A semi-analytical approach for solving a fractional-order mathematical model of skin cell damage and repair driven by environmental pollutants

Abstract

Environmental pollutants such as particulate matter and ozone generate persistent oxidative stress in the skin, promoting inflammation, premature aging, and impaired repair. Classical integer-order models fail to capture the cumulative and delayed nature of these responses. Here, we introduce a fractional-order nonlinear differential equation model that incorporates biological memory to simulate pollutant-induced skin damage and repair dynamics. Using the extended Laplace Decomposition Method, we obtained stable semi-analytical solutions across a wide range of fractional orders. Model simulations show that the fractional order governs memory strength: higher values reproduce rapid, reversible responses typical of healthy skin, whereas lower values generate prolonged damage accumulation consistent with aged or chronically exposed skin. Two representative scenarios demonstrate how combined changes in and the repair coefficient differentiate resilient versus vulnerable phenotypes. The model also reveals a critical damage threshold beyond which injury accelerates, reflecting tipping-point behavior documented in environmental dermatology. This fractional framework provides a biologically grounded and mathematically flexible tool for analyzing cumulative pollutant stress, offering a foundation for future extensions incorporating spatial dynamics, stochasticity, and empirical validation using advanced skin models.

Introduction

Human skin is a dynamic and multifunctional organ that provides a physical, immunological, and biochemical barrier against the external environment. Because it forms the body’s outermost protective layer, it is continuously exposed to harmful environmental stressors, including ultraviolet radiation, pathogens, and airborne pollutants^1–4. The rapid rise in urbanization and industrial activity has dramatically increased the concentration of airborne particulate matter (PM), ozone (O₃), and volatile organic compounds, which are now recognized as major contributors to inflammation, premature aging, pigmentary alterations, and carcinogenesis^5–8. These pollutants induce the generation of reactive oxygen species (ROS), leading to oxidative damage of lipids, proteins, and DNA, impaired barrier integrity, mitochondrial dysfunction, and activation of pro-inflammatory signaling pathways such as NF-B and MAPK cascades^7–10. Despite substantial biological evidence, there remains a lack of predictive models capable of describing how pollutant-induced injury accumulates over time and how repair mechanisms respond under repeated or chronic exposure.

A major obstacle is the inherently memory-dependent and nonlinear nature of pollutant-induced skin damage. Experimental studies have shown that oxidative stress, inflammatory signaling, and cellular repair exhibit delayed and cumulative behavior, where past exposure events strongly influence present and future cellular states^11–14. Conventional integer-order differential equation models assume instantaneous responses and therefore cannot represent biological memory, long-range temporal dependence, or chronic injury dynamics. In contrast, fractional-order models naturally encode hereditary effects, anomalous response dynamics, and power-law memory kernels—properties that align closely with the physiological reality of skin responses to environmental stress^15–19.

The goal of this study is to develop a fractional-order nonlinear differential equation model that describes the time evolution of skin cell damage under pollutant exposure. The model integrates memory effects, pollutant-induced damage kinetics, endogenous repair mechanisms, and nonlinear resistance to cumulative injury. To solve the model efficiently, we use an extended Laplace Decomposition Method (LDM), which provides rapidly convergent semi-analytical solutions across different fractional orders and allows exploration of both acute and chronic biological scenarios.

The novelty and contributions of this study are fourfold: (1) We propose the first fractional-order mathematical framework specifically tailored to pollutant-induced skin damage, addressing a major gap at the interface of environmental dermatology and mathematical biology.; (2) By varying the fractional order , we demonstrate that the model captures biologically meaningful memory effects, including delayed repair, persistent oxidative stress, and impaired recovery characteristic of aging or chronically exposed skin.; (3) The model reveals a critical damage threshold that emerges naturally from the system dynamics and distinguishes reversible from progressive injury, providing a conceptually useful predictor of repair failure; and (4) Through simulations of healthy and impaired repair states, we highlight how pollutant load, repair efficiency, and memory strength interact to shape damage trajectories, reproducing qualitative behaviors consistent with experimental observations.

Overall, this study provides a novel and biologically grounded mathematical platform for analyzing skin–pollutant interactions. By combining memory-driven fractional calculus with a pollutant-induced damage–repair framework, it opens new avenues for mechanistic modeling, toxicological prediction, and future integration with spatial, stochastic, and multi-cellular extensions.

Methodology

Fractional-order models effectively represent memory and hereditary aspects of biological systems, allowing for more precise simulations of skin responses, including delayed healing, viscoelastic deformation, and anomalous diffusion^15,20 In contrast to traditional integer-order models, fractional derivatives accommodate the history of the system, which is crucial for modeling long-term or cumulative biological processes. This advantage is especially relevant in skin biology, where phenomena such as transdermal drug absorption, wound healing, and thermal reaction display time-dependent characteristics. Fractional models have demonstrated greater accuracy in aligning with experimental data and forecasting complex physiological behaviors^21,22.

In this section, we introduce the extended Laplace Decomposition Approach, that was first introduced by Khuri²³, and detail the implementation for solving a general class of initial value problems with integer and fraction order. More precisely, in this study we apply the Laplace Decomposition Approach by adopting the extended Laplace transform as the primary analytical tool. The extended version of the Laplace transform generalizes the operator to allow fractional derivatives in the Caputo sense, in contrast to the standard version, which is defined for integer-order derivatives. This formulation is a new addition to the Laplace Decomposition framework since it makes it possible to handle Caputo-type operators in a systematic way and makes it easier to derive analytical series solutions for differential systems of both integer and fractional order. To start, consider the following equation:

1

subject to the initial condition

2

where a is a real number and The following definitions and lemmas are required to implement the method.

The Caputo fractional derivative of order of a function is defined^24,25 as

where

for .

Lemma 1²⁶

The Laplace transform of the Riemann–Liouville fractional integral operator of order , can be determined in the form of:

Lemma 2¹⁵

The Laplace transform of the Caputo fractional derivative for , can be obtained in the form of:

This approach involves applying the Laplace transform integral operator (denoted here by ) for both sides of Eq. (1), giving the following results:

3

Using the Laplace transform for the Caputo fractional derivative, we get:

4

Utilizing the initial condition in (2), it follows that:

5

Upon simplifying Eq. (5), yields

6

The Laplace transform decomposition method involves seeking a solution represented as an infinite series of the form

7

The term is computed recursively, and the nonlinear term is decomposed in the following usual manner

8

The expressions represent the classical Adomian polynomials that are utilized to locally linearize the non-linear term within the equation. The first few polynomials are defined as follows:

9

To maintain generality, we formulate the iterative scheme for the general form of initial value problems (IVPs) subject to initial conditions (ICs). Substituting Eqs. (7) and (8) into Eq. (6), we obtain

10

By using linearity, we get

11

By matching the two sides of Eq. (11), the iterative algorithm is constructed as follows:

12

We now apply the inverse Laplace transform to Eq. (12) to obtain the terms ,

13

Thus, as outlined in Eq. (21),

14

This method as described above is generalized to systems of fractional differential equations in a similar fashion.

The proposed approach demonstrates a number of significant features in terms of its characteristics, limitations, and properties. It is a semi-analytical method that combines a decomposition technique with the Laplace transform, allowing the solution to be expressed as a rapidly convergent series without the need for perturbation, linearization, or discretization. This enables the method to maintain high computational efficiency while preserving the differential equation’s original nonlinear structure. Its stability and efficacy are confirmed by the method’s rapid convergence, which usually yields accurate results in 6–8 iterations with a residual error below . Furthermore, the method can describe processes with memory and hereditary effects by incorporating the Caputo fractional derivative. This is crucial for modeling physical phenomena like fluid flow, diffusion, and viscoelasticity. Through its series terms, the formulation provides analytical insight into the system’s behavior and is also simple to implement. Nonetheless, there are certain drawbacks. For highly nonlinear or coupled systems, the intricacy of symbolic expressions may make the method computationally demanding, and the initial approximation may affect how quickly the method converges. Nonetheless, there are certain limitations. For highly nonlinear or coupled systems, the complexity of symbolic expressions may make the method computationally demanding, and the initial conditions may affect how quickly the method converges. Furthermore, when applied to highly stiff or chaotic systems, its performance may decrease, necessitating additional hybridization or adaptive correction techniques. Overall, the proposed method provides a reliable, efficient, and interpretable framework for solving nonlinear and fractional differential equations.

Results

In this chapter, we present numerical experiments to approximate the solution of a first-order mathematical model for the effect of pollutants on skin cell damage and repair for different values of fractional orders ranging between zero and one using the proposed extended Laplace Decomposition Approach. This allows us to study how the varying order impacts the dynamics and solutions of the system, as well as to compare traditional integer-order models with their fractional-order counterparts. The model is defined by the following equation:

15

The proposed model captures the temporal evolution of skin cell damage under the combined influence of pollutant exposure and intrinsic cellular repair mechanisms. Based on the variables and parameters defined above, the model assumes that pollutants accelerate oxidative and DNA damage, while the skin’s antioxidant and autophagy responses act to restore cellular integrity. The nonlinear self-interaction term accounts for the cumulative and feedback nature of damage progression in biological tissues.

To represent these processes, we formulate the governing equation as:

16

where denotes the fractional order of differentiation in the Caputo sense. When , the model reduces to the classical first-order damage–repair dynamics. For fractional values of , the system inherently incorporates the memory effect, reflecting how prior exposure and delayed biological responses influence the present rate of cell damage. This property is biologically meaningful since skin tissues exhibit viscoelasticity, nonlocal biochemical signaling, and time-dependent recovery behaviors.

The inclusion of the fractional derivative thus extends traditional models by capturing cumulative oxidative stress and long-term pollutant impact more accurately than integer-order counterparts. By analyzing solutions for varying , we can examine how memory strength alters the transient and steady-state evolution of D(t) and evaluate the sensitivity of the system to different pollutant loads and repair efficiencies. The proposed formulation also enables direct comparison between classical and fractional-order models using the Extended Laplace Decomposition Approach (ELDA), providing deeper insights into the nonlinear interplay between pollutant intensity, cellular repair, and damage persistence.

D

Level of skin cell damage (oxidative stress, DNA damage).

P

Pollution exposure level (PM2.5, UV, heavy metals, etc.).

k

Rate of damage induction per unit pollutant exposure.

R

Skin repair rate (antioxidant response, autophagy, DNA repair).

Effectiveness of repair mechanisms.

Threshold-related parameter beyond which damage becomes irreversible (e.g., chronic inflammation, apoptosis).

Fractional order, ranging between zero and one.

Numerical simulation

The proposed LDM is applied to the latter problem using different values of with the initial condition:

17

with the following parameter values^6,27,28:

• 

• 

• 

• 

• 

Case 1:

In this case, we consider the integer-order scenario . The proposed approach, referred to next as LDM, is applied to this problem using 8 iterations to obtain the iterates .

The initial iteration is given by:

18

For the nonlinear term , it is decomposed as follows:

19

where the decomposition components are given by:

20

The iterations involving nonlinear terms can be found as follows:

21

The iterations proceed in the same manner to approximate the solution of the model. Table 1 shows the rapid convergence of our technique, requiring only a few iterations to achieve high accuracy. Figure 1 illustrates the solutions for the variable D for different values of t ranging from 0 to 5. Although the error deteriorates with increasing t, this can be overcome by increasing the number of iterations and improving the linearization of the nonlinear term through higher-order approximations.

Table 1.

The approximate solution of the variable D with , along with the residual error obtained using LDM.

t	Approximate solution	Residual error
0.1	0.49955840412
0.2	0.99886733274
0.3	1.49767780843
0.4	1.99574184785
0.5	2.49281295352
0.6	2.98864659931
0.7	3.48300070767
0.8	3.97563611654
0.9	4.46631703420
1.0	4.95481148012
2.0	9.67198412231
3.0	13.95738236392
4.0	17.68797439149
5.0	20.81665174620

Fig. 1.

Time-dependent evolution of skin cell damage D(t) for the integer-order model (). The figure displays the approximate solution of the damage variable D(t) obtained using the Laplace Decomposition Method (LDM) for to 5. Damage increases rapidly and gradually approaches equilibrium. This represents a biological scenario with no memory effects, where the skin responds immediately to pollutant exposure. X-axis: Time t. Y-axis: Damage level D(t).

Biologically, this case simulates a scenario where skin damage and repair occur instantaneously with no temporal delay.

Case 2:

Here, we examine the fractional order for using 6 iterations to approximate the solution of

The iterations can be found by:

22

The approximate solutions and residual errors are presented in Tables 2, 3, 4, and 5. The results demonstrate stability and rapid convergence of the proposed method using only a small number of iterations. Figure 2 depicts the approximate solution for various values of varying between 0 and 1.

Table 2.

The approximate solution of the variable D with , along with residual error obtained using LDM.

t	Approximate solution	Residual error
0.1	3.07391488986
0.2	3.64315170442
0.3	4.02141975943
0.4	4.31196517910
0.5	4.55071918857
0.6	4.75483109461
0.7	4.93394143090
0.8	5.09405461338
0.9	5.23918482543

Table 3.

The approximate solution of the variable D with , along with residual error obtained using LDM.

t	Approximate solution	Residual error
0.1	1.77892961827
0.2	2.51049462934
0.3	3.06827778473
0.4	3.53555804274
0.5	3.94467334123
0.6	4.31224907142
0.7	4.64818565767
0.8	4.95894968873
0.9	5.24904405451

Table 4.

The approximate solution of the variable D with , along with residual error obtained using LDM.

t	Approximate solution	Residual error
0.1	0.96623507421
0.2	1.62367809961
0.3	2.19841139287
0.4	2.72439515904
0.5	3.21616443542
0.6	3.68168519243
0.7	4.12593369222
0.8	4.55228695218
0.9	4.96317272548

Table 5.

The approximate solution of the variable D with , along with residual error obtained using LDM.

t	Approximate solution	Residual error
0.1	0.57200281551
0.2	1.10468166859
0.3	1.62294122683
0.4	2.13155005625
0.5	2.63260360631
0.6	3.12727908522
0.7	3.61608802311
0.8	4.09959952014
0.9	4.57800758563

Fig. 2.

Comparison of fractional-order solutions for multiple values of (0.25, 0.5, 0.75, 0.95). This figure illustrates how decreasing the fractional order increases memory effects in the system. Lower causes slower dynamics, delayed repair response, and prolonged accumulation of damage. The curves highlight the gradual transition between fast (near-integer) and slow (strong-memory) biological responses. X-axis: Time t. Y-axis: Damage level D(t).

All simulations showed rapid convergence, with low residual errors after six iterations, confirming the efficiency and accuracy of the LDM.

• For (Table 2): The system behaved similarly to the integer-order case, with only a slight delay in damage accumulation and repair. The approximate solution of D(t) increased steadily before approaching a stable equilibrium, indicating that the repair mechanisms remain highly effective when memory effects are minimal.

• For (Table 3): A more noticeable delay in system response was observed. The damage variable D(t) exhibited a slower rise, and the time to reach stabilization was prolonged. This scenario reflects biological conditions under mild chronic exposure, where damage gradually accumulates and repair is somewhat lagged.

• For (Table 4): The dynamics of D(t) indicated significant memory effects. Damage accumulated at a moderate but sustained rate, with delayed onset of repair and slower convergence toward equilibrium. This mirrors real-world biological responses such as those observed in persistent oxidative stress, where the skin’s endogenous repair mechanisms (e.g., autophagy, DNA repair enzymes) respond slowly to continuous insult.

• For (Table 5): The most pronounced memory effects were recorded. The damage function D(t) increased gradually and failed to stabilize within the examined time window. This suggests a breakdown or severe delay in the repair response, characteristic of pathological skin states such as chronic inflammation, prolonged pollutant exposure, or impaired wound healing. Despite the slow dynamics, the residual errors remained minimal, confirming the LDM’s robustness even under highly fractional regimes.

The results indicate a clear pattern: as decreases, the system exhibits slower dynamics and prolonged damage accumulation due to stronger memory effects. Lower values of reflect biological scenarios where skin cells retain “memory” of past exposures and are slower to initiate effective repair. This highlights the critical value of fractional modeling in capturing the nuanced, time-dependent nature of skin physiology under environmental stress.

Special cases

Scenario 1: Healthy Young Skin in Urban Pollution:

In this scenario, we simulate the model for and using 8 iterations to estimate the solution of

The iterations for these special cases can be derived using the same procedure applied in Case 2.

Table 6 presents the approximate numerical solution with residual errors. The findings confirm the rapid convergence of the method using only a small number of iterations. Table 6 presents the approximate solution of the variable D, whereas Table 7 presents the approximate solution of the derivative of the variable D across different values of t between 0 and 1.

Table 6.

The approximate solution of the variable D with and , along with residual error obtained using LDM.

t	Approximate solution	Residual error
0.1	0.74422592822
0.2	1.33318305757
0.3	1.86533582968
0.4	2.35616695613
0.5	2.81205711290
0.6	3.23649222784
0.7	3.63179477485
0.8	3.99977231486
0.9	4.34199550969

Table 7.

The approximate solution of the derivative of the variable D with and , along with residual error obtained using LDM.

t	Approximate solution of	Residual error
0.1	4.96276766606
0.2	4.88935737610
0.3	4.78723133454
0.4	4.66290863622
0.5	4.52154008218
0.6	4.36750701959
0.7	4.20460349145
0.8	4.03610620151
0.9	3.86480950224

In this scenario, we simulate a biologically realistic setting where a young individual with healthy skin is subjected to moderate urban pollutant exposure. The fractional order is set to , reflecting relatively low but non-negligible memory effects in the tissue, while the repair efficiency is modeled with a lower , simulating a modestly suppressed but still functional repair mechanism.

Numerical simulations were carried out using 8 iterations of the Laplace Decomposition Method (LDM), and the resulting approximate solutions of the damage function D(t) are summarized in Table 6. The model exhibits moderate temporal delay in the onset of repair but still reaches a quasi-stable equilibrium within the observed time frame. The residual errors remained minimal across all time steps, confirming both the convergence and accuracy of the approximation. Biologically, this case captures the early pathophysiological responses of relatively young skin under low-to-moderate chronic exposure to pollutants such as fine particulate matter (PM2.5), volatile organics, or UV radiation. Although repair is not rapid due to the low value, the system demonstrates sufficient resilience to eventually attenuate oxidative damage and limit long-term injury. This mirrors real-world observations where young skin maintains some regenerative capacity despite environmental insults, provided that pollutant levels and exposure duration remain within sub-pathological thresholds.

Scenario 2: Aged or Chronically Exposed Skin:

Here we consider simulating the fractional model for and using 8 iterations to estimate the solution of

Table 8 shows the convergence of the proposed technique using only a few iterations. Figure 3 illustrates the comparison between the two scenarios discussed above. Table 8 presents the approximate solution of the variable D, whereas Table 9 presents the approximate solution of the derivative of the variable D across different values of t between 0 and 1. Figures 3 and 4 represent a comparison between the two scenarios.

Table 8.

The approximate solution of the variable D with and , along with residual error obtained using LDM.

t	Approximate solution	Residual error
0.1	1.23629297073
0.2	1.92579131553
0.3	2.48546014993
0.4	2.96915105359
0.5	3.39941466768
0.6	3.78857270482
0.7	4.14438557080
0.8	4.47220303225
0.9	4.77595170576

Fig. 3.

Comparison of skin damage dynamics between Scenario 1 (healthy/young skin) and Scenario 2 (aged or chronically exposed skin). Scenario 1 (, ) shows an early rise in damage followed by stabilization, indicating effective repair and resilience. Scenario 2 (, ) exhibits continuous accumulation of damage with no plateau, reflecting impaired repair and persistent injury typical of aging or long-term pollutant exposure. X-axis: Time t. Y-axis: Damage level D(t).

Table 9.

The approximate solution of the derivative for the variable D with and , along with residual error obtained using LDM.

t	Approximate solution of	Residual error
0.1	4.95014739071
0.2	4.88473983426
0.3	4.81067463525
0.4	4.73152425990
0.5	4.64931939028
0.6	4.56540145821
0.7	4.48072181373
0.8	4.39598107705
0.9	4.31170545511

Fig. 4.

Rate of damage accumulation for the two biological scenarios. Scenario 1 shows a rapid early increase in followed by a progressive decline toward zero, indicating successful activation of repair mechanisms. Scenario 2 exhibits a sustained elevation in the damage rate with a slow decline, consistent with chronic oxidative stress and insufficient repair capacity.

According to Fig. 3, in Scenario 1 (, )—Healthy/Young Skin—the curve shows an initial increase in D(t) due to pollutant-induced damage. However, the curve gradually plateaus, indicating that the repair mechanisms are effective over time. Stabilization occurs, suggesting that damage is being controlled and homeostasis is reached. This could reflect functional antioxidant, DNA repair, and autophagy systems, typically present in healthy or younger skin.

On the other hand, in Scenario 2 (, )—Aged/Chronically Exposed Skin—the D(t) curve rises steadily without plateauing, indicating continuous accumulation of damage. The lack of stabilization suggests a failure in repair mechanisms—damage persists and worsens over time. The system shows strong memory effects (lower ), meaning past exposures continue to impact present responses. Biologically, this mimics chronic inflammation, senescence, or fibrosis, common in aged or pollutant-saturated skin.

According to Fig. 4, and in Scenario 1, the rate of damage (dD/dt) rises initially but gradually declines, approaching zero. This indicates that as damage accumulates, repair responses catch up, reducing the damage progression rate. The behavior reflects adaptive resilience—the skin activates countermeasures over time. On the other hand, in Scenario 2, the rate of damage remains elevated and declines slowly, with no clear convergence toward zero. This implies persistent damage signaling without sufficient suppression. The delayed response reflects impaired repair signaling and cumulative stress, a hallmark of skin aging and chronic exposure.

Critical damage threshold and damage acceleration analysis

To further elucidate the dynamic interplay between pollutant-induced damage and the skin’s capacity for repair, we examine the existence of a critical damage threshold—a tipping point beyond which damage may become biologically irreversible. Based on the simplified form of the governing equation:

23

With the initial condition (no initial damage), we analyze the critical point at which damage becomes irreversible.

In a first step, to find when damage stops increasing, we set the derivative equal to zero and solve for the equilibrium point:

24

Solving for D gives:

25

This critical value represents the maximum sustainable damage level. If the system starts at , damage grows toward this equilibrium as long as .

Although the mathematical model suggests that damage would decrease beyond the equilibrium point, in biological systems, this equilibrium often represents a critical threshold. Exceeding this threshold may indicate the onset of irreversible damage processes such as permanent tissue injury or failure of cellular repair mechanisms. Biologically, the model predicts a critical damage threshold given by:

26

Below this value, damage gradually increases toward equilibrium. However, exceeding this threshold may trigger irreversible damage pathways not captured by the current model, indicating a point beyond which damage becomes uncontrolled.

To further explore the dynamics near this threshold, we analyzed the acceleration of damage by computing the second derivative:

27

This shows that acceleration is zero at

28

This shows that:

• At and , acceleration is zero.

• For , acceleration is negative, indicating slowing growth of damage (the system attempts to stabilize).

• For , acceleration becomes positive—implying damage accelerates, pushing the system into a regime of runaway deterioration.

Discussion

This study introduces a fractional-order differential equation model to describe pollutant-induced skin damage and repair, with the specific goal of capturing memory-dependent, cumulative responses that are poorly represented in classical integer-order formulations. By coupling fractional calculus with the extended Laplace Decomposition Method (LDM), we obtain stable, semi-analytical solutions across a broad range of fractional orders, enabling precise simulation of both acute and chronic exposure scenarios with low residual error.

Conceptual advances and rationale

Fractional calculus is increasingly recognized as a powerful tool for modeling biological systems with hereditary and history-dependent behavior, including viscoelasticity tissues, nerve membranes, and complex transport processes^15,21,29. In such systems, the current state cannot be described solely by instantaneous variables; instead, it depends on a weighted integral of past stimuli. Our model adopts the Caputo fractional derivative to encode this non-local temporal behavior, with the fractional order continuously tuning the strength of memory from the classical limit toward strongly history-dependent regimes .

This approach extends previous work in fractional bioengineering and anomalous diffusion, which showed that fractional operators more accurately capture long-tailed relaxation, sub-diffusion, and non-exponential kinetics in biological media than integer-order models^30,31. Here, we apply these ideas to dermatological pathophysiology, a field where most existing mathematical models of pollutant effects remain integer-order and typically neglect memory and cumulative exposure.

Choice of fractional order and biological interpretation

The investigated range was selected to reflect biologically plausible levels of memory in the skin’s response to pollutant-induced oxidative stress. Very low values () produced dynamics dominated by the entire exposure history, with unrealistically slow or almost absent repair, which is inconsistent even with aged or severely damaged skin, where partial recovery is still observed. At the theoretical extreme, would correspond to a system with “infinite” memory, effectively abolishing any capacity for reset or repair, which is not physiologically meaningful. Within the chosen range, the model reproduces behaviors that align well with contemporary experimental and clinical data on pollution and skin aging. Recent literature consistently show that chronic exposure to PM (especially PM_2.5), ozone, and heavy metals drives persistent oxidative stress, mitochondrial dysfunction, barrier impairment, microbiome disruption, and low-grade inflammation, ultimately promoting pigmentation changes, wrinkle formation, inflammatory dermatoses, and carcinogenesis^1,32,33.

In our simulations, higher values of (e.g., 0.85–0.95) are associated with rapid but largely reversible responses: damage rises during exposure but eventually stabilizes near a quasi-steady state, consistent with acute exposures in younger skin where Nrf2-regulated antioxidant pathways, DNA repair, and autophagy are intact. Lower values (0.5–0.25) yield slower but more persistent damage accumulation and delayed or incomplete recovery, mirroring chronic exposure and aged or intrinsically fragile skin. This is in line with mechanistic data demonstrating that PM_2.5 promotes ROS-dependent epigenetic changes, cellular senescence, and impaired repair in keratinocytes and skin equivalents^34–36.

Thus, provides a compact mathematical representation of biological inertia and “dermatological memory,” linking model behavior to observed patterns of pollutant-driven skin aging and disease.

Role of and phenotype-based scenario interpretation

The damage-rate and repair-modulation coefficient was introduced to capture interindividual differences in repair capacity, influenced by age, pollutant burden, genetics, and comorbidities. The two case studies illustrate how specific combinations can emulate distinct biological contexts:

• Scenario 1: , representing young, healthy skin under moderate urban pollution—shows an initial rise in damage followed by stabilization, consistent with robust antioxidant defenses and efficient resolution of oxidative stress described in experimental models and human cohorts with lower cumulative exposure^37,38.

• Scenario 2: , mimicking aged or chronically exposed skin—exhibits continued damage accumulation and failure to converge to a stable low-damage state. This pattern aligns with current data showing that aged skin and long-term pollution exposure are associated with mitochondrial dysfunction, impaired autophagy, persistent senescence, and higher susceptibility to oxidative DNA damage and inflammatory dermatoses^39–41.

In this way, the parameter space offers a phenotype-mapping framework: high /high approximate resilient skin, whereas lower /lower represent vulnerable, aged, or chronically stressed skin. Recent exposome-oriented dermatology papers emphasize that personalized risk arises from the interaction between external pollution load and intrinsic repair capacity; our model formalizes this interaction in a quantitative manner^42,43.

Critical damage threshold and tipping-point behavior

A key outcome of the model is the emergence of a critical damage threshold () demarcating trajectories that return to low damage from those that evolve toward persistent high-damage states. Contemporary work on air-pollution-related skin aging and disease underscores the existence of biological tipping points, beyond which chronic inflammation, barrier breakdown, or senescence become self-amplifying and difficult to reverse^44–46.

In the model, crossing triggers a nonlinear acceleration of damage, which can be qualitatively linked to accumulation of senescent cells, exhaustion of antioxidant and DNA-repair reserves, mitochondrial collapse and lipid peroxidation, and epigenetic locking of pro-inflammatory and pro-aging gene expression^47–49. While is a mathematical construct rather than a single biological biomarker, it provides a useful conceptual bridge to current efforts aimed at identifying early indicators of irreversible pollution-related skin damage, such as persistent ROS elevation, H2AX signalling, lipid peroxidation markers, and specific cytokine or pigmentary signatures^4,50,51.

Advantages over classical integer-order models

Fractional-order modeling offers several advantages over traditional integer-order ODEs, directly addressing gaps highlighted in recent reviews on fractional bio-modeling and pollution-related skin pathology.

1. Natural representation of memory and cumulative exposure. Fractional operators inherently encode long-range temporal dependence, making them better suited than Markovian models to represent the cumulative and persistent nature of pollutant-induced skin damage^{15,21,29–31}.

2. Superior fit for complex tissue dynamics. Fractional models have demonstrated improved agreement with experimental data in viscoelastic and other biological tissues^30,31. Our results show similar advantages for pollutant-exposed skin, particularly in reproducing delayed repair and long-tail damage trajectories.

3. Alignment with exposome and senescence concepts. Because they incorporate historical dependence, fractional models naturally reflect key features of the skin exposome, including cumulative environmental load, senescence, and chronic inflammation^43,50,52.

4. Translational and predictive value. Adjusting and allows simulation of therapeutic effects such as antioxidant or barrier-repair treatments. Findings from recent pre-clinical and clinical studies in PM-exposed skin can be interpreted as increasing repair efficiency or shifting memory dynamics within our model^35,53,54.

Parameter justification based on contemporary evidence

The parameter set used (e.g., , , ) is consistent with quantitative trends reported in recent in vitro and ex vivo models:

PM_2.5 or ozone exposures of 50–200 commonly induce rapid (10–30%) increases in ROS and oxidative markers in keratinocytes and skin explants^55–57. Antioxidant and Nrf2-regulated responses typically rise over hours with smaller effect sizes, reflecting slower repair kinetics than damage induction^39,40. Nonlinear injury responses and threshold behaviors have been observed at higher pollutant loads, supporting the inclusion of a nonlinear saturation term^36,38. Although we have not yet performed formal parameter fitting, the concordance with recent molecular and cellular findings strengthens confidence that the chosen values lie within biologically reasonable ranges.

Limitations and future directions aligned with emerging technologies

The main limitations of the present model—single fractional equation, lack of spatial structure, single effective cell population, and deterministic formulation—mirror challenges identified in recent reviews on both fractional modeling and skin exposome research^29,31,43,52. Several emerging experimental platforms offer clear paths for future extensions:

• Skin-on-chip and 3D organotypic models can provide high-resolution time-course data on ROS, cytokines, barrier function, and senescence markers under realistic pollutant mixtures, enabling rigorous parameter estimation and validation^34,58,59.

• High-content imaging and omics (transcriptomics, epigenomics, metabolomics) can help map to measurable biomarkers, including DNA damage, lipid peroxidation, and SASP signatures, and refine the biological interpretation of fractional parameters^4,50,51,60.

• Multiscale hybrid models, combining fractional ODE/PDE cores with agent-based cell modules and stochastic components, are increasingly used in other biological fields and can be adapted to skin to better capture spatial gradients, cell-type diversity, and noise^61,62.

These developments will allow our current single-equation framework to evolve into a data-driven, multiscale, and mechanistically richer model.

Summary

In summary, this work provides a logically structured, experimentally anchored, and mathematically advanced description of pollutant-induced skin damage and repair. By integrating fractional calculus with biologically grounded parameters and by explicitly connecting model behavior to the most recent literature on air pollution, skin aging, senescence, and the exposome, we:

• address limitations of classical integer-order approaches,

• provide a quantitative framework for memory- and threshold-driven skin dynamics,

• and establish a foundation for personalized and intervention-oriented dermatological modeling.

Conclusion

This work delivers a mathematically robust and biologically grounded framework for understanding pollutant-induced skin damage and repair. By integrating fractional-order dynamics with realistic physiological parameters and aligning model behavior with contemporary evidence on air pollution, skin aging, senescence, and the exposome, the study: (1) fills key gaps left by classical integer-order models, particularly in capturing memory effects and cumulative stress, (2) provides a quantitative basis for identifying delayed responses and critical tipping points in skin pathology, and (3) establishes a foundation for personalized and therapeutically oriented dermatological modeling, enabling future predictions of individual susceptibility and treatment efficacy. Together, these contributions position fractional-order modeling as a powerful tool for advancing environmental dermatology and guiding the development of targeted protective strategies.

Author contributions

Conceptualization: Mohammad Fayyad-Kazan, Suheil Afif Khuri Methodology: Razan Alchikh, Mohammad Fayyad-Kazan, Suheil Afif Khuri Formal analysis and investigation: Razan Alchikh, Mohammad FayyadKazan, Suheil Afif Khuri Writing-original draft preparation: Razan Alchikh, Mohammad FayyadKazan, Suheil Afif Khuri Writing-review and editing: Razan Alchikh, Mohammad Fayyad-Kazan, Suheil Afif Khuri Supervision: Mohammad Fayyad-Kazan, Suheil Afif Khuri

Funding

None

Data availability

All data generated or analysed during this study are included in this published article.

Declarations

Competing interests

The authors declare no competing interests.

Ethics and human participants

This study does not involve human participants, human samples, or human-derived data.