Stability Of Epidemic Model Research Articles

Stability of epidemic models over directed graphs: A positive systems approach

We study the stability properties of a susceptible–infected–susceptible (SIS) diffusion model, so-called the n-intertwined Markov model, over arbitrary directed network topologies. As in the majority of the work on infection spread dynamics, this model exhibits a threshold phenomenon. When the curing rates in the network are high, the disease-free equilibrium is the unique equilibrium over the network. Otherwise, an endemic equilibrium state emerges, where some infection remains within the network. Using notions from positive systems theory, we provide novel proofs for the global asymptotic stability of the equilibrium points in both cases over strongly connected networks based on the value of the basic reproduction number, a fundamental quantity in the study of epidemics. When the network topology is weakly connected, we provide conditions for the existence, uniqueness, and global asymptotic stability of an endemic state, and study the stability of the disease-free equilibrium. Finally, we demonstrate that the n-intertwined Markov model can be viewed as a best-response dynamical system of a concave game among the nodes. This characterization allows us to cast new infection spread dynamics; additionally, we provide a sufficient condition for global convergence to the disease-free equilibrium, which can be checked in a distributed fashion.

Stability of epidemic models with waning immunity

SIRS type epidemiological model has a fundamental form to study the role of temporal immunity of recovered individuals in disease transmission dynamics and several variant models have been considered in the last century, but up to now dynamical aspects of the model are not fully elucidated. We here look over previous studies concerning qualitative analysis for a family of SIRS type epidemiological models. To this aim we construct a general model in a form of delay equation, a coupled system of a renewal equation and delay differential equations, structuring infected population by infection-age (time elapsed since infection). We re-examine the structure of equilibria and stability of the disease free equilibrium. We then introduce slightly improved stability conditions for an endemic equilibrium. Specifying modelling ingredients we derive two special cases where the model can be represented as a system of ordinary and delay differential equations that have appeared in the literature. For those models we have a powerful tool, namely Lyapunov function, to study global stability of the endemic equilibrium. Relating epidemic models are also discussed.

Stability of Epidemic Models with Evolution

Epidemic models are considered where the diffusion of individuals is influenced by intraspecific competition pressures and are weakly affected by different classes. The nonlinear stability of some of the models is proved by introducing new Liapunov functions.

M-matrices and local stability in epidemic models

We apply the next-generator operator concept to the analysis of nondisease free equilibrium points in multigroup models in infectious diseases. In particular, we apply our results to the analysis of the local stability of a multigroup SEIR model.

Stability of epidemic model with time delays influenced by stochastic perturbations

Many processes in automatic regulation, physics, mechanics, biology, economy, ecology etc. can be modelled by hereditary equations (see, e.g. [1–6]). One of the main problems for the theory of stochastic hereditary equations and their applications is connected with stability. Many stability results were obtained by the construction of appropriate Lyapunov functionals. In [7–11], the procedure is proposed, allowing, in some sense, to formalize the algorithm of the corresponding Lyapunov functionals construction for stochastic functional differential equations, for stochastic difference equations. In this paper, stability conditions are obtained by using this procedure for the mathematical model of the spread of infections diseases with delays influenced by stochastic perturbations.