Competitive dual-strain SIS epidemiological models with awareness programs in heterogeneous networks: two modeling approaches.

Epidemic diseases and media campaigns are closely associated with each other. Considering most epidemics have multiple pathogenic strains, in this paper, we take the lead in proposing two multi-strain SIS epidemic models in heterogeneous networks incorporating awareness programs due to media. For the first model, we assume that the transmission rates for strain 1 and strain 2 depend on the level of awareness campaigns. For the second one, we further suppose that awareness divides susceptible population into two different subclasses. After defining the basic reproductive numbers for the whole model and each strain, we obtain the analytical conditions that determine the extinction, coexistence and absolute dominance of two strains. Moreover, we also formulate its optimal control problem and identify an optimal implementation pair of awareness campaigns using optimal control theory. Given the complexity of the second model, we use the numerical simulations to visualize its different types of dynamical behaviors. Through theoretical and numerical analysis of these two models, we discover some new phenomena. For example, during the persistence analysis of the first model, we find that the characteristic polynomials of two boundary equilibria may have a pair of pure imaginary roots, implying that Hopf bifurcation and periodic solutions may appear. Most strikingly, multistability occurs in the second model and the growth rate of awareness programs (triggered by the infection prevalence) has a multistage impact on the final size of two strains. The numerical results suggest that the spread of a two-strain epidemic can be controlled (even be eradicated) by taking the measures of enhancing awareness transmission, reducing memory fading of aware individuals and ensuring high-level influx and rapid growth of awareness programs appropriately.

Asymmetrical dynamics of epidemic propagation and awareness diffusion in multiplex networks.

To better explore asymmetrical interaction between epidemic spreading and awareness diffusion in multiplex networks, we distinguish susceptibility and infectivity between aware and unaware individuals, relax the degree of immunization, and take into account three types of generation mechanisms of individual awareness. We use the probability trees to depict the transitions between distinct states for nodes and then write the evolution equation of each state by means of the microscopic Markovian chain approach (MMCA). Based on the MMCA, we theoretically analyze the possible steady states and calculate the critical threshold of epidemics, related to the structure of epidemic networks, the awareness diffusion, and their coupling configuration. The achieved analytical results of the mean-field approach are consistent with those of the numerical Monte Carlo simulations. Through the theoretical analysis and numerical simulations, we find that global awareness can reduce the final scale of infection when the regulatory factor of the global awareness ratio is less than the average degree of the epidemic network but it cannot alter the onset of epidemics. Furthermore, the introduction of self-awareness originating from infected individuals not only reduces the epidemic prevalence but also raises the epidemic threshold, which tells us that it is crucial to enhance the early warning of symptomatic individuals during pandemic outbreaks. These results give us a more comprehensive and deep understanding of the complicated interaction between epidemic transmission and awareness diffusion and also provide some practical and effective recommendations for the prevention and control of epidemics.

Traveling Waves and Estimation of Minimal Wave Speed for a Diffusive Influenza Model with Multiple Strains.

Antiviral treatment remains one of the key pharmacological interventions against influenza pandemic. However, widespread use of antiviral drugs brings with it the danger of drug resistance evolution. To assess the risk of the emergence and diffusion of resistance, in this paper, we develop a diffusive influenza model where influenza infection involves both drug-sensitive and drug-resistant strains. We first analyze its corresponding reaction model, whose reproduction numbers and equilibria are derived. The results show that the sensitive strains can be eliminated by treatment. Then, we establish the existence of the three kinds of traveling waves starting from the disease-free equilibrium, i.e., semi-traveling waves, strong traveling waves and persistent traveling waves, from which we can get some useful information (such as whether influenza will spread, asymptotic speed of propagation, the final state of the wavefront). On the other hand, we discuss three situations in which semi-traveling waves do not exist. When the control reproduction number [Formula: see text] is larger than 1, the conditions for the existence and nonexistence of traveling waves are determined completely by the reproduction numbers [Formula: see text], [Formula: see text] and the wave speed c. Meanwhile, we give an interval estimation of minimal wave speed for influenza transmission, which has important guiding significance for the control of influenza in reality. Our findings demonstrate that the control of influenza depends not only on the rates of resistance emergence and transmission during treatment, but also on the diffusion rates of influenza strains, which have been overlooked in previous modeling studies. This suggests that antiviral treatment should be implemented appropriately, and infected individuals (especially with the resistant strain) should be tested and controlled effectively. Finally, we outline some future directions that deserve further investigation.

Interaction between epidemic spread and collective behavior in scale-free networks with community structure.

Many real-world networks exhibit community structure: the connections within each community are dense, while connections between communities are sparser. Moreover, there is a common but non-negligible phenomenon, collective behaviors, during the outbreak of epidemics, are induced by the emergence of epidemics and in turn influence the process of epidemic spread. In this paper, we explore the interaction between epidemic spread and collective behavior in scale-free networks with community structure, by constructing a mathematical model that embeds community structure, behavioral evolution and epidemic transmission. In view of the differences among individuals' responses in different communities to epidemics, we use nonidentical functions to describe the inherent dynamics of individuals. In practice, with the progress of epidemics, individual behaviors in different communities may tend to cluster synchronization, which is indicated by the analysis of our model. By using comparison principle and Gers˘gorin theorem, we investigate the epidemic threshold of the model. By constructing an appropriate Lyapunov function, we present the stability analysis of behavioral evolution and epidemic dynamics. Some numerical simulations are performed to illustrate and complement our theoretical results. It is expected that our work can deepen the understanding of interaction between cluster synchronization and epidemic dynamics in scale-free community networks.

Behavioral synchronization induced by epidemic spread in complex networks.

During the spread of an epidemic, individuals in realistic networks may exhibit collective behaviors. In order to characterize this kind of phenomenon and explore the correlation between collective behaviors and epidemic spread, in this paper, we construct several mathematical models (including without delay, with a coupling delay, and with double delays) of epidemic synchronization by applying the adaptive feedback motivated by real observations. By using Lyapunov function methods, we obtain the conditions for local and global stability of these epidemic synchronization models. Then, we illustrate that quenched mean-field theory is more accurate than heterogeneous mean-field theory in the prediction of epidemic synchronization. Finally, some numerical simulations are performed to complement our theoretical results, which also reveal some unexpected phenomena, for example, the coupling delay and epidemic delay influence the speed of epidemic synchronization. This work makes further exploration on the relationship between epidemic dynamics and synchronization dynamics, in the hope of being helpful to the study of other dynamical phenomena in the process of epidemic spread.

Spread of competing viruses on heterogeneous networks.

In this paper, we propose a model where two strains compete with each other at the expense of common susceptible individuals on heterogeneous networks by using pair-wise approximation closed by the probability-generating function (PGF). All of the strains obey the susceptible-infected-recovered (SIR) mechanism. From a special perspective, we first study the dynamical behaviour of an SIR model closed by the PGF, and obtain the basic reproduction number via two methods. Then we build a model to study the spreading dynamics of competing viruses and discuss the conditions for the local stability of equilibria, which is different from the condition obtained by using the heterogeneous mean-field approach. Finally, we perform numerical simulations on Barabási-Albert networks to complement our theoretical research, and show some dynamical properties of the model with competing viruses.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'.

Epidemic spreading on adaptively weighted scale-free networks.

We introduce three modified SIS models on scale-free networks that take into account variable population size, nonlinear infectivity, adaptive weights, behavior inertia and time delay, so as to better characterize the actual spread of epidemics. We develop new mathematical methods and techniques to study the dynamics of the models, including the basic reproduction number, and the global asymptotic stability of the disease-free and endemic equilibria. We show the disease-free equilibrium cannot undergo a Hopf bifurcation. We further analyze the effects of local information of diseases and various immunization schemes on epidemic dynamics. We also perform some stochastic network simulations which yield quantitative agreement with the deterministic mean-field approach.