Fractional Dynamics of a Measles Epidemic Model

Fractional order modeling of hepatitis B virus transmission with imperfect vaccine efficacy

This study aims to develop and analyze a model of hepatitis B virus transmission dynamics using integer and fractional derivatives in the Caputo sense. After formulating the models, we conduct an asymptotic stability analysis of the disease-free equilibrium point of both models. The Lyapunov technique demonstrates that under specific conditions, the disease-free equilibrium point in both models remains globally asymptotically stable. The study demonstrates that both models can have at least one endemic equilibrium when [Formula: see text], using the vaccination coverage parameter to identify positive equilibrium points. The Banach contraction principle is used to establish the uniqueness and existence of each fractional model's solutions, followed by demonstrating their global stability using the Ulam-Hyers technique. The model is calibrated using reported hepatitis B cases in Nigeria, allowing for parameter estimations. The study indicates that the disease is endemic in this country, as [Formula: see text], indicating a higher level of endemicity. The Adams-Bashforth approach is used to develop a numerical scheme, which is then validated through numerical simulations and evaluated under fractional order parameter variations.

A numerical investigation of fractional measles epidemic model using Chebyshev spectral collocation method

This paper investigates a fractional-order SEIR epidemic model for measles formulated as a system of four first-order differential equations using the spectral collocation technique based on the Chebyshev polynomials. The influencing parameters on each compartment is established using PRCC approach. Moreover, the key epidemiological parameters such as the drug therapy, recovery, infection and conversion rate, are thoroughly analyzed to evaluate their influence on the disease's spread. The findings are illustrated through tables and figures. This study also underscores the effectiveness and accuracy of the spectral method, offering valuable insights into the control and understanding of measles epidemics.

Fractional-Order Epidemic Model for Measles Infection

In this study, a nonlinear dynamic SEVIQR measles epidemic model is constructed and analyzed using the novel Caputo fractional-order derivative operator. The model's existence and uniqueness are established. In addition, the model equilibria are determined, and the novel Jacobian determinant method recently constructed in the literature of epidemiological modeling of infectious diseases is applied to determine the threshold quantity, ℛ 0. Furthermore, we construct appropriate Lyapunov functions to establish the global asymptotic stability of the disease-free and endemic equilibrium points. Finally, the numerical solution of the model is executed employing the efficient and widely known Adams-type predictor-corrector iterative scheme, and simulation is conducted to investigate the impact of memory index and diverse preventive measures on the occurrence of the disease. Numerical simulation of the model indicates that quarantine, vaccination, and treatment can reduce the numbers of infectious and exposed populations, thereby controlling the disease. Therefore, it is recommended that the government provide financial assistance for vaccine distribution.

Dengue dynamics in Nepal: A Caputo fractional model with optimal control strategies

An infectious disease called dengue is a significant health concern nowadays. The dengue outbreak occurred with a single serotype all over Nepal in 2023. In the tropical and subtropical regions, dengue fever is a leading cause of sickness and death. Currently, there is no specified treatment for dengue fever. Avoiding mosquito bites is strongly advised to reduce the likelihood of controlling this disease. In underdeveloped countries like Nepal, the implementation of appropriate control measures is the most important factor in preventing and controlling the spread of dengue illness. The Caputo fractional dengue model with optimum control variables, including mosquito repellent and insecticide use, investigates the impact of alternative control strategies to minimize dengue prevalence. Using the fixed point theorem, the existence and uniqueness of a solution will be demonstrated for the problem. Ulam-Hyers stability, disease-free equilibrium point stability, and basic reproduction number are studied for the proposed model. The model is simulated using a two-step Lagrange interpolation technique, and the least squares method is used to estimate parameter values using real monthly infected data. We then analyze the sensitivity analysis to determine influencing parameters and the control measure effects on the basic reproduction number. The Pontryagin Maximum Principle is used to determine the optimal control variable in the dengue model for control strategies. The present study suggests that the deployment of control measures is extremely successful in lowering infectious disease incidences. Which facilitates the decision-makers to practice rigorous evaluation of such an epidemiological scenario while implementing appropriate control measures to prevent dengue disease transmission in Nepal.

Modelling and stability analysis of the dynamics of measles with application to Ethiopian data

Measles, a highly contagious airborne disease, remains endemic in many developing countries with low vaccination coverage. In this paper, we present a deterministic mathematical compartmental model to analyze the dynamics of measles. We establish global stability conditions for both disease-free and endemic equilibria using the Lyapunov functional stability method. By using arbitrary parameters, we find that the proposed model exhibits forward bifurcation. To simulate the solution of the model for the forward problem, we perform numerical integration using MATLAB software. Moreover, we calibrate the model with real data from Ethiopia and estimate the parameters along with a 95 percent confidence interval (CI) by formulating an inverse problem. It is noteworthy that our model fits well with the actual data from Ethiopia. The estimated basic reproduction number (R 0 ) is determined to beR 0 = 1.3973 , demonstrating the endemic status of the disease. Additionally, our local sensitivity analysis indicates that reducing the transmission rate and increasing vaccination coverage can effectively minimizeR 0 .

Mathematical modeling and dynamics of immunological exhaustion caused by measles transmissibility interaction with HIV host

This paper mainly addressed the study of the transmission dynamics of infectious diseases and analysed the effect of two different types of viruses simultaneously that cause immunodeficiency in the host. The two infectious diseases that often spread in the populace are HIV and measles. The interaction between measles and HIV can cause severe illness and even fatal patient cases. The effects of the measles virus on the host with HIV infection are studied using a mathematical model and their dynamics. Analysing the dynamics of infectious diseases in communities requires the use of mathematical models. Decisions about public health policy are influenced by mathematical modeling, which sheds light on the efficacy of various control measures, immunization plans, and interventions. We build a mathematical model for disease spread through vertical and horizontal human population transmission, including six coupled nonlinear differential equations with logistic growth. The fundamental reproduction number is examined, which serves as a cutoff point for determining the degree to which a disease will persist or die. We look at the various disease equilibrium points and investigate the regional stability of the disease-free and endemic equilibrium points in the feasible region of the epidemic model. Concurrently, the global stability of the equilibrium points is investigated using the Lyapunov functional approach. Finally, the Runge-Kutta method is utilised for numerical simulation, and graphic illustrations are used to evaluate the impact of different factors on the spread of the illness. Critical factors that effect the dynamics of disease transmission and greatly affect the rate and range of the disease's spread in the population have been determined through a thorough analysis. These factors are crucial in determining the expansion of the disease.

Modelling and Analysis of a Measles Epidemic Model with the Constant Proportional Caputo Operator

Despite the existence of a secure and reliable immunization, measles, also known as rubeola, continues to be a leading cause of fatalities globally, especially in underdeveloped nations. For investigation and observation of the dynamical transmission of the disease with the influence of vaccination, we proposed a novel fractional order measles model with a constant proportional (CP) Caputo operator. We analysed the proposed model’s positivity, boundedness, well-posedness, and biological viability. Reproductive and strength numbers were also verified to examine how the illness dynamically behaves in society. For local and global stability analysis, we introduced the Lyapunov function with first and second derivatives. In order to evaluate the fractional integral operator, we used different techniques to invert the PC and CPC operators. We also used our suggested model’s fractional differential equations to derive the eigenfunctions of the CPC operator. There is a detailed discussion of additional analysis on the CPC and Hilfer generalised proportional operators. Employing the Laplace with the Adomian decomposition technique, we simulated a system of fractional differential equations numerically. Finally, numerical results and simulations were derived with the proposed measles model. The intricate and vital study of systems with symmetry is one of the many applications of contemporary fractional mathematical control. A strong tool that makes it possible to create numerical answers to a given fractional differential equation methodically is symmetry analysis. It is discovered that the proposed fractional order model provides a more realistic way of understanding the dynamics of a measles epidemic.