The explicit series solution of SIR and SIS epidemic models

In this paper the SIR and SIS epidemic models in biology are solved by means of an analytic technique for nonlinear problems, namely the homotopy analysis method (HAM). Both of the SIR and SIS models are described by coupled nonlinear differential equations. A one-parameter family of explicit series solutions are obtained for both models. This parameter has no physical meaning but provides us with a simple way to ensure convergent series solutions to the epidemic models. Our analytic results agree well with the numerical ones. This analytic approach is general and can be applied to get convergent series solutions of some other coupled nonlinear differential equations in biology.     

Stability techniques in SIR epidemic models

In biological communities interaction among different species affect their stability. In this paper, we consider a nonlinear SIR model, which describes the dynamics of the interaction between susceptible and infected individuals in population. We establish stability techniques to find out the equilibria for the model and their numerical results are given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models

Lyapunov functions for classical SIR, SIRS, and SIS epidemiological models are introduced. Global stability of the endemic equilibrium states of the models is thereby established.     

The topological defense in SIS epidemic models

The spreading of dangerous malware or faults in inter-dependent networks of electronics devices has raised deep concern, because from the ICT networks infections may propagate to other Critical Infrastructures producing the well-known domino or cascading effect. Researchers are attempting to develop a high level analysis of malware propagation discarding software details, in order to generalize to the maximum extent the defensive strategies. For example, it has been suggested that the maximum eigenvalue of the network adjacency matrix could act as a threshold for the malware’s spreading. This leads naturally to use the spectral graph theory to identify the most critical and influential nodes in technological networks. Many well-known graph parameters have been studied in the past years to accomplish the task. In this work, we test our AV11 algorithm showing that outperforms degree, closeness, betweenness centrality and the dynamical importance.     

Lyapunov functions for SIR and SIRS epidemic models

In this paper, we construct a new Lyapunov function for a variety of SIR and SIRS models in epidemiology. Global stability of the endemic equilibrium states of these systems is established.     

Global dynamics and bifurcation in delayed SIR epidemic model

An SIR epidemic model with time delay, information variable and saturated incidence rate, where the susceptibles are assumed to satisfy the logistic equation and the incidence term, is of saturated form with the susceptibles. This model exhibits two bifurcations, one is transcritical bifurcation and the other is Hopf bifurcation. The local and global stability of endemic equilibrium is also discussed. Finally, numerical simulations are carried out to explain the mathematical conclusions.     

Global stability of an SIS epidemic model with delays

In this paper, we consider the global dynamics of the S(E)IS model with delays denoting an incubation time. By constructing a Lyapunov functional, we prove stability of a disease‐free equilibrium E₀ under a condition different from that in the recent paper. Then we claim that R₀≤1 is a necessary and sufficient condition under which E₀ is globally asymptotically stable. We also propose a discrete model preserving positivity and global stability of the same equilibria as the continuous model with distributed delays, by means of discrete analogs of the Lyapunov functional.     

Hybrid stochastic epidemic SIR models with hidden states

This paper focuses on realistic hybrid SIR models that take into account stochasticity. The proposed systems are applicable to most incidence rates that are used in the literature including the bilinear incidence rate, the Beddington–DeAngelis incidence rate, and a Holling type II functional response. Given that many diseases can lead to asymptomatic infections, this paper looks at a system of stochastic differential equations that also includes a class of hidden state individuals, for which the infection status is unknown. Assuming that the direct observation of the percentage of hidden state individuals being infected, $\alpha \left(t\right)$, is not given and only a noise-corrupted observation process is available. Using nonlinear filtering techniques in conjunction with an invasion type analysis, this paper shows that the long-term behavior of the disease is governed by a threshold $\lambda \in \mathbb{R}$ that depends on the model parameters. It turns out that if $\lambda <0$ the number $I\left(t\right)$ of infected individuals converges to zero exponentially fast (extinction). However, if $\lambda >0$, the infection is endemic and the system is persistent. We showcase our theorems by applying them in some illuminating examples.     

Global convergence analysis of a class of epidemic models

This paper addresses the global convergence of the epidemic models whose infected subsystems are monotone in the sense of Hirsch (1984). By invoking results from monotone system theory and nonlinear control theory, a simple method is proposed for determining the global asymptotic stability of a disease free equilibrium (DFE) and the global convergence to an endemic equilibrium (EE). Typical epidemic models are studied to illustrate the applicability of the proposed methodology.     

SIR epidemic models with general infectious period distribution

We show how epidemics in which individuals’ infectious periods are not necessarily exponentially distributed may be naturally modelled as piecewise deterministic Markov processes. For the standard susceptible–infective–removed (SIR) model, we exhibit a family of martingales which may be used to derive the joint distribution of the number of survivors of the epidemic and the area under the trajectory of infectives. We also show how these results may be extended to a model in which the rate at which an infective generates infectious contacts may be an arbitrary function of the number of susceptible individuals present.     

Global behaviour of an SIR epidemic model with time delay

We study the stability of a delay susceptible–infective–recovered epidemic model with time delay. The model is formulated under the assumption that all individuals are susceptible, and we analyse the global stability via two methods—by Lyapunov functionals, and—in terms of the variance of the variables. The main theorem shows that the endemic equilibrium is stable. If the basic reproduction number ?₀ is less than unity, by LaSalle invariance principle, the disease‐free equilibrium E_s is globally stable and the disease always dies out. By applying the integral averaging theory, we also investigate the stability in variance of the model. Copyright © 2006 John Wiley & Sons, Ltd.     

Global dynamics of SIS models with transport-related infection

To understand the effect of transport-related infection on disease spread, an epidemic model for several regions which are connected by transportation of individuals has been proposed by Cui, Takeuchi and Saito [J. Cui, Y. Takeuchi, Y. Saito, Spreading disease with transport-related infection, J. Theoret. Biol. 239 (2006) 376-390]. Transportation among regions is one of the main factors which affects the outbreak of diseases. The purpose of this paper is the further study of the local asymptotic stability of the endemic equilibrium and the global dynamics of the system. Sufficient conditions are established for global asymptotic stability of the endemic equilibrium. Permanence is also discussed. It is shown that the disease is endemic in the sense of permanence if and only if the endemic equilibrium exists. This implies that transport-related infection on disease can make the disease endemic even if all the isolated regions are disease free.     

Global attractivity of a network-based epidemic SIS model with nonlinear infectivity

In this paper, a new epidemic SIS model with nonlinear infectivity, as well as birth and death of nodes and edges, is investigated on heterogeneous networks. The global behavior of the model is studied mathematically. When the basic reproductive number is less than or equal to unity, it is verified that the disease dies out; otherwise, the model solutions lead to positive steady states. This paper provides a concise mathematical analysis to verify the global dynamics of the model.     

Global behavior of a multi-group SIS epidemic model with age structure

We study global dynamics of a system of partial differential equations. The system is motivated by modelling the transmission dynamics of infectious diseases in a population with multiple groups and age-dependent transition rates. Existence and uniqueness of a positive (endemic) equilibrium are established under the quasi-irreducibility assumption, which is weaker than irreducibility, on the function representing the force of infection. We give a classification of initial values from which corresponding solutions converge to either the disease-free or the endemic equilibrium. The stability of each equilibrium is linked to the dominant eigenvalue s(A), where A is the infinitesimal generator of a “quasi-irreducible” semigroup generated by the model equations. In particular, we show that if s(A)<0 then the disease-free equilibrium is globally stable; if s(A)>0 then the unique endemic equilibrium is globally stable.