Deterministic Nonlinear Model Research Articles

Corrosion-induced thickness diminution of an ageing bulk carrier

The 2030 Agenda for Sustainable Development introduced 17 Sustainable Development Goals (SDGs). The International Maritime Organization (IMO) has endorsed connections between the maritime sector and SDGs, emphasising sustainable production in the marine environment. Consequently, technical considerations, such as ship structure sustainability, are crucial for marine safety. The impact of corrosion influences the structural degradation of vessels and the risk of structural failure and fuel oil spills significantly. The formulation of predictive corrosion models, grounded in historical data accumulated during vessel operations, emerges as an optimal strategy for designing structural elements and guaranteeing their structural integrity. This paper analyses non-linear deterministic and stochastic corrosion models, focusing on an ageing bulk carrier, to provide design-phase information for sustainable structures. Utilising data on measured steel plate thickness over a 20-year operational period, non-linear models are employed to calculate the millimetres of thickness diminution. Assuming corrosion onset at 7, 8, and 9 years, corresponding coefficients are computed, based on the developed model. A model is also considered, in which there were no assumptions about the beginning of corrosion initiation. The corresponding annual corrosion rate was observed separately in each of the models, as a deterministic variable and as a probabilistic random variable. This approach aims to enhance the understanding of the causal relationship between corrosion processes and structural performance, contributing to the development of sustainable design practices in the maritime industry.

Optimal control and cost-effectiveness analysis of nonlinear deterministic Zika virus model

Optimal control and cost-effectiveness analysis of nonlinear deterministic Zika virus model

Strategic approaches to mitigating Hookworm infection: An optimal control and cost-effectiveness analysis

Human hookworm infection remains a serious threat to public health, particularly in highly endemic regions. The high mortality rate associated with this infection emphasizes the urgent need for effective control measures and intervention strategies to curb its spread. A nonlinear deterministic hookworm model with saturated incidence is investigated in this paper. The model exhibits a unique disease-free and endemic equilibria, and the reproduction number is computed and explained. Sensitivity analysis shows that increasing the transmission rate, β, the hatching rate, α, the number of eggs excreted within the environment, Ne, and the rate of excretion of the eggs, γ, significantly increases the reproduction number. Based on this analysis, we extend the model to consider optimal control in the presence of three time-dependent controls namely: sanitation, preventative chemotherapy, and shoe-wearing. We define an objective function to be minimized and the conditions necessary for the optimal control are established and proved using Pontryagin’s maximum principle. We present a one-way analysis of variance to evaluate the impact of constant implementation of the control measures on the number of infected individuals. Numerical simulations also show that hookworm infection can be contained in the presence of all control measures. However, a cost-effectiveness analysis shows that combining shoe-wearing control with preventative chemotherapy is the most cost-effective measure for controlling hookworm infection. The results presented hold substantial implications for public health policy, especially in low-income regions.

Dynamical analysis of methicillin-resistant Staphylococcus aureus infection in North Cyprus with optimal control: prevalence and awareness

The number of Methicillin-resistant Staphylococcus aureus (MRSA) cases in communities and hospitals is on the rise worldwide. In this work, a nonlinear deterministic model for the dynamics of MRSA infection in society was developed to visualize the significance of awareness in interventions that could be applied in the prevention of transmission with and without optimal control. Positivity and uniqueness were verified for the proposed corruption model to identify the level of resolution of infection factors in society. Furthermore, how various parameters affect the reproductive number R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}$$\\end{document} and sensitivity analysis of the proposed model was explored through mathematical techniques and figures. The global stability of model equilibria analysis was established by using Lyapunov functions with the first derivative test. A total of seven years of data gathered from a private hospital consisting of inpatients and outpatients of MRSA were used in this model for numerical simulations and for observing the dynamics of infection by using a non-standard finite difference (NSFD) scheme. When optimal control was applied as a second model, it was determined that increasing awareness of hand hygiene and wearing a mask were the key controlling measures to prevent the spread of community-acquired MRSA (CA-MRSA) and hospital-acquired MRSA (HA-MRSA). Lastly, it was concluded that both CA-MRSA and HA-MRSA cases are on the rise in the community, and increasing awareness concerning transmission is extremely significant in preventing further spread.

Unraveling the importance of early awareness strategy on the dynamics of drug addiction using mathematical modeling approach.

A drug is any substance capable of altering the functioning of a person's body and mind. In this paper, a deterministic nonlinear model was adapted to investigate the behavior of drug abuse and addiction that incorporates intervention in the form of awareness and rehabilitation. In the mathematical analysis part, the positivity and boundedness of the solution and the existence of drug equilibria have been ascertained, which shows that the model consists of two equilibria: a drug-free equilibrium and a drug endemic equilibrium point. The drug-free equilibrium was found to be both globally and locally asymptotically stable if the effective reproduction number is less than or equal to one (Rc≤1). Furthermore, we were able to show the existence of a unique drug endemic equilibrium whenever Rc>1. Global asymptotic stability of a drug endemic equilibrium point has been ascertained using a nonlinear Lyapunov function of Go-Volterra type, which reveals that the drug endemic equilibrium point is globally asymptotically stable if an effective reproduction number is greater than unity and if there is an absence of a reversion rate of mended individuals (i.e., ω=0). In addition, an optimal control problem was formulated to investigate the optimal strategy for curtailing the spread of the behavior using control variables. The control variables are massive awareness and rehabilitation intervention of both public and secret addicted individuals. The optimal control simulation shows that massive awareness control is the best to control drug addiction in a society. In sensitivity analysis section, the proportion of those who are exposed publicly shows to be a must sensitive parameter that can reduce the reproduction number, and the effective contact rate shows to be a must sensitive parameter to increase the reproduction number. Numerical simulations reveal that the awareness rate of exposed publicly and the rehabilitation rate of addicted publicly are very important parameters to control drug addiction in a society.

Dynamical Analysis of Peptic Ulcer Disease Model in Nigeria with the Effect of Vaccination and Treatment Program

The study developed a non-linear deterministic mathematical model to investigate the transmission dynamics of peptic ulcer disease in human population by considering both direct and indirect contact transmission with vaccine and treatment as control. In developing the model, the population was compartmentalized into susceptible-vaccinated-exposed-infected-treated-helicobacter pylori concentration in the environment recovered. The model developed is a system of differential equations. The rate of change of the system, existence and uniqueness of solution, region of absolute stability and positivity of the solution was established. Existence of disease-free equilibrium state and basic reproduction number was also established. Mathematical analysis was determined by the basic reproduction number if , the disease-free equilibrium is locally asymptotically stable whereas if then the equilibrium is unstable whereas the numerical analysis for the estimated basic reproduction number of the model is greater than unity in perspective of Nigeria, for which the Herd Immunity Threshold indicated that vaccinating 99.47% of Nigeria population can control spreading of Peptic ulcer disease in the country. The paper recommended amongst other things that if significant changes concerning the issue of threshold target are observed to improve the rate at which peptic ulcer can be minimized from the population where Government and non-governmental organization should encourage the use of peptic ulcer drugs and vaccines among individuals.

All Road User Casualties (Killed) in Great Britain from 1926. Linear and Nonlinear Trends with Persistent Data

This paper deals with the analysis of road casualties in Great Britain, using annual data since 1926. Based on the persistent nature of the data, fractional integration methods are used that include linear and non-linear (structural breaks) models. The results indicate that when the whole data set is employed the series is nonstationary I(1) implying permanency of shocks. However, considering data starting in 1964 we observe a significant negative time trend along with a lower degree of integration that implies transitory shocks. In order to avoid the abrupt change produced by the break, a nonlinear deterministic trend model based on Chebyshev polynomials in time is also considered with the whole sample, and though the order of integration is much lower than 1, the unit root null hypothesis cannot yet be rejected.

Deterministic and Stochastic Nonlinear Model for Transmission Dynamics of COVID-19 with Vaccinations Following Bayesian-Type Procedure

We develop a mathematical model for the SARAS-CoV-2 double variant transmission characteristics with variant 1 vaccination to address this novel aspect of the disease. The model is theoretically examined, and adequate requirements are derived for the stability of its equilibrium points. The model includes the single variant 1 and variant 2 endemic equilibria in addition to the endemic and disease-free equilibria. Various approaches are used for the global and local stability of the model. For both strains, we determine the basic reproductive numbers R1 and R2. To investigate the occurrence of the layers (waves), we expand the model to include some analysis based on the second-order derivative. The model is then expanded to its stochastic form, and numerical outcomes are computed. For numerical purposes, we use the nonstandard finite difference method. Some error analysis is also recorded.

Unveiling the Complexity of HIV Transmission: Integrating Multi-Level Infections via Fractal-Fractional Analysis

This article presents a non-linear deterministic mathematical model that captures the evolving dynamics of HIV disease spread, considering three levels of infection in a population. The model integrates fractal-fractional order derivatives using the Caputo operator and undergoes qualitative analysis to establish the existence and uniqueness of solutions via fixed-point theory. Ulam-Hyer stability is confirmed through nonlinear functional analysis, accounting for small perturbations. Numerical solutions are obtained using the fractional Adam-Bashforth iterative scheme and corroborated through MATLAB simulations. The results, plotted across various fractional orders and fractal dimensions, are compared with integer orders, revealing trends towards HIV disease-free equilibrium points for infective and recovered populations. Meanwhile, susceptible individuals decrease towards this equilibrium state, indicating stability in HIV exposure. The study emphasizes the critical role of controlling transmission rates to mitigate fatalities, curb HIV transmission, and enhance recovery rates. This proposed strategy offers a competitive advantage, enhancing comprehension of the model’s intricate dynamics.

Mathematical insights of social media addiction: fractal-fractional perspectives

The excessive use of social media has become a growing concern in the current century, with dependence on these platforms developing into a complex behavioral addiction. Addressing this issue requires the employment of well-directed and inclusive efforts. In pursuit of continuous development in existent strategies, this article presents a non-linear deterministic mathematical model that encapsulates the dynamics of social media addiction within a population. The proposed model incorporates the fractal-fractional order derivative in the sense of the Caputo operator. The objectives of this research are attained by groping the dynamics of the social media addiction model through the stratification of the population into five compartments: susceptible individuals, exposed individuals, addicted individuals, recovery individuals, and those who have quit using social media. The validity of the devised model is established by proving the existence and uniqueness of the solution within the framework of the fixed-point theory. The Ulam-Hyer’s stability is established through nonlinear functional analysis, perturbing the problem with a small factor. Utilizing the Adam Bashforth numerical scheme, we obtain numerical solutions, which we validate through MATLAB simulations. Additionally, we explore the application of artificial neural networks (ANNs) to approximate solutions, presenting a significant innovation in this domain. We propose the adoption of this novel method for solving integral equations that elucidate the dynamics of social media addiction, surpassing traditional numerical methods. Numerical results are illustrated across various fractional orders and fractal dimensions, with comparisons made against integer orders. Our study indicates that ANN outperforms the Adams-Bashforth algorithm, offering a dependable approach to problem-solving. Throughout the article, we underscore the competitive advantage of our proposed strategy, providing a more nuanced understanding of the complex dynamics outlined in the model.

A Mathematical modelling for co-infection dynamics of Japanese encephalitis-Dengue and influence of JE Vaccine on Dengue disease

A non-linear deterministic mathematical model has been described with the co-infection dynamics of Japanese encephalitis(JE) and dengue disease, incorporating the JE vaccine. A basic reproduction number is discussed to study transmission potential of the co-infection model. In co-infection model, disease-free equilibrium points of Japanese encephalitis and dengue, along with endemic, are presented in system and investigated their stability with the help of their specified method. Our analysis suggests that vaccination against Japanese encephalitis positively affect control of co-infection. By using Center Manifold Theory, the model undergoes backwards bifurcation phenomenon and this has been occurred when basic reproduction number is smaller than unity. It is shown that by taking simultaneous preventive steps, the basic reproduction number of co-infection can be reduced to less than one after eliminating both infections. Sensitivity analysis has been performed to determine which parameters significantly affect disease dynamics. The effects of these parameters on transmission of disease were investigated using a numerical simulation. According to the findings, we obtain that JE-dengue co-infection can be managed with use of JE vaccine, also minimize JE transmission rate rapidly.

LOCAL AND GLOBAL STABILITY ANALYSIS OF MEASLES EPIDEMIC MODEL AT DISEASE-FREE EQUILIBRIUM

In this study, a continuous mathematical model for the dynamics of Measles (rubeola) outbreak at constant recruitment rate was formulated. In the model, we partitioned the population into Susceptible (S), Vaccinated (V), exposed (E), Infected (I) and recovered (R) individuals. We analyzed a SVEIR compartmental nonlinear deterministic mathematical model of measles epidemic in a community with constant population. Analytical studies were carried out on the model using the method of linearized stability. The basic reproductive number R0 that governs the disease transmission is obtained from the largest eigenvalue of the next-generation matrix. The disease-free equilibrium is computed and proved to be locally and globally asymptotically stable if R0<1 and unstable if R0 >1 respectively. Finally, we simulate the model system in MATLAB and obtained the graphical behavior of each compartment. From the simulation, we observed that the measles infection was eradicated in the environment when R0<1.

Stability analysis of the corruption dynamics under fractional-order interventions

Abstract This article proposes a nonlinear deterministic mathematical model that encapsulates the dynamics of the prevailing degree of corruption in a population. The objectives are attained by exploring the dynamics of the corruption model under fractional-order derivative in the Caputo sense. The outcomes of the research are facilitated by stratifying the population into five compartments: susceptible class, exposed class, corrupted class, recovered class, and honest class. The developed model is validated by proving pivotal delicacies such as positivity, invariant region, basic reproduction number, and stability analysis. The Ulam–Hyers stability technique is used to prove the stable solution. The Adam–Bashforth numerical scheme is employed to estimate the numerical solution. Moreover, the research environment is further enriched by studying each compartment with respect to a wide range of relevant parametric settings. The realizations of this study indicate that susceptible individuals remain subject to being influenced by corrupt individuals. In addition, it is observed that the population of exposed individuals, recovered individuals, and honest individuals asymptotically approach toward the corruption equilibrium point, whereas the magnitudes of susceptible individuals and corrupted individuals decrease asymptotically to the corruption equilibrium state. The compartment dynamics are witnessed to be sensitive for various fractional-orders indicating the utility of the fractional approach. The findings of this study support the fundamental understanding of conceptualizing corruption in accordance with the viral transmission of infectious disease.

Dynamic Relationship Between Informal Sector and Unemployment: A Mathematical Model

Shortage of formal jobs, lack of skills in workforce and increasing human population proliferate the informal sector. This sector provides an opportunity to unskilled workers to gain skills along with earnings. In this paper, a deterministic nonlinear mathematical model is developed to study the effects of informal skill learning and job generation on unemployment. For the formulated system, feasibility of equilibria and their stability properties are discussed. A pertinent quantity ([Formula: see text]), known as the reproduction number, is calculated and it is shown that the formulated system undergoes transcritical, saddle-node, Hopf and Bogdanov–Takens bifurcations on the variation of [Formula: see text]. The analytically obtained results are validated through numerical simulations. The results obtained from this study indicate that a substantial rate of job generation by self-employed individuals has a stabilizing effect on the system. Moreover, self-employment along with informal skill acquisition through engaging in informal work proves to be an effective measure in curbing the issue of unemployment in society.

Computational analysis of corruption dynamics insight into fractional structures

The fractional derivative that is used to compute the solution of the corruption system with Power-Law Kernel, Mittag-Leffler Kernel, and Exponential Decay Kernel. It is important to study and analyse corruption dynamics, because it is an act that has a direct effect on public rights, and because of this the right of the rightful owner, just got destroyed. Using hypothesis theory for differential equation, this work suggests and assesses a nonlinear deterministic model for the dynamics of corruption. Positivity and boundedness are verified for the proposed corruption model to identify the level of resolution of corruption factor in society. Fractional-order corruption model is investigated with different kernels for efficient results. The necessary criteria for the best control of corruption transmission were identified using Pontryagin's maximal concept. The numerical simulation showed that corruption must be resisted by an integrated control strategy. Numerical simulations are used to demonstrate the correctness of the proposed approaches. Finally, simulations are derived for the proposed schemes to check the effectiveness of the results and to analyse the corruption behaviour in society as well as dynamically highlight the propagation of corruption group.

Pneumonia and COVID-19 co-infection modeling with optimal control analysis

In this study, we present a nonlinear deterministic mathematical model for co-infection of pneumonia and COVID-19 transmission dynamics. To understand the dynamics of the co-infection of COVID-19 and pneumonia sickness, we developed and examined a compartmental based ordinary differential equation type mathematical model. Firstly, we showed the limited region and non-negativity of the solution, which demonstrate that the model is biologically relevant and mathematically well-posed. Secondly, the Jacobian matrix and the Lyapunov function are used to illustrate the local and global stability of the equilibrium locations. If the related reproduction numbers R0c, R0p, and R0 are smaller than unity, then pneumonia, COVID-19, and their co-infection have disease-free equilibrium points that are both locally and globally asymptotically stable otherwise the endemic equilibrium points are stable. Sensitivity analysis is used to determine how each parameter affects the spread or control of the illnesses. Moreover, we applied the optimal control theory to describe the optimal control model that incorporates four controls, namely, prevention of pneumonia, prevention of COVID-19, treatment of infected pneumonia and treatment of infected COVID-19. Then the Pontryagin's maximum principle is introduced to obtain the necessary condition for the optimal control problem. Finally, the numerical simulation of optimality system reveals that the combination of treatment and prevention is the most optimal to minimize the diseases.

Unveiling the dynamics of drug transmission: A fractal-fractional approach integrating criminal law perspectives

<abstract><p>The excessive use of drugs has become a growing concern in the current century, with the global toll of drug-related deaths and disabilities posing a significant public health challenge in both developed and developing countries. In pursuit of continuous improvement in existing strategies, this article presented a nonlinear deterministic mathematical model that encapsulates the dynamics of drug addiction transmission while considering the legal implications imposed by criminal law within a population. The proposed model incorporated the fractal-fractional order derivative using the Atangana-Baleanu-Caputo ($ \mathbb{ABC} $) operator. The objectives of this research were achieved by examining the dynamics of the drug transmission model, which stratifies the population into six compartments: The susceptible class to drug addicts, the number of individuals receiving drug misuse education, the count of mild drug addicts, the population of heavy-level drug addicts, individuals subjected to criminal law, and those who have ceased drug use. The qualitative analysis of the devised model established the existence and uniqueness of solutions within the framework of fixed-point theory. Furthermore, Ulam-Hyer's stability was established through nonlinear functional analysis. To obtain numerical solutions, the fractional Adam-Bashforth iterative scheme was employed, and the results were validated through simulations conducted using MATLAB. Additionally, numerical results were plotted for various fractional orders and fractal dimensions, with comparisons made against integer orders. The findings underscored the necessity of controlling the effective transmission rate to halt drug transmission effectively. The newly proposed strategy demonstrated a competitive advantage, providing a more nuanced understanding of the complex dynamics outlined in the model.</p></abstract>

Mathematical Analysis of COVID-19 model with Vaccination and Partial Immunity to Reinfection

COVID-19 is an infectious respiratory disease caused by a new virus, called SARS-CoV-2. Since itsinception, it has been a major cause of deaths and illnesses in the general population across the globe. Inthis paper, we have formulated and theoretically analyzed a non-linear deterministic model for COVID-19transmission dynamics by incorporating vaccination of the susceptible population. The system properties,such as the boundedness of solutions, the basic reproduction number R0, the local stability of disease-freeequilibrium(DFE), and endemic equilibrium (EE) points, are explored. Besides, the Lyapunov function isutilized to prove the global stability of both DFE and EE. The bifurcation analysis was carried out by utilizingthe center manifold theory. Then, the model is fitted with real COVID-19 cumulative data of infected casesin Kenya as from March 30, 2020, to March 30, 2022. Furthermore, sensitivity analysis was performed forthe proposed model to ascertain the relative significance of model parameters to COVID-19 transmissiondynamics. The simulations revealed that the spread of COVID-19 can be curtailed not only via vaccinationof susceptible populations but also increased administration of COVID-19 booster vaccine to the vaccinatedpersons and early detection and treatment of asymptomatic individuals.

Global stability analysis of transmission dynamics in a compartmental model of coronavirus disease with environmental reservoirs

This work proposes and analyzes a non-linear deterministic model which describes the outbreak of Coronavirus disease with fomites (environmental reservoirs) propagating the spread of the infection in human population. Suitable Lyapunov functions are constructed to demonstrate the Global Asymptotic Stability (GAS) of both the Endemic Equilibrium (EE) and Disease Free Equilibrium (DFE) of the system. The EE was also found to be epidemiologically relevant when the basic reproduction number of new infections is greater than unity.

Modeling and Optimizing Multi-Faceted Strategies for HPV Control: Insights from a Nonlinear Mathematical Approach

This study presents a nonlinear deterministic mathematical model of Human Papillomavirus (HPV) dynamics, emphasizing qualitative analysis through the stability theory of differential equations. Our investigation focuses on validating the existence and stability of disease-free and endemic equilibrium points using the basic reproduction number governing HPV transmission. Notably, the model exhibits a backward bifurcation, prompting a sensitivity analysis. We address the challenge of HPV control by formulating an optimal control problem, applying the Pontryagin maximum principle with three distinct strategies: prevention, treatment, and screening. Numerical simulations reveal that a combined approach involving prevention, screening, and treatment emerges as the most effective strategy for eradicating HPV within the community. In conclusion, this study offers a comprehensive mathematical model for HPV dynamics and underscores the importance of multi-faceted intervention strategies. Our findings have significant implications for public health initiatives aiming to combat HPV infections and reduce their impact on communities.