Understanding the complex interactions among tumor cells, immune responses, and cytokines, such as interleukin-2 (IL-2), is crucial for advancing cancer immunotherapy. We introduce a four-dimensional mathematical model that includes healthy cells, tumor cells, effector immune cells, and IL-2 dynamics. This study explores the continuous-time and discrete-time formulations of the proposed model to examine the nonlinear behaviors of tumor-immune interactions. A detailed bifurcation analysis explores the system’s qualitative changes in response to variations in key immunological parameters. The analytical finding suggests the existence of Hopf and saddle-node bifurcations in the continuous-time model, while the discrete-time analog indicates the presence of Neimark-Sacker bifurcations. We further investigate the sensitivity of the system to the choice of step size (h), demonstrating that the qualitative dynamics remain robust within a biologically reasonable range of h. We employ a data-fitting approach to estimate model parameters using real-world tumor growth data with and without IL-2 treatment. The analytical findings are validated through numerical simulations demonstrating Hopf, saddle-node, torus, and generalized bifurcations, resulting in stable limit cycles and quasi-periodic behavior. In contrast to the continuous counterpart, the discrete model exhibits more complex dynamics, including Neimark-Sacker bifurcations, strong resonance, and chaotic attractors, validated through maximum Lyapunov coefficient, phase space and period-density maps. The findings underscore the significance of IL-2-mediated feedback in regulating immune activation and tumor suppression. The comparison reveals that discrete-time modeling captures additional dynamical complexity absent in the continuous-time system, offering deeper insights into potential outcomes of immunotherapy interventions. This study enhances the understanding of tumor-immune-cytokine interactions and provides a computational framework for exploring stability and chaos in cancer-immune regulation.

Malaria transmission is strongly influenced by the seasonal dynamics of mosquito populations. However, existing climate-based malaria models do not include humidity or seasonal optimal control to guide timely interventions. We present a climate-based, non-autonomous malaria transmission model that incorporates temperature, rainfall, and humidity-dependent parameters. Analysis of the autonomous version exhibits a backward bifurcation, implying that reducing R0 below unity does not guarantee malaria elimination. In the non-autonomous framework, we establish that the disease-free equilibrium is globally attractive for R0(t)<1, while R0(t)>1 guarantees at least one positive periodic sustained transmission. The model is fitted using real climate and epidemiological data from Kolkata, India. We propose a season-based optimal control model and analyze seven intervention strategies combining bed nets, treatment, and insecticide spraying. The single-control strategy with bed nets is the least expensive, but the combined strategy reduces infections by 96.98%. The findings emphasize the importance of synchronizing intervention intensity with seasonal and climatic variations, offering a quantitative framework to inform region-specific malaria control policies and efficient resource allocation.

The release of Wolbachia-infected mosquitoes has emerged as a promising biocontrol strategy to suppress mosquito-borne diseases. However, the success of such interventions is highly sensitive to ecological conditions, release timing, and seasonal climate variability. In this study, we develop a climate-driven compartmental model that explores the interplay between mosquito life cycle dynamics, human disease transmission, and Wolbachia-based biocontrol. Analysis of the autonomous version of the system shows that the Wolbachia-free subsystem exhibits a backward bifurcation. In contrast, the Wolbachia-invaded subsystem shows a forward bifurcation, which supports disease elimination. In the non-autonomous framework, we establish that disease-free equilibrium is globally attractive for R_0 < 1 , whereas the system admits at least one positive periodic solution for R_0 > 1 , implying disease persistence. Numerical simulations validate the analytical results using real climate data from Niterói, Brazil. We find that the consistent presence of Wolbachia-carrying mosquitoes successfully invades the Wolbachia-free population and effectively reduces disease prevalence. To evaluate the efficacy of Wolbachia deployment, we analyze five release strategies, namely: three-pulse, four-pulse, five-pulse, continuous exponentially decreasing, and quadratically increasing releases. In addition to comparing release modes, we examine release timing and gaps, showing that aligning strategies with seasonal climate patterns ensures sustained Wolbachia establishment and disease control. Additionally, a global sensitivity analysis using partial rank correlation coefficients identifies key parameters of infection dynamics. Parameters such as mosquito biting rate and egg maturation rate positively influence infection, while recruitment of humans, probability of cytoplasmic incompatibility, and recovery of humans exert strong negative effects. These insights help prioritize key biological parameters for an accurate intervention strategy. The model, supported by sensitivity analysis, provides a practical tool for designing adaptive Wolbachia release strategies personalized for local temperature and rainfall conditions.

Obesity has emerged as a risk factor for several malignancies, primarily due to the dysregulation of adipokines, particularly adiponectin. We develop a nonlinear mathematical model to examine the effects of plasma adiponectin on interactions among tumor cells, healthy cells, and immune cells. Stability and bifurcation analysis reveal a Hopf bifurcation, indicating periodic dynamics in tumor cell behavior. The numerical simulations demonstrate the local stability of equilibrium points and confirm the presence of bifurcations. A two-stage parameter estimation method is used to estimate model parameters using experimental tumor-growth data associated with adiponectin-enhancing treatment. The global sensitivity analysis underscores the adiponectin production rate and tumor immune competition coefficient as the most sensitive parameters. The model suggests a potential threshold of 8.66 μg/mL for achieving tumor suppression. An independent data-driven analysis on clinical data from 2708 individuals yields an optimal threshold of 8.95 μg/mL, showing strong concordance with the model's prediction. We propose an optimal control model that combines immune stimulation and adiponectin-boosting therapy. The model suggests that starting with strong immune activation followed by sustained adiponectin treatment provides the most effective approach. These findings provide a scientific basis for developing cancer treatments that target adiponectin pathways, particularly for obesity-related cancers.

We introduce the definition of triple new Laplace type integral transform in q-calculus namely q-triple new Laplace type integral transform, by using the functions of several variables and establish some properties. Furthermore, several theorems dealing with the properties of the q-triple new Laplace type Integral transform are proved. We also give some applications of q-triple new Laplace type Integral transform for solving non-homogeneous Mboctara partial differential equation and diffusion partial differential equation with initial and boundary values problems to its effectiveness and performance of the proposed method.

We introduce three dimensional fractional Mellin transform and establish analytic theorem, boundedness theorem, inversion theorem and uniqueness theorem for three dimensional fractional Mellin transform. We present propositions of 3-dimensional fractional Mellin transform. We give some applications of 3-dimensional fractional Mellin transform for solving PDE’s.

The Marburg virus is a serious global health threat due to its high mortality rate and rapid transmission. Effective control measures, such as hospital beds, are vital but often limited by inadequate healthcare resources. This study aims to address this challenge by developing a fractional-order epidemic model for Marburg virus, which considers the effects of limited hospital beds on transmission dynamics. We present a model to provide a more accurate understanding of Marburg virus transmission patterns and prevalence incorporating the memory effect through a fractional-order approach. The study explores the impact of constrained healthcare resources on virus progression and calculates the basic reproduction number using the next-generation matrix technique. Further analysis of the model's global dynamics is conducted using reproduction numbers, the Lyapunov functional approach, and the Routh-Hurwitz criterion, shedding light on how hospital bed availability influences disease progression.We also apply Hyers-Ulam stability criterion to find the stability of the model and obtain numerical solutions through a fractional Lagrange two-step interpolation method. The fractional-order Marburg virus model, by accounting for memory effects, offers a more nuanced understanding of the disease dynamics compared to classical models. Our results demonstrate that increasing hospital bed availability significantly reduces Marburg virus infection rates. This approach highlights the value of fractional calculus in epidemiological modeling, offering significant insights into optimal control measures and strategies to improve public health outcomes during Marburg virus outbreaks.

In this study, we present a novel conformable fractional-order SIQR (Susceptible, Infected, Quarantined, Recovered) model that incorporates awareness programs to investigate the spread of infectious diseases. The novelty of this work lies in employing the conformable fractional derivative a local operator that, unlike classical non-local fractional derivatives, does not account for memory but allows fractional-order modeling with simplified mathematical structure. This approach is particularly suited for capturing short-term behavioral responses, such as immediate effects of awareness campaigns and rapid quarantine actions. We establish the model’s mathematical soundness by proving the existence, uniqueness, positivity, and boundedness of solutions. The basic reproduction number is derived to analyze the stability of the disease-free equilibrium, followed by a sensitivity analysis to identify influential parameters. The system is numerically solved using the generalized Euler method, and the scheme’s accuracy is validated through error and convergence analysis. Additionally, we compare our results with those obtained from the classical integer-order Ode45 method and validate the model using real epidemiological data. A bifurcation analysis further explores qualitative changes in disease dynamics under varying parameters. Our findings demonstrate that awareness efforts and timely quarantine significantly reduce disease transmission, and highlight the novelty and practicality of using conformable fractional-order modeling in contexts where memory effects are negligible.

Malaria has remained a global health burden over the past few decades. Remote regions with limited healthcare resources are significant contributors to malaria cases worldwide. In the present study, we propose a deterministic compartmental model to explore the dynamics of malaria transmission in the presence of Wolbachia. The nonlinear recovery rate is incorporated to elucidate the impact of available public healthcare resources. The analytical result of the model exhibits the existence of multiple malaria-present endemic equilibria. We observe the coexistence of a malaria-present endemic equilibria with a stable malaria-free equilibria. Sensitivity analysis is performed to explore the relative importance of different parameters. Additionally, the phenomenon of backward bifurcation exists in the proposed model. Numerical simulation validates the analytical results of the model and confirms the existence of backward bifurcation. We demonstrate the inhibition of malaria transmission with the release of Wolbachia-infected mosquitoes in the region with limited availability of public health resources. The simulation suggests the possible increment in the availability of the healthcare system to ensure malaria-free equilibria. We validate the model by fitting it to the reported human infection data from Niter & oacute;i, Brazil, using 16 months of data collected before and after the release of Wolbachia. These findings will be helpful to healthcare professionals in planning the control strategy of malaria in remote or hard-to-reach locations in the tropical and subtropical regions of the world.

The Laplace transform is widely used in science and technology to deal with complex problemsin stability and control systems. The modified Laplace transform has been applied in physics andmathematics to solve boundary layer equations in ordinary differential equations with variablecoefficients. The q-calculus appeared as a connection between mathematics and physics. It hasmany applications in different mathematical areas, such as number theory, combinatory theory,orthogonal polynomials, essential hyper-geometric functions, quantum mechanics, and relativity.Laplace transform, and its several extended versions are used frequently. The double Laplacetransform applies to solving some q-functional and partial q-differential equations. Q-calculushas been used to solve complex and more potentially typical problems in a larger domain toinvestigate the calculus without limits for getting more generalizations. In the paper, weintroduce the double-modified Laplace transform in q-calculus, namely the q-double modifiedLaplace transform, and establish some properties. Furthermore, several propositions concernedwith q-double modified Laplace transform are explored.

The paper has developed an efficient and accurate Chebyshev wavelet-based numerical method (CWNM) for solving singular perturbed nonlinear Benjamin–Bona–Mahony equation. The key idea of the CWNM is based on the expansion of unknown function into a series of the basis of shifted Chebyshev wavelets; that is, it reduces the underlying problem to a system of algebraic equations. The proposed method is straightforward and accurate with a small computational cost. We have proved that the proposed method is convergent, and CWNM is very high estimation accuracy. The numerical results are more precise than other existing methods available in the literature and very close to exact solutions. Finally, we have shown that the proposed method’s CPU time is compared with the CPU time taken by other existing methods.

In this paper, the concept of dual framelets on manifolds and its characterization are introduced. The accuracy of the proposed dual framelets transform is determined by sparse representation on graphs. If any pair of the framelet system is associated with filter-bank transform, then compactly supported refinable functions can have vanishing moments at most one and framelet approximation is the order of at most two. An algorithm of decomposition and reconstruction for the dual framelets transform on graph is presented. A new method called dual framelets filter-bank transform (DFFT) is employed, which is faster than the existing method spectral graph wavelet transform (SGWT). The theoretical results along with algorithms for accurate and efficient computation of the DFFT on discrete data sets are provided. Subsequently, some numerical examples are provided to show the importance of DFFT over SGWT on graphs.

Mixed invasive ductal and lobular carcinoma (Mi-DLC) is a subtype of breast cancer having both ductal and lobular morphology. The coexistence of both morphologies together is not sufficiently addressed in earlier research, so the metastatic behavior of Mi-DLC is not known to healthcare professionals. We introduce the mathematical model to analyze the metastatic behavior of Mi-DLC for controlling the proliferation of tumors. Breast cancer is highly sensitive to estrogen receptors. We incorporate estrogen in the competition among the population of healthy cells, lobular cells, ductal cells, and immune cells. The ketogenic diet sensitizes cancer cells and alters their metabolism. We obtain the increase in the ketogenic diet with the use of anticancer drugs in humans, resulting in a stable cancer-free equilibrium state. We find that the immune response of humans is incapable of fighting the disease when the initial volume of Mi-DLC is high and the ketogenic diet rate is low in the body. The model has four cancer invasion equilibriums: lobular cancer invasion, ductal cancer invasion, mixed ductal and lobular invasion, and dead equilibrium state. We obtain all four cancer invasion equilibriums are stable when we consider the low rate of the ketogenic diet in the body. We calculate that with the proper use of anticancer drugs and a ketogenic diet in humans, the Mi-DLC moves to its dormant stage, and the co-existing equilibrium is stable. The numerical simulation supports the mathematical analysis of the model and establish that the 10

The corona virus, which causes COVID-19 disease, is constantly changing its genetic characteristics, and new variants of the virus are expected to occur as the virus spreads. New variants may become more difficult to stop. Numerous variants of the corona virus that causes COVID-19 are being tracked globally during this pandemic. To break the spread of this virus, people around the globe must practice some prevention methods; one of the most effective methods is to quarantine the infected population. This paper represents the effect of quarantine in India by analysing the mathematical model in the applicable timeframe. This model shows the spread of infection in India with contemporary norms of both restrictions of home quarantine and quarantine facilities provided by the government. We validate the model with the actual data, and statistics calculates the model's accuracy is 91.4% with actual known data. By applying the rough set method to the known real data, we observe that the rough set supports the statistical interference of the data from the mathematical model. Also, this paper shows that the number of active cases in India decreases by applying stricter norms to both home quarantine and quarantine facilities provided by the government.

Monkeypox is a deadly disease from the Orthopox family. The paper explores a nonlinear trend in the dynamics of Monkeypox propagation. In this model, we include vaccination, treatment, and level of awareness as the main parameters of the propagation of Monkeypox. We also include the quarantined infected individuals with severe complications in the Monkeypox dynamics. We show the boundedness and positivity of the model and also demonstrate the stability of the model using the Banach fixed point theory and the Picard successive approximation method to ensure that it is meaningful mathematically and epidemiologically. We obtain the unique solution of the model under appropriate conditions. This work finds three equilibrium points: Monkeypox free, Rodents free, and endemic. We show the local stability of monkeypox free equilibria and rodent-free endemic equilibria through reproduction number. Furthermore, Dulac’s function technique demonstrates the global stability of endemic equilibrium. We show the transformation from endemic equilibria to Monkeypox-free equilibria via rodent-free. We also obtain the condition of transcritical bifurcation. We analyze the system’s dynamic behavior to develop efficient infection control strategies. We find that if we increase awareness, vaccination, treatment, and quarantine rates, we effectively control the transmission. The model is connected with a continuous model using an ordinary differential equation. We examine the complex dynamics of Monkeypox infection under diverse system input factors through numerical simulation of the proposed model with variable input parameters in reducing Monkeypox. We present the theory of Monkeypox disease control in humankind as an application in real-world problems. The work can be helpful in the vector-borne disease control system.

Malaria is an infectious vector-borne disease with a high fatality rate among infants. Malaria causes anaemia, slow fatal growth, preterm birth, and low birth weight. Intermittent preventive treatment is an intervention for treating and preventing malaria in pregnant women, infants, children, and schoolchildren using antimalarial drugs. This study introduces a host-vector mathematical model of vertical and horizontal malaria transmission with intermittent preventive treatment in pregnancy. The model is well-posed due to the positivity and boundedness of the solution. The basic reproduction number is determined using the next-generation matrix method. Stability analysis reveals that the malaria-free equilibrium point is locally and globally stable if the reproduction number is less than one and unstable if it exceeds one. The existence and stability of the endemic equilibrium points are established using bifurcation analysis and the Lyapunov function. The validation is carried out on a benchmark dataset to assess the efficacy of the proposed model with parameter estimation. The numerical simulation presents the effect of intermittent preventive treatment programs, the efficacy of antimalarial drugs, vertical transmission, and treatment. The most influential parameters are identified by sensitivity analysis. The proposed model asserts that malaria infections in humans are reduced by 41.51

A rough set is a method to approximate the decision classes in the dataset using only a single attribute set. The boundary region of the rough set finds objects whose precise classification into any of the decision classes is not possible and is the uncertainty region of the rough set. This work presents a framework for minimizing the uncertainty region of the rough set using multiple attribute subsets. We propose an uncertainty optimization-based rough set (UOBRS) to reduce the uncertainty region of a rough set. We give the application of UOBRS for feature subset selection and decision rule generation. The average classification accuracy found by the proposed algorithm is up to 96.82

Female Anopheles mosquitoes are the primary vectors responsible for malaria transmission. The control of malaria is essential for humankind. In this study, we introduce a nonlinear deterministic model to control malaria disease by deploying Wolbachia bacteria into the Anopheles mosquitoes. Further, we study the effect of Wol-bachia on temperature variation in tropical and subtropical regions of the world. We analyze the model for its positivity and boundedness with an initial condition in a specific set so that the model is well-defined mathe-matically and meaningful epidemiologically. We find the malaria-free equilibrium points of the model and analyze the local stability of all the equilibrium points. We calculate the basic reproduction number by the next -generation matrix method, which shows the stability of malaria-free equilibrium. Then we explore malaria present endemic equilibrium in both cases: Wolbachia-infected and Wolbachia-free possibilities. We observe that the number of infections is less in the Wolbachia-infected equilibrium case, and we find that Wolbachia can reduce malaria transmission in all temperature conditions. The sensitivity analysis of all different parameters is also calculated. We demonstrate the optimum temperature (from 22 degrees C to 28 degrees C) by numerical simulation where the malaria exposure and transmission rate is high. We can apply the work in all subtropical and tropical regions to control malaria transmission worldwide.

In this paper, we develop a collocation method for solving three-dimensional partial differential equations using Haar wavelet and Kronecker tensor product. The approach is based on a series of Haar wavelet basis functions to approximate sixth-order mixed derivatives. The proposed method is mathematically fast, less error and straightforward for the numerical solution of many types of three-dimensional Poisson, biharmonic and Helmholtz equations. Some numerical examples verify the accuracy and efficiency of the proposed method. Finally, we conclude that numerical results computed by our proposed method are more accurate than numerical results obtained in the existing methods in the literature. We find that the CPU time consumed by the suggested approach is lesser than the CPU time of existing methods. Thus, the process is fast, efficient and has a low numerical error.