Global Stability Of Endemic Equilibrium Research Articles

Exploring Tuberculosis: A Theoretical Framework for Infection Regulation and Eradication

In this paper we explored the study of tuberculosis, throughout human history, tuberculosis (TB) has been a persistent challenge due to its severe consequences for society. It is an infectious disease transmitted by Mycobacterium tuberculosis (MT),more than 150 million years ago, it was thought to have originated from the genus Mycobacterium. Our theoretical framework presents methods for regulating and eradicating tuberculosis infections. This has led to the compartmentalization of the model population and the analytical solution of the ensuing model equations. To validate the outcomes of the theoretical approach, a numerical simulation has been deployed. This model carried out six compartment stage, susceptible, latent for a short time period and latent for long time period, infected in hospital, random infected at public place and the recovered class. In this model we examined Disease free equilibrium points (DFEP), Basic reproduction number R_0. Positive boundaries solution is used to describe the infection at particular stage. If R_0<1, the DFE is locally asymptotically stable, using Jacobian matrix we explored the negative Eigen values. Lyapunov function is used to examined the global stability of endemic equilibrium. We used Carrying out a simulation using random data in a selected region,using random values in MATLAB, to simulate the result of the model.

Modeling and stability analysis of the transmission dynamics of Monkeypox with control intervention

Monkeypox is a deadly disease from the Orthopox family. The paper explores a nonlinear trend in the dynamics of Monkeypox propagation. In this model, we include vaccination, treatment, and level of awareness as the main parameters of the propagation of Monkeypox. We also include the quarantined infected individuals with severe complications in the Monkeypox dynamics. We show the boundedness and positivity of the model and also demonstrate the stability of the model using the Banach fixed point theory and the Picard successive approximation method to ensure that it is meaningful mathematically and epidemiologically. We obtain the unique solution of the model under appropriate conditions. This work finds three equilibrium points: Monkeypox free, Rodents free, and endemic. We show the local stability of monkeypox free equilibria and rodent-free endemic equilibria through reproduction number. Furthermore, Dulac’s function technique demonstrates the global stability of endemic equilibrium. We show the transformation from endemic equilibria to Monkeypox-free equilibria via rodent-free. We also obtain the condition of transcritical bifurcation. We analyze the system’s dynamic behavior to develop efficient infection control strategies. We find that if we increase awareness, vaccination, treatment, and quarantine rates, we effectively control the transmission. The model is connected with a continuous model using an ordinary differential equation. We examine the complex dynamics of Monkeypox infection under diverse system input factors through numerical simulation of the proposed model with variable input parameters in reducing Monkeypox. We present the theory of Monkeypox disease control in humankind as an application in real-world problems. The work can be helpful in the vector-borne disease control system.

Influence of incubation delays on COVID-19 transmission in diabetic and non-diabetic populations – an endemic prevalence case

Abstract The study of dynamics of diabetic population infected by COVID-19 is of pressing concern as people with diabetes are considered to be at higher risk of severe illness from COVID-19. A three-compartment mathematical model to describe the interactions of diabetic population and non-diabetic population both infected by COVID-19 with a susceptible population is considered. Time delays in incubation periods of COVID-19 in diabetic and non-diabetic populations are introduced. Besides the basic properties of such a dynamical system, both local and global stability of endemic equilibrium, are studied. The lengths of time delays are estimated for which the stability of the system is preserved locally, while sufficient conditions on system parameters are obtained for global stability. Numerical examples are provided to establish the theory, and simulations are provided to visualize the examples. It is noted that an increase in length of time delay in either of infected populations leads to oscillations in susceptible population but has no impact on infected populations.

MATHEMATICAL ANALYSIS OF A TUBERCULOSIS MODEL WITH TWO DIFFERENT STAGES OF INFECTION

Tuberculosis is an infectious disease. This disease causes death and the world notes that Tuberculosis has a high mortality rate. A mathematical model of Tuberculosis with two infection stages of individuals, pre infected and actively infected, is studied in this paper. The rate of treatment considered in this model. The stability analysis of the equilibrium is determined by the basic reproduction ratio. Routh Hurwitz linearization is used for investigate the local stability of uninfected equilibrium. While the global stability of endemic equilibrium is investigated by construct Lyapunov function. The effect of treatment in pre infected and actively infected stages can reduce the spread rate of Tuberculosis as shown in numerical simulation.

Dynamics of a fractional COVID-19 model with immunity using harmonic incidence mean-type.

The transmission dynamics of COVID-19 is investigated through the prism of the Atangana-Baleanu fractional model with acquired immunity. Harmonic incidence mean-type aims to drive exposed and infected populations towards extinction in a finite time frame. The reproduction number is calculated based on the next-generation matrix. A disease-free equilibrium point can be achieved globally using the Castillo-Chavez approach. Using the additive compound matrix approach, the global stability of endemic equilibrium can be demonstrated. Utilizing Pontryagin's maximum principle, we introduce three control variables to obtain the optimal control strategies. Laplace transform allows simulating the fractional-order derivatives analytically. Analysis of the graphical results led to a better understanding of the transmission dynamics.

Effect of partially and fully vaccinated individuals in some regions of India: A mathematical study on COVID-19 outbreak

In this paper, we investigate the effect of partially vaccinated and fully vaccinated individuals in preventing the transmit of COVID-19, especially in the regions of Tamil Nadu, Maharashtra, West Bengal and Delhi. Here we construct an SEIR model and analyse the behaviour. We obtained R0 by using next generation matrix approach. Also, our system shows two types of equilibria, namely disease free and endemic equilibrium. For both disease free and endemic equilibrium, local and global stability is obtained here. Our disease-free equilibrium is locally asymptotically stable whenever R0 is less than one, whereas the endemic equilibrium is locally asymptotically stable whenever R0 is greater than one. Furthermore, the global stability of disease-free equilibrium has been proven by using Lyapunov function and the global stability of endemic equilibrium has been obtained by using Poincare Bendixson technique. Also, we enhance our analytic results by numerical simulation. At the end we have attempted to fit our proposed model with the real-world data.

A Mathematical Model for a Transmissible Disease with a Variant

The outbreak of the Coronavirus (COVID-19) pandemic around the world has caused many health and socioeconomic problems, and the identification of variants like Delta and Omicron with similar and often even more transmissible modes of transmission has motivated us to do this study. In this article, we have proposed and analyzed a mathematical model in order to study the effect of health precautions and treatment for a disease transmitted by contact in a constant population. We determined the four equilibria of the system of ordinary differential equations representing the model and characterized their existence using exact methods of algebraic geometry and computer algebra. The model is studied using the stability theory for systems of differential equations and the basic reproduction number R 0 . The stability of the equilibria is analyzed using the Lienard-Chipart criterion and Lyapunov functions. The asymptotic or global stability of endemic equilibria is established, and the disease-free equilibrium is globally asymptotically stable if R 0 < 1 . Model simulation is done with Python software to study the effects of health precautions and treatment, and the results are analyzed. It is observed that if the rate of treatment and compliance with health precautions are high, the number of infections decreases in the classes of infectious and is canceled out over time. It is concluded that the high treatment rate accompanied by a suitable rate of compliance with health precautions allows for the control the disease.

Global Dynamics of a Multi-group SEIR Epidemic Model with Infection Age

Consider the heterogeneity (e.g., heterogeneous social behaviour, heterogeneity due to different geography, contrasting contact patterns and different numbers of sexual partners etc.) of host population, in this paper, the authors propose an infection age multi-group SEIR epidemic model. The model system also incorporates the feedback variables, where the infectivity of infected individuals may depend on the infection age. In the direction of mathematical analysis of model, the basic reproduction number R0 has been computed. The global stability of disease-free equilibrium and endemic equilibrium have been established in the term of R0. More precisely, for R0 ≤ 1, the disease-free equilibrium is globally asymptotically stable and for R0 > 1, they establish global stability of endemic equilibrium using some graph theoretic techniques to Lyapunov function method. By considering a numerical example, they investigate the effects of infection age and feedback on the prevalence of the disease. Their result shows that feedback parameters have different and even opposite effects on different groups. However, by choosing an appropriate value of feedback parameters, the disease could be eradicated or maintained at endemic level. Besides, the infection age of infected individuals may also change the behaviour of the disease, global stable to damped oscillations or damped oscillations to global stable.

Dynamics of coronavirus pandemic: effects of community awareness and global information campaigns.

The effects of social media advertisements together with local awareness in controlling COVID-19 are explored in the present investigation by means of a mathematical model. The expression for the basic reproduction number is derived. Sufficient conditions for the global stability of endemic equilibrium are obtained. We perform sensitivity analysis to identify the key parameters of the model having great impacts on the prevalence and control of COVID-19. We calibrate the proposed model to fit the data set of COVID-19 cases for India. Our simulation results show that dissemination rate of awareness among susceptible individuals at community level and individual level plays pivotal role in curtailing the COVID-19 disease. Moreover, we observe that the global information distributing from social media and local awareness coming from mouth-to-mouth communication between unaware susceptible and aware people, together with hospitalization of symptomatic individuals and quarantine of asymptomatic individuals, are much beneficial in reducing COVID-19 cases in India. Our study suggests that both global and local awareness must be implemented effectively to manage the burden of COVID-19 pandemic.

On the local and global stability of an sirs epidemic model with logistic growth and information intervention

In this study, we investigate an susceptible-infected-recovered-susceptible (SIRS) epidemic model with logistic growth and information intervention. Firstly, the basic reproduction number $R_0$ is defined and the main results are given in terms of local stability. Then, sufficient conditions for the global stability of endemic equilibrium are obtained. Finally, some numerical simulations are given to validate our theoretical conclusions.

Threshold dynamics and optimal control on an age-structured SIRS epidemic model with vaccination.

We consider a vaccination control into a age-structured susceptible-infective-recovered-susceptible (SIRS) model and study the global stability of the endemic equilibrium by the iterative method. The basic reproduction number $ R_0 $ is obtained. It is shown that if $ R_0 < 1 $, then the disease-free equilibrium is globally asymptotically stable, if $ R_0 > 1 $, then the disease-free and endemic equilibrium coexist simultaneously, and the global asymptotic stability of endemic equilibrium is also shown. Additionally, the Hamilton-Jacobi-Bellman (HJB) equation is given by employing the Bellman's principle of optimality. Through proving the existence of viscosity solution for HJB equation, we obtain the optimal vaccination control strategy. Finally, numerical simulations are performed to illustrate the corresponding analytical results.

A new way of constructing Lyapunov functions with application to an SI epidemic model

We reconsider an SI epidemic model studied by Li et al. (2007). The model is shown to have complicated dynamics including backward bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation. But for the case where the basic reproductive number is larger than one, the global stability of the unique endemic equilibrium is not investigated. In this paper, using a novel function, we construct a new type of Lyapunov functions to derive a sufficient condition on the global stability of the endemic equilibrium, which is weaker than the one obtained by using the traditional way of constructing Lyapunov functions. We also propose some other ways of constructing Lyapunov functions in the discussion.

Dynamical analysis model of HIV-1 infection in CD4+ T cells with antibody response

The spread of HIV infection one of which is affected by cell to cell transmission. A dynamical model of HIV-1 infection in CD4+T cells with considering viral transmission from cell to cell and antibody response is constructed in this paper. Antibody response determines viral load in early HIV-1 infection. The existence and stability of the equilibrium is investigated by the basic reproduction ratio. The local stability of uninfected equilibrium is analyzed using Routh Hurwitz linearization. After investigating the local stability, we construct a Lyapunov function to investigate the global stability of endemic equilibrium. Numerical simulation is given to illustrate the control of HIV-1 infection based on the effect of antibody response and the effectiveness of antiretroviral treatment.

Impact of variability of reproductive ageing and rate on childhood infectious disease prevention and control: insights from stage-structured population models.

We propose a stage-structured model of childhood infectious disease transmission dynamics, with the population demographics dynamics governed by a certain family and population planning strategy giving rise to nonlinear feedback delayed effects on the reproduction ageing and rate. We first describe the long-term aging-profile of the population by describing the pattern and stability of equilibrium of the demographic model. We also investigate the disease transmission dynamics, using the epidemic model when the population reaches the positive equilibrium (limiting equation). We establish conditions for the existence, uniqueness and global stability of the disease endemic equilibrium. We then prove the global stability of the endemic equilibrium for the original epidemic model with varying population demographics. The global stability of the endemic equilibrium allows us to examine the effects of reproduction ageing and rate, under different family planning strategies, on the childhood infectious disease transmission dynamics. We also examine demographic distribution, diseases reproductive number, infant disease rate and age distribution of disease, and as such, the work can be potentially used to inform targeted age group for optimal vaccine booster programs.

Dynamics of a waterborne pathogen model with spatial heterogeneity and general incidence rate

This paper is concerned with the dynamics of a waterborne pathogen model with spatial heterogeneity and general incidence rate. We first establish the well-posedness of this model. Then we clarify the relationship between the local basic reproduction number R̃ and the basic reproduction number R0. It could be seen that R0 plays an important role in determining the global dynamics of this model. In fact, we show that the disease-free equilibrium is globally asymptotically stable when R0<1. If R0=1, then the disease-free equilibrium is globally asymptotically stable under some assumptions. In addition, the phenomena of uniform persistence occurs when R0>1. We also consider the local and global stability of endemic equilibrium when all the parameters of this model are constant. In the case R0>1, we further establish the existence of traveling wave solutions of this model. Moreover, we provide an example and numerical simulations to support our theoretical results. Our model extended some known results.

The global dynamics for an age-structured tuberculosis transmission model with the exponential progression rate

• We fixed the exponential function to describe the progression rate of TB. • The threshold value of disease persistent or not is given. • The existence and global stability of disease free equilibrium and endemic equilibrium both are proved. • The existence of global attractor, and persistence of disease are studied. This paper deals with the global dynamic analysis for a tuberculosis transmission model with age-structure and the exponential progress rate. The time delay in the progression from the latent individuals to becoming the infectious individuals is also considered in our model. The basic reproduction number R 0 of the model is defined. It is proved that R 0 = 1 is a threshold to determine the disease extinction or persistence. The disease free equilibrium is globally stable (unstable) if R 0 < 1 (if R 0 > 1). There exists an endemic equilibrium and the system is uniformly persistent if R 0 > 1. The global stability of the endemic equilibrium is given when R 0 > 1. At end, the model is applied to describe tuberculosis transmission in China. The number of total population, the number of the annual new TB cases, the annual PPD positive rate, and the prevalent rate all match the statistical data well.

Dynamics and asymptotic profiles of endemic equilibrium for SIS epidemic patch models.

In this paper, we perform qualitative analysis to two SIS epidemic models in a patchy environment, without and with linear recruitment. The model without linear recruitment was proposed and studied by Allen et al. (SIAM J Appl Math 67(5):1283-1309, 2007). This model possesses a conserved total population number, whereas the model with linear recruitment has a varying total population. However, both models have the same basic reproduction number. For both models, we establish the global stability of endemic equilibrium in a special case, which partially solves an open problem. Then we investigate the asymptotic behavior of endemic equilibrium as the mobility of infected and/or susceptible population tends to zero. Though the basic reproduction number is a well-known critical index, our theoretical results strongly suggest that other factors such as the variation of total population number and individual movement may also play vital roles in disease prediction and control. In particular, our results imply that the variation of total population number can cause infectious disease to become more threatening and difficult to control.

Analysis of mathematical model of HIV-1 infection of CD4+ T cells with CTL response and antiretroviral treatment

A mathematical model is developed to capture the spread of HIV-1 infection within host cells caused by the contact of cell to cell and CTL response. In this paper, we propose a mathematical model of HIV-1 infection in CD4+T cells taking into account viral transmission from cell to cell and CTL response. The HIV transmission from cell to cell is one of the main factor in the spread of HIV infection and CTL response determines viral set point. We analyse the model to investigate the existence and stability of the equilibria. We analyse the local stability of disease free equilibrium by linearization, while the global stability of endemic equilibrium of the system by constructing Lyapunov function. Numerical simulations are presented to find the effectiveness of antiretroviral treatment in different scenarios and to the implication of CTL response in controlling the progression of HIV-1 infection.

Asymptotic analysis of endemic equilibrium to a brucellosis model.

Brucellosis is one of the worlds major infectious and contagious bacterial disease. In order to study different types of brucellosis transmission models among sheep, we propose a deterministic model to investigate the transmission dynamics of brucellosis with the flock of sheep divided into basic ewes and other sheep. The global dynamical behavior of this model is given: including the basic repro-duction number, the existence and uniqueness of positive equilibrium, the global asymptotic stability of the equilibrium. We prove the uniqueness of positive endemic equilibrium through using proof by contradiction, and the global stability of endemic equilibrium by using Lyapunov function. Especially, we give the specific coefficients of global Lyapunov function, and show the calculation method of these specific coefficients. By running numerical simulations for the cases with the basic reproduction number to demonstrate the global stability of the equilibria and the unique endemic equilibrium, re-spectively. By some sensitivity analysis of the basic reproduction number on parameters, we find that vaccination rate of sheep and seropositive detection rate of recessive infected sheep are very important factor for brucellosis.

Application of Homotopy Analysis Method for Solving an SEIRS Epidemic Model

In this paper, we modified the model of [23] and then applied a new semi-analytic technique namely the Homotopy Analysis Method (HAM) in solving the SEIRS Epidemic Mathematical Model. The modified SEIRS model wasfirst formulated and adequately analyzed. We investigated the basic properties of the model by proving the positivity of the solutions and establishing the invariant region. We further obtained the steady states: disease-free equilibrium (DFE) and endemic equilibrium (EE), then we went further to determine the local stability of the DEF and EE using the basic reproduction number which was calculated. We also applied Lyaponuv method to prove the global stability of endemic equilibrium, The HAM was applied to obtain an accurate solution to the model in few iterations. Finally, a numerical solution (simulation) of the model was obtained using MAPLE 15 computation software.