Disease-free Equilibrium Point Research Articles

Mathematical Model and Optimal Control Strategy for the Dynamics of Hepatitis B Virus Disease Incorporating Treatment Failure and Advanced Stage Compartments

In this paper, mathematical model for the dynamic of Hepatitis B virus infection was developed taking into cognizance the effect of rate of progression to the end stage (cirrhosis and HCC) by individuals who experience treatment failure in the treatment class and chronically infected class. In order to track the transmission of these diseases, the basic reproduction number which determines the rate of new secondary infection when an infected individual is introduced into a totally susceptible population was computed by next generation matrix method (NGM). The stability of the disease-free equilibrium point was also investigated using Routh-Hurwitz Criterion and the result proven to be Locally Asymptotically Stable (LAS) when the basic reproduction number less than unity. The global stability was also investigated by Castillo-Chavez method and the results show that the model is Globally Asymptotically (GAS) when reproduction number is less than unity. Sensitivity analysis was carried out on the parameters of the reproduction number, using forward normalized sensitivity index to ascertain those parameters of high impact on the reproduction number to aid the incorporation of the control variables appropriately. Numerical simulation was also carried using Forward-backward Runge-Kutta fourth order scheme to study the effect of these control strategies on the dynamics of the model and the results shows that the best strategy to implement to help eradicate HBV disease in the system is strategy E which is the combination of effective condom use, HBV vaccination, HBV Treatment and Behavioral change

Mathematical Modeling of Infectious Disease and Prey-Predator Interaction with Optimal Control

In this paper, the impact of viral illnesses on the predator-prey relationship with an optimal control analysis is studied. An ecoepidemiological model of four compartments, namely, susceptible prey, susceptible predator, infected prey, and infected predator populations, in the interaction of the prey-predator system is formulated. The fundamental tenet of our ecoepidemiology model is that sick predators do not engage in predation. It is confirmed that the system’s solution exists, is positive, and is bounded. The system’s equilibrium points are determined and computed. Lyapunov functions and a linearizing form are used for local and global stability analysis, respectively. The next generation matrix approach is used to calculate the threshold value for diseased predators and prey at the disease-free equilibrium point. Optimal treatment options for vulnerable and infected populations are established by applying optimal control theory to the ecoepidemiology model of a prey-predator system. MATLAB software is utilized to obtain numerical simulations that validate the analytical outcomes. The optimal control problem simulations demonstrate that the number of infected populations in a given prey-predator system can be decreased by implementing control measures.

ANALYSIS AND OPTIMAL CONTROL OF TUBERCULOSIS DISEASE SPREAD MODEL WITH VACCINATION AND CASE FINDING CONTROL (CASE STUDY: SURABAYA CITY)

Tuberculosis is a contagious disease that infects humans. It is caused by the bacterium Mycobacterium tuberculosis (M.tb). It is the largest infectious disease in the world and has become a major global health problem. Therefore, efforts are being made to control the spread of tuberculosis disease through vaccination and case finding, with the aim of reducing the population of latently infected and actively infected individuals. This study discusses the mathematical model of tuberculosis disease spread, disease-free and endemic equilibrium points, and stability analysis around the equilibrium points. Then, using Pontryagin's minimum principle, the optimal control problem is solved numerically by the 4th-order Runge-Kutta method. Based on the analysis and simulation results, the system is asymptotically stable around the disease-free and endemic equilibrium points. Furthermore, optimal control in the form of vaccination of susceptible individuals is required to further suppress the rate of change of susceptible individuals into latent individuals, while control in the form of case finding on latently infected individuals is required until the 9th year to minimize the population size, while on actively infected individuals, it is required until the 8th year to minimize the population size. Providing optimal control resulted in a 100% increase in the susceptible population and a 100% reduction in the latent and infected populations.

MODELING THE SPREAD OF COVID-19 DISEASE WITH TIME DELAY IN PONTIANAK CITY

Coronavirus disease 2019 (COVID-19) is an infectious disease caused by a coronavirus originating from the city of Wuhan in 2019. This disease affects the respiratory system. The city of Pontianak has the highest population density in West Kalimantan. This density results in a higher spread of Covid-19. In this article, the spread of COVID-19 is formulated into a mathematical model, equilibrium points are sought, stability is analyzed, and a delay time is introduced to reduce the spread of COVID-19. The magnitude of the delay time given during quarantine complies with health protocols, which is between 2 – 14 days. This article aims to analyze the influence of the delay time in modeling the spread of Covid-19. The problem of COVID-19 spread is constructed into an SIQR model, with a sub-population of recovered individuals returning to the susceptible sub-population. The population is divided into four sub-populations: susceptible (S), Infected (I), Quarantined (Q), and Recovered (R). The parameters used include the natural birth rate ( ), the rate of susceptibility to infection ( ), the rate of infection under quarantine ( ), the recovery rate from infection ( ), the recovery rate from infection under quarantine ( ), the death rate from infection ( ), the death rate under quarantine ( ), the delay time from infection to quarantine process ( ), the natural death rate ( ), and the rate of recovered immunity returning to susceptibility ( ). The simulation results show that when the basic reproduction number is less than , the disease-free equilibrium is stable, and when the basic reproduction number is greater than , the endemic equilibrium point is stable. The addition of a time delay ( ) in the SIQR model affects the stability of the endemic equilibrium point but does not affect the stability of the disease-free equilibrium point.

A new co-infection model for HBV and HIV with vaccination and asymptomatic transmission using actual data from Taiwan

The co-infection of Human Immunodeficiency Virus (HIV) and Hepatitis B virus (HBV) poses a major threat to public health due to their combined negative impacts on health and increased risk of complications. A novel fractional mathematical model of the dynamics of co-infection between HBV and HIV for Taiwan is presented in this paper. Detailed analyses are conducted on the possible impact of HBV vaccination on the dynamics of HBV and HIV co-infection. The next-generation matrix technique is used to calculate the fundamental reproduction number R 0 = max{R 1, R 2}, where R 1 and R 2 are the reproduction numbers for HBV and HIV, respectively. The disease-free and endemic equilibria of the co-infection model are calculated. An extensive investigation is carried out to determine the local and global stability of the disease-free equilibrium point through Rough Hurtwiz criteria and the construction of Lyapunov function, respectively. We demonstrate that when R 1 < 1 < R 2, HBV infection is eradicated, but HIV remains prevalent. If R 2 < 1 < R 1, the opposite outcome occurs. The real data from 2000-2023 for Taiwan is used to fit the model. The fitting results show how effectively our model handles the data. In addition, numerical simulations are run for different scenarios to observe how the vaccine and fractional parameters changed the model state variables, as well as how the solutions behaved and how quickly they reached the model’s equilibrium points. According to the model’s numerical analysis, greater vaccination efforts against HBV have a positive effect on the propagation of co-infection.

Study of chronic myeloid leukemia with T-cell under fractal-fractional order model

Abstract This research work is devoted to investigate myeloid leukemia mathematical model. We give some details about the existence of trivial and nontrivial equilibrium points and their stability. Also, local asymptotical stability of disease-free and endemic equilibrium points is discussed. Also, positivity of the solution has been discussed. Some sufficient results are achieved to study the local existence and uniqueness of solution to the considered model for Mittag–Leffler kernel using the Banach contraction theorem. Three numerical algorithms are derived in obtaining the numerical solution of suggested model under three different kernels using Adams–Basforth technique. Numerical results have been presented for different fractals and fractional orders to show the behavior of the proposed model.

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.

A Stochastic Semi-Parametric SEIR Model with Infectivity in an Incubation Period

This paper introduces stochastic disturbances into a semi-parametric SEIR model with infectivity in an incubation period. The model combines the randomness of disease transmission and the nonlinearity of transmission rate, providing a flexible framework for more accurate description of the process of infectious disease transmission. On the basis of the discussion of the deterministic model, the stochastic semi-parametric SEIR model is studied. Firstly, we use Lyapunov analysis to prove the existence and uniqueness of global positive solutions for the model. Secondly, the conditions for disease extinction are established, and appropriate stochastic Lyapunov functions are constructed to discuss the asymptotic behavior of the model’s solution at the disease-free equilibrium point of the deterministic model. Finally, the specific transmission functions are enumerated, and the accuracy of the results are demonstrated through numerical simulations.

Mathematical Models of Human Papillomavirus Transmission with Vaccination

The human papillomavirus (HPV) is one of the viruses that can cause cervical cancer. However, vaccination can prevent such a virus. The spread of the virus in cervical cancer can be modelled in the SI model, however, such a model has not produced accurate results. The development and extension of the model into S1S2IRC, with S1 denoting the population aged 0–10 years and S2 representing the population over 10 years, which is susceptible to HPV infection. The results produce disease-free and endemic equilibrium points. The analysis of the equilibrium points yields that the disease-free equilibrium point will be asymptotically stable for R0 1 and R0 1 for the endemic equilibrium point. The results indicate that the higher the probability of humans being prone to the HPV virus, the greater the chance of these individuals being infected by the virus. Therefore, vaccination is required to protect against the virus infection.

Mathematical Modeling of HIV/AIDS Disease Spread with Public Awareness

This study develops mathematical model for the spread of HIV/AIDS by the population is divided into seven sub-populations, namely the susceptible unaware HIV subpopulation, the susceptible aware HIV sub-population, the infected sub-population, the pre-AIDS sub-population, the ARV treatment sub-population, the AIDS sub-population, and unlikely to be infected with HIV/AIDS sub-population. In this mathematical model, two equilibrium points are obtained, namely the disease-free equilibrium point and the disease-endemic equilibrium point and the basic reproduction number . The stability analysis shows that the disease-free equilibrium point is locally asymptotically stable if and the disease-endemic equilibrium point is locally asymptotically stable if . Numerical simulations of the equilibrium points are carried out to provide an overview of the analyzed results with parameter values from several sources. Based on the sensitivity analysis, the parameters that significantly affect the spread of HIV/AIDS are the contact rate of HIV-unaware individuals with infected individuals and the transmission rate of HIV infection

SEIR Mathematical Model with the Use of Hand Sanitizers to Prevent the Spread of Covid-19 Disease

The SARS-CoV-2 coronavirus can spread through contact with contaminated surfaces. The use of hand sanitizer is claimed to reduce the risk of transmission. For this reason, this study aims to develop a model of the spread of COVID-19 using the SEIR model with the use of hand sanitizer for infected individuals. The individual population is divided into six compartments, namely two compartments for susceptible individuals who using hand sanitizer and not, one compartment for exposed individuals, two compartments for infected individuals who using hand sanitizer and not, and one compartment for individuals died and recovered. The results obtained two equilibrium points: the disease-free and endemic equilibrium point, and also the basic reproduction number. The existence of a disease-free equilibrium point is unconditional, while the endemic there exist when the basic reproduction number is more than one. Stability analysis of the disease-free equilibrium point is locally asymptotic stable when the basic reproduction number is less than one. Numerical simulations carried out also strengthen them. Finally, the results of basic reproduction number sensitivity analysis show that the basic reproduction number is strongly influenced by contact of the susceptible individuals with exposed and infected individuals, neglecting of hand sanitizer use, mortality and cure rates.

Wiener and Lévy processes to prevent disease outbreaks: Predictable vs stochastic analysis

The study considers stochastic behavior along with modeling, mathematical analysis, theory development, and numerical simulation of the COVID-19 virus. To evaluate current trends and make future projections regarding the basic reproduction number and infection case, we have taken the modified five-compartment SEIRD mathematical model. Since the basic reproduction number R0 is not sufficient to predict the outbreak, we applied the Itô stochastic differential equations (SDEs) along with the Weiner process and Lévy jump to investigate the nature of the disease outbreak. Since COVID-19 is an infectious disease caused by the severe acute respiratory syndrome coronavirus 2 “(SARS-CoV-2)”, that has common symptoms including fever, cough, and difficulty breathing, we have illustrated the boundedness and positivity of solutions and investigated the stability of endemic and disease-free equilibrium points. The findings show that the dynamics of disease epidemics are influenced by contact patterns. For all the models, the threshold quantity, R0 is calculated which is the key reason to prove the global and local stability analysis of disease-free equilibrium and endemic equilibrium points. Using the least squares method for data fitting, we have performed a case study on COVID-19 data in Italy for this article. A sensitivity analysis is part of our investigation to find important factors. This paper investigates a stochastic epidemic model (SDE) using Lévy jumps and Weiner processes. Local and global stability in a stochastic setting are examined in the analysis. To view the analytical study, a multitude of numerical results are achieved. The study is important to comprehend the stochastic behavior of the virus in the contagious model in order to prevent similar disease outbreaks in the future.

Modeling and analysis of a fractional order spatio-temporal SEIR model: Stability and prediction

This study introduces a novel fractional-order spatio-temporal SEIR model for epidemic modeling, providing an advanced approach to understanding disease dynamics. Our model, categorizing the population into Susceptible (S), Exposed (E), Infected (I), and Recovered (R), incorporates fractional calculus to accurately reflect the complex, non-linear nature of infectious diseases. Key findings include the confirmation of the existence and uniqueness of the model’s solutions, ensuring reliability for epidemiological predictions. Through rigorous stability analysis at both disease-free and endemic equilibrium points, we identified critical parameters influencing epidemic outcomes. Numerical simulations reveal that the fractional order significantly impacts disease progression, offering valuable insights for intervention strategies.

Dynamics analysis and optimal control of delayed SEIR model in COVID-19 epidemic

The coronavirus disease 2019 (COVID-19) remains serious around the world and causes huge deaths and economic losses. Understanding the transmission dynamics of diseases and providing effective control strategies play important roles in the prevention of epidemic diseases. In this paper, to investigate the effect of delays on the transmission of COVID-19, we propose a delayed SEIR model to describe COVID-19 virus transmission, where two delays indicating the incubation and recovery periods are introduced. For this system, we prove its solutions are nonnegative and ultimately bounded with the nonnegative initial conditions. Furthermore, we calculate the disease-free and endemic equilibrium points and analyze the asymptotical stability and the existence of Hopf bifurcations at these equilibrium points. Then, by taking the weighted sum of the opposite number of recovered individuals at the terminal time, the number of exposed and infected individuals during the time horizon, and the system cost of control measures as the cost function, we present a delay optimal control problem, where two controls represent the social contact and the pharmaceutical intervention. Necessary optimality conditions of this optimal control problem are exploited to characterize the optimal control strategies. Finally, numerical simulations are performed to verify the theoretical analysis of the stability and Hopf bifurcations at the equilibrium points and to illustrate the effectiveness of the obtained optimal strategies in controlling the COVID-19 epidemic.

Comparative analysis of a fractional co-infection model using nonstandard finite difference and two-step Lagrange polynomial methods

In this work, Caputo fractional model for co-dynamics of two-strains (Alpha and Delta variants) of SARS-CoV-2 and tuberculosis (TB) is formulated. We investigates the basic mathematical analysis, existence of unique solution, stability analysis, sensitivity analysis and simulations of the proposed model. In the basic mathematical analysis, it is proved that the model’s solutions are non-negative, bounded and locally asymptotically stable at disease free equilibrium (DFE) point. The existence of unique solution is proved by using Banach fixed point theorem (BFPT). Stability analysis is carried out by employing Hyers–Ulam stability criteria. Sensitivity analysis is performed to assess the impact of the model parameters on the general dynamics of the model by implementing partial rank correlation coefficient (PRCC) technique. Simulations of the proposed model obtained from non-standard finite difference scheme (NSFDs) are compared with the simulations obtained from two-step Lagrange polynomial method (TLPM). It is concluded that NSFDs gives more accurate and realistic results than TLPM. To assess some control measures necessary for reducing the co-spread of both diseases, comparative simulations are carried out for the infected classes. Also, numerical assessments showed that preventive efforts against SARS-CoV-2 variants could results in the reduction of TB prevalence and the co-infection of both diseases.

A deterministic compartment model for analyzing tuberculosis dynamics considering vaccination and reinfection

Tuberculosis is a pressing global health concern, particularly pervasive in many developing nations. This study investigates the influence of treatment failure on tuberculosis control strategies, incorporating vaccination interventions using a deterministic compartmental epidemiological model. Mathematical analysis unveils disease-free and endemic equilibrium points, with the control reproduction number determined using next-generation methods. Identifying endemic equilibrium points and determining the control reproduction number provide essential metrics for assessing the effectiveness of control strategies and guiding policy decisions. The model exhibits a backward bifurcation phenomenon, leading to multiple endemic equilibria despite a reproduction number below one due to reinfection. Sensitivity analysis using Latin Hypercube Sampling/Partial Rank Correlation Coefficient elucidates parameter impacts on the control reproduction number. Vaccination efficacy is crucial for quality and validity, with superior quality and longer validity yielding more significant effects. While reinfection may not directly affect the reproduction number, its influence is pivotal in determining tuberculosis persistence or extinction. This study underscores the intricate interplay of factors in tuberculosis control strategies, providing insights vital for effective interventions and policy formulation.

On Theoretical and Numerical Results of Serum Hepatitis Disease Using Piecewise Fractal–Fractional Perspectives

In the present research, we consider a biological model of serum hepatitis disease. We carry out a detailed analysis of the mentioned model along with a class with asymptomatic carriers to explore its theoretical and numerical aspects. We initiate the study by using the piecewise fractal–fractional derivative (FFD) by which the crossover effects within the model are examined. We split the time interval into subintervals. In one subinterval, FFD with a power law kernel is taken, while in the second one, FFD with an exponential decay kernel of the proposed model is considered. This model is then studied for its disease-free equilibrium point, existence, and Hyers–Ulam (H-U) stability results. For numerical results of the model and a visual presentation, we apply the Lagrange interpolation method and an extended Adams–Bashforth–Moulton (ABM) method, respectively.

An infectious disease epidemic model with migration and stochastic transmission in deterministic and stochastic environments

Understanding population migration is essential for controlling highly infectious diseases. This paper studies the global dynamics of an infectious disease epidemic model incorporating population migration and a stochastic transmission rate. Our findings demonstrate that in deterministic and stochastic environments, the models exhibit global Lyapunov stability near the disease-free equilibrium point, determined by a threshold parameter. Furthermore, we analyze the effect of migration on infectious diseases. We discover that the number of infections and the peak value of the infection curve increase with a higher level of population migration. These results are supported by numerical illustrations that hold epidemiological relevance. Additionally, the disease-free equilibrium of the associated time delay model is linearly asymptotically stable, and the endemic equilibrium exhibits more bifurcation for larger time delay values.

Nonlinear dynamics and optimal control strategies of a novel fractional-order lumpy skin disease model

Fractional order differential equations are often employed to better and accurately model several natural and physical processes with memory characteristics than the integer-order models. In this work, a proposed fractional-order model is presented to simulate one of the most dangerous viral cattle diseases. The present fractional-order Lumpy Skin Disease model considers the interactions between different compartments of cattle and virus vectors. The solution of the model is examined in terms of existence, uniqueness, positivity and boundedness. Then, the stability analysis of the different equilibrium points in the model is carried out. The basic reproduction number, R0, has been obtained too. The dependence of stability regions on the values of key parameters is demonstrated and the sensitivity of R0 to different parameters in the system is also investigated. Finally, a proposed optimal control scheme is adopted to successfully diminish the disease spread. It is found that the stability of disease-free equilibrium point is verified at small values of the infection rate of cattle caused by infected cattle or insects. In addition, the disease-free equilibrium point maintains its stability in cases where the recovery rate of cattle or the vaccination rate of cattle are increased. Moreover, the high-efficiency vaccination program helps in stabilizing the disease-free equilibrium point. Also, the fractional-order parameter is found to provide additional degree of freedom in the model to represent different scenarios of evolution for solution orbits. The optimal control parameters of the model are examined in different scenarios to limit or suppress the spread of the disease. It is demonstrated that two control parameters are sufficient to achieve the goals of disease elimination and stabilization of an infection-free steady state. Numerical simulations are verified to confirm our theoretical findings and estimations.

Analysis of COVID-19 Disease Model: Backward Bifurcation and Impact of Pharmaceutical and Nonpharmaceutical Interventions

The SEIQHR model, introduced in this study, serves as a valuable tool for anticipating the emergence of various infectious diseases, such as COVID-19 and illnesses transmitted by insects. An analysis of the model’s qualitative features was conducted, encompassing the computation of the fundamental reproduction number, R0. It was observed that the disease-free equilibrium point remains singular and locally asymptotically stable when R0&lt;1, while the endemic equilibrium point exhibits uniqueness when R0&gt;1. Additionally, specific conditions were outlined to guarantee the local asymptotic stability of both equilibrium points. Employing numerical simulations, the graphical representation illustrated the influence of model parameters on disease dynamics and the potential for its eradication across different noninteger orders of the Caputo derivative. In essence, the adoption of a fractional epidemic model contributes to a deeper comprehension and enhanced biological insights into the dynamics of diseases.