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Quarantine generated phase transition in epidemic spreading
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We study the critical effect of quarantine on the propagation of epidemics on an adaptive network of social contacts. For this purpose, we analyze the susceptible-infected-recovered model in the presence of quarantine, where susceptible individuals protect themselves by disconnecting their links to infected neighbors with probability w and reconnecting them to other susceptible individuals chosen at random. Starting from a single infected individual, we show by an analytical approach and simulations that there is a phase transition at a critical rewiring (quarantine) threshold w(c) separating a phase (w<w(c)) where the disease reaches a large fraction of the population from a phase (w≥w(c)) where the disease does not spread out. We find that in our model the topology of the network strongly affects the size of the propagation and that w(c) increases with the mean degree and heterogeneity of the network. We also find that w(c) is reduced if we perform a preferential rewiring, in which the rewiring probability is proportional to the degree of infected nodes.
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... These models have been solved in the past both by Mean Field approaches, by writing down a set of coupled differential equations which take into account all paths of the disease with the proper rate constants, as well as by Monte-Carlo computer simulations. The properties of interest include the distribution of the infected population [12], the methodology of immunization in order to contain the spreading [13] [14], the inclusion and effects of several social measures, such as quarantine [15] [16], social distancing [17], etc. These theoretical models are sometimes coupled with the medical efforts to find a cure [18], or find the most effective vaccine [19]. ...
Infectious diseases, such as the current COVID-19, have a huge economic and societal impact. The ability to model its transmission characteristics is critical to minimize its impact. In fact, predicting how fast an infection is spreading could be a major factor in deciding on the severity, extent and strictness of the applied mitigation measures, such as the recent lockdowns. Even though modelling epidemics is a well studied subject, usually models do not include quarantine or other social measures, such as those imposed in the recent pandemic. The current work builds upon a recent paper by Maier and Brockmann (2020), where a compartmental SIRX model was implemented. That model included social or individual behavioral changes during quarantine, by introducing state X, in which symptomatic quarantined individuals are not transmitting the infection anymore, and described well the transmission in the initial stages of the infection. The results of the model were applied to real data from several provinces in China, quite successfully. In our approach we use a Monte-Carlo simulation model on networks. Individuals are network nodes and the links are their contacts. We use a spreading mechanism from the initially infected nodes to their nearest neighbors, as has been done previously. Initially, we find the values of the rate constants (parameters) the same way as in Maier and Brockmann (2020) for the confirmed cases of a country, on a daily basis, as given by the Johns Hopkins University. We then use different types of networks (random Erdős-Rényi, Small World, and Barabási-Albert Scale-Free) with various characteristics in an effort to find the best fit with the real data for the same geographical regions as reported in Maier and Brockmann (2020). Our simulations show that the best fit comes with the Erdős-Rényi random networks. We then apply this method to several other countries, both for large-size countries, and small size ones. In all cases investigated we find the same result, i.e. best agreement for the evolution of the pandemic with time is for the Erdős-Rényi networks. Furthermore, our results indicate that the best fit occurs for a random network with an average degree of the order of ≈ 10–25, for all countries tested. Scale Free and Small World networks fail to fit the real data convincingly.
... We consider that infection and recovery rates are allowed to change over time, and that are homogeneous among the population, instead of heterogeneous rates as in [4,7]. We also assume a mean-field hypothesis that implies random interactions between any pair of agents, unlike the works in [8,13,17,24,25] where interactions are mediated by an underlying network of contacts. ...
We study first order necessary conditions for an optimal control problem of a Susceptible-Infected-Recovered (SIR) model with limitations on the duration of the quarantine. The control is done by means of the reproduction number, i.e., the number of secondary infections produced by a primary infection, which represents an external intervention that we assume time-dependent. Moreover, the control function can only be applied over a finite time interval, and the duration of the most strict quarantine (smallest possible reproduction number) is also bounded. We consider a maximization problem where the cost functional has two terms: one is the number of susceptible individuals in the long-term and the other depends on the cost of interventions. When the intervention term is linear with respect to the control, we obtain that the optimal solution is bang-bang, and we characterize the times to begin and end the strict quarantine. In the general case, when the cost functional includes the term that measures the intervention cost, we analyze the optimality of controls through numerical computations.
... Versions of an SEIQR model have been used in a number of epidemiological studies, e.g. [10,[25][26][27][28]. ...
This paper contains a theoretical study of epidemic control. It is inspired by current events but not intended to be an accurate depiction of the SARS-CoV-2 pandemic. We consider the emergence of a highly transmissible pathogen, focusing on metropolitan areas. To ensure some degree of realism, we present a conceptual model of the outbreak and early attempts to stave off the onslaught, including the use of lockdowns. Model outputs show strong qualitative—in some respects even quantitative—resemblance to the events of Spring 2020 in many cities worldwide. We then use this model to project forward in time to examine different paths in epidemic control after the initial surge is tamed and before the arrival of vaccines. Three very different control strategies are analyzed, leading to vastly different outcomes in terms of economic recovery and total infected population (or progress toward herd immunity). Our model, which is a version of the SEIQR model, is a time-dependent dynamical system with feedback-control. One of the main conclusions of this analysis is that the course of the epidemic is not entirely dictated by the virus: how the population responds to it can play an equally important role in determining the eventual outcome.
... The structure of social networks critically affects epidemic spreading [1,2]. Research has shown that properties that are universally associated with social networks, such as small characteristic path lengths, high clustering [3], and broad-scale degree distributions [4], often work together to provide an environment where epidemics spread fast and virtually uninhibited across the population [5][6][7][8][9][10][11]. This spreading is often accelerated further by the temporal component of social networks [12], where traveling and mobility, in particular, play an important role [13][14][15][16][17][18][19][20]. ...
Community lockdowns and travel restrictions are commonly employed to decelerate epidemic spreading.We here use a stochastic susceptible-infectious-recovered model on different social networks to determine when and to what degree such lockdowns are likely to be effective. Our research shows that community lockdowns are effective only if the links outside of the communities are virtually completely sealed off. The benefits of targeting specifically these links, as opposed to links uniformly at random across the whole network, are inferable only beyond 90% lockdown effectiveness. And even then the peak of the infected curve decreases by only 20% and its onset is delayed by a factor of 1.5. This holds for static and temporal social networks, regardless of their size and structural particularities. Networks derived from cell phone location data and online location-based social platforms yield the same results as a large family of hyperbolic geometric network models where characteristic path lengths, clustering, and community structure can be arbitrarily adjusted. The complex connectedness of modern human societies, which enables the ease of global communication and the lightning speeds at which news and information spread, thus makes it very difficult to halt epidemic spreading with top-down measures. We therefore emphasize the outstanding importance of endogenous self-isolation and social distancing for successfully arresting epidemic spreading.
... Recent works also investigated other mitigation strategies such as awareness campaigns [61], wealth differences [62,63], economic incentives [64], social distancing [65,66], information spreading [67,68], multi-layer contact networks [69], dynamic contacts [70] and others [71][72][73][74][75]. A general overview of these investigations shows the presence of a cycle, where effective mitigation measures lead to a low risk perception, which in turn weakens said mitigation strategies, bringing the disease back [2]. ...
During pandemic events, strategies such as social distancing can be fundamental to reduce simultaneous infections and mitigate the disease spreading, which is very relevant to the risk of a healthcare system collapse. Although these strategies can be recommended, or even imposed, their actual implementation may depend on the population perception of the risks associated with a potential infection. The current COVID-19 crisis, for instance, is showing that some individuals are much more prone than others to remain isolated. To better understand these dynamics, we propose an epidemiological SIR model that uses evolutionary game theory for combining in a single process social strategies, individual risk perception, and viral spreading. In particular, we consider a disease spreading through a population, whose agents can choose between self-isolation and a lifestyle careless of any epidemic risk. The strategy adoption is individual and depends on the perceived disease risk compared to the quarantine cost. The game payoff governs the strategy adoption, while the epidemic process governs the agent's health state. At the same time, the infection rate depends on the agent's strategy while the perceived disease risk depends on the fraction of infected agents. Our results show recurrent infection waves, which are usually seen in previous historic epidemic scenarios with voluntary quarantine. In particular, such waves re-occur as the population reduces disease awareness. Notably, the risk perception is found to be fundamental for controlling the magnitude of the infection peak, while the final infection size is mainly dictated by the infection rates. Low awareness leads to a single and strong infection peak, while a greater disease risk leads to shorter, although more frequent, peaks. The proposed model spontaneously captures relevant aspects of a pandemic event, highlighting the fundamental role of social strategies.
... Reference [55] tried to quantify the effectiveness of a quarantine strategy, where healthy people are advised to avoid contacts with individuals that might carry the disease, using network based models. Here, the parameter has a direct interpretation, since it is related to the average rate of contact between individuals. ...
An extended SIR model, including several features of the recent COVID-19 outbreak, is considered: the infected and recovered individuals can either be detected or undetected and we also integrate an intensive care unit (ICU) capacity. We identify the optimal policy for controlling the epidemic dynamics using both lockdown and detection intervention levers, and taking into account the trade-off between the sanitary and the socio-economic cost of the pandemic, together with the limited capacity level of ICU. With parametric specification based on the COVID-19 literature, we investigate the sensitivities of various quantities on the optimal strategies. The optimal lockdown policy is structured into 4 phases: First a quick and strong lockdown intervention to stop the exponential growth of the contagion; second a short transition to reduce the prevalence of the virus; third a long period with full ICU capacity and stable virus prevalence; finally a return to normal social interactions with disappearance of the virus. The optimal scenario avoids the second wave of infection, provided the lockdown is released sufficiently slowly. Whenever massive resources are introduced to detect infected individuals, the pressure on social distancing can be released, whereas the impact of detection of immune individuals reveals to be more moderate.
Distancing is one of the barrier measures in mitigating epidemics. We aimed to investigate the typology, effectiveness, and side effects of distancing rules during epidemics. Electronic searches were conducted on MEDLINE, PubMed in April 2020, using Mesh-Terms representing various forms of distancing (‘social isolation’, ‘social distancing’, ‘quarantine’) combining with ‘epidemics’. PRISMA-ScR statement was consulted to report this review. A total of 314 titles were identified and 93 were finally included. 2009 influenza A and SARS-CoV-2 epidemics were the most studied. Distancing measures were mostly classified as case-based and community-based interventions. The combination of distancing rules, like school closure, home working, isolation and quarantine, has proven to be effective in reducing R0 and flattening the epidemic curve, also when initiated early at a high rate and combined with other non-pharmaceutical interventions. Epidemiological and modeling studies showed that Isolation and quarantine in the 2009 Influenza pandemic were effective measures to decrease attack rate also with high level of compliance but there was an increased risk of household transmission. lockdown was also effective to reduce R0 from 2.6 to 0.6 and to increase doubling time from 2 to 4 days in the covid-19 pandemic. The evidence for school closure and workplace distancing was moderate as single intervention. Psychological disorder, unhealthy behaviors, disruption of economic activities, social discrimination, and stigmatization were the main side effects of distancing measures. Earlier implementation of combined distancing measures leads to greater effectiveness in containing outbreaks. Their indication must be relevant and based on evidence to avoid adverse effects on the community. These results would help decision-makers to develop response plans based on the required experience and strengthen the capacity of countries to fight against future epidemics. Mesh words: Physical Distancing, Quarantine, Epidemics, Public Health, Scoping Review.
Social distance, quarantines and total lock-downs are non-pharmaceutical interventions that policymakers have used to mitigate the spread of the COVID-19 virus. However, these measures could be harmful to societies in terms of social and economic costs, and they can be maintained only for a short period of time. Here we investigate the optimal strategies that minimize the impact of an epidemic, by studying the conditions for an optimal control of a Susceptible-Infected-Recovered model with a limitation on the total duration of the quarantine. The control is done by means of the reproduction number σ(t), i.e., the number of secondary infections produced by a primary infection, which can be arbitrarily varied in time over a quarantine period T to account for external interventions. We also assume that the most strict quarantine (lower bound of σ) cannot last for a period longer than a value τ. The aim is to minimize the cumulative number of ever-infected individuals (recovered) and the socioeconomic cost of interventions in the long term, by finding the optimal way to vary σ(t). We show that the optimal solution is a single bang-bang, i.e., the strict quarantine is turned on only once, and is turned off after the maximum allowed time τ. Besides, we calculate the optimal time to begin and end the strict quarantine, which depends on T, τ and the initial conditions. We provide rigorous proofs of these results and check that are in perfect agreement with numerical computations.
One of the most effective strategies to mitigate the global spreading of a pandemic (e.g. coronavirus disease 2019) is to shut down international airports. From a network theory perspective, this is since international airports and flights, essentially playing the roles of bridge nodes and bridge links between countries as individual communities, dominate the epidemic spreading characteristics in the whole multi-community system. Among all epidemic characteristics, the peak fraction of infected, $I_{\max}$, is a decisive factor in evaluating an epidemic strategy given limited capacity of medical resources but is seldom considered in multi-community models. In this article, we study a general two-community system interconnected by a fraction $r$ of bridge nodes and its dynamic properties, especially $I_{\max}$, under the evolution of the susceptible-infected-recovered model. Comparing the characteristic time scales of different parts of the system allows us to analytically derive the asymptotic behaviour of $I_{\max}$ with $r$, as $r\rightarrow 0$, which follows different power-law relations in each regime of the phase diagram. We also detect crossovers when $I_{\max}$ changes from one power law to another, crossing different power-law regimes as driven by $r$. Our results enable a better prediction of the effectiveness of strategies acting on bridge nodes, denoted by the power-law exponent $\epsilon_I$ as in $I_{\max}\propto r^{1/\epsilon_I}$.
Vaccination, if available, is the best preventive measure against infectious diseases. It is, however, needed to prudently design vaccination strategies to successfully mitigate the disease spreading, especially in a time when vaccine scarcity is inevitable. Here we investigate a vaccination strategy on a scale-free network where susceptible individuals, who have social connections with infected people, are being detected and given vaccination before having any physical contact with the infected one. Nevertheless, detecting susceptible (also infected ones) may not be perfect due to the lack of information. Also, vaccines do not confer perfect immunity in reality. We incorporate these pragmatic hindrances in our analysis. We find that if vaccines are highly efficacious, and the detecting error is low, then it is possible to confine the disease spreading—by administering a less amount of vaccination—within a short period. In a situation where tracing susceptible seems difficult, then expanding the range for vaccination targets can be socially advantageous only if vaccines are effective enough. Our analysis further reveals that a more frequent screening for vaccination can reduce the effect of detecting errors. In the end, we present a link percolation-based analytic method to approximate the results of our simulation.
Despite its increasing role in communication, the world wide web remains the least controlled medium: any individual or institution can create websites with unrestricted number of documents and links. While great efforts are made to map and characterize the Internet's infrastructure, little is known about the topology of the web. Here we take a first step to fill this gap: we use local connectivity measurements to construct a topological model of the world wide web, allowing us to explore and characterize its large scale properties. Comment: 5 pages, 1 figure, updated with most recent results on the size of the www
Examining important results and analytical techniques, this graduate-level textbook is a step-by-step presentation of the structure and function of complex networks. Using a range of examples, from the stability of the internet to efficient methods of immunizing populations, and from epidemic spreading to how one might efficiently search for individuals, this textbook explains the theoretical methods that can be used, and the experimental and analytical results obtained in the study and research of complex networks. Giving detailed derivations of many results in complex networks theory, this is an ideal text to be used by graduate students entering the field. End-of-chapter review questions help students monitor their own understanding of the materials presented.
The idea behind adaptive behavioral epidemiology is that groups and individuals respond to the knowledge of a disease threat by changing their habits to avoid interactions with those who are contagious. Network-based models take this adaptive behavior into account by allowing the network to "rewire" its connections.
A model for epidemics on an adaptive network is considered. Nodes follow a susceptible-infective-recovered-susceptible pattern. Connections are rewired to break links from noninfected nodes to infected nodes and are reformed to connect to other noninfected nodes, as the nodes that are not infected try to avoid the infection. Monte Carlo simulation and numerical solution of a mean field model are employed. The introduction of rewiring affects both the network structure and the epidemic dynamics. Degree distributions are altered, and the average distance from a node to the nearest infective increases. The rewiring leads to regions of bistability where either an endemic or a disease-free steady state can exist. Fluctuations around the endemic state and the lifetime of the endemic state are considered. The fluctuations are found to exhibit power law behavior.
In 2007, adults in Australia were interviewed about their willingness to comply with potential health interventions during a hypothetical influenza outbreak. After the first wave of pandemic (H1N1) 2009 in Australia, many of the same respondents were interviewed about behavior and protection measures they actually adopted. Of the original 1,155 respondents, follow-up interviews were conducted for 830 (71.9%). Overall, 20.4% of respondents in 2009 had recently experienced influenza-like illness, 77.7% perceived pandemic (H1N1) 2009 to be mild, and 77.8% reported low anxiety. Only 14.5% could correctly answer 4 questions about influenza virus transmission, symptoms, and infection control. Some reported increasing handwashing (46.6%) and covering coughs and sneezes (27.8%) to reduce transmission. Compared with intentions reported in 2007, stated compliance with quarantine or isolation measures in 2009 remained high. However, only respondents who perceived pandemic (H1N1) 2009 as serious or who had attained higher educational levels expressed intention to comply with social distancing measures.
Social networks are dynamic: We make new friends and loose touch with old ones, depending on the interactions with them. Most analytic studies of social networks assume that links remain unchanged at all times. In this case, individuals have no control over the number, frequency or duration of their interactions with others. Here, we discuss analytical and numerical models in which individuals can break links and create new ones. Interactions are modeled as general symmetric twoplayer games. Once a link between two individuals has formed, the productivity of this link is evaluated. Links can be broken off at different rates. In the limiting cases where linking dynamics is much faster than evolutionary dynamics or vice-versa, the system can be tackled analytically.We show how the individual capacity of forming new links or severing inconvenient ones can change the nature of the game. If the linking rules are local, numerical simulations show that networks emerge that have several features of real-world social networks.
We study the spreading of an infection within an SIS epidemiological model on a network. Susceptible agents are given the opportunity of breaking their links with infected agents. Broken links are either permanently removed or reconnected with the rest of the population. Thus, the network coevolves with the population as the infection progresses. We show that a moderate reconnection frequency is enough to completely suppress the infection. A partial, rather weak isolation of infected agents suffices to eliminate the endemic state.