Stability analysis for differential infectivity epidemic models

We present several differential infectivity (DI) epidemic models under different assumptions. As the number of contacts is assumed to be constant or a linear function of the total population size, either standard or bilinear incidence of infection is resulted. We establish global stability of the infection-free equilibrium and the endemic equilibrium for DI models of SIR (susceptible/infected/removed) type with bilinear incidence and standard incidence but no disease-induced death, respectively. We also obtain global stability of the two equilibria for a DI SIS (susceptible/infected/susceptible) model with population-density-dependent birth and death functions. For completeness, we extend the stability of the infection-free equilibrium for the standard DI SIR model previously proposed.     

Effects of population density on the spread of disease

The effect of population density on the epidemic outbreak of measles or measles-like infectious diseases was evaluated. Using average-number contacts with susceptible individuals per infectious individual as a measure of population density, an analytical model for the distribution of the nonstationary stochastic process of susceptible contact is presented. A 5-dimensional lattice simulation model of disease spread was used to evaluate the effects of four different population densities. A zero-inflated Poisson probability model was used to quantify the nonstationarity of the contact rate in the stochastic epidemic process. Analysis of the simulation results identified a decrease in a susceptible contact rate from four to three, resulted in a dramatic effect on the distribution of contacts over time, the magnitude of the outbreak, and, ultimately, the spread of disease. © 2001 John Wiley & Sons, Inc.     

离散的SI和SIS传染病模型的研究

为了描述个体的死亡、染病者的恢复以及疾病的传染,引入了相应的概率.基于总种群中个体数量为常数的假设,根据染病者能否恢复分别建立了具有生命动力学的离散SI和SIS传染病模型.所得到的结果显示:它们具有与相应连续模型相同的动力学性态,并确定了各自的阈值.在它们的阈值之下,传染病最终将灭绝;在它们的阈值之上,传染病将会发展成为地方病,染病者的数量将趋向于一确定的正常数.     

Competing spreading processes and immunization in multiplex networks

Epidemic spreading on physical contact network will naturally introduce the human awareness information diffusion on virtual contact network, and the awareness diffusion will in turn depress the epidemic spreading, thus forming the competing spreading processes of epidemic and awareness in a multiplex networks. In this paper, we study the competing dynamics of epidemic and awareness, both of which follow the SIR process, in a two-layer networks based on microscopic Markov chain approach and numerical simulations. We find that strong capacities of awareness diffusion and self-protection of individuals could lead to a much higher epidemic threshold and a smaller outbreak size. However, the self-awareness of individuals has no obvious effect on the epidemic threshold and outbreak size. In addition, the immunization of the physical contact network under the interplay between of epidemic and awareness spreading is also investigated. The targeted immunization is found performs much better than random immunization, and the awareness diffusion could reduce the immunization threshold for both type of random and targeted immunization significantly.     

A note on the global behaviour of the network-based SIS epidemic model

In this note we introduce the study of the global behaviour of the network-based SIS epidemic model recently proposed by Pastor-Satorras and Vespignani [Epidemic spreading in scale-free networks, Phys. Rev. Lett. 86 (2001) 3200], characterized in case of homogeneous scale-free networks by a very small epidemic threshold, and extended by Olinky and Stone [Unexpected epidemic threshold in heterogeneous networks: the role of disease transmission, Phys. Rev. E 70 (2004) 03902(r)]. We show that the above model may be read as a particular case of the classical multi-group SIS model proposed by Lajmainovitch and Yorke [A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28 (1976) 221] and extended by Aronsson and Mellander [A deterministic model in biomathematics. Asymptotic behaviour and threshold conditions, Math. Biosci. 49 (1980) 207]. Thus, by applying the methods used for SIS multi-group models, we straightforwardly show, for the first time, that the local conditions identified in the physics literature also determine the global behaviour of a disease spreading on a network. Finally, we briefly study the case in which the force of infection is non-linear, by showing that multiple coexisting equilibria are possible, and by giving a global threshold condition for the extinction.     

Estimation of the parameters of an infectious disease model using neural networks

In this paper, we propose a realistic mathematical model taking into account the mutual interference among the interacting populations. This model attempts to describe the control (vaccination) function as a function of the number of infective individuals, which is an improvement over the existing susceptible–infective epidemic models. Regarding the growth of the epidemic as a nonlinear phenomenon we have developed a neural network architecture to estimate the vital parameters associated with this model. This architecture is based on a recently developed new class of neural networks known as co-operative and supportive neural networks. The application of this architecture to the present study involves preprocessing of the input data, and this renders an efficient estimation of the rate of spread of the epidemic. It is observed that the proposed new neural network outperforms a simple feed-forward neural network and polynomial regression.     

Global analysis of a delayed epidemic dynamical system with pulse vaccination and nonlinear incidence rate

This study explores the influence of epidemics by numerical simulations and analytical techniques. Pulse vaccination is an effective strategy for the treatment of epidemics. Usually, an infectious disease is discovered after the latent period, H1N1 for instance. The vaccinees (susceptible individuals who have started the vaccination process) are different from both susceptible and recovered individuals. So we put forward a SVEIRS epidemic model with two time delays and nonlinear incidence rate, and analyze the dynamical behavior of the model under pulse vaccination. The global attractivity of ‘infection-free’ periodic solution and the existence, uniqueness, permanence of the endemic periodic solution are investigated. We obtain sufficient condition for the permanence of the epidemic model with pulse vaccination. The main feature of this study is to introduce two discrete time delays and impulse into SVEIRS epidemic model and to give pulse vaccination strategies.     

Dynamics of SIS Epidemic Model with Varying Total Population and Multivaccination Control Strategies

In this paper, two susceptible‐infected‐susceptible epidemic models with varying total population size, continuous vaccination, and state‐dependent pulse vaccination are formulated to describe the transmission of infectious diseases, such as diphtheria, measles, rubella, pertussis, and so on. The first model incorporates the proportion of infected individuals in population as monitoring threshold value; we analytically show the existence and orbital asymptotical stability of positive order‐1 periodic solution for this control model. The other model determines control strategy by monitoring the proportion of susceptible individuals in population; we also investigate the existence and global orbital asymptotical stability of the disease‐free periodic solution. Theoretical results imply that the disease dies out in the second case. Finally, using realistic parameter values, we carry out some numerical simulations to illustrate the main theoretical results and the feasibility of state‐dependent pulse control strategy.     

Spatial epidemics: critical behavior in one dimension

In the simple mean-field SIS and SIR epidemic models, infection is transmitted from infectious to susceptible members of a finite population by independent p-coin tosses. Spatial variants of these models are considered, in which finite populations of size N are situated at the sites of a lattice and infectious contacts are limited to individuals at neighboring sites. Scaling laws for these models are given when the infection parameter p is such that the epidemics are critical. It is shown that in all cases there is a critical threshold for the numbers initially infected: below the threshold, the epidemic evolves in essentially the same manner as its branching envelope, but at the threshold evolves like a branching process with a size-dependent drift. The corresponding scaling limits are super-Brownian motions and Dawson–Watanabe processes with killing, respectively.     

Spreading dynamics and global stability of a generalized epidemic model on complex heterogeneous networks

In this paper, a generalized epidemic model on complex heterogeneous networks is proposed. To give a theoretical explanation for the simulation results established on networks, mathematical analysis of the epidemic dynamics is presented via mean-field approximation. Stabilities of the disease-free equilibrium and the endemic equilibrium are studied. The results explain why the heterogeneous connectivity patterns impact the epidemic threshold and reveal how the host parameters and the underlying network structures determine disease propagation.     

An epidemic model to evaluate the homogeneous mixing assumption

Many epidemic models are written in terms of ordinary differential equations (ODE). This approach relies on the homogeneous mixing assumption; that is, the topological structure of the contact network established by the individuals of the host population is not relevant to predict the spread of a pathogen in this population. Here, we propose an epidemic model based on ODE to study the propagation of contagious diseases conferring no immunity. The state variables of this model are the percentages of susceptible individuals, infectious individuals and empty space. We show that this dynamical system can experience transcritical and Hopf bifurcations. Then, we employ this model to evaluate the validity of the homogeneous mixing assumption by using real data related to the transmission of gonorrhea, hepatitis C virus, human immunodeficiency virus, and obesity.     

Epidemic spreading of an SEIRS model in scale-free networks

An SEIRS epidemic model on the scale-free networks is presented, where the active contact number of each vertex is assumed to be either constant or proportional to its degree for this model. Using the analytical method, we obtain the two threshold values for above two cases and find that the threshold value for constant contact is independent of the topology of the underlying networks. The existence of positive equilibrium is determined by threshold value. For a finite size of scale-free network, we prove the local stability of disease-free equilibrium and the permanence of the disease on the network. Furthermore, we investigate two major immunization strategies, random immunization and targeted immunization, some similar results are obtained. The simulation shows the positive equilibrium is stable.     

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.     

A linked risk group model for investigating the spread of HIV

This paper describes a model that simulates the spread of HIV and progression to AIDS. The model is based on classical models of disease transmission. It consists of six linked risk groups and tracks the numbers of infectives, AIDS cases, AIDS related deaths, and other deaths of infected persons in each risk group. Parametric functions are used to represent risk-group-specific and time-dependent average contact rates. Contacts are needle sharing, sexual contacts, or blood product transfers.

An important feature of the model is that the contact rate parameters are estimated by minimizing differences between AIDS incidence and reported AIDS cases adjusted for undercounting biases. This feature results in an HIV epidemic curve that is analogous to one estimated by backcalculation models but whose dynamics are determined by simulating disease transmission. The model exhibits characteristics of both the disease transmission and the backcalculation approaches, i.e., the model:

• reconstructs the historical behavior patterns of the different risk groups,

• includes separate effects of treatment and changes in average contact rates,

• accounts for other mortality risks for persons infected with HIV,

• calculates short-term projections of AIDS incidence, HIV incidence, and HIV prevalence,

• calculates cumulative HIV infections (the quantity calculated by backcalculation approaches) and HIV prevalence (the quantity measured by seroprevalence and sentinel surveys). This latter feature permits the validation of the estimates generated by two distinct approaches.

We demonstrate the use of the model with an application to U.S. AIDS data through 1991.     

Complex dynamics of epidemic models on adaptive networks

There has been a substantial amount of well mixing epidemic models devoted to characterizing the observed complex phenomena (such as bistability, hysteresis, oscillations, etc.) during the transmission of many infectious diseases. A comprehensive explanation of these phenomena by epidemic models on complex networks is still lacking. In this paper we study epidemic dynamics in an adaptive network proposed by Gross et al., where the susceptibles are able to avoid contact with the infectious by rewiring their network connections. Such rewiring of the local connections changes the topology of the network, and inevitably has a profound effect on the transmission of the disease, which in turn influences the rewiring process. We rigorously prove that the adaptive epidemic model investigated in this paper exhibits degenerate Hopf bifurcation, homoclinic bifurcation and Bogdanov–Takens bifurcation. Our study shows that adaptive behaviors during an epidemic may induce complex dynamics of disease transmission, including bistability, transient and sustained oscillations, which contrast sharply to the dynamics of classical network models. Our results yield deeper insights into the interplay between topology of networks and the dynamics of disease transmission on networks.     

Analysis of epidemic spreading of an SIRS model in complex heterogeneous networks

In this paper, we study the spreading of infections in complex heterogeneous networks based on an SIRS epidemic model with birth and death rates. We find that the dynamics of the network-based SIRS model is completely determined by a threshold value. If the value is less than or equal to one, then the disease-free equilibrium is globally attractive and the disease dies out. Otherwise, the disease-free equilibrium becomes unstable and in the meantime there exists uniquely an endemic equilibrium which is globally asymptotically stable. A series of numerical experiments are given to illustrate the theoretical results. We also consider the SIRS model in the clustered scale-free networks to examine the effect of network community structure on the epidemic dynamics.     

一类SIRS传染病模型

This paper considers an SIRS epidemic model that incorporates constant immigration rate, a general population-size dependent contact rate and proportional transfer rate from the infective class to susceptible class. A threshold parameter a is identified. If σ≤1, the disease-free equilibrium is globally stable. If σ＞1, a unique endemic equilibrium is locally asymptotically stable. For two important special cases of mass action incidence and standard incidence,global stability of the endemic equilibrium is proved provided the threshold is larger than unity. Some previous results are extended and improved.     

Viral conductance: Quantifying the robustness of networks with respect to spread of epidemics

In this paper, we propose a novel measure, viral conductance (VC), to assess the robustness of complex networks with respect to the spread of SIS epidemics. In contrast to classical measures that assess the robustness of networks based on the epidemic threshold above which an epidemic takes place, the new measure incorporates the fraction of infected nodes at steady state for all possible effective infection strengths. Through examples, we show that VC provides more insight about the robustness of networks than does the epidemic threshold. We also address the paradoxical robustness of Barabási–Albert preferential attachment networks. Even though this class of networks is characterized by a vanishing epidemic threshold, the epidemic requires high effective infection strength to cause a major outbreak. On the contrary, in homogeneous networks the effective infection strength does not need to be very much beyond the epidemic threshold to cause a major outbreak. To overcome computational complexities, we propose a heuristic to compute the VC for large networks with high accuracy. Simulations show that the heuristic gives an accurate approximation of the exact value of the VC. Moreover, we derive upper and lower bounds of the new measure. We also apply the new measure to assess the robustness of different types of network structures, i.e. Watts–Strogatz small world, Barabási–Albert, correlated preferential attachment, Internet AS-level, and social networks. The extensive simulations show that in Watts–Strogatz small world networks, the increase in probability of rewiring decreases the robustness of networks. Additionally, VC confirms that the irregularity in node degrees decreases the robustness of the network. Furthermore, the new measure reveals insights about design and mitigation strategies of infrastructure and social networks.     

Nonlinear dynamical analysis and control strategies of a network-based SIS epidemic model with time delay

In this paper, we establish a susceptible-infected-susceptible (SIS) epidemic model with nonlinear incidence rate and time delay on complex networks. Firstly, according to the existence of a positive equilibrium point, we work out the threshold values R₀ and λ_c of disease propagation. Secondly, we demonstrate the stabilities of the disease-free equilibrium point and the disease-spreading equilibrium point by constructing Lyapunov function and applying delay differential equations theorem. Thirdly, four different control strategies are investigated and compared, including uniform immunization control, acquaintance immunization control, active immunization control and optimal control. Finally, we perform representative numerical simulations to illustrate the theoretical results and further discover that the nonlinear incidence rate can more accurately reflect individual psychological activities when a certain disease outbreaks at a high level.