Malaria continues to pose a significant global health challenge, with its persistent transmission creating major difficulties for healthcare systems worldwide. Tackling this problem calls for innovative and effective methods to enhance understanding and control of the disease. In this work, we proposed a fractional-order mathematical model to study the dynamics of malaria transmission, integrating essential control measures such as treatment of humans and management of mosquito populations. The model employed three different types of non-integer order differential operators: the Caputo operator, the Caputo-Fabrizio operator with exponential decay, and the Atangana-Baleanu operator with an extended Mittag-Leffler kernel. Using fixed-point theory, we proved the existence and uniqueness of solutions for the proposed model. Numerical simulations are carried out to assess the impact of varying fractional orders on the progression of the disease. The results revealed that increasing the fractional order slows down the spread of malaria, reduces the peak number of infections, and prolongs the duration of outbreaks highlighting the memory-dependent nature of fractional systems. Our findings demonstrated that fractional-order models offer a more accurate and flexible approach to capturing the complex dynamics of malaria transmission. The study underscores the importance of integrating both therapeutic interventions and vector control strategies in reducing disease burden. Based on the findings of this study, we recommended the integration of fractional order modeling into malaria control strategies, as it captures the memory effects and long-term dynamics of disease transmission more accurately than classical models. Public health programs should adopt combined intervention approaches incorporating both effective treatment and vector control measures to significantly reduce infection rates. Furthermore, control efforts should be sustained over time, as fractional models reveal that short-term interventions may not be sufficient in curbing prolonged outbreaks. Policymakers are encouraged to use insights from these models to design adaptive, data-driven strategies that enhance the efficiency and sustainability of malaria control programs.
Malaria remains one of the world's deadliest infectious diseases, caused by Plasmodium parasites and transmitted through the bites of female Anopheles mosquitoes. Despite decades of global efforts, malaria continues to pose a major public health threat, particularly in sub-Saharan Africa where conditions support year-round transmission. In 2023, there were an estimated 263 million malaria cases and approximately 597,000 deaths globally. Most of these deaths occurred among children under five, highlighting the vulnerability of this age group. The human impact of malaria goes beyond statistics. Millions of families face repeated illness, loss of income, and the emotional toll of preventable deaths. Social determinants such as poverty, poor housing, inadequate access to healthcare, and lack of education increase exposure and reduce the chances of early treatment. In many rural regions, communities still lack basic tools such as insecticide-treated bed nets and rapid diagnostic tests. Nonetheless, progress has been made. Since 2000, global malaria control initiatives have averted over 2.2 billion cases and 12.7 million deaths. However, progress has stalled in recent years due to emerging resistance to artemisinin-based therapies and insecticides, as well as underfunded health programs. Between 2022 and 2023 alone, malaria cases rose by 11 million, demonstrating a worrying reversal of previous gains.
Encouragingly, scientific breakthroughs are offering renewed hope. Two WHO-approved malaria vaccines RTS,S/AS01 and R21/Matrix-M have been introduced into immunization programs across multiple African countries. These vaccines have been shown to significantly reduce severe malaria cases in young children. Additionally, interventions like seasonal chemoprevention, improved diagnostic tools, and next-generation insecticide-treated nets are strengthening malaria control. Despite these advances, substantial challenges remain. Artemisinin resistance is spreading in East Africa, jeopardizing the efficacy of front-line treatments. Climate change is expanding mosquito habitats, leading to increased transmission in previously low-risk areas. Furthermore, the global funding gap continues to limit the scale-up of life-saving interventions only about $4 billion was available in 2023, far short of the $8.1 billion required annually. To achieve the 2030 malaria targets, sustained investment, innovative tools, and community-based strategies will be essential.
Malaria remains a major public health concern in Nigeria, accounting for one of the highest burdens of the disease globally. In 2021, Nigeria alone contributed 26.6% of global malaria cases and 31.3% of deaths, highlighting the country's critical role in the global malaria fight. The disease is endemic throughout Nigeria, with seasonal variations in transmission intensity influenced by ecological zones and rainfall patterns. Despite the availability of control strategies such as insecticide-treated nets (ITNs), indoor residual spraying (IRS), and artemisinin-based combination therapies (ACTs), malaria continues to exert significant socio-economic pressure on households and the healthcare system. Several factors hinder effective control, including drug resistance, poor health-seeking behavior, and inadequate funding for malaria programs. Moreover, gaps in healthcare infrastructure and limited access to diagnostic tools in rural communities exacerbate the challenge. Malaria prevalence remains highest among children under five and pregnant women, who face the greatest risk of severe disease and death. Climate variability and environmental changes have also been shown to influence vector population and disease transmission. Additionally, weak community participation and low usage of preventive measures like ITNs have undermined progress. Although progress has been made through various donor-supported initiatives, sustaining long-term control requires domestic investment and policy commitment. Recent modeling studies emphasize the need for integrated interventions, including environmental management and vaccination, to accelerate progress toward malaria elimination in Nigeria.
Yusuf et al. developed a fractional-order model to investigate the transmission dynamics of malaria using Caputo derivatives. The study incorporated key compartments such as susceptible, exposed, infectious, and recovered classes for both human and mosquito populations. The model was analyzed for existence, uniqueness, and stability of solutions. Numerical simulations demonstrated that the fractional-order system captured memory effects more realistically, showing slower convergence to equilibrium compared to the classical integer-order model.
Ngonhala et al. formulated a fractional-order malaria model that integrated environmental effects, including seasonal variation in mosquito biting rates. They applied the Atangana-Baleanu fractional derivative in the Caputo sense to account for the memory and hereditary properties of the disease. Their results indicated that fractional derivatives provided a more accurate fit to real data than classical models, and that incorporating environmental variability led to improved understanding of malaria outbreaks in endemic regions. Omame et al. proposed a fractional-order malaria model incorporating treatment, prevention, and relapse mechanisms. The model used the Caputo fractional operator and included control strategies such as bed net use and drug administration. Stability analysis was conducted using the Mittag-Leffler function, and numerical simulations showed that lower fractional orders led to prolonged disease persistence, emphasizing the importance of early intervention. Kumar and Agrawal analyzed a fractional malaria model using a generalized Mittag-Leffler kernel to reflect the memory effect of disease transmission. They introduced a new numerical method to solve the system and compared the results with real epidemiological data. Their findings showed that the model provided a better fit and richer dynamics compared to the classical models, particularly in representing long-term behavior. Abdelrazec et al. developed a fractional-order model focusing on the combined impact of vaccination and vector control strategies. They used Caputo-Fabrizio derivatives to eliminate singularities and observed smoother trajectories. Their simulation results suggested that fractional models with memory kernels provided more realistic predictions, and they recommended such models for guiding public health policies in malaria-endemic countries.
Sweilam et al. focused on optimal control of malaria using a single fractional operator, emphasizing the cost-effectiveness of intervention strategies. The recent work extended this by comparing Caputo, Caputo-Fabrizio, and Atangana-Baleanu operators, which improved upon Sweilam et al.'s model by linking different kernel structures to distinct epidemiological memory processes, thereby enhancing interpretability and practical applicability. Boulkroune et al. applied adaptive fuzzy control techniques to achieve fixed-time synchronization of fractional-order chaotic systems, demonstrating robustness and efficiency in engineering control problems. In contrast, the recent malaria study focused on comparing Caputo, Caputo-Fabrizio, and Atangana-Baleanu operators, thereby improving upon works like Boulkroune et al. by linking the choice of fractional kernel to specific biological memory processes. While the fuzzy control study advanced synchronization performance in chaotic systems, the recent work advanced interpretability in epidemiological modeling, offering practical insights into how memory effects shape intervention outcomes in malaria control. Zouarin developed a neural network-based adaptive control for chemotherapy, ensuring stability through Lyapunov methods. The recent malaria study improved upon this by using fractional-order models with Caputo, Caputo-Fabrizio, and Atangana-Baleanu derivatives to better capture complex dynamics and prove global stability. While Zouari focused on cancer treatment, the malaria study applied advanced mathematical methods to epidemiological modeling. Gassem et al. introduced a generalized fractional derivative framework using power non-local kernels, unifying Caputo-Fabrizio, Atangana-Baleanu, and Hattaf derivatives, focusing on theoretical modeling of nonlinear fractional systems. The recent malaria study improved on this by applying fractional-order modeling to real-world malaria transmission, using Caputo, Caputo-Fabrizio, and Atangana-Baleanu operators. It showed how varying fractional orders affected disease progression, proved existence and uniqueness of solutions, and provided insights for treatment and vector control strategies. This study bridged the gap between theoretical fractional calculus and practical epidemiological applications.
Khan et al. showed how memory effects improve epidemic modeling. The new malaria study went further by comparing multiple operators, incorporating treatment and mosquito control strategies, and linking results to public health policies, making it more comprehensive and practically relevant. Khan et al. used a single modified-ABC fractional derivative to establish theoretical results for smoking behavior, focusing on existence, uniqueness, and stability. The new malaria study advanced further by comparing multiple fractional operators, integrating treatment and mosquito control, and linking results directly to public health applications, making it more comprehensive and policy-relevant. Alzabut et al. focused on a discrete-time fractional system using a Caputo-type operator, where existence, uniqueness, and synchronization were proven and validated with numerical results. The new malaria study also established existence and uniqueness through fixed-point theory but advanced further by comparing multiple fractional operators, integrating treatment and mosquito control, and demonstrating through simulations how fractional orders affected malaria outbreaks, making it more comprehensive and policy-relevant.
Khan et al. developed a fractal fractional model for tuberculosis and proved the existence of solutions, showing that such models can capture the persistence and long-term effects of the disease more accurately than classical approaches. Similarly, Eiman et al. studied a rotavirus infection model using a piecewise modified ABC fractional derivative, demonstrating how different operators influence disease dynamics and control strategies. In another application, Shah proposed a fractal-fractional model for multiple sclerosis, a chronic disease, to highlight how fractional operators account for long-term memory effects inherent in progressive illnesses. Abidemi et al. developed non-fractional and fractional-order models to examine Lassa fever transmission, including nosocomial infections. The authors demonstrated that increasing the fractional order slowed disease spread, lowered peak infections, and prolonged outbreaks. They highlighted the importance of memory effects and suggested that fractional-order models could improve predictions and public health strategies.
Abidemi et al. proposed a nonlinear model of Lassa fever transmission with vertical transmission and nonlinear incidence rates. The study showed that vertical transmission and hygiene practices significantly influenced outbreak dynamics. The model identified the basic reproduction number and offered insights for targeted interventions to reduce transmission.
Shyamsunder et al. introduced a fractional-order model to investigate vaccination effects on COVID-19 dynamics. It incorporated memory effects and different vaccination stages. Simulations revealed that accounting for fractional-order memory improved prediction accuracy and demonstrated that vaccination strategies could more effectively control outbreaks. Abidemi et al. assessed Lassa fever dynamics with a focus on environmental sanitation using an optimal control approach. The study found that combined interventions, particularly sanitation and rodent control, were most effective and cost-efficient in reducing disease burden. The results provided guidance for designing practical, resource-efficient control strategies. Nisar et al. focused on typhoid fever and showed that mass vaccination effectively reduced disease spread. Abboubakar et al. studied typhoid fever in Cameroon, incorporating vaccination and calibrating models with real clinical data, identifying key factors influencing transmission. Nabi et al. modeled COVID-19, demonstrating that fractional derivatives captured memory effects and provided more accurate forecasts than classical integer-order models.
The choice of fractional operator carries biological significance in malaria modeling. The Caputo-Fabrizio operator, with its exponential decay kernel, is well suited for short-term processes such as the rapid decline of mosquito infectivity or the temporary impact of interventions. The Atangana-Baleanu operator, based on the Mittag-Leffler kernel, better captures long-range memory effects like waning immunity, relapse cycles, and seasonal vector dynamics. In contrast, the Caputo operator, with its power-law kernel, reflects cumulative and persistent effects characteristic of repeated exposures in endemic settings. This comparison underscores that different operators correspond to distinct epidemiological processes, enhancing both the interpretability and practical relevance of fractional-order models.
The model assumed constant total human and mosquito populations, homogeneous mixing between individuals, and no seasonal variation in mosquito dynamics. While these assumptions were necessary to focus on the influence of fractional operators and establish theoretical results, they simplify the biological complexity of malaria transmission. In reality, human and vector populations fluctuate, contact patterns are heterogeneous, and mosquito abundance is strongly affected by environmental and seasonal factors. Future research will aim to relax these assumptions by incorporating variable populations, seasonality, and spatial heterogeneity, thereby enhancing the biological realism and applicability of the model.
While the introduction of fractional orders increases model flexibility by capturing long-term dependencies and memory effects, it also raises practical challenges in real-world applications. Parameter estimation can become more difficult when working with limited or noisy epidemiological data, as fractional parameters often require numerical fitting rather than direct biological measurement. Although fractional-order models are computationally more demanding than classical integer-order models, modern numerical schemes make them tractable for simulation. Importantly, a carefully calibrated integer-order model may perform comparably in short-term predictions of intervention outcomes, particularly when data are sparse. The main strength of fractional models lies in their ability to represent delayed effects, long-term persistence, and cumulative memory processes, which are often underestimated by classical models. Thus, fractional models should be regarded as complementary tools: while classical models are useful for operational planning with limited data, fractional models provide deeper insight into the memory-driven dynamics of malaria when high-quality longitudinal data are available.
The fractional-order nature of the model allows it to capture memory-dependent effects inherent in malaria transmission. Specifically, the Caputo, Caputo-Fabrizio, and Atangana-Baleanu operators enable the model to account for the influence of past states on current dynamics. This feature reflects biological processes such as delayed immunity development in humans, time-lagged mosquito life cycles, and environmental factors that affect vector population persistence. Although these mechanisms are not explicitly modeled as separate compartments, the memory property of fractional derivatives effectively represents their cumulative impact on the progression and spread of malaria, providing a more realistic and biologically interpretable modeling framework.
This study is novel in its comparative use of three distinct fractional-order differential operators Caputo, Caputo-Fabrizio, and Atangana-Baleanu to model malaria transmission dynamics. Unlike previous models that often rely on a single operator or classical integer-order systems, this work uniquely explores how different memory kernels affect the trajectory of disease spread. Additionally, the integration of key control strategies such as treatment and mosquito management within a fractional-order framework provides a more flexible and realistic tool for analyzing long-term disease dynamics. The use of fixed-point theory to establish existence and uniqueness further adds mathematical rigor to the modeling approach. Traditional malaria models typically assume memory less transmission dynamics and instantaneous intervention effects, limiting their ability to capture the delayed and cumulative impacts of control measures. This study fills that gap by incorporating memory-dependent behavior using fractional calculus, thereby offering a more accurate representation of real-world malaria progression. Moreover, while some previous studies have used fractional models, few have analyzed and compared multiple fractional operators in the context of malaria with integrated intervention strategies. This study addresses that deficiency and provides new insights for long-term, adaptive public health planning.
The aim of this study is to develop and analyze a fractional-order mathematical model that captures the transmission dynamics of malaria while integrating key intervention strategies such as treatment and vector control. The objectives of the study are to formulate a novel malaria model using three different fractional derivatives Caputo, Caputo-Fabrizio, and Atangana-Baleanu operators to explore the role of memory and hereditary effects in disease progression. The study further seeks to establish the existence and uniqueness of the model's solution using fixed-point theory, to perform numerical simulations assessing the impact of varying fractional orders on malaria dynamics, and to evaluate how combined control measures influence infection rates and outbreak duration.