SIR dynamics in random networks with communities

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.

Effects of Community Connectivity on the Spreading Process of Epidemics

Community structure exists widely in real social networks. To investigate the effect of community structure on the spreading of infectious diseases, this paper proposes a community network model that considers both the connection rate and the number of connected edges. Based on the presented community network, a new SIRS transmission model is constructed via the mean-field theory. Furthermore, the basic reproduction number of the model is calculated via the next-generation matrix method. The results reveal that the connection rate and the number of connected edges of the community nodes play crucial roles in the spreading process of infectious diseases. Specifically, it is demonstrated that the basic reproduction number of the model decreases as the community strength increases. However, the density of infected individuals within the community increases as the community strength increases. For community networks with weak strength, infectious diseases are likely not to be eradicated and eventually will become endemic. Therefore, controlling the frequency and range of intercommunity contact will be an effective initiative to curb outbreaks of infectious diseases throughout the network. Our results can provide a theoretical basis for preventing and controlling the spreading of infectious diseases.

Pattern mechanism in stochastic SIR networks with ER connectivity

The diffusion of the susceptible and infected is a vital factor in spreading infectious diseases. However, the previous SIR networks cannot explain the dynamical mechanism of periodic behavior and endemic diseases. Here, we incorporate the diffusion and network effect into the SIR model and describes the mechanism of periodic behavior and endemic diseases through wavenumber and saddle-node bifurcation. We also introduce the standard network structured entropy (NSE) and demonstrate diffusion effect could induce the saddle-node bifurcation and Turing instability. Then we reveal the mechanism of the periodic outbreak and endemic diseases by the mean-field method. We provide the Turing instability condition through wavenumber in this network-organized SIR model. In the end, the data from COVID-19 authenticated the theoretical results.

Large deviations of a susceptible-infected-recovered model around the epidemic threshold

We numerically study the dynamics of the SIR disease model on small-world networks by using a large-deviation approach. This allows us to obtain the probability density function of the total fraction of infected nodes and of the maximum fraction of simultaneously infected nodes down to very small probability densities like 10^{-2500}. We analyze the structure of the disease dynamics and observed three regimes in all probability density functions, which correspond to quick mild, quick extremely severe, and sustained severe dynamical evolutions, respectively. Furthermore, the mathematical rate functions of the densities are investigated. The results indicate that the so-called large-deviation property holds for the SIR model. Finally, we measured correlations with other quantities like the duration of an outbreak or the peak position of the fraction of infections, also in the rare regions which are not accessible by standard simulation techniques.

A SIRD epidemic model with community structure

The study of epidemics spreading with community structure has become a hot topic. The classic SIR epidemic model does not distinguish between dead and recovered individuals. It is inappropriate to classify dead individuals as recovered individuals because the real-world epidemic spread processes show different recovery rates and death rates in different communities. In the present work, a SIRD epidemic model with different recovery rates is proposed. We pay more attention to the changes in the number of dead individuals. The basic reproductive number is obtained. The stationary solutions of a disease-free state and an endemic state are given. We show that quarantining communities can decrease the basic reproductive number, and the total number of dead individuals decreases in a disease-free steady state with an increase in the number of quarantined communities. The most effective quarantining strategy is to preferentially quarantine some communities/cities with a greater population size and a fraction of initially infected individuals. Furthermore, we show that the population flows from a low recovery rate and high population density community/city/country to some high recovery rate and low population density communities/cities/countries, which helps to reduce the total number of dead individuals and prevent the prevalence of epidemics. The numerical simulations on the real-world network and the synthetic network further support our conclusions.

Epidemic in networked population with recurrent mobility pattern

The novel Coronavirus (COVID-19) has caused a global crisis and many governments have taken social measures, such as home quarantine and maintaining social distance. Many recent studies show that network structure and human mobility greatly influence the dynamics of epidemic spreading. In this paper, we utilize a discrete-time Markov chain approach and propose an epidemic model to describe virus propagation in the heterogeneous graph, which is used to represent individuals with intra social connections and mobility between individuals and common locations. There are two types of nodes, individuals and public places, and disease can spread by social contacts among individuals and people gathering in common areas. We give theoretical results about epidemic threshold and influence of isolation factor. Several numerical simulations are performed and experimental results further demonstrate the correctness of proposed model. Non-monotonic relationship between mobility possibility and epidemic threshold and differences between Erdös-Rényi and power-law social connections are revealed. In summary, our proposed approach and findings are helpful to analyse and prevent the epidemic spreading in networked population with recurrent mobility pattern.

Epidemic spreading in heterogeneous networks with recurrent mobility patterns

Much recent research has shown that network structure and human mobility have great influences on epidemic spreading. In this paper, we propose a discrete-time Markov chain method to model susceptible-infected-susceptible epidemic dynamics in heterogeneous networks. There are two types of locations, residences and common places, for which different infection mechanisms are adopted. We also give theoretical results about the impacts of important factors, such as mobility probability and isolation, on epidemic threshold. Numerical simulations are conducted, and experimental results support our analysis. In addition, we find that the dominations of different types of residences might reverse when mobility probability varies for some networks. In summary, the findings are helpful for policy making to prevent the spreading of epidemics.