Stability Analysis of an SVLI Epidemic Model

Despite all efforts to curb and exterminate the menace of Tuberculosis (TB) epidemics on the human population, the disease still remains one of the major causes of death, with one –third of the world's population infected. In this paper, we study a deterministic mathematical model to have a better insight in the transmission dynamics of TB. The model is shown to have disease-free and endemic equilibria and their local stabilities are established using the basic reproduction number, R o. If R o < 1, the infection can be controlled and then eradicated and when R o > 1, the disease will persist. Numerical simulations are performed to validate the theoretical results.

2014

ComTech: Computer, Mathematics and Engineering Applications, 2019

This research focused on the modification of deterministic mathematical models for tuberculosis with vaccination. It also aimed to see the effect of giving the vaccine. It was done by adding vaccine compartments to people who were given the vaccine in the susceptible compartment. The population was divided into nine different groups. Those were susceptible individuals (S), vaccine (V), new latently infected (E 1), diagnosed latently infected (E 2), undiagnosed latently infected (E 3), undiagnosed actively infected (l), diagnosed actively infected with prompt treatment (D r), diagnosed actively infected with delay treatment (D p), and treated (T). Basic reproduction number was constructed using next-generation matrix. Sensitivity analysis was also conducted. The results show that the model comprises two equilibriums: diseasefree equilibrium (T 0) and endemic equilibrium (T *). It also shows that there is a relationship between R 0 and two equilibriums. Moreover, the disease-free equilibrium point is asymptotically stable local when it is R 0 < 1. Then, the disease-endemic equilibrium point is asymptotically stable local when it is R 0 > 1. Furthermore, the parameters of β, ρ, and γ are the most important parameter.

In this paper, a deterministic Tuberculosis (TB) model is formulated with the aim of assessing the effects of vaccination, screening and treatment on the transmission of TB infections. The analysis of the model shows that its dynamics are completely determined by the effective reproduction number, R eff. If R eff &lt; 1 the disease-free equilibrium exists and is locally and globally asymptotically stable whereas an endemic equilibrium exists if R eff &gt; 1 and is globally asymptotically stable, the disease persistence occurs. Furthermore, when the effective reproduction number is equal to one, that is R eff = 1, a backward bifurcation occurs. Numerical results are presented for the justifications of theoretical results.

This paper considers the mathematical model for dynamics of TB disease with vaccination, taking into consideration the passively immune infants (M) and the vaccination of the susceptible. We considered a Susceptible-Exposed-Infectious- Recovered (SEIR) model by introduced the passively immune infants resulting to an MSEIR model. The dynamics of the compartments were described by system of ordinary differential equations which were solved algebraically, and analyzed for stability. It was established that the disease free equilibrium state of the model is stable, when the basic reproduction number R<1, it was also established that the endemic states for the modified model is stable using Bellman and Cooke theorem and that if efforts are made to ensure that more susceptible infants are vaccinated, the breakdown of the susceptible and progression to infectious state is reduced and hence we recommend more vaccination of the susceptible and the treatment of the infected.

Journal of Biological Dynamics, 2019

The long and binding treatment of tuberculosis (TB) at least 6-8 months for the new cases, the partial immunity given by BCG vaccine, the loss of immunity after a few years doing that strategy of TB control via vaccination and treatment of infectious are not sufficient to eradicate TB. TB is an infectious disease caused by the bacillus Mycobacterium tuberculosis. Adults are principally attacked. In this work, we assess the impact of vaccination in the spread of TB via a deterministic epidemic model (SV ELI) (Susceptible, Vaccinated, Early latent, Late latent, Infectious). Using the Lyapunov-Lasalle method, we analyse the stability of epidemic system (SV ELI) around the equilibriums (disease-free and endemic). The global asymptotic stability of the unique endemic equilibrium whenever R 0 > 1 is proved, where R 0 is the reproduction number. We prove also that when R 0 is less than 1, TB can be eradicated. Numerical simulations, using some TB data found in the literature in relation with Cameroon, are conducted to approve analytic results, and to show that vaccination coverage is not sufficient to control TB, effective contact rate has a high impact in the spread of TB.

IOSR Journals , 2019

In this paper we considered a nonlinear deterministic dynamical system to study the effect of post exposure vaccination on fast and slow latent infection stages. We found that there are two equilibrium points exist. These are disease free equilibrium point and endemic equilibrium point. Their local stability and global stability analysis investigated using nonlinear stability methods.

Journal of Theoretical Biology, 2008

Epidemic control strategies alter the spread of the disease in the host population. In this paper, we describe and discuss mathematical models that can be used to explore the potential of pre-exposure and post-exposure vaccines currently under development in the control of tuberculosis. A model with bacille Calmette-Guerin (BCG) vaccination for the susceptibles and treatment for the infectives is first presented. The epidemic thresholds known as the basic reproduction numbers and equilibria for the models are determined and stabilities are investigated. The reproduction numbers for the models are compared to assess the impact of the vaccines currently under development. The centre manifold theory is used to show the existence of backward bifurcation when the associated reproduction number is less than unity and that the unique endemic equilibrium is locally asymptotically stable when the associated reproduction number is greater than unity. From the study we conclude that the preexposure vaccine currently under development coupled with chemoprophylaxis for the latently infected and treatment of infectives is more effective when compared to the post-exposure vaccine currently under development for the latently infected coupled with treatment of the infectives.

IOSR Journals , 2019

This article considers nonlinear dynamical system to study the dynamics of tuberculosis through vaccination and dual treatments. By dual treatments we studychemoprophylaxis and therapeuticstreatments of latent and active tuberculosis respectively. The total population is divided in to ten compartments. We found the dynamical system has disease free equilibrium point and endemic equilibrium point.

Mathematical Biosciences and Engineering, 2004

The reemergence of tuberculosis (TB) from the 1980s to the early 1990s instigated extensive researches on the mechanisms behind the transmission dynamics of TB epidemics. This article provides a detailed review of the work on the dynamics and control of TB. The earliest mathematical models describing the TB dynamics appeared in the 1960s and focused on the prediction and control strategies using simulation approaches. Most recently developed models not only pay attention to simulations but also take care of dynamical analysis using modern knowledge of dynamical systems. Questions addressed by these models mainly concentrate on TB control strategies, optimal vaccination policies, approaches toward the elimination of TB in the U.S.A., TB co-infection with HIV/AIDS, drug-resistant TB, responses of the immune system, impacts of demography, the role of public transportation systems, and the impact of contact patterns. Model formulations involve a variety of mathematical areas, such as ODEs (Ordinary Differential Equations) (both autonomous and non-autonomous systems), PDEs (Partial Differential Equations), system of difference equations, system of integro-differential equations, Markov chain model, and simulation models.