EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5670 ISSN 1307-5543 – ejpam.com Published by New York Business Global Modeling the Response of the Immune System to HIV-Tumor Interaction via a Fractional Framework Rashid Jan1,2, Mansoor Alsulami3,∗, Normy Norfiza Abdul Razak1 1 Institute of Energy Infrastructure (IEI), Department of Civil Engineering, College of Engineering, Universiti Tenaga Nasional (UNITEN), Putrajaya Campus, Jalan IKRAM-UNITEN, 43000 Kajang, Selangor, Malaysia 2 Mathematics Research Center, Near East University TRNC, Mersin 10, Nicosia 99138, Turkey 3 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia Abstract. The interaction between cancer and HIV underscores the paramount significance of immune response mechanisms in both diseases, illuminating the necessity for tailored management and treatment approaches to address cancer in individuals living with HIV. In this study, a math- ematical model is formulated to conceptualize the interaction between HIV and tumors in relation to the immune system’s response. The basic theory and concepts of the Caputo-Fabrizio operator are presented to analyze the recommended dynamics. The dynamics of HIV and tumor interactions in the context of immune response are systematically investigated. A numerical scheme has been developed to analyze the proposed system and evaluate the impact of various input factors on its dynamics. The numerical findings highlight the key parameters driving the intricate interactions between HIV, tumors, and immune dynamics, providing valuable insights to guide public health interventions and treatment strategies. 2020 Mathematics Subject Classifications: 92D25, 92D30 Key Words and Phrases: Fractional calculus, HIV-tumor dynamics, Immune response, Numer- ical results, Dynamical behavior 1. Introduction The progression of HIV in the presence of a tumor involves a complex interplay between viral dynamics, immune suppression, and cancer biology [8]. HIV targets CD4+ T cells, leading to significant immunosuppression, which weakens the body’s ability to detect and combat cancerous cells. This compromised immune system allows tumors to ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5670 Email addresses: rashid ash2000@yahoo.com (R. Jan), maalsulami2@kau.edu.sa (M. Alsulami), normy@uniten.edu.my (N. N. A. Razak) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) R. Jan, M. Alsulami, N. N. A. Razak / Eur. J. Pure Appl. Math, 18 (1) (2025), 5670 2 of 17 grow more aggressively and evade immune surveillance more easily. Chronic inflammation caused by HIV further alters the tumor microenvironment, promoting tumor growth and metastasis. The introduction of antiretroviral therapy (ART) has improved the immune function of HIV-positive individuals, reducing the incidence of AIDS-defining cancers [10]. However, the increased life expectancy due to ART has led to a higher prevalence of non- AIDS-defining cancers, which can be more challenging to manage in the presence of HIV. The interaction between ART and cancer therapies, coupled with the underlying immuno- suppression, complicates treatment strategies and can adversely affect the prognosis. As such, managing the progression of HIV in the presence of a tumor requires a multidisci- plinary approach that addresses both the viral infection and the cancer while navigating the challenges posed by drug interactions and immune recovery. HIV still affects millions of people worldwide, making it a serious global health prob- lem [23]. HIV continues to place a heavy financial and health burden on afflicted indi- viduals and communities even in the face of tremendous advancements in antiretroviral therapy, which have changed the status of the infection. HIV disproportionately affects marginalized populations, exacerbating social and economic inequalities. Consequently, HIV remains not just a medical issue but a complex social and public health challenge that demands sustained global efforts and resources [29]. Mathematical modeling is crucial for understanding and managing viral infections because it allows researchers to simulate the spread and progression of viruses within individuals and across populations [17, 24]. These models provide insights into key factors such as transmission dynamics, immune re- sponse, and the impact of interventions like vaccination or antiviral treatments [6, 28]. By capturing the complex interactions between a virus and its host, as well as the broader pop- ulation, mathematical models enable the prediction of disease outcomes, the assessment of potential public health strategies, and the optimization of treatment protocols [11]. Even though HIV is still a serious worldwide health concern, current studies and developments in biomathematics present encouraging opportunities for enhancing the management and treatment of this fatal illness. The intricate phenomena of the HIV viruses and CD4+ T-cells have been studied pre- viously using mathematical modeling, as mentioned in [25]. By clarifying the relationship between HIV and host cells, these findings contribute to our understanding of the effects on immune response, depletion of CD4+ T-cells, and viral replication rates. Several clin- ical aspects of HIV, such as the decrease in CD4+ T-cells, lower amounts of free virus, and the extended latency period inside the patients, are successfully captured by the work introduced by Perelson and Nelson [26]. First of all, the model was presented by Perelson et al. [26] which was then examined by the authors in their work [7]. They presented the interaction of T cells and HIV in more precious way. This streamlined formulation enhances the understanding of HIV dynamics. In [4], the authors investigated the stability of the equilibria of HIV model. The research by Bushnaq and his friend [4] enhanced the existing models by including memory related factor in the dynamics. Later studies by other researchers [22, 30] used a variety of techniques and approaches to look at the pro- gression of HIV. This study focuses on the progression of HIV in the presence of a tumor, aiming to elucidate the interactions between HIV, the tumor, and the immune system. R. Jan, M. Alsulami, N. N. A. Razak / Eur. J. Pure Appl. Math, 18 (1) (2025), 5670 3 of 17 Fractional calculus is employed to model the dynamics of infectious diseases with greater precision and accuracy [14, 16]. This approach allows the model to capture mem- ory effects and non-local interactions, which are often present in biological systems but are not well-represented by traditional integer-order models [9, 13]. Fractional epidemic models provide a more comprehensive and accurate framework for modeling the dynamics of vector-borne infections [15]. They incorporate memory effects, capture complex and heterogeneous dynamics, and offer better fits to real-world data. These advantages make fractional models a powerful tool for understanding, predicting, and controlling vector- borne diseases, ultimately leading to more effective public health interventions and better outcomes [3]. Recent advancements in the theory of fractional calculus have significantly expanded its mathematical foundations as well as its applications [1, 20, 21]. The ability of FO derivatives is to incorporate memory effects and non-local interactions provides a more accurate representation of the complexities inherent in the transmission and pro- gression of HIV and tumor in the response of immune. The fractional framework provides a more effective means of capturing the intricate dynamics of diseases compared to tradi- tional integer-order models, enabling improved predictions and the development of more effective control strategies. In this study, we employ fractional derivatives to investigate the in-vivo dynamics of HIV progression and its interaction with the immune system, influenced by the presence of a tumor. This study is arranged as: In Section 2, we formulate the dynamics of HIV-tumor interaction in the response of immune. Section 3 presents the key concepts and idea of Caputo-Fabrizio derivative. A numerical method has been outlined in Section 4 for the model analysis. In Section 4, the impact of input factors has been investigated. The conclusion and final remarks are stated in Section 5. 2. HIV and tumor-immune dynamics Here, we have structured a mathematical model to represent the progression of HIV in the presence of tumor to immune response. Numerous studies [2, 19, 30, 31] have significantly advanced the understanding of the intricate dynamics governing HIV infec- tion. These models have examined diverse facets of HIV pathogenesis, identifying various factors that contribute to the infection’s progression. Notably, the work in [30] provided detailed insights into the dynamics of HIV as: dT dt = s− ηTT+ rT(1− T+I Tmax )− kVT, dI dt = kVT− ηII, dV dt = NηII− ηV V, where particles of HIV, uninfected and infected T-cells, indicated by V, T, and I, re- spectively. In addition, the factor s represents the addition of new T-cells, whereas ηT measures the rate of T-cell death. Within the framework given, k indicates the infection rate, whereas r represents the proliferation rate of healthy CD4+ T-cell. The dynamics of HIV with saturated incidence that was first put forward by Perelson and Nelson [27] R. Jan, M. Alsulami, N. N. A. Razak / Eur. J. Pure Appl. Math, 18 (1) (2025), 5670 4 of 17 looks like this mathematically: dT dt = rT(1− T Tmax )− βVT 1+αV , dI dt = βVT 1+αV − ηII, dV dt = NηII− ηV V. Additionally, we have considered the case where CD4+ T-cells that are not yet infected can infect previously infected T-cells; this is denoted by the mass infection terms αTI. Furthermore, α is supposed to equal the recovery rate. Consequently, the dynamics of HIV in light of the above described factors can be stated as follows: dT dt = s− ηTT+ rT(1− T Tmax )− kVT− αIT, dI dt = kVT+ αIT− ηII, dV dt = NηII− ηV V − kVT, where ν denotes the protease inhibitor’s efficiency and α indicates the rate of cellular infection. Fractional epidemic models have attracted interest in recent years because they effectively capture the intricate dynamics involved in the transmission of infectious diseases [12]. These models serve as a robust and adaptable approach for analyzing and forecasting the transmission of infectious diseases, offering benefits that enhance and broaden the capabilities of traditional integer-order models. Here, we assume tumor cells, health T cells, infected T cells and HIV particles by T, H, I and V, respectively. Then, the dynamics of HIV and tumor in the presence of immune response is as: CF 0 Dξ tT(t) = ℏ+ rT(1− T ℵ )− k1TH, CF 0 Dξ tH(t) = s− k1HT− k2HI− k3HV+ r1H(1− H Hmax )− δHH, CF 0 Dξ t I(t) = k2HI + k3HV − δII, CF 0 Dξ tV(t) = NcI− kHV − δV V, (1) where CF 0 Dξ t denotes the Caputo-Fabrizio operator which is a relatively new concept. Its unique properties make it more attractive for modeling real-world phenomena. 3. Fractional theory and results Here, we present the following concepts and ides of Caputo-Fabrizio (CF) operator to investigate our model. Definition 1. Consider g ∈ H1(a, b), then the CF fractional operator [5] with normality W(ℏ) can be stated as Dℏ k(g(k)) = W(ℏ) 1− ℏ ∫ k a g′(x) exp [ −ℏ k − x 1− ℏ ] dx, (2) where b > a and ℏ ∈ [0, 1], additionally, W(ℏ) represents normalization function with W(0) = W(1) = 1. If g /∈ H1(a, b), then we get Dℏ k(g(k)) = ℏW(ℏ) 1− ℏ ∫ k a (g(k)− g(x)) exp [ −ℏ k − x 1− ℏ ] dx. (3) R. Jan, M. Alsulami, N. N. A. Razak / Eur. J. Pure Appl. Math, 18 (1) (2025), 5670 5 of 17 Remark 1. If σ = 1−ℏ ℏ ∈ [0,∞) and ℏ = 1 1+σ ∈ [0, 1], then from (3), one can get Dℏ k(g(k)) = M(σ) σ ∫ k a g′(x)e[− k−x σ ]dx, (4) furthermore, the below is obtained lim σ−→0 1 σ exp [ − k − x σ ] = δ(x− k), (5) where σ ∈ [0,∞] and M(σ) is the normalized term of W(ℏ) with M(0) = M(∞) = 1. Moreover, σ → 0 as ℏ → 1. Definition 2. [18], Integral for the above fractional operator can be stated as Iℏk(g(k)) = 2(1− ℏ) (2− ℏ)W(ℏ) g(k) + 2ℏ (2− ℏ)W(ℏ) ∫ k 0 g(u)du, k ≥ 0, (6) with the order ℏ and 0 < ℏ < 1. Remark 2. From the above Definition (2), we have 2(1− ℏ) (2− ℏ)W(ℏ) + 2ℏ (2− ℏ)W(ℏ) = 1, (7) where W(ℏ) = 2 2−ℏ . From [18], we have Dℏ k(g(k)) = 1 1− ℏ ∫ k 0 g′(x) exp [ ℏ k − x 1− ℏ ] dx, k ≥ 0. (8) 4. Numerical scheme for the dynamics Here, we will present the dynamics of the recommended through a numerical method to analyze the significance of different scenario. We use the Adams-Bashforth technique of two step for solution of the model. For the first equation of the model, we have y1(t)− y1(0) = 1− ξ U (ξ) H1(t, y1) + ξ U (ξ) ∫ t 0 H1(ϑ, y1)dϑ. (9) In the next step, by take t = tm+1,m = 0, 1, . . . ,, we have y1(tm+1)− y1(0) = 1− ξ U (ξ) H1(tm, y1(tm)) + ξ U (ξ) ∫ tm+1 0 H1(t, y1)dt. (10) and y1(tm)− y1(0) = 1− ξ U (ξ) H1(tm−1, y1(tm−1)) + ξ U (ξ) ∫ tm 0 H1(t, y1)dt. (11) R. Jan, M. Alsulami, N. N. A. Razak / Eur. J. Pure Appl. Math, 18 (1) (2025), 5670 6 of 17 Next, the difference of successive terms are y1m+1 − y1m = 1− ξ U (ξ) ( H1(tm, y1m)− H1(tm−1, y1m−1) ) + ξ U (ξ) ∫ tm+1 m H1(t, y1)dt. (12) Here, the function H 1(t, y1) is approximated over [tk, tk + 1] using an interpolation poly- nomial, yielding Pk(t) ∼= H1(tk, yk) h (t− tk−1)− H1(tk−1, yk−1) h (t− tk), (13) where h = tm − tm−1 is the time spent. Moreover, Pk(t) is utilized to evaluate the below integral∫ tm+1 m H1(t, y1)dt = ∫ tm+1 m ( H1(tm, y1m) h (t− tm−1)− H1(tm−1, y1m−1) h (t− tm) ) dt, = 3h 2 H1(tm, y1m)− h 2 H1(tm−1, y1m−1). (14) Now, put (14) into (12), we have y1m+1 = y1m + ( 1− ξ U (ξ) + 3ξh 2U (ξ) ) H1(tm, y1m) − ( 1− ξ U (ξ) + ξh 2U (ξ) ) H1(tm−1, y1m−1), (15) which is the necessary scheme for the first equation of the model (1). By following a similar procedure, the corresponding schemes for other equations of (1) of HIV-tumor can also be determined, as given by y2m+1 = y2m + ( 1− ξ U (ξ) + 3ξh 2U (ξ) ) H2(tm, y2m) − ( 1− ξ U (ξ) + ξh 2U (ξ) ) H2(tm−1, y2m−1), (16) y3m+1 = y3m + ( 1− ξ U (ξ) + 3ξh 2U (ξ) ) H3(tm, y3m) − ( 1− ξ U (ξ) + ξh 2U (ξ) ) H3(tm−1, y3m−1), (17) and y4m+1 = y4m + ( 1− ξ U (ξ) + 3ξh 2U (ξ) ) H4(tm, y4m) − ( 1− ξ U (ξ) + ξh 2U (ξ) ) H4(tm−1, y4m−1). (18) This approach uses an exponential decay rule for the CF operator. R. Jan, M. Alsulami, N. N. A. Razak / Eur. J. Pure Appl. Math, 18 (1) (2025), 5670 7 of 17 Time in days 0 50 100 150 200 250 300 350 400 T u m o r ce lls T 120 120.5 121 121.5 122 122.5 123 123.5 (a) Time in days 0 50 100 150 200 250 300 350 400 H e a lth T -c e lls H 0 50 100 150 200 250 300 350 400 450 (b) Time in days 0 50 100 150 200 250 300 350 400 In fe ct e d T -c e lls I 0 50 100 150 200 250 300 350 400 450 (c) Time in days 0 50 100 150 200 250 300 350 400 F re e H IV v ir u s p a rt ic le V ×104 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 (d) Figure 1: Time series analysis of the model (1) HIV and tumor in the response of immune system by taking ℏ = 1, r = 0.3 and ξ = 1.00. 5. Results and discussions The significance of analyzing the progression of HIV in the presence of a tumor using a fractional approach lies in the ability to model complex interactions between the im- mune system, viral load, and tumor dynamics more accurately. HIV and tumor cells both impact the immune system, particularly CD4+ T-cells. Traditional integer-order models may fall short in capturing the long-term memory and hereditary properties inherent in bi- ological systems. By applying fractional calculus, especially with non-singular kernels like Atangana-Baleanu operator, the intricate feedback loops and interactions between HIV, immune cells, and tumors can be better represented. This offers a more realistic simulation of the disease progression, accounting for the time-dependent changes in immune system R. Jan, M. Alsulami, N. N. A. Razak / Eur. J. Pure Appl. Math, 18 (1) (2025), 5670 8 of 17 Time in days 0 50 100 150 200 250 300 350 400 T u m o r ce lls T 120 120.5 121 121.5 122 122.5 123 123.5 (a) Time in days 0 50 100 150 200 250 300 350 400 H e a lth T -c e lls H 0 50 100 150 200 250 300 350 400 450 (b) Time in days 0 50 100 150 200 250 300 350 400 In fe ct e d T -c e lls I 0 50 100 150 200 250 300 350 400 450 (c) Time in days 0 50 100 150 200 250 300 350 400 F re e H IV v ir u s p a rt ic le V ×104 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 (d) Figure 2: Time series analysis of the model (1) HIV and tumor in the response of immune system by taking ℏ = 1, r = 0.3, s = 1 and ξ = 0.90. efficacy and viral load, which are vital for designing effective treatment strategies. Moreover, a fractional approach to immune response dynamics allows researchers to explore the oscillatory and potentially chaotic behaviors observed in the immune system under the dual pressure of HIV infection and tumor growth. This mathematical frame- work provides deeper insights into how both HIV and tumor cells may exploit the immune system’s vulnerabilities, causing destabilization in immune function. Such modeling is not only significant for understanding the co-progression of HIV and cancer but also for optimizing therapeutic interventions. It enables the development of combined treatment strategies that take into account both infections, targeting not just the viral load or tu- mor growth individually but considering the overall immune system dynamics, leading to improved patient outcomes in co-infection scenarios. R. Jan, M. Alsulami, N. N. A. Razak / Eur. J. Pure Appl. Math, 18 (1) (2025), 5670 9 of 17 Time in days 0 50 100 150 200 250 300 350 400 T u m o r ce lls T 117 118 119 120 121 122 123 124 (a) Time in days 0 50 100 150 200 250 300 350 400 H e a lth T -c e lls H 0 100 200 300 400 500 600 700 800 900 (b) Time in days 0 50 100 150 200 250 300 350 400 In fe ct e d T -c e lls I 0 100 200 300 400 500 600 700 800 900 (c) Time in days 0 50 100 150 200 250 300 350 400 F re e H IV v ir u s p a rt ic le V ×104 0 2 4 6 8 10 12 (d) Figure 3: Representation of the tracking paths of the proposed system (1) by taking the value of s = 0.1, ξ = 0.85 and r = 0.3. Here, various simulations are performed to conceptualize the intricate phenomena of the dynamics of HIV and tumor with the response of immune. For simulation purposes, we assumed the values T = 120, H = 200, I = 300 and V = 150. We conducted various simulations to demonstrate the influence of various factors on the tracking paths of the system and its chaotic behavior. In the first simulation, we set the parameter values to ℏ = 1.00, s = 1, and r = 0.3, while the value of ξ is assumed to be 1.00 in Figure 1 and 0.90 in Figure 2. We observed the influence of the fractional order on the solution pathways. The simulation outcomes clearly reveal the substantial effect of the fractional order on the infection dynamics. Importantly, υ shows potential as an effective instrument for controlling the infection spread. Consequently, we advise policymakers to undertake a more thorough examination of this fractional parameter to obtain a better knowledge of R. Jan, M. Alsulami, N. N. A. Razak / Eur. J. Pure Appl. Math, 18 (1) (2025), 5670 10 of 17 Time in days 0 50 100 150 200 250 300 350 400 T u m o r ce lls T 114 115 116 117 118 119 120 121 122 123 124 (a) Time in days 0 50 100 150 200 250 300 350 400 H e a lth T -c e lls H 0 200 400 600 800 1000 1200 1400 (b) Time in days 0 50 100 150 200 250 300 350 400 In fe ct e d T -c e lls I 0 200 400 600 800 1000 1200 1400 (c) Time in days 0 50 100 150 200 250 300 350 400 F re e H IV v ir u s p a rt ic le V ×104 0 2 4 6 8 10 12 14 16 18 (d) Figure 4: Visualization of the tracking paths of the proposed system (1) by taking the value of s = 0.01, ξ = 0.85 and r = 0.3. its ability to mitigate the infection. This thorough examination could provide important insights for developing focused policies for infection prevention and control. In the second simulation, shown in Figures 3 and 4, we illustrate the effect of the CD4+ T-cell source term on solution pathways of the recommended system. In Figure 3, the source term is set to s = 0.1, while in Figure 4, the value of s is reduced to 0.01. The variation in the oscillatory behavior of the system has been noticed with the variation of the source term of the CD4+ T-cell. CD4+ T-cells influence the tumor microenvironment, promoting anti-tumor immunity by activating macrophages and other immune cells that can inhibit tumor growth. An increased source term of CD4+ T-cells leads to a stronger immune response, potentially reducing tumor growth. CD4+ T-cells are the primary target of HIV, and the progressive depletion of these cells leads to immune system failure. R. Jan, M. Alsulami, N. N. A. Razak / Eur. J. Pure Appl. Math, 18 (1) (2025), 5670 11 of 17 Time in days 0 50 100 150 200 250 300 350 400 T u m o r ce lls T 118.6 118.8 119 119.2 119.4 119.6 119.8 120 120.2 120.4 (a) Time in days 0 50 100 150 200 250 300 350 400 H e a lth T -c e lls H 0 500 1000 1500 2000 2500 (b) Time in days 0 50 100 150 200 250 300 350 400 In fe ct e d T -c e lls I 0 1000 2000 3000 4000 5000 6000 (c) Time in days 0 50 100 150 200 250 300 350 400 F re e H IV v ir u s p a rt ic le V ×105 0 1 2 3 4 5 6 (d) Figure 5: Graphical view analysis of the time series of the proposed system (1) by taking the value of r1 = 3, ξ = 0.85 and s = 1. The source term reflects the replenishment rate of CD4+ T-cells, which is critical for maintaining immune function in HIV-positive individuals. The source term of CD4+ T- cells is essential for maintaining a robust immune response, and its modulation can have significant therapeutic implications for both tumor reduction and HIV management. In Figure 5, we illustrate the effect of tumor growth rate on the dynamics of HIV-tumor interactions in response to the immune system. In this simulation, the tumor growth rate r is varied from 0.3 to 3.0 to highlight the resulting changes in the system’s time series. The results indicate that this input parameter amplifies oscillatory behavior within the system, thereby increasing the risk of infection. The chaotic nature of HIV and tumor dynamics in the presence of an immune response is characterized by complex, unpredictable behavior that arises from the nonlinear interactions between viral replication, tumor growth, and R. Jan, M. Alsulami, N. N. A. Razak / Eur. J. Pure Appl. Math, 18 (1) (2025), 5670 12 of 17 Tumor cells T 117 117.5 118 118.5 119 119.5 120 120.5 H e a lth y T -c e lls H 0 50 100 150 200 250 300 350 400 450 (a) Health T-cells H 0 50 100 150 200 250 300 350 400 450 In fe ct e d T -c e lls I 0 50 100 150 200 250 300 350 400 450 (b) Infected T-cells I 0 50 100 150 200 250 300 350 400 450 F re e H IV v ir u s p a rt ic le V ×104 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 (c) Free HIV virus particle V ×104 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 T u m o r ce lls T 117 117.5 118 118.5 119 119.5 120 120.5 (d) Figure 6: Chaotic nature of the recommended model (1) of the HIV and tumor in the response of immune by considering the input values ξ = 0.9, ℏ = 0.1, s = 1.00 and r1 = 0.35. the body’s immune defense. In Figures 6-8, the chaotic behavior of the proposed system is depicted. In Figure 6, the parameters are set to ξ = 0.90, ℏ = 0.1, s = 1.0, and r = 0.3 to demonstrate the system’s chaotic dynamics. We assume N = 2000, Hmax = 1000, r = 0.3 and 0.85 in Figure 7 while the value of ℏ = 0.1, s = 2.0 and k1 = k1 = 1 ∗ 10(−5) in Figure 8 to conceptualize the chaotic behavior of the system. The interactions between HIV, tumor cells, and immune components exhibit nonlinear feedback loops. These feedback mechanisms can lead to sensitive dependence on initial conditions, where small variations in the system’s parameters or state can result in vastly different outcomes over time. Chaotic dynamics often manifest as irregular oscillations in viral load, tumor cell populations, and immune cell levels. These oscillations are driven by the immune system’s efforts to control HIV and tumor growth, as well as the adaptive R. Jan, M. Alsulami, N. N. A. Razak / Eur. J. Pure Appl. Math, 18 (1) (2025), 5670 13 of 17 Tumor cells T 120 120.5 121 121.5 122 122.5 123 123.5 H e a lth y T -c e lls H 0 50 100 150 200 250 (a) Health T-cells H 0 50 100 150 200 250 In fe ct e d T -c e lls I 0 50 100 150 200 250 300 350 400 (b) Infected T-cells I 0 50 100 150 200 250 300 350 400 F re e H IV v ir u s p a rt ic le V ×104 0 1 2 3 4 5 6 7 8 9 (c) Infectious virus particle V ×104 0 1 2 3 4 5 6 7 8 9 T u m o r ce lls T 120 120.5 121 121.5 122 122.5 123 123.5 (d) Figure 7: Dynamical behavior of the model (1) of HIV and tumor in the response of immune by considering the input values N = 2000, r = 0.3, Hmax = 1000 and ξ = 0.85 to visualize the chaotic nature of the system. strategies of the virus and tumor to evade immune detection. Moreover, HIV weakens the immune response by infecting CD4+ T-cells, while tumors can evade immune surveil- lance through various mechanisms, such as altering antigen expression or promoting an immunosuppressive microenvironment. This dual interference from HIV and tumors can lead to complex and chaotic dynamics, where the immune system’s response becomes highly variable and unpredictable. The chaotic nature of these dynamics presents chal- lenges for treatment strategies, as the system may respond in unexpected ways to inter- ventions like antiretroviral therapy, immunotherapy, or chemotherapy. Effective treatment must account for the potential for sudden shifts in the behavior of the system due to the sensitivity of chaotic dynamics. We demonstrated distinct patterns in HIV viral load and tumor growth in response to R. Jan, M. Alsulami, N. N. A. Razak / Eur. J. Pure Appl. Math, 18 (1) (2025), 5670 14 of 17 Tumor cells T 119.85 119.9 119.95 120 120.05 120.1 120.15 120.2 120.25 120.3 H e a lth y T -c e lls H 0 50 100 150 200 250 (a) Health T-cells H 0 50 100 150 200 250 In fe ct e d T -c e lls I 0 50 100 150 200 250 300 350 400 (b) Infected T-cells I 0 50 100 150 200 250 300 350 400 F re e H IV v ir u s p a rt ic le V ×104 0 1 2 3 4 5 6 7 8 9 (c) Infectious virus particle V ×104 0 1 2 3 4 5 6 7 8 9 T u m o r ce lls T 119.85 119.9 119.95 120 120.05 120.1 120.15 120.2 120.25 120.3 (d) Figure 8: Dynamical behavior of the model (1) of HIV and tumor in the response of immune by considering the input values s = 2, ℏ = 0.1, Hmax = 1000 and k1 = 1 ∗ 10(−5) to visualize the chaotic nature of the system. variations in immune system dynamics. Specifically, changes in immune response parame- ters led to observable shifts in both HIV and tumor trajectories, highlighting the sensitivity of these dynamics to immune modulation. Variations in CD4+ T-cell dynamics, such as changes in the source term and growth rate, influenced the system’s stability and os- cillatory behavior. Higher source terms for CD4+ T-cells were associated with a more robust immune response, which in turn impacted both HIV control and tumor growth. The system exhibited chaotic behavior under certain parameter settings, as evidenced by irregular oscillations in HIV viral load and tumor cell populations. This chaos was at- tributed to the complex interactions between HIV, tumor cells, and the immune response. Numerical simulations revealed sensitive dependence on initial conditions and parameter values, underscoring the importance of precise and adaptive treatment strategies. Future R. Jan, M. Alsulami, N. N. A. Razak / Eur. J. Pure Appl. Math, 18 (1) (2025), 5670 15 of 17 research directions should aim to refine mathematical models, explore novel therapeutic approaches, integrate diverse data sources, and address global health challenges to advance our understanding and treatment of HIV and tumors in the context of immune responses. 6. Concluding Remarks The study on HIV and tumor dynamics within the framework of immune responses holds substantial importance. It deepens our understanding of the intricate interactions between these diseases, facilitates the optimization of treatment strategies, enhances dis- ease management, and guides public health policies. This comprehensive approach not only contributes to better patient outcomes but also drives advancements in scientific knowledge. In this study, a mathematical model was developed to conceptualize the in- teraction between HIV and tumors in relation to the immune system’s response. The foundational theory and concepts of the Caputo-Fabrizio derivative were presented to in- vestigate the proposed dynamics. The dynamics of HIV-tumor interactions in the context of immune response were systematically explored. We presented a numerical scheme to conceptualize the dynamics through numerical results. Numerical results are presented to illustrate the impact of various parameters on the infection dynamics. The system’s trajectories have been analyzed numerically, revealing distinct patterns in HIV viral load and tumor growth under varying immune system dynamics. Specifically, alterations in im- mune response parameters resulted in notable shifts in both HIV and tumor progression, underscoring the sensitivity of these dynamics to immune modulation. Our findings high- light key parameters that critically influence system behavior, offering valuable insights for designing targeted therapeutic interventions. References [1] Sultan Alyobi and Rashid Jan. Qualitative and quantitative analysis of fractional dy- namics of infectious diseases with control measures. Fractal and Fractional, 7(5):400, 2023. [2] Sadia Arshad, Dumitru Baleanu, Weiping Bu, and Yifa Tang. Effects of hiv infection on cd4+ t-cell population based on a fractional-order model. Advances in Difference Equations, 2017:1–14, 2017. [3] Abdon Atangana and Sania Qureshi. Mathematical modeling of an autonomous non- linear dynamical system for malaria transmission using caputo derivative. Fractional order analysis: Theory, methods and applications, pages 225–252, 2020. [4] SAMIA Bushnaq, Sajjad Ali Khan, Kamal Shah, and Gul Zaman. Existence theory of hiv-1 infection model by using arbitrary order derivative of without singular kernel type. J. Math. Anal, 9(1):16–28, 2018. [5] Michele Caputo and Mauro Fabrizio. A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation & Applications, 1(2):73–85, 2015. R. Jan, M. Alsulami, N. N. A. Razak / Eur. J. Pure Appl. Math, 18 (1) (2025), 5670 16 of 17 [6] Anil Chavada, Nimisha Pathak, and Sagar R Khirsariya. A fractional mathematical model for assessing cancer risk due to smoking habits. Math. Model. Control, 4(3):246– 259, 2024. [7] Rebecca V Culshaw and Shigui Ruan. A delay-differential equation model of hiv infection of cd4+ t-cells. Mathematical biosciences, 165(1):27–39, 2000. [8] Dima Dandachi and Fanny Morón. Effects of hiv on the tumor microenvironment. Tumor Microenvironment: State of the Science, pages 45–54, 2020. [9] Ihtisham Ul Haq, Nigar Ali, and Shabir Ahmad. A fractional mathematical model for covid-19 outbreak transmission dynamics with the impact of isolation and social distancing. Math. Model. Control, 2:228–242, 2022. [10] Corrilynn O Hileman and Nicholas T Funderburg. Inflammation, immune activation, and antiretroviral therapy in hiv. Current Hiv/aids reports, 14:93–100, 2017. [11] Rashid Jan, Salah Boulaaras, Hussain Ahmad, Muhammad Jawad, Sulima Zubair, and Mohamed Abdalla. A robust study of tumor-immune cells dynamics through non-integer derivative. Fractal and Fractional, 7(2):164, 2023. [12] Rashid Jan, Salah Boulaaras, Asma Alharbi, and Normy Norfiza Abdul Razak. Non- linear dynamics of a zoonotic disease with control interventions through fractional derivative. European Journal of Pure and Applied Mathematics, 17(4):3781–3800, 2024. [13] Rashid Jan, Salah Boulaaras, Mohammad Alnegga, and Farah Aini Abdullah. Fractional-calculus analysis of the dynamics of typhoid fever with the effect of vaccina- tion and carriers. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 37(2):e3184, 2024. [14] Rashid Jan, Salah Boulaaras, Sultan Alyobi, and Muhammad Jawad. Transmission dynamics of hand–foot–mouth disease with partial immunity through non-integer derivative. International Journal of Biomathematics, 16(06):2250115, 2023. [15] Rashid Jan, Salah Boulaaras, Sultan Alyobi, Karthikeyan Rajagopal, and Muhammad Jawad. Fractional dynamics of the transmission phenomena of dengue infection with vaccination. Discrete Contin. Dyn. Syst. Ser. S, 2022. [16] Rashid Jan, Normy Norfiza Abdul Razak, Salah Boulaaras, Ziad Ur Rehman, and Salma Bahramand. Mathematical analysis of the transmission dynamics of viral in- fection with effective control policies via fractional derivative. Nonlinear Engineering, 12(1):20220342, 2023. [17] Rashid Jan, Normy Norfiza Abdul Razak, Sania Qureshi, Imtiaz Ahmad, and Salma Bahramand. Modeling rift valley fever transmission: insights from fractal-fractional dynamics with the caputo derivative. Mathematical Modelling and Control, 4(2):163– 177, 2024. [18] Jorge Losada and Juan J Nieto. Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl, 1(2):87–92, 2015. [19] Bobisa Mobisa, GO Lawi, and JK Nthiiri. Modelling in vivo hiv dynamics under combined antiretroviral treatment. Journal of Applied Mathematics, 2018(1):8276317, 2018. [20] Pshtiwan Othman Mohammed, Ravi P Agarwal, Majeed A Yousif, Eman Al-Sarairah, R. Jan, M. Alsulami, N. N. A. Razak / Eur. J. Pure Appl. Math, 18 (1) (2025), 5670 17 of 17 Alina Alb Lupas, and Mohamed Abdelwahed. Theoretical results on positive solutions in delta riemann–liouville setting. Mathematics, 12(18):2864, 2024. [21] Pshtiwan Othman Mohammed, Alina Alb Lupas, Ravi P Agarwal, Majeed A Yousif, Eman Al-Sarairah, and Mohamed Abdelwahed. Theoretical investigation of fractional estimations in liouville–caputo operators of mixed order with applications. Axioms, 13(8):570, 2024. [22] Mevlüde Yakit Ongun. The laplace adomian decomposition method for solving a model for hiv infection of cd4+ t cells. Mathematical and Computer Modelling, 53(5- 6):597–603, 2011. [23] Abhishek Pandey and Alison P Galvani. The global burden of hiv and prospects for control. The Lancet HIV, 6(12):e809–e811, 2019. [24] Bengisen Pekmen and Ummuhan Yirmili. Numerical and statistical approach on chemotaxis-haptotaxis model for cancer cell invasion of tissue. Mathematical Mod- elling and Control, 4(2):195–207, 2024. [25] Alan S Perelson, Denise E Kirschner, and Rob De Boer. Dynamics of hiv infection of cd4+ t cells. Mathematical biosciences, 114(1):81–125, 1993. [26] Alan S Perelson and Patrick W Nelson. Mathematical analysis of hiv-1 dynamics in vivo. SIAM review, 41(1):3–44, 1999. [27] Alan S Perelson and Patrick W Nelson. Mathematical analysis of hiv-1 dynamics in vivo. SIAM review, 41(1):3–44, 1999. [28] Tao-Qian Tang, Zahir Shah, Rashid Jan, Wejdan Deebani, and Meshal Shutaywi. A robust study to conceptualize the interactions of cd4+ t-cells and human immunod- eficiency virus via fractional-calculus. Physica Scripta, 96(12):125231, 2021. [29] Marta Trapero-Bertran and Juan Oliva-Moreno. Economic impact of hiv/aids: a systematic review in five european countries. Health economics review, 4:1–16, 2014. [30] H Vazquez-Leal, L Hernandez-Martinez, Y Khan, VM Jimenez-Fernandez, U Filobello-Nino, A Diaz-Sanchez, AL Herrera-May, R Castaneda-Sheissa, A Marin- Hernandez, F Rabago-Bernal, et al. Multistage hpm applied to path tracking damped oscillations of a model for hiv infection of cd4+ t cells. British Journal of Mathematics & Computer Science, 4(8):1035–1047, 2014. [31] Xueyong Zhou, Xinyu Song, and Xiangyun Shi. A differential equation model of hiv infection of cd4+ t-cells with cure rate. Journal of Mathematical Analysis and Applications, 342(2):1342–1355, 2008.