrained fuzzy arithmetic that solves the dependency problem in interval-based uncertainty quantification; (ii) well-posedness analysis that uses the Banach fixed-point theorem to show positivity, boundedness, Lipschitz continuity, and existence-uniqueness with verifiable contraction conditions ($L_{\overrightarrow{\mathbb{G}}}=0.0247$); (iii) stability analysis that uses fuzzy differential inclusions to find threshold conditions based on ${\mathcal{R}}_0=[{\underline{\mathcal{R}}}_0,{\overline{\mathcal{R}}}_0]$; (iv) range-based sensitivity analysis that respects the possibilistic nature of fuzzy parameters; and (v) numerical schemes that keep fuzzy number properties through constrained arithmetic.

The methodology creates uncertainty bands that show how parametric imprecision spreads through the system. Simulations display qualitative characteristics aligned with established tumor biology: angiogenic switch, prolonged immune suppression, and hypoxia-induced adaptability. All outcomes are based on mathematical analysis and numerical simulation of parameter values from the existing literature; experimental validation is essential.

The mathematical framework establishes a basis for future enhancements, including the integration of therapeutic interventions, adaptation to patient data, and the incorporation of spatial impacts via partial differential equations.