On Global Stability of Disease-Free Equilibrium in Epidemiological Models (original) (raw)
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Epidemiological Models and Lyapunov Functions
Mathematical Modelling of Natural Phenomena, 2007
We give a survey of results on global stability for deterministic compartmental epidemiological models. Using Lyapunov techniques we revisit a classical result, and give a simple proof. By the same methods we also give a new result on differential susceptibility and infectivity models with mass action and an arbitrary number of compartments. These models encompass the so-called differential infectivity and staged progression models. In the two cases we prove that if the basic reproduction ratio R 0 ≤ 1, then the disease free equilibrium is globally asymptotically stable. If R 0 > 1, there exists an unique endemic equilibrium which is asymptotically stable on the positive orthant.
Stability of Disease Free Equilibria in Epidemiological Models
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In this paper we study the structural properties of epidemiological models and we derive a necessary and sufficient condition for the stability of their disease free equilibria. We then show that for a large class of models our condition may be explicitly obtained by using symbolic computation tools. We also give a sufficient condition for the global stability of the disease free equilibrium.
An Approach for the Global Stability of Mathematical Model of an Infectious Disease
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The global stability analysis for the mathematical model of an infectious disease is discussed here. The endemic equilibrium is shown to be globally stable by using a modification of the Volterra–Lyapunov matrix method. The basis of the method is the combination of Lyapunov functions and the Volterra–Lyapunov matrices. By reducing the dimensions of the matrices and under some conditions, we can easily show the global stability of the endemic equilibrium. To prove the stability based on Volterra–Lyapunov matrices, we use matrices with the symmetry properties (symmetric positive definite). The results developed in this paper can be applied in more complex systems with nonlinear incidence rates. Numerical simulations are presented to illustrate the analytical results.
Mathematical Biosciences, 2008
One goal of this paper is to give an algorithm for computing a threshold condition for epidemiological systems arising from compartmental deterministic modeling. We calculate a threshold condition T 0 of the parameters of the system such that if T 0 < 1 the diseasefree equilibrium (DFE) is locally asymptotically stable (LAS), and if T 0 > 1, the DFE is unstable. The second objective, by adding some reasonable assumptions, is to give, depending on the model, necessary and sufficient conditions for global asymptotic stability (GAS) of the DFE. In many cases, we can prove that a necessary and sufficient condition for the global asymptotic stability of the DFE is R 0 6 1, where R 0 is the basic reproduction number [O. Diekmann, J.A. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, New York, 2000]. To illustrate our results, we apply our techniques to examples taken from the literature. In these examples we improve the results already obtained for the GAS of the DFE. We show that our algorithm is relevant for high dimensional epidemiological models.
Global stability of a two-stage epidemic model with generalized non-linear incidence
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A multi-stage model of disease transmission, which incorporates a generalized non-linear incidence function, is developed and analysed qualitatively. The model exhibits two steady states namely: a disease-free state and a unique endemic state. A global stability of the model reveals that the disease-free equilibrium is globally asymptotically stable (and therefore the disease can be eradicated) provided a certain threshold R 0 (known as the basic reproductive number) is less than unity. On the other hand, the unique endemic equilibrium is globally asymptotically stable for R 0 > 1.
Global stability of an SAIRD epidemiological model with negative feedback
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In this work, we study the dynamical behavior of a modified SIR epidemiological model by introducing negative feedback and a nonpharmaceutical intervention. The first model to be defined is the usceptible–Isolated–Infected–Recovered–Dead (SAIRD) epidemics model and then the S-A-I-R-D-Information Index (SAIRDM) model that corresponds to coupling the SAIRD model with the negative feedback. Controlling the information about nonpharmaceutical interventions is considered by the addition of a new variable that measures how the behavioral changes about isolation influence pandemics. An analytic expression of a replacement ratio that depends on the absence of the negative feedback is determined. The results obtained show that the global stability of the disease-free equilibrium is determined by the value of a certain threshold parameter called the basic reproductive number mathcalR0\mathcal{R}_{0}mathcalR0 R 0 and the local stability of the free disease equilibrium depends on the replacement ratios. A Hopf...
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In this paper, we study an epidemic model describing two strains with non-monotone incidence rates. The model consists of six ordinary differential equations illustrating the interaction between the susceptible, the exposed, the infected and the removed individuals. The system of equations has four equilibrium points, disease-free equilibrium, endemic equilibrium with respect to strain 1, endemic equilibrium with respect to strain 2, and the last endemic equilibrium with respect to both strains. The global stability analysis of the equilibrium points was carried out through the use of suitable Lyapunov functions. Two basic reproduction numbers R 1 0 and R 2 0 are found; we have shown that if both are less than one, the disease dies out. It was established that the global stability of each endemic equilibrium depends on both basic reproduction numbers and also on the strain inhibitory effect reproduction number(s) R m and/or R k. It was also shown that any strain with highest basic reproduction number will automatically dominate the other strain. Numerical simulations were carried out to support the analytic results and to show the effect of different problem parameters on the infection spread.
Local and Global Stability of Host-Vector Disease Models
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Global Stability of an Epidemic Model with Two Infected Stages and Mass-Action Incidence
Mathematical theory and modeling, 2020
In this research work, we study the global stability of the SIR model which describes the dynamics of infectious disease with two classes of infected stages and varying total population size. The incidence used in the mathematical modeling was the mass-action incidence. The basic reproduction number R0 is computed. If the basic reproduction number is less than one, then the disease-free equilibrium point is locally and globally asymptotically stable. Existence and uniqueness of the endemic equilibrium is established when the basic reproduction number is greater than one and locally stable. We prove that global stability of the disease free equilibrium point using Lyapunov function. Numerical simulations have been carried out applying mat lab. Our result show that if the basic reproduction number R0 is below one the disease free equilibrium point is locally and globally stable in the feasible region, so that the disease dies out. If the basic reproduction number R0 is greater than on...