Epidemic dynamics on scale-free networks with piecewise linear infectivity and immunization

Co-evolution model of traffic travel and disease transmission under limited resources

The co-evolution mechanisms between traffic mobility and disease transmission under resource constraints remain poorly understood. This study proposes a two-layer transportation network model integrating the Susceptible-Infectious-Susceptible (SIS) epidemic framework to address this gap. The model incorporates critical factors such as total medical resources, inter-network infection delays, travel willingness, and network topology. Through simulations, we demonstrate that increasing medical resources significantly reduces infection scale during outbreaks, while prolonging inter-network delays slows transmission rates but extends epidemic persistence. Complex network topologies amplify the impact of travel behavior on disease spread, and multi-factor interventions (e.g., combined resource allocation and delay extension) outperform single-factor controls in suppressing transmission. Furthermore, reducing network connectivity (lower average degree) proves effective in mitigating outbreaks, especially under low travel willingness. These findings highlight the necessity of coordinated policies that leverage resource optimization, travel regulation, and network simplification to manage epidemics. This work provides actionable insights for policymakers to design efficient epidemic control strategies in transportation-dependent societies.

A hierarchical intervention scheme based on epidemic severity in a community network

As there are no targeted medicines or vaccines for newly emerging infectious diseases, isolation among communities (villages, cities, or countries) is one of the most effective intervention measures. As such, the number of intercommunity edges ([Formula: see text]) becomes one of the most important factor in isolating a place since it is closely related to normal life. Unfortunately, how [Formula: see text] affects epidemic spread is still poorly understood. In this paper, we quantitatively analyzed the impact of [Formula: see text] on infectious disease transmission by establishing a four-dimensional [Formula: see text] edge-based compartmental model with two communities. The basic reproduction number [Formula: see text] is explicitly obtained subject to [Formula: see text] [Formula: see text]. Furthermore, according to [Formula: see text] with zero [Formula: see text], epidemics spread could be classified into two cases. When [Formula: see text] for the case 2, epidemics occur with at least one of the reproduction numbers within communities greater than one, and otherwise when [Formula: see text] for case 1, both reproduction numbers within communities are less than one. Remarkably, in case 1, whether epidemics break out strongly depends on intercommunity edges. Then, the outbreak threshold in regard to [Formula: see text] is also explicitly obtained, below which epidemics vanish, and otherwise break out. The above two cases form a severity-based hierarchical intervention scheme for epidemics. It is then applied to the SARS outbreak in Singapore, verifying the validity of our scheme. In addition, the final size of the system is gained by demonstrating the existence of positive equilibrium in a four-dimensional coupled system. Theoretical results are also validated through numerical simulation in networks with the Poisson and Power law distributions, respectively. Our results provide a new insight into controlling epidemics.

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.

Heterogeneous pair-approximation analysis for susceptible-infectious-susceptible epidemics on networks

The pair heterogeneous mean-field (PHMF) model has been used extensively in previous studies to investigate the dynamics of susceptible-infectious-susceptible epidemics on complex networks. However, the approximate treatment of the classical or reduced PHMF models lacks a rigorous theoretical analysis. By means of the standard and full PHMF models, we first derived the equivalent conditions for the approximate model treatment. Furthermore, we analytically derived a novel epidemic threshold for the PHMF model, and we demonstrated via numerical simulations that this threshold condition differs from all those reported in earlier studies. Our findings indicate that both the reduced and full PHMF models agree well with continuous-time stochastic simulations, especially when infection is spreading at considerably higher rates.

A swarm-optimizer-assisted simulation and prediction model for emerging infectious diseases based on SEIR

Mechanism-driven models based on transmission dynamics and statistic models driven by public health data are two main methods for simulating and predicting emerging infectious diseases. In this paper, we intend to combine these two methods to develop a more comprehensive model for the simulation and prediction of emerging infectious diseases. First, we combine a standard epidemic dynamic, the susceptible–exposed–infected–recovered (SEIR) model with population migration. This model can provide a biological spread process for emerging infectious diseases. Second, to determine suitable parameters for the model, we propose a data-driven approach, in which the public health data and population migration data are assembled. Moreover, an objective function is defined to minimize the error based on these data. Third, based on the proposed model, we further develop a swarm-optimizer-assisted simulation and prediction method, which contains two modules. In the first module, we use a level-based learning swarm optimizer to optimize the parameters required in the epidemic mechanism. In the second module, the optimized parameters are used to predicate the spread of emerging infectious diseases. Finally, various experiments are conducted to validate the effectiveness of the proposed model and method.

Dynamic analysis of disturbance propagation in ecological networks with quarantine items and proportional migration

In order to study the stability of the ecosystem under external attack, we regard the ecosystem as a complex network and the species disturbance after the attack as an infectious disease. We establish an ecological network disturbance propagation model based on the infectious disease model, and analyze its dynamics with the above ideas. In this paper, the species are regarded as nodes in the network, and the predator–prey relationship is regarded as the edge of the network. When the ecosystem is attacked by external forces, the disturbance can be transmitted from a species to its predator or prey through the food chain, and the disturbed species can recover themselves and then return to a stable state. At the same time, we consider adding human quarantine and protection of disturbed species. In this way, all species in the ecosystem are divided into four states: undisturbed, disturbed, quarantine and recovered. By analyzing the dynamics of disturbance propagation, the critical threshold and equilibrium point of disturbance diffusion are determined, and the local and global stability of disease-free equilibrium and endemic equilibrium are analyzed. The results show that the existence of endemic equilibrium depends on the critical threshold of disturbance propagation, which is related to the structure of food web, the propagation proportion of disturbance and the recovery proportion of species after being attacked. The larger the propagation proportion is, the weaker the resistance stability is, and the easier the disturbance propagates in the system. The higher the recovery proportion of the disturbed species, the stronger the stability of the recovery rate, and the more difficult it is for the disturbance to propagate in the system. Then we regard human protection of species as species immunity, and choose the most effective species protection measures by comparing and analyzing the threshold changes under the three protection strategies. The results show that the moderately large neighbor nodes of the disturbed species should be protected. This kind of protection measure is the most effective and it is easier to restrain the propagation of disturbance. Finally, the food webs of 85 species in a pine forest in Otago, New Zealand is selected to analyze the propagation process of disturbance by numerical simulation.

Optimal Investment in Prevention and Recovery for Mitigating Epidemic Risks

The worldwide healthcare and economic crisis caused by the COVID-19 pandemic highlights the need for a deeper understanding of investing in the mitigation of epidemic risks. To address this, we built a mathematical model to optimize investments into two types of measures for mitigating the risks of epidemic propagation: prevention/containment measures and treatment/recovery measures. The new model explicitly accounts for the characteristics of networks of individuals, as a critical element of epidemic propagation. Subsequent analysis shows that, to combat an epidemic that can cause significant negative impact, optimal investment in either category increases with a higher level of connectivity and intrinsic loss, but it is limited to a fraction of that total potential loss. However, when a fixed and limited mitigation investment is to be apportioned among the two types of measures, the optimal proportion of investment for prevention and containment increases when the investment limit goes up, and when the network connectivity decreases. Our results are consistent with existing studies and can be used to properly interpret what happened in past pandemics as well as to shed light on future and ongoing events such as COVID-19.

Global dynamics of a network-based SIQS epidemic model with nonmonotone incidence rate

The risk of propagation of infectious diseases such as avian influenza and COVID-19 is generally controlled or reduced by quarantine measures. Considering this situation, a network-based SIQS (susceptible-infected-quarantined-susceptible) infectious disease model with nonmonotone incidence rate is established and analyzed in this paper. The psychological impact of the transmission of certain diseases in heterogeneous networks at high levels of infection may be characterized by the related nonmonotone incidence rate. The expressions of the basic reproduction number and equilibria of the model are determined analytically. We demonstrate in detail the uniform persistence of system and the global asymptotic stability of the disease-free equilibrium. The global attractivity of the unique endemic equilibrium is discussed by using monotone iteration technique. We obtain that the endemic equilibrium is globally asymptotically stable under certain conditions by constructing appropriate Lyapunov function. In addition, numerical simulations are performed to indicate the theoretical results.

A two-step vaccination technique to limit COVID-19 spread using mobile data

Vaccination is one of the most effective methods to prevent the spread of infectious diseases, but due to limitations in vaccines' availability, especially when faced with a new disease such as COVID-19, not all individuals in the community can be vaccinated. A limited number of candidates should be selected when the supply of vaccines is limited. In this paper, a method is introduced to prioritize the individuals for vaccination in order to achieve the best results in preventing the spread of COVID-19. We divide this problem into two steps: vaccine allocation and targeted vaccination. In vaccine allocation, vaccines are allocated among different population. An algorithm is proposed by defining the maximization of the total immunity among populations as an optimization problem. The aim of the targeted vaccination step is to select the individuals in each population that when vaccinated, create the greatest reduction in the transmission paths of the disease. The contact tracing data for this step is obtained from wireless communication networks and is modeled using graph theory. A metric is presented for selection of the candidates, based on centrality metrics. Simulations indicate that a 30% drop in infection rate could be achieved compared to random vaccination.

Building epidemic models for living populations and computer networks

Accurate modeling of viral outbreaks in living populations and computer networks is a prominent research field. Many researchers are in search for simple and realistic models to manage preventive resources and implement effective measures against hazardous circumstances. The ongoing Covid-19 pandemic has revealed the fact about deficiencies in health resource planning of some countries having relatively high case count and death toll. A unique epidemic model incorporating stochastic processes and queuing theory is presented, which was evaluated by computer simulation using pre-processed data obtained from an urban clinic providing family health services. Covid-19 data from a local corona-center was used as the initial model parameters (e.g. R 0 , infection rate, local population size, number of contacts with infected individuals, and recovery rate). A long-run trend analysis for 1 year was simulated. The results fit well to the current case data of the sample corona center. Effective preventive and reactive resource planning basically depends on accurately designed models, tools, and techniques needed for the prediction of feature threats, risks, and mitigation costs. In order to sufficiently analyze the transmission and recovery dynamics of epidemics it is important to choose concise mathematical models. Hence, a unique stochastic modeling approach tied to queueing theory and computer simulation has been chosen. The methods used here can also serve as a guidance for accurate modeling and classification of stages (or compartments) of epidemics in general.

Modelling Strong Control Measures for Epidemic Propagation With Networks-A COVID-19 Case Study

We show that precise knowledge of epidemic transmission parameters is not required to build an informative model of the spread of disease. We propose a detailed model of the topology of the contact network under various external control regimes and demonstrate that this is sufficient to capture the salient dynamical characteristics and to inform decisions. Contact between individuals in the community is characterised by a contact graph, the structure of that contact graph is selected to mimic community control measures. Our model of city-level transmission of an infectious agent (SEIR model) characterises spread via a (a) scale-free contact network (no control); (b) a random graph (elimination of mass gatherings); and (c) small world lattice (partial to full lockdown-"social" distancing). This model exhibits good qualitative agreement between simulation and data from the 2020 pandemic spread of a novel coronavirus. Estimates of the relevant rate parameters of the SEIR model are obtained and we demonstrate the robustness of our model predictions under uncertainty of those estimates. The social context and utility of this work is identified, contributing to a highly effective pandemic response in Western Australia.

Immunization strategies in directed networks

Many complex systems can be modeled as directed networks, which can be regarded as a generalization of undirected networks. In this paper, epidemic dynamics and immunization strategies in directed networks are studied. First, a Susceptible-Infected-Susceptible (SIS) model on a directed network is established employing the mean-field method, and its dynamics and epidemic threshold of the network are studied. Then based on the continuous degree technique, namely, considering the degree of a node as a continuous variable, we propose a method to calculate the epidemic threshold of the immunized network. Besides, some immunization strategies, including optimal immunization, random immunization, combined targeted immunization, and combined acquaintance immunization, and three special networks are considered. Finally, through numerical analysis, all immunization strategies are simulated and compared on different types of networks. We find that the nodes with the largest product of in-degree and out-degree are the most worthy of being immunized.

DR-SCIR Public Opinion Propagation Model with Direct Immunity and Social Reinforcement Effect

The DR-SCIR network public opinion propagation model was employed to study the characters of S-state users stopping transmitting information for the first time and secondary transmission of immune users. The model takes into account symmetry and complexity such as direct immunization and social reinforcement effect, proposes the probability of direct immunity Psr and the probability of transform from the immune state to the hesitant state Prc, and divides public opinion information into positive public opinion and negative public opinion based on whether the public opinion information is confirmed. Simulation results show that, when direct immunity Psr = 0.5, the density of I-state nodes in the model decreased by 54.12% at the peak index; when the positive social reinforcement effect factor b = 10, the density of I-state nodes in the model increased by 16.67% at the peak index; and when the negative social reinforcement effect factor b = -10, the density of I-state nodes in the model decreased by 55.36% at the peak index. It shows that increasing the positive social reinforcement effect factor b can promote the spread of positive public opinion, reducing the negative social reinforcement effect factor b can control the spread of negative public opinion, and direct immunization can effectively suppress the spread of public opinion. This model can help us better analyze the rules of public opinion on social networks, so as to maintain a healthy and harmonious network and social environment.

Epidemic Spread on One-Way Circular-Coupled Networks

Real epidemic spreading networks are often composed of several kinds of complex networks interconnected with each other, such as Lyme disease, and the interrelated networks may have different topologies and epidemic dynamics. Moreover, most human infectious diseases are derived from animals, and zoonotic infections always spread on directed interconnected networks. So, in this article, we consider the epidemic dynamics of zoonotic infections on a unidirectional circular-coupled network. Here, we construct two unidirectional three-layer circular interactive networks, one model has direct contact between interactive networks, the other model describes diseases transmitted through vectors between interactive networks, which are established by introducing the heterogeneous mean-field approach method. Then we obtain the basic reproduction numbers and stability of equilibria of the two models. Through mathematical analysis and numerical simulations, it is found that basic reproduction numbers of the models depend on the infection rates, infection periods, average degrees, and degree ratios. Numerical simulations illustrate and expand these theoretical results very well.

Suppression of epidemic spreading process on multiplex networks via active immunization

Spatial epidemic spreading, a fundamental dynamical process upon complex networks, attracts huge research interest during the past few decades. To suppress the spreading of epidemic, a couple of effective methods have been proposed, including node vaccination. Under such a scenario, nodes are immunized passively and fail to reveal the mechanisms of active activity. Here, we suggest one novel model of an observer node, which can identify infection through interacting with infected neighbors and inform the other neighbors for vaccination, on multiplex networks, consisting of epidemic spreading layer and information spreading layer. In detail, the epidemic spreading layer supports susceptible-infected-recovered process, while observer nodes will be selected according to several algorithms derived from percolation theory. Numerical simulation results show that the algorithm based on large degree performs better than random placement, while the algorithm based on nodes' degree in the information spreading layer performs the best (i.e., the best suppression efficacy is guaranteed when placing observer nodes based on nodes' degree in the information spreading layer). With the help of state probability transition equation, the above phenomena can be validated accurately. Our work thus may shed new light into understanding control of empirical epidemic control.

A colored mean-field model for analyzing the effects of awareness on epidemic spreading in multiplex networks

We study the impact of susceptible nodes' awareness on epidemic spreading in social systems, where the systems are modeled as multiplex networks coupled with an information layer and a contact layer. We develop a colored heterogeneous mean-field model taking into account the portion of the overlapping neighbors in the two layers. With theoretical analysis and numerical simulations, we derive the epidemic threshold which determines whether the epidemic can prevail in the population and find that the impacts of awareness on threshold value only depend on epidemic information being available in network nodes' overlapping neighborhood. When there is no link overlap between the two network layers, the awareness cannot help one to raise the epidemic threshold. Such an observation is different from that in a single-layer network, where the existence of awareness almost always helps.

Transmission Dynamics of an SIS Model with Age Structure on Heterogeneous Networks

Infection age is often an important factor in epidemic dynamics. In order to realistically analyze the spreading mechanism and dynamical behavior of epidemic diseases, in this paper, a generalized disease transmission model of SIS type with age-dependent infection and birth and death on a heterogeneous network is discussed. The model allows the infection and recovery rates to vary and depend on the age of infection, the time since an individual becomes infected. We address uniform persistence and find that the model has the sharp threshold property, that is, for the basic reproduction number less than one, the disease-free equilibrium is globally asymptotically stable, while for the basic reproduction number is above one, a Lyapunov functional is used to show that the endemic equilibrium is globally stable. Finally, some numerical simulations are carried out to illustrate and complement the main results. The disease dynamics rely not only on the network structure, but also on an age-dependent factor (for some key functions concerned in the model).

Susceptible-infected-recovered epidemics in random networks with population awareness

The influence of epidemic information-based awareness on the spread of infectious diseases on networks cannot be ignored. Within the effective degree modeling framework, we discuss the susceptible-infected-recovered model in complex networks with general awareness and general degree distribution. By performing the linear stability analysis, the conditions of epidemic outbreak can be deduced and the results of the previous research can be further expanded. Results show that the local awareness can suppress significantly the epidemic spreading on complex networks via raising the epidemic threshold and such effects are closely related to the formulation of awareness functions. In addition, our results suggest that the recovered information-based awareness has no effect on the critical condition of epidemic outbreak.

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.

DEMOGRAPHICS INDUCE EXTINCTION OF DISEASE IN AN SIS MODEL BASED ON CONDITIONAL MARKOV CHAIN

Demographics have significant effects on disease spread in populations and the topological evolution of the underlying networks that represent the populations. In the context of network-based epidemic modeling, Markov chain-based approach and pairwise approximation are two powerful tools — the former can capture stochastic effects of disease transmission dynamics and the latter can characterize the dynamical correlations in each pair of connected individuals. However, to our best knowledge, the study on epidemic spreading in networks relying on these two techniques is still lacking. To fill this gap, in this paper, a deterministic pairwise susceptible–infected–susceptible (SIS) epidemic model with demographics on complex networks with arbitrary degree distributions is studied based on a continuous time conditional Markov chain. This deterministic model is rigorously derived — using the moment generating function — from the Kolmogorov differential equations for the evolution of individuals and pairs. It is found that demographics will induce the extinction of the disease by reducing the basic reproduction number or lowering the epidemic prevalence after the disease prevails. Moreover, due to the demographical effects, the resulting network tends to a homogeneous network with a degree distribution similar to Poisson distribution, irrespective of the initial network structure. Additionally, we find excellent agreement between numerical solutions and individual-based stochastic simulations using both Erdös–Renyi (ER) random and Barabási–Albert (BA) scale-free initial networks. Our results may provide new insights on the understanding of the influence of demographics on epidemic dynamics and network evolution.

Optimal Control of Multi-strain Epidemic Processes in Complex Networks

The emergence of new diseases, such as HIV/AIDS, SARS, and Ebola, represent serious problems for the public health and medical science research to address. Despite the rapid development of vaccines and drugs, one challenge in disease control is the fact that one pathogen sometimes generates many strains with different spreading features. Hence it is of critical importance to investigate multi-strain epidemic dynamics and its associated epidemic control strategies. In this paper, we investigate two controlled multi-strain epidemic models for heterogeneous populations over a large complex network and obtain the structure of optimal control policies for both models. Numerical examples are used to corroborate the analytical results.

DYNAMICS OF COMPETING STRAINS WITH SATURATED INFECTIVITY AND MUTATION ON NETWORKS

This paper deals with the dynamical behavior of a two-strain epidemic model in a complex network with saturated infectivity and mutation. The model is shown to exhibit the phenomena of strain coexistence where two epidemic strains co-exist and strain replacement where the strain with smaller basic reproduction number can become predominant. By using the stability and persistence theory of dynamical systems, we calculate the epidemic threshold, and show that above which at least one strain persists in a population, and also obtain the critical values to discriminate the strain coexistence and replacement. The results suggest that the newly mutated strain can spread in the population, even if it has a very small transmissivity.

Global stability of a network-based SIS epidemic model with a general nonlinear incidence rate

In this paper, we develop and analyze an SIS epidemic model with a general nonlinear incidence rate, as well as degree-dependent birth and natural death, on heterogeneous networks. We analytically derive the epidemic threshold R0 which completely governs the disease dynamics: when R0 < 1, the disease-free equilibrium is globally asymptotically stable, i.e., the disease will die out; when R0 > 1, the disease is permanent. It is interesting that the threshold value R0 bears no relation to the functional form of the nonlinear incidence rate and degree-dependent birth. Furthermore, by applying an iteration scheme and the theory of cooperative system respectively, we obtain sufficient conditions under which the endemic equilibrium is globally asymptotically stable. Our results improve and generalize some known results. To illustrate the theoretical results, the corresponding numerical simulations are also given.

Crossover phenomena of percolation transition in evolution networks with hybrid attachment

A first-order percolation transition, called explosive percolation, was recently discovered in evolution networks with random edge selection under a certain restriction. For many real world networks, the mechanism of preferential attachment plays a significant role in the formation of heterogeneous structures, but the network percolation in evolution process with preferential attachment has not yet been concerned. We propose a tunable network percolation model by introducing a hybrid mechanism of edge selection into the Bohman-Frieze-Wormald model, in which a parameter adjusts the relative weights between random and preferential selections. A large number of simulations indicate that there exist crossover phenomena of percolation transition by adjusting the parameter in the evolution processes. When the strategy of selecting a candidate edge is dominated by random selection, a single discontinuous percolation transition occurs. When a candidate edge is selected more preferentially based on nodes degree, the size of the largest component undergoes multiple discontinuous jumps, which exhibits a peculiar difference from the network percolation of random selection with a certain restriction. Besides, the percolation transition becomes continuous when the candidate edge is selected completely preferentially.

A biologically inspired immunization strategy for network epidemiology

Well-known immunization strategies, based on degree centrality, betweenness centrality, or closeness centrality, either neglect the structural significance of a node or require global information about the network. We propose a biologically inspired immunization strategy that circumvents both of these problems by considering the number of links of a focal node and the way the neighbors are connected among themselves. The strategy thus measures the dependence of the neighbors on the focal node, identifying the ability of this node to spread the disease. Nodes with the highest ability in the network are the first to be immunized. To test the performance of our method, we conduct numerical simulations on several computer-generated and empirical networks, using the susceptible-infected-recovered (SIR) model. The results show that the proposed strategy largely outperforms the existing well-known strategies.

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We study an efficient, bio-inspired immunization strategy for network epidemiology.

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Inspiration stems from a single-celled, ameba-like organism, Physarum polycephalum.

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Our strategy goes beyond the node degree in selecting targets for immunization.

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The strategy performs considerably better than several well-known competitors.

Epidemic outbreak for an SIS model in multiplex networks with immunization

With the aim of understanding epidemic spreading in a general multiplex network and designing optimal immunization strategies, a mathematical model based on multiple degree is built to analyze the threshold condition for epidemic outbreak. Two kinds of strategies, the multiplex node-based immunization and the layer node-based immunization, are examined. Theoretical results show that the general framework proposed here can illustrate the effect of diverse correlations and immunizations on the outbreak condition in multiplex networks. Under a set of conditions on uncorrelated coefficients, the specific epidemic thresholds are shown to be only dependent on the respective degree distribution in each layer.

Local immunization program for susceptible-infected-recovered network epidemic model

The immunization strategies through contact tracing on the susceptible-infected-recovered framework in social networks are modelled to evaluate the cost-effectiveness of information-based vaccination programs with particular focus on the scenario where individuals belonging to a specific set can get vaccinated due to the vaccine shortages and other economic or humanity constraints. By using the block heterogeneous mean-field approach, a series of discrete-time dynamical models is formulated and the condition for epidemic outbreaks can be established which is shown to be not only dependent on the network structure but also closely related to the immunization control parameters. Results show that increasing the immunization strength can effectively raise the epidemic threshold, which is different from the predictions obtained through the susceptible-infected-susceptible network framework, where epidemic threshold is independent of the vaccination strength. Furthermore, a significant decrease of vaccine use to control the infectious disease is observed for the local vaccination strategy, which shows the promising applications of the local immunization programs to disease control while calls for accurate local information during the process of disease outbreak.

Infectious Disease Containment Based on a Wireless Sensor System

Infectious diseases pose a serious threat to public health due to its high infectivity and potentially high mortality. One of the most effective ways to protect people from being infected by these diseases is through vaccination. However, due to various resource constraints, vaccinating all the people in a community is not practical. Therefore, targeted vaccination, which vaccinates a small group of people, is an alternative approach to contain infectious diseases. Since many infectious diseases spread among people by droplet transmission within a certain range, we deploy a wireless sensor system in a high school to collect contacts happened within the disease transmission distance. Based on the collected traces, a graph is constructed to model the disease propagation, and a new metric (called connectivity centrality) is presented to find the important nodes in the constructed graph for disease containment. Connectivity centrality considers both a node's local and global effect to measure its importance in disease propagation. Centrality based algorithms are presented and further enhanced by exploiting the information of the known infected nodes, which can be detected during targeted vaccination. Simulation results show that our algorithms can effectively contain infectious diseases and outperform other schemes under various conditions.

Local immunization strategy based on the scores of nodes

The problem of finding a better immunization strategy for controlling the spreading of the epidemic with limited resources has attracted much attention because of its great theoretical significance and wide application. In this paper, we propose a successful immunization strategy only depending on local information. Our strategy initializes the scores of nodes with the values of their degree and recalculates the score of a certain immunized node based on its local information, and then replaces the certain immunized node with its nonimmunized higher-score neighbor. To test the effectiveness of the proposed strategy, we conduct the experiments on several synthetic networks and real-world networks. The results show that the proposed strategy outperforms the existing well-known local strategies, even the degree centrality targeted strategy.

EPIDEMIC SPREADING AND GLOBAL STABILITY OF A NEW SIS MODEL WITH DELAY ON HETEROGENEOUS NETWORKS

In this paper, we study a susceptible-infected-susceptible (SIS) model with time delay on complex heterogeneous networks. Here, the delay describes the incubation period in the vector population. We calculate the epidemic threshold by using a Lyapunov functional and some analytical methods, and find that adding delay increases the epidemic threshold. Then, we prove the global stability of disease-free and endemic equilibria by using the theory of functional differential equations. Furthermore, we show numerically that the epidemic threshold of the new model may change along with other factors, such as the infectivity function, the heterogeneity of the network, and the degrees of nodes. Finally, we find numerically that the delay can affect the convergence speed at which the disease reaches equilibria.

Immunization strategy based on the critical node in percolation transition

The problem of finding a better immunization strategy for controlling the spreading of the epidemic with limited resources has attracted much attention since its great theoretical significance and wide application. In this letter, we propose a novel and successful targeted immunization strategy based on percolation transition. Our strategy repeatedly looks for the critical nodes for immunizing. The critical node, which leads to the emergence of the giant connected component as the degree threshold increases, is determined when the maximal second-largest connected component disappears. To test the effectiveness of the proposed method, we conduct the experiments on several artificial networks and real-world networks. The results show that the proposed method outperforms the degree centrality strategy, the betweenness centrality strategy and the adaptive degree centrality strategy with 18% to 50% fewer immunized nodes for same amount of immunization.

Dynamic analysis of an SEIRS model with nonlinear infectivity on complex networks

In this paper, we study the spreading of infections on complex heterogeneous networks based on an SEIRS epidemic model with nonlinear infectivity. By mathematical analysis, the basic reproduction number R0 is obtained. When R0 is less than one, the disease-free equilibrium is globally asymptotically stable and the disease dies out, while R0 is greater than one, the disease-free equilibrium becomes unstable and the disease is permanent, and in the meantime there exists a unique endemic equilibrium which is globally attractive under certain conditions. Finally, the effects of various immunization schemes are studied. To verify our theoretical results, the corresponding numerical simulations are also included.

Immunization against the Spread of Rumors in Homogenous Networks

Since most rumors are harmful, how to control the spread of such rumors is important. In this paper, we studied the process of "immunization" against rumors by modeling the process of rumor spreading and changing the termination mechanism for the spread of rumors to make the model more realistic. We derived mean-field equations to describe the dynamics of the rumor spread. By carrying out steady-state analysis, we derived the spreading threshold value that must be exceeded for the rumor to spread. We further discuss a possible strategy for immunization against rumors and obtain an immunization threshold value that represents the minimum level required to stop the rumor from spreading. Numerical simulations revealed that the average degree of the network and parameters of transformation probability significantly influence the spread of rumors. More importantly, the simulations revealed that immunizing a higher proportion of individuals is not necessarily better because of the waste of resources and the generation of unnecessary information. So the optimal immunization rate should be the immunization threshold.

Impact of media coverage on epidemic spreading in complex networks

An SIS network model incorporating the influence of media coverage on transmission rate is formulated and analyzed. We calculate the basic reproduction number R 0 by utilizing the local stability of the disease-free equilibrium. Our results show that the disease-free equilibrium is globally asymptotically stable and that the disease dies out if R 0 is below 1; otherwise, the disease will persist and converge to a unique positive stationary state. This result may suggest effective control strategies to prevent disease through media coverage and education activities in finite-size scale-free networks. Numerical simulations are also performed to illustrate our results and to give more insights into the dynamical process.

Effects of delayed recovery and nonuniform transmission on the spreading of diseases in complex networks

We investigate the effects of delaying the time to recovery (delayed recovery) and of nonuniform transmission on the propagation of diseases on structured populations. Through a mean-field approximation and large-scale numerical simulations, we find that postponing the transition from the infectious to the recovered states can largely reduce the epidemic threshold, therefore promoting the outbreak of epidemics. On the other hand, if we consider nonuniform transmission among individuals, the epidemic threshold increases, thus inhibiting the spreading process. When both mechanisms are at work, the latter might prevail, hence resulting in an increase of the epidemic threshold with respect to the standard case, in which both ingredients are absent. Our findings are of interest for a better understanding of how diseases propagate on structured populations and to a further design of efficient immunization strategies.

Interplay between collective behavior and spreading dynamics on complex networks

There are certain correlations between collective behavior and spreading dynamics on some real complex networks. Based on the dynamical characteristics and traditional physical models, we construct several new bidirectional network models of spreading phenomena. By theoretical and numerical analysis of these models, we find that the collective behavior can inhibit spreading behavior, but, conversely, this spreading behavior can accelerate collective behavior. The spread threshold of spreading network is obtained by using the Lyapunov function method. The results show that an effective spreading control method is to enhance the individual awareness to collective behavior. Many real-world complex networks can be thought of in terms of both collective behavior and spreading dynamics and therefore to better understand and control such complex networks systems, our work may provide a basic framework.

Epidemic spreading on contact networks with adaptive weights

The heterogeneous patterns of interactions within a population are often described by contact networks, but the variety and adaptivity of contact strengths are usually ignored. This paper proposes a modified epidemic SIS model with a birth-death process and nonlinear infectivity on an adaptive and weighted contact network. The links' weights, named as 'adaptive weights', which indicate the intimacy or familiarity between two connected individuals, will reduce as the disease develops. Through mathematical and numerical analyses, conditions are established for population extermination, disease extinction and infection persistence. Particularly, it is found that the fixed weights setting can trigger the epidemic incidence, and that the adaptivity of weights cannot change the epidemic threshold but it can accelerate the disease decay and lower the endemic level. Finally, some corresponding control measures are suggested.

GLOBAL ATTRACTIVENESS OF DISCRETE-TIME EPIDEMIC OUTBREAK IN NETWORKS

Epidemic dynamics in networks have attracted a great deal of attention from researchers of many fields. In this paper, we mainly study the global behaviors of discrete-time epidemic model in heterogenous networks. By theoretical analysis, we show that the model can be characterized by the basic reproduction number R0. When R0 is smaller than unit, the disease-free equilibrium is globally stable, while R0 is larger than unit, the unique positive equilibrium is globally attractive.

Effects of degree-biased transmission rate and nonlinear infectivity on rumor spreading in complex social networks

We introduce a generalized rumor spreading model and analytically investigate the spreading of rumors on scale-free (SF) networks. In the standard rumor spreading model, each node has an infectivity equal to its degree, and connectivity is uniform across all links. To generalize this model, we introduce an infectivity function that determines the number of simultaneous contacts that a given node (individual) may establish with its connected neighbors and a connectivity strength function (CSF) for the direct link between two connected nodes. These lead to a degree-biased propagation of rumors. For nonlinear functions, this generalization is reflected in the infectivity's exponent α and the CSF's exponent β. We show that, by adjusting exponents α and β, the epidemic threshold can be controlled. This feature is absent in the standard rumor spreading model. In addition, we obtain a critical threshold. We show that the critical threshold for our generalized model is greater than that of the standard model on a finite SF network. Theoretically, we show that β=-1 leads to a maximum spreading of rumors, and computation results on different networks verify our theoretical prediction. Also, we show that a smaller α leads to a larger spreading of rumors. Our results are interesting since we obtain these results regardless of the network topology and configuration.

The impact of awareness on epidemic spreading in networks

We explore the impact of awareness on epidemic spreading through a population represented by a scale-free network. Using a network mean-field approach, a mathematical model for epidemic spreading with awareness reactions is proposed and analyzed. We focus on the role of three forms of awareness including local, global, and contact awareness. By theoretical analysis and simulation, we show that the global awareness cannot decrease the likelihood of an epidemic outbreak while both the local awareness and the contact awareness can. Also, the influence degree of the local awareness on disease dynamics is closely related with the contact awareness.

WEAK MEAN-FIELD APPROXIMATION FOR DISCRETE EPIDEMIC MODELS IN SCALE-FREE NETWORKS

Epidemic dynamics in networks have attracted a great deal of attention from many fields. Based on the previous work, we propose a weak discrete mean-field approximation, and the difference with the previous approximation approach is that it can result in more simple difference equations. We mainly consider the minimal SIS epidemic model in complex networks, and make comparisons amongst two kinds of approximation formulations on the prediction of epidemic prevalence and find that they are effective to model epidemic spreading. Moreover, we investigate its application to the risk feedback case and simulations indicate its effectiveness.

Adaptive mechanism between dynamical synchronization and epidemic behavior on complex networks

Many realistic epidemic networks display statistically synchronous behavior which we will refer to as epidemic synchronization. However, to the best of our knowledge, there has been no theoretical study of epidemic synchronization. In fact, in many cases, synchronization and epidemic behavior can arise simultaneously and interplay adaptively. In this paper, we first construct mathematical models of epidemic synchronization, based on traditional dynamical models on complex networks, by applying the adaptive mechanisms observed in real networks. Then, we study the relationship between the epidemic rate and synchronization stability of these models and, in particular, obtain the conditions of local and global stability for epidemic synchronization. Finally, we perform numerical analysis to verify our theoretical results. This work is the first to draw a theoretical bridge between epidemic transmission and synchronization dynamics and will be beneficial to the study of control and the analysis of the epidemics on complex networks.

Staged progression model for epidemic spread on homogeneous and heterogeneous networks

In this paper, epidemic spread with the staged progression model on homogeneous and heterogeneous networks is studied. First, the epidemic threshold of the simple staged progression model is given. Then the staged progression model with birth and death is also considered. The case where infectivity is a nonlinear function of the nodes' degree is discussed, too. Finally, the analytical results are verified by numerical simulations.

Modelling and analysis of influenza A (H1N1) on networks

Background

In April 2009, a new strain of H1N1 influenza virus, referred to as pandemic influenza A (H1N1) was first detected in humans in the United States, followed by an outbreak in the state of Veracruz, Mexico. Soon afterwards, this new virus kept spreading worldwide resulting in a global outbreak. In China, the second Circular of the Ministry of Health pointed out that as of December 31, 2009, the country’s 31 provinces had reported 120,000 confirmed cases of H1N1.

Methods

We formulate an epidemic model of influenza A based on networks. We calculate the basic reproduction number and study the effects of various immunization schemes. The final size relation is derived for the network epidemic model. The model parameters are estimated via least-squares fitting of the model solution to the observed data in China.

Results

For the network model, we prove that the disease-free equilibrium is globally asymptotically stable when the basic reproduction is less than one. The final size will depend on the vaccination starting time, T, the number of infective cases at time T and immunization schemes to follow. Our theoretical results are confirmed by numerical simulations. Using the parameter estimates based on the observation data of the cumulative number of hospital notifications, we estimate the basic reproduction number R₀ to be 1.6809 in China.

Conclusions

Network modelling supplies a useful tool for studying the transmission of H1N1 in China, capturing the main features of the spread of H1N1. While a uniform, mass-immunization strategy helps control the prevalence, a targeted immunization strategy focusing on specific groups with given connectivity may better control the endemic.