1,946 research outputs found
Global Stability of Infectious Disease Models Using Lyapunov Functions
Two systematic methods are presented to guide the construction of Lyapunov functions for general infectious disease models and are thus applicable to establish their global dynamics. Specifically, a matrix-theoretic method using the Perron eigenvector is applied to prove the global stability of the disease-free equilibrium, while a graph-theoretic method based on Kirchhoff\u27s matrix tree theorem and two new combinatorial identities are used to prove the global stability of the endemic equilibrium. Several disease models in the literature and two new cholera models are used to demonstrate the applications of these methods
Lyapunov Functions for Tuberculosis Models with Fast and Slow Progression
The spread of tuberculosis is studied through two models which include fast and slow progression to the infected class. For each model, Lyapunov functions are used to show that when the basic reproduction number is less than or equal to one, the disease-free equilibrium is globally asymptotically stable, and when it is greater than one there is an endemic equilibrium which is globally asymptotically stable
On the dynamics of a class of multi-group models for vector-borne diseases
The resurgence of vector-borne diseases is an increasing public health
concern, and there is a need for a better understanding of their dynamics. For
a number of diseases, e.g. dengue and chikungunya, this resurgence occurs
mostly in urban environments, which are naturally very heterogeneous,
particularly due to population circulation. In this scenario, there is an
increasing interest in both multi-patch and multi-group models for such
diseases. In this work, we study the dynamics of a vector borne disease within
a class of multi-group models that extends the classical Bailey-Dietz model.
This class includes many of the proposed models in the literature, and it can
accommodate various functional forms of the infection force. For such models,
the vector-host/host-vector contact network topology gives rise to a bipartite
graph which has different properties from the ones usually found in directly
transmitted diseases. Under the assumption that the contact network is strongly
connected, we can define the basic reproductive number and show
that this system has only two equilibria: the so called disease free
equilibrium (DFE); and a unique interior equilibrium---usually termed the
endemic equilibrium (EE)---that exists if, and only if, . We
also show that, if , then the DFE equilibrium is globally
asymptotically stable, while when , we have that the EE is
globally asymptotically stable
Global Stability for an SEIR Epidemiological Model with Varying Infectivity and Infinite Delay
A recent paper (Math. Biosci. and Eng. (2008) 5:389-402) presented an SEIR model using an infinite delay to account for varying infectivity. The analysis in that paper did not resolve the global dynamics for R0 \u3e 1. Here, we show that the endemic equilibrium is globally stable for R0 \u3e 1. The proof uses a Lyapunov functional that includes an integral over all previous states
Global Stability of an SIR Epidemic Model with Delay and General Nonlinear Incidence
An SIR model with distributed delay and a general incidence function is studied. Conditions are given under which the system exhibits threshold behaviour: the disease-free equilibrium is globally asymptotically stable if R0 \u3c 1 and globally attracting if R0 = 1; if R0 \u3e 1, then the unique endemic equilibrium is globally asymptotically stable. The global stability proofs use a Lyapunov functional and do not require uniform persistence to be shown a priori. It is shown that the given conditions are satisfied by several common forms of the incidence function
Nonlinear incidence and stability of infectious disease models
In this paper we consider the impact of the form of the non-linearity of the infectious disease incidence rate on the dynamics of epidemiological models. We consider a very general form of the non-linear incidence rate (in fact, we assumed that the incidence rate is given by an arbitrary function f (S, I, N) constrained by a few biologically feasible conditions) and a variety of epidemiological models. We show that under the constant population size assumption, these models exhibit asymptotically stable steady states. Precisely, we demonstrate that the concavity of the incidence rate with respect to the number of infective individuals is a sufficient condition for stability. If the incidence rate is concave in the number of the infectives, the models we consider have either a unique and stable endemic equilibrium state or no endemic equilibrium state at all; in the latter case the infection-free equilibrium state is stable. For the incidence rate of the form g(I)h(S), we prove global stability, constructing a Lyapunov function and using the direct Lyapunov method. It is remarkable that the system dynamics is independent of how the incidence rate depends on the number of susceptible individuals. We demonstrate this result using a SIRS model and a SEIRS model as case studies. For other compartment epidemic models, the analysis is quite similar, and the same conclusion, namely stability of the equilibrium states, holds
A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence
Explicit Lyapunov functions for SIR and SEIR compartmental epidemic models with nonlinear incidence of the form for the case p<1 are constructed. Global stability of the models is thereby established
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