https://orcid.org/0000-0003-4316-5895
Figures
Abstract
This study presents a novel approach to modeling breast cancer dynamics, one of the most significant health threats to women worldwide. Utilizing a piecewise mathematical framework, we incorporate both deterministic and stochastic elements of cancer progression. The model is divided into three distinct phases: (1) initial growth, characterized by a constant-order Caputo proportional operator (CPC), (2) intermediate growth, modeled by a variable-order CPC, and (3) advanced stages, capturing stochastic fluctuations in cancer cell populations using a stochastic operator. Theoretical analysis, employing fixed-point theory for the fractional-order phases and Ito calculus for the stochastic phase, establishes the existence and uniqueness of solutions. A robust numerical scheme, combining the nonstandard finite difference method for fractional models and the Euler-Maruyama method for the stochastic system, enables simulations of breast cancer progression under various scenarios. Critically, the model is validated against real breast cancer data from Saudi Arabia spanning 2004-2016. Numerical simulations accurately capture observed trends, demonstrating the model’s predictive capabilities. Further, we investigate the impact of chemotherapy and its associated cardiotoxicity, illustrating different treatment response scenarios through graphical representations. This piecewise fractional-stochastic model offers a powerful tool for understanding and predicting breast cancer dynamics, potentially informing more effective treatment strategies.
Citation: Aldwoah K, Louati H, Eljaneid N, Aljaaidi T, Alqarni F, Elsayed A (2025) Fractional and stochastic modeling of breast cancer progression with real data validation. PLoS ONE 20(1): e0313676. https://doi.org/10.1371/journal.pone.0313676
Editor: Ahmad Al-Omari, Yarmouk University Hijjawi Faculty for Engineering Technology, JORDAN
Received: August 8, 2024; Accepted: October 29, 2024; Published: January 10, 2025
Copyright: © 2025 Aldwoah et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript and its Supporting information files.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
Breast cancer is a complex and diverse illness that originates from the uncontrolled growth of cells in the breast tissue. It is one of the most common cancers among women worldwide [1]. In 2004, the World Health Organization (WHO) ranked breast cancer as the second most common cancer worldwide. As per WHO statistics, breast cancer affects 8 to 9 percent of women globally. Global health researchers remain uncertain about the precise causes of breast cancer [2]. In 2020, breast cancer impacted 2.3 million women worldwide and led to 685,000 deaths [3]. In 2022, breast cancer caused 670,000 deaths globally [4]. Certain factors increase the risk of breast cancer such as tobacco use, age, family history and genetics, hormonal factors, obesity, alcohol consumption, and radiation exposure. Common symptoms of breast cancer include a lump in the breast, changes in breast size and shape, skin changes on the breast, nipple discharge, inverted nipple, and redness of the breast skin. For prediction and forecasting of breast cancer, mathematicians develop several mathematical models in the literature [5–7].
Over time, there have been significant advancements in understanding the diagnosis, development, and cure of breast cancer. Despite significant progress in understanding breast cancer over the past 50 years, it remains a big public health concern worldwide. Breast cancer is the most frequently occurring invasive cancer affecting women worldwide [8]. Breast cancer ranked as the most commonly diagnosed cancer in Saudi Arabia in 2018, following leukemia as the second-leading cause of death [9]. While Saudi Arabia has historically had a lower incidence of breast cancer compared to different Western nations, increasing evidence suggests a rapid rise in the incidence rates. Investigating breast cancer features, fashions, age distributions, and regional differences is considered a leading research area in developing countries like Saudi Arabia, where over 70 percent of women are below 39 years old.
In the paper [10], the authors have formulated the mathematical model for breast cancer. As per the medical records, admitted patients are categorized into subgroups ranging from stage 1 to stage 4. The population is categorized into five subclasses:
(1)
with initial values X > 0 B ≥ 0, C ≥ 0, R ≥ 0 and E ≥ 0. The class X denotes the patients hospitalized due to the impact of cancer who have stage 1, the class B describes individuals identified as patients with stage 3X and stage 3B, the class C contains the people who undergo testing and are diagnosed as cancer patients, the class R denotes individuals who are in disease-free states following chemotherapy treatment and the class E represents cancer patients experiencing cardiotoxic effects.
In recent years, fractional calculus (FC) has received tremendous attention of researchers due to several applications in different fields of science and engineering [11]. FC has been used in the analysis of physical system which occurs in mathematical biology [12, 13], physics and engineering [14] and many more [15, 16]. Scientists have been actively exploring novel fractional operators to tackle challenges related to memory, locality, and singularity. The Caputo fractional derivative (CFD) has proven to be a highly valuable operator for representing non-local behaviors through fractional differential equations. In reference [17], the authors proposed a novel hybrid fractional operator by innovatively combining the concepts of the CFD and the proportional derivative (CPC operator). This hybrid fractional operator is designed as a linear combination of the CFD and the Riemann–Liouville integral of non-integer order. Only few works have been published using CPC operator [18, 19].
Disease often exhibit random behaviour due to various factors such as genetic mutations, environmental influences, and stochastic processes within biological systems, Stochastic differential equations (SDEs) allow for the inclusion of stochastic components, making them suitable for capturing the inherent uncertainty in disease dynamics. Several researchers have used SDEs for modelling various kinds of diseases such as COVID-19 [20], HIV [21] and some others [22, 23].
Recently, Abdon and Seda [24] introduced a hybrid non-local operator based on a piece-wise (PW) function approach. This PW operator involves dividing the interval into two or more sub-intervals, and the authors have proposed various types of PW operators by combining different operators within these sub intervals. The PW operator is specially used for analysis of crossover behaviour in the dynamics of a physical system. Some disease models have been investigated using PW operators [25–27]. Motivated by works on PW operator, we aim to study the considered breast cancer model under PW. Consider the considered model under PW operator as:
(2)
where
represents a PW differential operator, which will be defined later section for three subintervals. In this paper, we utilize the PW operator in a unique manner, employing the CPC operator in the first two sub-intervals with constant and variable order, respectively. The stochastic operator in the other sub interval, to analyze random behaviour the proposed model 1. This type of pattern is used only for COVID-19 model [28]. The PW formulation of the model is expressed as follows:
(3)
with initial values is
(4)
For variable order, we can write the above model 3 as given below:
(5)
with initial values is
The stochastic version of system 2 is expressed by:
(6)
with initial values is
in the above model
were the densities and
are the noise intensities, while H⋆ is the Hurst index.
2 Basic concepts
Presently, we define some basic concepts which will be use in our proposed work.
Definition 2.1 [11] Suppose is a function which is continuous given by Ψ = [ω, ν], −∞ < ω < ν < +∞, θ ∈ C, R(θ) > 0. The Riemann-Liouville derivative is defined under, both left and right for order θ:
(7) where
.
Definition 2.2 [11] Suppose is a function which is continuous given by Ψ = [ω, ν], −∞ < ω < ν < +∞, θ ∈ C, R(θ) > 0. The Riemann-Liouville integrals are defined under, both left and right for order θ:
(8) where 0 < θ < 1.
Definition 2.3 [11] Let be function in C, then the Caputo derivative for order θ is defined below,
(9) where
.
Definition 2.4 [17] Let the Caputo proportional Fractional Hybrid operator is presented by (CP) then (CP) is defined as;
(10)
where
, and 0 < θ < 1.
Definition 2.5 [17] Let the Fractional hybrid operator along with Caputo proportional Constant is presented by (CPC), and is defined as;
(11)
where Z0(θ) = θ Q1−θ, Z1(θ) = (1 − θ)Qθ, are kernels and Q is constant also 0 < θ < 1.
Definition 2.6 [17] Fractional Caputo proportional operator (CP) for variable order as defined as:
(12)
where
, and
. Consequently, the variable order CPC can be written as:
(13) where
, and Q is constant also
.
3 Theoretical and numerical analysis
We prove some results for the existence and uniqueness of the proposed model 2. We also present a numerical scheme for this model (2).
3.1 The existence and uniqueness of model 3 and 5
In this section, we elaborate on the existence and uniqueness of considered models 3 and 5. So we will prove the perovs theorem discussed in [29]. To prove this theorem first we will discuss some basic concepts concerning this theorem.
Definition 3.1 Let K be a vector space with field V, and it be C or R. In this context, someone defined a function (generalized norm) on K as:
having the following properties:
(i) For all Φ ∈ K; if , then Φ = 0K.
(ii) ||b||M = |b|||Φ||M ∀ Φ ∈ K and b ∈ V, and
(iii) ||Φ + λ||M ≤ ||Φ||M + ||λ||M for all Φ, λ ∈ K.
In the above the generalized norm space is (K, ||.||M). Also if the given metric valued space is complete, then the space (K, ||.||M) will form generalized Banach space, having the property ωM(Φ1, Φ2) = ||Φ1 − Φ2||G.
Definition 3.2 Suppose (K, λM) be a generalized metric space and let P be a mapping from K → K, then the operator as known as a Λ contraction along with matrix from Gn×n(R+) which goes to 0n, suggested that, ∀ ω, μ ∈ K, the following holds:
The result given below is the extension of Banach contraction principle.
Theorem 1 [29] Suppose K is a full generalized metric space also P: K → K is a M-contraction operator, then, P will have a single fixed point belong to K. Further, the models 3 and 5 can be written in the following form,
(15)
in the above equation,
is vector which is equal to
also Z operator is given below,
(16)
Let
be a generalized Banach space if we define it along with the generalized norm given below:
(17)
Now we have to prove that the models 3 and 5 possess unique solutions, so we are shifting them into fixed point problems having the operator Q, which is defined as ,
(18)
Further, we need to prove the following lemma.
Lemma 3.1 Let Λ is a vector belongs to R5 and fulfill the following conditions:
(19)
So, Z maps U(Y0, Λ) ⊂ K into itself, where U(Y0, Λ) is a generalized ball. Further we prove that Z is G-Lipschitz.
Proof 3.1 Suppose Y ∈ U(Y0, Λ), then
(20)
Hence the proof.
Lemma 3.2 The operator defined in theorem 1 Z is M lipschitz i.e
satisfying the following condition:
where Θ is a square matrix.
Similarly,
writing the above equation in matrix form, we get:
(21)
where
(22)
Theorem 2 Let us consider the matrix (23) and it tends to O5, so system 3 and 5 will have a unique solution and the solution will be true for all
in
.
Proof 3.3 Moreover, for any Y, , further with the help of lemma 3.2, we have:
Where
tends to 0, and the operator fulfills the criteria of G-contraction, so by using theorem 1 which is known as perov’s fixed point theorem, the systems 3 and 5 has only one solution lies in
. Hence, the proof is complete.
4 Existence and uniqueness of solution for our stochastic model 6
Theorem 4.1 We can find a non negative solution
for our proposed model with in the interval
along with initial values
and the solution will be in
having probability one.
Proof 4.1 If the coefficients of the model (6) fulfil the Lipschitz condition, a non-negative solution be existent for the consider model over the time interval
, where
denotes the time of occurrence. The initial value is given by
.
In order to inaugurate the global nature of the solution for (6), it is essential to reveal that Te = ∞, demonstrating an infinite time limit without an explosion.
Given
as a non-negative real number sufficiently large to guarantee that all initial values lie within the interval
, we can establish a stopping time for
.
(24)
We assume that inf(⌀) = ∞, where ⌀ represents the empty set. The notation is defined as increasing as
. Additionally, we suppose that
so
. Our goal is to demonstrate that for all
, T∞ = ∞. This indicates that Te = ∞, and simultaneously confirms that
. If our supposition happens to be false, it implies the existence of a F > 0 and ξ ∈ (0, 1) so that:
(25)
Further, let’s take
, which is a function belonging to the C2 space, defined as:
Utilizing the inequality
for all
, we can demonstrate that
. Now, assuming
and F ≥ F0, applying Ito’s results to Eq (26), we obtain:
(26)
From the above equation, we obtain:
The significance of
is that it remains constant, as it does not depend on any state or variable.
(28)
Applying integration to the above Eq (28), we obtained
(29)
Let
be well-defined as the set of all ϖ in Φ such that
. This condition holds for
. Eq 25 can then be uttered as
. It can be shown that, for any specific value of ϖ, there exists at least one corresponding
,
,
,
or
that is equal to either
or
.
(30)
In the light of Eqs (29) and (30), we have
The function
is certain about Φ(t) and functions as an indicator function. In the situation where
approaches infinity, then
(31)
Here a contradiction arises, leading to
, the proof can be considered complete, given the context of the argument. This signifies a critical point where further logical progression may be impeded, leading to the fulfillment of the proof’s objectives.
5 Numerical analysis
In this portion, we propose a numerical scheme based on CPC Grunwald-Letnikov nonstandard FDM. This scheme is for fractional and variable order, is given below:
(32)
(33)
(34)
in the above equation
represent the Brownian motion and ς is the intensity while H⋆ represent Hurst index. The system 3 can be written as:
(35)
here, V0(θ) and V1(θ) both are kernels which is depending on θ also V0(θ) = θ Q1−θ, V1(θ) = (1 − θ)Qθ, where Q represent a constant, and ϖ0 = 1. Applying GLNFDM approach, 35 can be discretized as bellow:
where:
(36)
Furthermore, Eq 32 can also be discretized as bellow:
(37)
in the above expression,
represent a natural number.
,
, where
. Further, we extend our work according to the assumption given in [28]
So, if V0(θ) = 1 and V1(θ) = 0 in Eq 37 then one can easily do the discretization for Caputo operator using finite difference method.
Now discretizing Eq 33 we get:
(38)
here,
. Moreover we discretized Eq 34 by using Euler-Maruyama method as given below:
(39)
6 Numerical illustrations
This section illustrates the visualization of the numerical findings using the proposed approach of the model outlined in Eq (6), showcasing a range of fractional orders alongside the modulation of key parameters. The time span is set from 0 to 100, with a step size of dt = 0.01. Here the interval is sub divided as [0, 30), [30, 60) and [60, 100], where the first one presents the fractional dynamics while the second interval presents the dynamics with variable order and the third interval presents the stochastic dynamics. The initial values are assumed to follow the pattern X(0) = 30000, B(0) = 12300, C(0) = 738, R(0) = 334, E(0) = 10. The parameters outlined in Table 1 are employed throughout the simulations. The behavior of state compartment X is visualized in Fig 1. It’s noticeable that the number of individuals admitted to hospitals and diagnosed with cancer is on the rise. The dynamics of state compartment B, is depicted in Fig 2. We notice in Fig 2, that in those people suffering from cancer with stages 3X and 3B, there’s a rapid initial decrease in their number of individuals, followed by a steady increase over time. Moreover, Figs 3 and 4 depict the evolution of individuals in classes C and R, respectively. From Fig 3, the trend shows a continuous rise in individuals with stage 4 cancer over time until t = 20, then we see that the number is large when θ(t) is large. Additionally, Fig 4 illustrates a growing number of individuals achieving disease-free status through chemotherapy as time progresses. In similar way, Fig 5 shows the dynamics of E, it’s evident that the population of cardiotoxic patients is steadily increasing over time which increases more rapidly in the second interval followed by stochastic dynamics.
Furthermore, the behavior of X is depicted in Fig 6 with the variations in fractional order θ. It’s noticeable that the number of individuals admitted to hospitals and diagnosed with cancer increases as fractional order decreases. The dynamics of B, is visualized in Fig 7. We see in Fig 7, that those people affected by cancer with stages 3X and 3B, there’s a rapid initial decrease in their number of individuals, then increase with time. Moreover, the Figs 8 and 9, depict the evolution of the individuals in C and R, respectively. From Fig 8, the trend shows a continuous rise in individuals with stage 4 cancer over time, then we see that there are more individuals when θ is small. Additionally, Fig 9 illustrates a growing number of individuals achieving disease-free status through chemotherapy as time progresses. In similar fashion, Fig 10 shows the dynamics of E, it’s evident that the population of cardiotoxic patients is steadily increasing over time which increases slightly faster in the second interval followed by stochastic dynamics.
Fig 11 illustrates the comparison between real and simulated results, as examined in [31]. The time is considered to be [0, 12] with step size 0.01. The time interval is sub-divided as [0, 3), [3, 5) and [5, 12], where the first one presents the fractional dynamics with fractional order 0.9 while the second interval presents the behavior with variable order as variable order 0.01 − 0.001 * cos(t), and the third one shows stochastic dynamics. The data spans a period of 12 years, covering the years 2004 to 2016. The comparison highlights a significant overlap between the data points and the simulated results exhibiting stochastic behavior. This suggests that these operators possess the capability to provide improved predictions for cancer disease.
6.1 Parameters’ influence on the dynamics of the proposed model
This section of the current study is dedicated to examining the impact of various parameters on distinct state variables within the system. The objective is to ascertain whether augmenting specific parameters correlates with an increase in the population count within a particular class or if it yields an alternative effect. In the subsequent simulations, a fractional order of θ = 0.9 is utilized along with variable order as variable order 0.01 − 0.001 * cos(t). Henceforth, our initial focus revolves around exploring a range of values for the recovery rate during stages 1 and 2, attributed to chemotherapy, denoted as ϕ1. We examine values of 0.03, 0.06, 0.09, and 0.12 for ϕ1. From the figures, it is evident that as more individuals recover during stages 1 and 2, there is a notable decrease in the number of patients, highlighting the significance and efficacy of this outcome. Figs 12 and 13, demonstrate the number of people in X and B is decreased as the ϕ1 is increased. In same way, Figs 14–16, depicts that the individuals in the state variables ,
and
increases with increase in ϕ1. These figures illustrate that augmenting ϕ1 leads to an increase in individuals afflicted with stage 4, while concurrently resulting in a higher count of individuals recovering from the disease. Further, the impact of the rate of recovery at stage 4 with chemotherapy is investigated. For ϕ3, the values are utilized to be 0.03, 0.07, 0.11, 0.15. As depicted in Fig 17, a decrease in the number of individuals afflicted with stage 4 cancer is evident with an increase in chemotherapy at stage 4, particularly noticeable when the value of ϕ3 is elevated, as demonstrated in Fig 17. Furthermore, it has been observed that as chemotherapy is intensified at stage 4, there is a corresponding increase in the number of individuals who recover from the disease Fig 18. Now we show the dynamics of the influence of cardiotoxicity caused by intensive chemotherapy γ1 on B and R. From Fig 19, it can be seen that an increase in γ1 directly decreases the population in the presented state variables. The influence of increased cardiotoxicity in patients at stage 4 chemotherapy γ2, is demonstrated in Fig 20 with values considered as 0.05, 0.10, 0.15, 0.20. Figs 21 and 22 show that as γ2 increases, the number of individuals in C decreases, while the population and E increase, indicating that fewer people will recover from the disease. Cancer therapy has advanced significantly in recent years, substantially increasing the cure rate and preventing recurrences of breast cancer. However, the risk of cardiotoxicity reduces the usage of these medications. Cardiotoxicity, one of the most important side impacts of chemotherapy, substantially increases both mortality and morbidity rates, contributing to the slight increase observed in Fig 22.
7 Conclusion
Many operators have been defined in the literature to depict the dynamics of physical system. The PW operator which is helpful in the analysis of cross over dynamics of disease model, has been proposed very recently by Abdon and Seda. When the disease shows deterministic behaviour in one interval and stochastic dynamics in the other interval, the PW operator can be helpful for analysis of such types of behaviours. In this work, breast cancer, one of the most dangerous diseases affecting women, has been investigated using a piecewise differential operator. The considered operator has been divided into three sub-intervals. In the first sub-interval, a constant Caputo proportional operator (CPC) with a constant order has been applied. In the second sub-interval, the CPC operator has been used with a variable order. In the third sub-interval, a stochastic operator has been used to examine the stochastic dynamics of cancer cells. The study has had two primary focuses: theoretical and numerical analysis. For the theoretical analysis of the fractional order system with constant and variable fractional orders, fixed point theory has been employed to explore the existence and uniqueness of solutions in the first two sub-intervals. For the stochastic system, Ito calculus and nonlinear analysis have been used to demonstrate the existence of solutions in the third sub-interval. In terms of numerical analysis, the nonstandard finite difference technique has been applied to the fractional order models, and Euler-Maruyama method has been used for the stochastic system. Numerical simulations have been performed for the proposed operators, and the results have been compared with real data from Saudi Arabia covering the period from 2004 to 2016. Additionally, the effects of chemotherapy and cardiotoxicity on breast cancer cells in different scenarios are illustrated through graphs.
Supporting information
S1 File. The Supplementary data to this article is attached as a word file.
https://doi.org/10.1371/journal.pone.0313676.s001
(DOCX)
Acknowledgments
The authors wish to extend their sincere gratitude to the Deanship of Scientific Research at the Islamic University of Madinah.
References
- 1. Feng Yixiao, Spezia Mia, Huang Shifeng, Yuan Chengfu, Zeng Zongyue, Zhang Linghuan, et al. “Breast cancer development and progression: Risk factors, cancer stem cells, signaling pathways, genomics, and molecular pathogenesis.” Genes & diseases 5, no. 2 (2018): 77–106. pmid:30258937
- 2.
https://www.who.int/news-room/fact-sheets/detail/breast-cancer.
- 3. Sedeta Ephrem Tadele, Jobre Bilain, and Avezbakiyev Boris. “Breast cancer: Global patterns of incidence, mortality, and trends.” (2023): 10528–10528.
- 4.
https://www.bcrf.org/breast-cancer-statistics-and-resources.
- 5. Miniere Hugo JM, Lima Ernesto ABF, Lorenzo Guillermo, Hormuth David A. II, Ty Sophia, Brock Amy, et al. “A mathematical model for predicting the spatiotemporal response of breast cancer cells treated with doxorubicin.” Cancer Biology & Therapy 25, no. 1 (2024): 2321769. pmid:38411436
- 6. Wei Hsiu-Chuan. “Mathematical modeling of tumor growth and treatment: Triple negative breast cancer.” Mathematics and Computers in Simulation 204 (2023): 645–659.
- 7. Davenport Angelica A., Lu Yun, Gallegos Carlos A., Massicano Adriana VF, Heinzman Katherine A., Song Patrick N., et al. “Mathematical model of triple-negative breast cancer in response to combination chemotherapies.” Bulletin of Mathematical Biology 85, no. 1 (2023): 7.
- 8. Houghton Serena C., and Hankinson Susan E. “Cancer progress and priorities: breast cancer.” Cancer epidemiology, biomarkers & prevention 30, no. 5 (2021): 822–844. pmid:33947744
- 9. Alqahtani Wedad Saeed, Almufareh Nawaf Abdulrahman, Domiaty Dalia Mostafa, Albasher Gadah, Alduwish Manal Abduallah, et al. “Epidemiology of cancer in Saudi Arabia thru 2010–2019: a systematic review with constrained meta-analysis.” AIMS public health 7, no. 3 (2020): 679. pmid:32968686
- 10. Alzahrani Ebraheem, El-Dessoky M. M., and Khan Muhammad Altaf. “Mathematical model for understanding the dynamics of cancer, prevention diagnosis, and therapy.” Mathematics 11, no. 9 (2023): 1975.
- 11.
Podlubny Igor. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution, and some of their applications. Elsevier, 1998.
- 12. Pinto Carla MA, and Carvalho Ana RM. “A latency fractional order model for HIV dynamics.” Journal of Computational and Applied Mathematics 312 (2017): 240–256.
- 13. Carvalho Ana RM, Pinto Carla MA, and de Carvalho João M. “Fractional model for type 1 diabetes.” Mathematical modelling and optimization of engineering problems (2020): 175–185.
- 14.
Chen Wen, Sun HongGuang, and Li Xicheng. Fractional derivative modeling in mechanics and engineering. Springer Nature, 2022.
- 15. Xu Changjin, Zhao Yingyan, Lin Jinting, Pang Yicheng, Liu Zixin, Shen Jianwei, et al. “Bifurcation investigation and control scheme of fractional neural networks owning multiple delays.” Computational and Applied Mathematics 43, no. 4 (2024): 1–33.
- 16. Xu Changjin, Farman Muhammad, and Shehzad Aamir. “Analysis and chaotic behavior of a fish farming model with singular and non-singular kernel.” International Journal of Biomathematics (2023): 2350105.
- 17. Baleanu Dumitru, Fernandez Arran, and Akgül Ali. “On a fractional operator combining proportional and classical differintegrals.” Mathematics 8, no. 3 (2020): 360.
- 18. Xu Changjin, Farman Muhammad, Liu Zixin, and Pang Yicheng. “Numerical approximation and analysis of epidemic model with constant proportional Caputo operator.” FRACTALS (fractals) 32, no. 02 (2024): 1–17.
- 19. Günerhan Hatıra, Dutta Hemen, Dokuyucu Mustafa Ali, and Adel Waleed. “Analysis of a fractional HIV model with Caputo and constant proportional Caputo operators.” Chaos, Solitons & Fractals 139 (2020): 110053.
- 20. Xu Changjin, Pang Yicheng, Liu Zixin, Shen Jianwei, Liao Maoxin, and Li Peiluan. “Insights into COVID-19 stochastic modelling with effects of various transmission rates: simulations with real statistical data from UK, Australia, Spain, and India.” Physica Scripta 99, no. 2 (2024): 025218.
- 21. Rao Feng, and Luo Junling. “Stochastic effects on an HIV/AIDS infection model with incomplete diagnosis.” Chaos, Solitons & Fractals 152 (2021): 111344.
- 22. Omame Andrew, Abbas Mujahid, and Din Anwarud. “Global asymptotic stability, extinction and ergodic stationary distribution in a stochastic model for dual variants of SARS-CoV-2.” Mathematics and Computers in Simulation 204 (2023): 302–336. pmid:36060108
- 23. Din Anwarud. “The stochastic bifurcation analysis and stochastic delayed optimal control for epidemic model with general incidence function.” Chaos: An Interdisciplinary Journal of Nonlinear Science 31, no. 12 (2021). pmid:34972335
- 24. Atangana Abdon, and Araz Seda İğret. “New concept in calculus: Piecewise differential and integral operators.” Chaos, Solitons & Fractals 145 (2021): 110638.
- 25. Zeb Anwar, Atangana Abdon, Khan Zareen A., and Djillali Salih. “A robust study of a piecewise fractional order COVID-19 mathematical model.” Alexandria Engineering Journal 61, no. 7 (2022): 5649–5665.
- 26. Ahmad Shabir, Yassen Mansour F., Alam Mohammad Mahtab, Alkhati Soliman, Jarad Fahd, and Riaz Muhammad Bilal. “A numerical study of dengue internal transmission model with fractional piecewise derivative.” Results in Physics 39 (2022): 105798.
- 27. Saifullah Sayed, Ahmad Shabir, and Jarad Fahd. “Study on the dynamics of a piecewise tumor–immune interaction model.” Fractals 30, no. 08 (2022): 2240233.
- 28. Alalhareth Fawaz K., Al-Mekhlafi Seham M., Boudaoui Ahmed, Laksaci Noura, and Alharbi Mohammed H. “Numerical treatment for a novel crossover mathematical model of the COVID-19 epidemic.” AIMS Mathematics 9, no. 3 (2024): 5376–5393.
- 29. Perov A. I. “On the Cauchy problem for a system of ordinary differential equations.” Pviblizhen. Met. Reshen. Differ. Uvavn 2, no. 1964 (1964): 115–134.
- 30.
Key Statistics for Breast Cancer. Available online: https://www.cancer.org/cancer/breast-cancer/about/how-common-is-breastcancer.html.
- 31. Albeshan Salman M., and Alashban Yazeed I. “Incidence trends of breast cancer in Saudi Arabia: A joinpoint regression analysis (2004–2016).” Journal of King Saud University-Science 33, no. 7 (2021): 101578.