Disease-free Equilibrium Point Research Articles

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.

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.

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<1, while the endemic equilibrium point exhibits uniqueness when R0>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.

A Co-infection Model for Monkeypox and HIV/AIDS: Sensitivity and Bifurcation Analyses

Monkeypox can make people very sick. The skin becomes infected with bacteria, thus causing severe skin damage. This can lead to corneal infection with loss of vision, pneumonia, difficulty swallowing, diarrhoea and vomiting leading to harsh malnutrition or dehydration, several organs inflammation or death. HIV-AIDS is a life-threatening and chronic condition. In HIV-infected individuals whose immune systems have been compromised, monkeypox mortality alone may be much higher. The co-infection of monkeypox and HIV/AIDS infections has been studied from a mathematical perspective by constructing a 13-compartment deterministic model. Basic mathematical analyses were performed on the co-infection model and the sub-models. The disease equilibrium points, the non-negativity of solutions, the basic reproduction numbers, the invariant region and the stability patterns. When the basic reproduction number is less than unity, the disease-free equilibrium points of each sub-model are globally asymptomatically stable. Certain calculations were done using the maple 18 programming language. The sensitivity analysis reveals that the parameters of the basic reproduction of the monkeypox sub-model with positive sensitivity indices are the probability of catching the monkeypox virus, the rate of effective contact, the compartment Im ’s coefficient of infection and the monkeypox vaccine’s waning rate, while the parameters of the basic reproduction of the HIV/AIDS sub-model with positive sensitivity indices are the probability of catching HIV virus, the rate effective contact, the compartment Ih ’s coefficient of infection and the compartment Ah ’s coefficient of infection. Via the centre manifold theorem, the bifurcation analysis reveals a forward bifurcation pattern for the monkeypox sub-model and the HIV/AIDS sub-model, and under a certain condition, a critical value of the monkeypox basic reproduction exists such that an effective management and possible elimination of the monkeypox infection would require that the monkeypox basic reproduction number should be kept below unity and above the critical value.

Mathematical Modelling of COVID-19 with Vaccination, Quarantine and Non-Pharmaceutical Interventions

This study formulated and analyzed a deterministic mathematical model of ten compartments for the transmission dynamics of COVID-19 infection using a system of non-linear ordinary differential equations. The system has disease-free and endemic equilibrium points. The basic reproduction number   was obtained using the next-generation matrix method and the stability of the equilibrium points were analyzed. From the qualitative analysis, the disease-free equilibrium point is both locally and globally asymptotically stable. Finally, numerical simulations of the model were carried out using MATLAB R2021a. Based on the result obtained, it was concluded that the implementation of vaccination, non-pharmaceutical intervention as well as contact tracing and quarantine can lead to effective control or elimination of the COVID-19 pandemic. And hence, we recommend that the responsible agencies should create a way of enlightening the public on the need of being vaccinated for COVID-19 so as to prevent themselves from getting infected; also, adequate and effective implementation of vaccination and non-pharmaceutical interventions is required in order to prevent or curtail the COVID-19 infection completely. COVID-19,  Non-pharmaceutical intervention, Quarantine, Stability analysis,  Vaccination

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.

ANALYSIS AND SIMULATION OF THE SIR MODEL ON THE SPREAD OF COVID-19 BY CONSIDERING THE VACCINATION FACTOR

Covid-19 is a serious respiratory disease that can be fatal for those affected. Governments have tried various strategies to conquer the Covid-19 pandemic. One of them is to vaccinate people with 6 years old and over. The vaccination program aims to form herd immunity so that the number of confirmed positive cases can be reduced. The purpose of this research is to form a mathematical model of the SIR (Susceptible-Infected-Recovery) spread of Covid-19 by considering vaccination factors. The SIR model is combined with a vaccination factor to forestall the unfold of Covid-19. The research method includes deriving models of nonlinear differential equation systems, solving qualitative models, deriving the basic reproduction ratio ( ), analysis of equilibrium points, and building simulation models. This model has an asymptotically stable disease-free equilibrium point. At the same time, the endemic equilibrium point is unstable. Model simulation is obtained by using different parameter values. This is proven through the outcomes of the model analysis vaccination coverage is a key parameter that can be controlled to reduce  so that the pandemic ends soon.

Mathematical modelling, analysis and numerical simulation of social media addiction and depression.

We formulate a mathematical model of social media addiction and depression (SMAD) in this study. Key aspects, such as social media addiction and depression disease-free equilibrium point (SMADDFEP), social media addiction and depression endemic equilibrium point (SMADEEP), and basic reproduction number (R0), have been analyzed qualitatively. The results indicate that if R0 < 1, the SMADDFEP is locally asymptotically stable. The global asymptotic stability of the SMADDFEP has been established using the Castillo-Chavez theorem. On the other hand, if R0 > 1, the unique endemic equilibrium point (SMADEEP) is locally asymptotically stable by Lyapunov theorem, and the model exhibits a forward bifurcation at R0 = 1 according to the Center Manifold theorem. To examine the model's sensitivity, we calculated the normalized forward sensitivity index and conducted a Partial Rank Correlation Coefficient (PRCC) analysis to describe the influence of parameters on the SMAD. The numerical results obtained using the Fourth-order Runge-Kutta (RK-4) scheme show that increasing the number of addicted individuals leads to an increase in the number of depressed individuals.

A tuberculosis model with three infected classes

The dynamics of tuberculosis within a population cannot be adequately represented by a single infectious class. Therefore, this study develops a compartmental model encompassing latent, active, and drug-resistant populations to better capture tuberculosis dynamics in a community. Model analysis reveals that the disease-free equilibrium point is locally asymptotically stable when the basic reproduction number is below one. Moreover, the use of a suitable Lyapunov function demonstrates global asymptotic stability of the disease-free equilibrium point. An endemic equilibrium emerges when the basic reproduction number exceeds one. Sensitivity analysis is conducted for each parameter associated with the basic reproduction number, and optimal control analysis is employed to assess the impact of various control strategies on disease containment. Numerical simulations are conducted to supplement theoretical findings, illustrating the practical implications of the proposed control strategies.

Impacts of optimal control strategies on the HBV and COVID-19 co-epidemic spreading dynamics

Different cross-sectional and clinical research studies investigated that chronic HBV infected individuals’ co-epidemic with COVID-19 infection will have more complicated liver infection than HBV infected individuals in the absence of COVID-19 infection. The main objective of this study is to investigate the optimal impacts of four time dependent control strategies on the HBV and COVID-19 co-epidemic transmission using compartmental modeling approach. The qualitative analyses of the model investigated the model solutions non-negativity and boundedness, calculated all the models effective reproduction numbers by applying the next generation operator approach, computed all the models disease-free equilibrium point (s) and endemic equilibrium point (s) and proved their local stability, shown the phenomenon of backward bifurcation by applying the Center Manifold criteria. By applied the Pontryagin’s Maximum principle, the study re-formulated and analyzed the co-epidemic model optimal control problem by incorporating four time dependent controlling variables. The study also carried out numerical simulations to verify the model qualitative results and to investigate the optimal impacts of the proposed optimal control strategies. The main finding of the study reveals that implementation of protections, COVID-19 vaccine, and treatment strategies simultaneously is the most effective optimal control strategy to tackle the HBV and COVID-19 co-epidemic spreading in the community.

Modeling the Impact of Vaccination on Epidemic Disease Variants with Hospitalization: A Case Study for the COVID-19 Pandemic in Turkey

The stability analysis of an epidemic model that takes into account the impact of vaccination and hospitalization is investigated in this study. Disease-free and endemic equilibrium points are obtained for the stability analysis. The necessary conditions for analyzing local stability at equilibrium points as well as global stability at the disease-free equilibrium point are also defined. Using data from three different periods corresponding to the emergence of three different variants of the COVID-19 outbreak in Turkey, the numerical simulation with graph fitting for the model is also taken into account. The analysis considers the efficacy of vaccination in restricting the virus's spread.

Numerical analysis of COVID-19 model with Caputo fractional order derivative

This paper focuses on the numerical solutions of a six-compartment fractional model with Caputo derivative. In this model, we obtain non-negative and bounded solutions, equilibrium points, and the basic reproduction number and analyze the stability of disease free equilibrium point. The existence and uniqueness of the solution are proven by employing the Picard–Lindelof approach and fixed point theory. The product–integral trapezoidal rule is employed to simulate the system of FODEs (fractional ordinary differential equations). The numerical results are presented in the form of graphs for each compartment. Finally, the sensitivity of the most important parameter (β) and its impact on COVID-19 dynamics and the basic reproduction number are reported.

Modeling the Mathematical Transmission of a Pneumonia Epidemic Model with Awareness

A new model for the transmission of a pneumonia epidemic, considering awareness and a mass action incidence function, is presented. Stability analyses for disease-free and endemic equilibrium points are conducted. The Basic Reproduction Number (Ro) for pneumonia with awareness is defined and analyzed, showing stability when (Ro &lt; 1)  and transitioning to an endemic state when (Ro &gt; 1)  . Additionally, a special case is highlighted where the Basic Reproduction Number (Ro*) (without awareness) is greater than  (Ro). Basic Reproduction Number with awareness, i.e. (Ro* = 1.0965 &gt; Ro = 0.8772). Furthermore, a numerical simulation is provided to depict how awareness influences the dynamic management of the disease. The results underscore the crucial role of awareness in educating the public about infection risks, ultimately contributing to a decrease in the health burden by mitigating the epidemic peak.

Mathematical modeling and stability analysis of the novel fractional model in the Caputo derivative operator: A case study

The fundamental goal of this research is to suggest a novel mathematical operator for modeling visceral leishmaniasis, specifically the Caputo fractional-order derivative. By utilizing the Fractional Euler Method, we were able to simulate the dynamics of the fractional visceral leishmaniasis model, evaluate the stability of the equilibrium point, and devise a treatment strategy for the disease. The endemic and disease-free equilibrium points are studied as symmetrical components of the proposed dynamical model, together with their stabilities. It was shown that the fractional calculus model was more accurate in representing the situation under investigation than the classical framework at α = 0.99 and α = 0.98. We provide justification for the usage of fractional models in mathematical modeling by comparing results to real-world data and finding that the new fractional formalism more accurately mimics reality than did the classical framework. Additional research in the future into the fractional model and the impact of vaccinations and medications is necessary to discover the most effective methods of disease control.

Feasibility and Stability Analysis for Basic Measles Model Using Fuzzy Parameter

In this article, we investigate the behaviors of the measles viral pandemic using fuzzy susceptible-infectiousrecovered (SIR) model. To examine the effects of various compartment phases, we analyze disease-free equilibrium points along with basic reproduction number. The measles model is stated to be globally asymptotically stable at the disease-free equilibrium point. In order to mathematically simulate the measles, we use a first-order nonlinear differential equation. The numerical solution is computed using the Runge-Kutta method, and the model’s feasibility is also covered.