The reason is that there are some individuals with bad immune effector protection, due to many reasons. We introduce a direct transfer from the class of susceptible individuals toward the compartment of infectious individuals in each patch. In the model given here, there are migrations between people of two patches, who have the same epidemiological characteristics. This is of particular importance for tuberculosis since latency may last for years or even decades. By allowing for an arbitrary number of latently infected compartments, the model allows for the approximation of a wide class of distributions of latency durations. Therefore, this fact cannot be neglected in the modeling of TB. For example, according to the Cameroonian National Program of Fight against Tuberculosis, about 10% of infectious individuals that begun their therapy treatment lost sight. These lost sight individuals usually occur in sub-Saharan Africa.
It is an extension with two patches and latent classes of the SEIL model in, where denotes the lost sight individuals. The issue of modeling tuberculosis motivates the model studied in this paper. Since epidemic models are Lotka-Volterra-like models, the pertinence of this function is not surprising. Volterra himself originally discovered this function, although he did not use the vocabulary and the theory of Lyapunov functions. The function has a long story in epidemiology. In this paper, the stability of a -dimensions system is investigated using Lyapunov-LaSalle functions and quadratic forms. When such a function can be found, then any isolated minimum of the function is a stable equilibrium of the flow. Lyapunov’s method requires finding a function such that the flow always crosses the level sets from higher values of to lower values. For higher-dimensional systems, the theory of quadratic forms or Lyapunov’s method can be used.
Another method of Li and Muldowney for demonstrating global stability in dimensions has been developed more recently, with applications in three and four dimensions. Equilibrium can be shown to be globally and asymptotically stable, using Poincaré-Bendixson theory, Bendixson’s negative criterion, or the generalized version of Dulac.
The simplest possible attractor is a globally and asymptotically stable equilibrium. Introductionįor a given system, the focus in qualitative mathematical epidemiology is the long-term dynamics. Numerical simulation results are provided to illustrate the theoretical results and analyze the influence of lost sight individuals.
The local stability of endemic equilibria can be illustrated using numerical simulations. When it is greater than one, there exists at least one endemic equilibrium. A Lyapunov function is used to show that when the basic reproduction ratio is less than one, the disease-free equilibrium is globally and asymptotically stable. The existence and uniqueness of the associated equilibria are discussed.
#Seil model Patch
The model incorporates migrations from one patch to another. This fact usually occurs in sub-Saharan Africa, due to many reasons. A fraction of infectious individuals that begun their treatment will not return to the hospital for the examination of sputum. A two-patch model,, , is used to analyze the spread of tuberculosis, with an arbitrary number of latently infected compartments in each patch.
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