Fractional-order modeling of vaccination strategies for measles transmission incorporating immune memory.

Analysis of tuberculosis infection dynamics using Caputo fractional-order models with diagnosis and treatment interventions.

The newest infection is a novel coronavirus named COVID-19, that initially appeared in December 2019, in Wuhan, China, and is still challenging to control. The main focus of this paper is to investigate a novel fractional-order mathematical model that explains the behavior of COVID-19 in Ethiopia. Within the proposed model, the entire population is divided into nine groups, each with its own set of parameters and initial values. A nonlinear system of fractional differential equations for the model is represented using Caputo fractional derivative. Legendre spectral collocation method is used to convert this system into an algebraic system of equations. An inexact Newton iterative method is used to solve the model system. The effective reproduction number (R0) is computed by the next-generation matrix approach. Positivity and boundedness, as well as the existence and uniqueness of solution, are all investigated. Both endemic and disease-free equilibrium points, as well as their stability, are carefully studied. We calculated the parameters and starting conditions (ICs) provided for our model using data from the Ethiopian Public Health Institute (EPHI) and the Ethiopian Ministry of Health from 22 June 2020 to 28 February 2021. The model parameters are determined using least squares curve fitting and MATLAB R2020a is used to run numerical results. The basic reproduction number is R0=1.4575. For this value, disease free equilibrium point is asymptotically unstable and endemic equilibrium point is asymptotically stable, both locally and globally.

Infectious illnesses like hepatitis place a heavy cost on global health, and precise mathematical models must be created in order to understand and manage them. The Adomian decomposition method (ADM) and an optimal control strategy are utilized to solve a fractional-order hepatitis model in this research. By adding fractional derivatives to account for memory effects and non-integer order dynamics, the fractional-order model expands the conventional compartmental model to take into account the complexity of hepatitis dynamics. The fractional-order hepatitis model is resolved using the ADM, a powerful and effective analytical approach. This approach offers a series solution that converges quickly, enabling the model’s precise analytical solution to be derived. To identify crucial criteria and enhance control mechanisms for the management of hepatitis, an optimal solution strategy is also introduced. The optimization procedure tries to lessen the disease’s spread and its negative effects on public health. We can find the best interventions, immunization schedules, and treatment regimens to effectively reduce the hepatitis pandemic by integrating the ADM solution with an optimization framework. The findings of this study show that the suggested method may be used to solve the fractional-order hepatitis model and optimize control measures. The analytical solution produced by ADM offers important insights into the underlying dynamics of hepatitis transmission, and the optimization process produces suggestions that public health professionals and politicians may put into practice. In the end, this research presents a promising direction for improving disease control efforts in a fractional-order context and contributes to a deeper understanding of hepatitis epidemiology. The importance of this method is that it gives solutions that coincide with that obtained using the numerical approach.

Presently, monkeypox virus infection has spread worldwide in the ongoing outbreak that began in the UK. To study the transmission dynamics of monkeypox, we formulate here a seven-compartmental (five compartments for the human population and two compartments for animals or rodents) fractional-order mathematical model. The existence and uniqueness of the solution of the proposed fractional order model are examined here. The basic reproduction number for humans ( R0h ) and animals ( R0a ) are obtained through the next-generation matrix approach. Depending on the values of R0h and R0a , we observed that the fractional order model has three equilibria, namely, monkeypox-free equilibrium, animal-free endemic equilibrium, and endemic equilibrium. Also, the stability of all equilibria is checked in this present article. We found that the model goes through transcritical bifurcation at R0a=1 for any values of R0h and at R0h=1 for R0a<1 . Best of our knowledge, this is the first work where the fractional order optimal control for monkeypox is formulated and solved considering vaccination and treatment controls. Several feasible parameter values are used in the simulations to visualize and verify the findings, from which the results show that fractional order is more appropriate. Finally, parameters involved in the expression of R0h and R0a are scaled using the sensitivity index approach.

A fractional-order tuberculosis model with efficient and cost-effective optimal control interventions

The co-dynamics of COVID-19 and human Metapneumovirus (HMPV) pose a public health threat that can be caused by severe respiratory illness in vulnerable groups such as the elderly, children, and immune-weakened individuals. In this study, we present a mathematical model with the Caputo fractional derivative and use a semi-analytical Laplace-Adomian decomposition method (LADM) to obtain the approximate solutions and simulate the co-dynamics of two respiratory pathogens. The results are validated with real COVID-19 data for Bangladesh, revealing that the fractional-order model demonstrates optimal agreement at $$\alpha =0.83$$ . The analysis highlights the impact of fractional-order dynamics on transmission rates, quarantine efficacy, and recovery trajectories. The study advances by integrating memory effects and providing a framework for evaluating intervention strategies. The results of this study suggest that the fractional-order model provides a more flexible framework with memory effects for multiple respiratory disease outbreaks.

Hepatitis B virus (HBV) remains a major global health challenge due to its high transmission potential, chronic progression, and significant morbidity. In this study, we formulate and analyze a novel fractional-order compartmental model for Hepatitis B dynamics, incorporating Atangana–Baleanu–Caputo (ABC) derivatives and a Beddington–DeAngelis-type incidence rate. The model accounts for preventive awareness efforts among susceptible individuals, as well as the suppressive impact of treatment on infected populations. Rigorous mathematical analysis is carried out, including the positively invariant region, existence and uniqueness of solutions, equilibrium points, local and global stability, and the computation of the basic reproduction number $${R}_{0}$$ using the next-generation matrix approach. Bifurcation analysis reveals the occurrence of a forward bifurcation at $${R}_{0}=1$$ , highlighting the transition from a disease-free state to an endemic state. A sensitivity analysis identifies key parameters influencing $${R}_{0}$$ , particularly the transmission rates from acutely and chronically infected individuals, and the rates of government-led awareness campaigns. Numerical simulations, based on a novel iterative scheme for fractional differential equations, demonstrate the significant role of memory effects in HBV transmission dynamics. Comparative results with the classical integer-order model show that the ABC fractional model more effectively captures the hereditary and long-term memory effects of the disease. The findings suggest that strengthening awareness efforts and improving their effectiveness are as critical as medical interventions in reducing Hepatitis B prevalence. This work provides both theoretical insights and practical guidance for policymakers, offering a comprehensive framework for the control and long-term management of Hepatitis B infection.

1 - Epidemic theory: Studying the effective and basic reproduction numbers, epidemic thresholds and techniques for the analysis of infectious diseases with particular emphasis on tuberculosis

Investigation of the dynamics of COVID-19 with a fractional mathematical model: A comparative study with actual data

This article studies the dynamical behavior of the analytical solutions of the system of fraction order model of HIV‐1 infection. For this purpose, first, the proposed integer order model is converted into fractional order model. Then, Laplace‐Adomian decomposition method (L‐ADM) is applied to solve this fractional order HIV model. Moreover, the convergence of this method is also discussed. It can be observed from the numerical solution that (L‐ADM) is very simple and accurate to solve fraction order HIV model.

The socio-economic burdens of onchocerciasis have prompted the formulation of several mathematical models to better comprehend the epidemic. However, existing models either use integer-order derivatives, which often do not capture the memory and non-local effects seen in infectious diseases, or fractional order with singularity kernels, which may inadequately represent memory effects due to their singularity kernels. Onchocerciasis has a prolonged incubation and slow progression, making past conditions impactful on the disease's current and future course. Fractional derivatives effectively capture this memory effect, providing a more realistic depiction of the infection dynamics than integer-order models. We propose a non-local, non-singular exponential kernel fractional-order onchocerciasis model in the Caputo-Fabrizio fractional derivative sense to capture the disease's memory effects. Our model incorporates early treatment of exposed individuals as a critical intervention parameter, and vector management strategies are also incorporated. Using fixed-point theorem and iterative methods, we establish the existence and uniqueness of solutions, derive conditions for onchocerciasis-free and endemic equilibrium points, and analyze their stability, confirming the model's biological feasibility. Numerical simulations are conducted using a three-step fractional Adams-Bashforth method. Sensitivity analyses indicate that vector management and early treatment effectively reduce the effective reproduction number, while increases in the human-to-vector contact rate elevate it. Numerical results demonstrate that early treatment and vector management can significantly control onchocerciasis. The fractional-order "memory effect" highlights the importance of continuous monitoring and consistent application of control measures to reduce the memory index and curb onchocerciasis prevalence over time.

Banana bunchy top disease (BBTD) significantly threatens banana production, considerably endangering food safety and security. Aphid vectors and the use of latently infected planting materials disseminate the disease. This paper uses a deterministic mathematical model to examine the BBTD dynamics while considering the Banana Bunchy Top Virus (BBTV)-resistance of the planting material. After model formulation, we establish the positivity and boundedness of the model solution. We derived the effective reproduction number via the next-generation matrix approach and used it to investigate the asymptotic stability of the model equilibrium points using the Lyapunov function. To support the stability results, we conducted a bifurcation analysis. The bifurcation analysis confirmed a forward bifurcation, implying that the disease-free equilibrium point is stable when the effective reproduction number is less than one and unstable when the effective reproduction number is greater than one. The endemic equilibrium point is also stable when the effective reproduction number is greater than one and unstable otherwise. Finally, we apply the fourth-order Runge-Kutta method to simulate the proposed model. One limitation of our research is the need for real data to support our findings. in this instance, we used simulated data from earlier studies to conduct numerical simulations in this study. The results revealed that replanting with BBTV-resistance planting material while the rate of removing symptomatic infected plants is reduces the number of latent and symptomatic infected banana plants by and , respectively, in two years. Moreover, it was observed that increasing the rate of roguing to and replanting with BBTV-resistant planting material remaining at cleared the diseased plants in 10 months. Hence, it eliminates the disease. Therefore, the numerical simulation results suggest that while virus-resistant planting materials alone can reduce disease prevalence, they are most effective when combined with a timely roguing strategy. The results indicate that increasing the number of resistant plants beyond a certain threshold can lead to disease elimination. It is recommended that scientists provide farmers with reliable BBTV-resistant planting material and farming education on the safe way to rogue infected plants and replant.

Arbitrary order calculus to model real phenomena has been applied to various fields such as physics, chemisty, biology, etc. Therefore, more and more researchers prefer to use fractional order to describe infectious disease models. This article mainly discusses the dynamics of one susceptible-exposed-infected-recovered (SEIR) model of fractional order in Caputo sense. The premise of this model is that there is no vaccination. Recovered person will lose immunity and then return to susceptible group after a period of time. The approximate solution will be obtained with Laplace Adomian decomposition method (LADM) which has been proved to be an effective and reliable approach. Approximate results can be obtained through fewer iterations, which shows the effectiveness and simplicity of the LADM. From the graphical results it is suggested the flexibility and practicability of fractional differential. It is also indicated that universal vaccination in time is essential, otherwise the number of infected people is likely to continue to increase for a long time.

A next-generation matrix approach using Routh–Hurwitz criterion and quadratic Lyapunov function for modeling animal rabies with infective immigrants

The highly infectious Marburg virus (MARV) spreads rapidly through contact with infected individuals and improperly handled deceased bodies. Although burial and cremation practices have been examined for other pathogens, their role in MARV transmission remains largely unquantified. This study introduces a novel fractional-order compartmental model that explicitly incorporates burial, cremation, and awareness-based interventions to assess their combined impact on MARV control. The model employs the Caputo fractional derivative to capture memory effects in disease progression, offering a more realistic representation of transmission dynamics compared to traditional integer-order models. The basic reproduction number is derived via the next-generation matrix (NGM) approach, and the local asymptotic stability of the Marburg-free equilibrium is established. The Marburg present equilibrium point is obtained, and its global stability is rigorously proved using a Lyapunov function framework. Furthermore, bifurcation analysis is conducted to explore qualitative changes in system dynamics and confirm the occurrence of a forward bifurcation at $$\textrm{R}_0=1$$ . Global sensitivity analysis identifies the most influential parameters affecting disease spread, highlighting that increasing awareness and reducing contact with infectious or deceased individuals markedly suppress transmission. Theoretical contributions include proofs of positivity, boundedness, existence, and uniqueness of solutions, and Ulam–Hyers stability. To validate the proposed approach, results from the generalized Euler method (GEM) are compared with those obtained using Runge–Kutta methods for integer-order models and the Adams-Bashforth-Moulton (ABM) scheme for fractional-order models, showing strong agreement. The findings underscore the novel application of fractional-order modeling to MARV, demonstrating its superior capability to capture long-term effects and guide culturally sensitive, evidence-based public health strategies for outbreak mitigation.