Chapter

# Importance of Interaction Structure and Stochasticity for Epidemic Spreading: A COVID-19 Case Study

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## Abstract

In the recent COVID-19 pandemic, computer simulations are used to predict the evolution of the virus propagation and to evaluate the prospective effectiveness of non-pharmaceutical interventions. As such, the corresponding mathematical models and their simulations are central tools to guide political decision-making. Typically, ODE-based models are considered, in which fractions of infected and healthy individuals change deterministically and continuously over time. In this work, we translate an ODE-based COVID-19 spreading model from literature to a stochastic multi-agent system and use a contact network to mimic complex interaction structures. We observe a large dependency of the epidemic's dynamics on the structure of the underlying contact graph, which is not adequately captured by existing ODE-models. For instance, existence of super-spreaders leads to a higher infection peak but a lower death toll compared to interaction structures without super-spreaders. Overall, we observe that the interaction structure has a crucial impact on the spreading dynamics, which exceeds the effects of other parameters such as the basic reproduction number R0. We conclude that deterministic models fitted to COVID-19 outbreak data have limited predictive power or may even lead to wrong conclusions while stochastic models taking interaction structure into account offer different and probably more realistic epidemiological insights.

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... Heterogeneity in the transmission structure has been found to play a key role in infection spread, e.g. by application of infection models on networks [31] or agestratified contact matrices [32]. In order to implement a relevant heterogeneous transmission structure into our model, the four agent classes-patients, low-risk staff, Recovered. ...
... Indeed, we could see in our study that the impact of the parameter controlling the extent of heterogeneity in the contact structure was negligible. Therefore, only the variation of individual transmissibility needed to be incorporated into the model, but not necessarily an explicit contact structure as employed in many other studies [7][8][9]31] including commonly encountered models with contact structure dictated by a network. This can be attributed to the fact that on average, the first few transmissions rely on the more general epidemiological parameters such as the reproduction number rather than the contact structure. ...
Article
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Background Surveillance testing within healthcare facilities provides an opportunity to prevent severe outbreaks of coronavirus disease 2019 (COVID-19). However, the quantitative impact of different available surveillance strategies and their potential to decrease the frequency of outbreaks are not well-understood. Methods We establish an individual-based model representative of a mental health hospital yielding generalizable results. Attributes and features of this facility were derived from a prototypical hospital, which provides psychiatric, psychosomatic and psychotherapeutic treatment. We estimate the relative reduction of outbreak probability for three test strategies (entry test, once-weekly test and twice-weekly test) relative to a symptom-based baseline strategy. Based on our findings, we propose determinants of successful surveillance measures. Results Entry Testing reduced the outbreak probability by 26%, additionally testing once or twice weekly reduced the outbreak probability by 49% or 67% respectively. We found that fast diagnostic test results and adequate compliance of the clinic population are mandatory for conducting effective surveillance. The robustness of these results towards uncertainties is demonstrated via comprehensive sensitivity analyses. Conclusions We conclude that active testing in mental health hospitals and similar facilities considerably reduces the number of COVID-19 outbreaks compared to symptom-based surveillance only.
... Network based models have also been studied in the literature for analyzing disease spread [59,25] and optimized vaccine allocation [48,10]. In these papers, a network based ODE model called the N-Intertwined model is proposed for analyzing the spread/transmission of COVID-19 among population. ...
... In these papers, a network based ODE model called the N-Intertwined model is proposed for analyzing the spread/transmission of COVID-19 among population. In [25], the state of the nodes is assumed to be in one of the predefined compartments, while in [59,48,10] the states are stochastic. The network is assumed to be a static random graph in these models. ...
Article
In this study, we address three important challenges related to disease transmissions such as the COVID-19 pandemic, namely, (a) providing an early warning to likely exposed individuals, (b) identifying individuals who are asymptomatic, and (c) prescription of optimal testing when testing capacity is limited. First, we present a dynamic-graph based SEIR epidemiological model in order to describe the dynamics of the disease propagation. Our model considers a dynamic graph/network that accounts for the interactions between individuals over time, such as the ones obtained by manual or automated contact tracing, and uses a diffusion-reaction mechanism to describe the state dynamics. This dynamic graph model helps identify likely exposed/infected individuals to whom we can provide early warnings, even before they display any symptoms and/or are asymptomatic. Moreover, when the testing capacity is limited compared to the population size, reliable estimation of individual’s health state and disease transmissibility using epidemiological models is extremely challenging. Thus, estimation of state uncertainty is paramount for both eminent risk assessment, as well as for closing the tracing-testing loop by optimal testing prescription. Therefore, we propose the use of arbitrary Polynomial Chaos Expansion, a popular technique used for uncertainty quantification, to represent the states, and quantify the uncertainties in the dynamic model. This design enables us to assign uncertainty of the state of each individual, and consequently optimize the testing as to reduce the overall uncertainty given a constrained testing budget. These tools can also be used to optimize vaccine distribution to curb the disease spread when limited vaccines are available. We present a few simulation results that illustrate the performance of the proposed framework, and estimate the impact of incomplete contact tracing data.
... We calibrate the only unknown parameter λ accordingly (the relationships from the previous section remain valid). We explain the relation to R 0 when taking C and I into account in the Appendix (available at [16]). Substituting β by µ c gives ...
... Independent of the network, R 0 = 1.8 yields R 0 ≈ 2.05 (cf. Appendix [16]). ...
Preprint
Full-text available
In the recent COVID-19 pandemic, computer simulations are used to predict the evolution of the virus propagation and to evaluate the prospective effectiveness of non-pharmaceutical interventions. As such, the corresponding mathematical models and their simulations are central tools to guide political decision-making. Typically, ODE-based models are considered, in which fractions of infected and healthy individuals change deterministically and continuously over time. In this work, we translate an ODE-based COVID-19 spreading model from literature to a stochastic multi-agent system and use a contact network to mimic complex interaction structures. We observe a large dependency of the epidemic's dynamics on the structure of the underlying contact graph, which is not adequately captured by existing ODE-models. For instance, existence of super-spreaders leads to a higher infection peak but a lower death toll compared to interaction structures without super-spreaders. Overall, we observe that the interaction structure has a crucial impact on the spreading dynamics, which exceeds the effects of other parameters such as the basic reproduction number R0. We conclude that deterministic models fitted to COVID-19 outbreak data have limited predictive power or may even lead to wrong conclusions while stochastic models taking interaction structure into account offer different and probably more realistic epidemiological insights.
... 14, 15,16] and, on the other hand, using stochastic models [e.g. 17,18]. Dynamic stochastic models for COVID-19 spread prediction can be broadly categorized into: i) stochastic differential equations based in classical SIR models [17,8], and ii) compartmental models combined with Mote Carlo methods [19,20,21,6]. ...
Article
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The level of unpredictability of the COVID-19 pandemics poses a challenge to effectively model its dynamic evolution. In this study we incorporate the inherent stochasticity of the SARS-CoV-2 virus spread by reinterpreting the classical compartmental models of infectious diseases (SIR type) as chemical reaction systems modelled via the Chemical Master Equation and solved by Monte Carlo Methods. Our model predicts the evolution of the pandemics at the level of municipalities, incorporating for the first time (i) a variable infection rate to capture the effect of mitigation policies on the dynamic evolution of the pandemics (ii) SIR-with-jumps taking into account the possibility of multiple infections from a single infected person and (iii) data of viral load quantified by RT-qPCR from samples taken from Wastewater Treatment Plants. The model has been successfully employed for the prediction of the COVID-19 pandemics evolution in small and medium size municipalities of Galicia (Northwest of Spain).
... As shown in Table 8, the agent-based model (ABM) predicts considerably fewer cases than the SEIR model (where by a case we mean any agent either exposed or infected). This highlights the impact of heterogeneity and stochasticity [93], with spatial clustering limiting the reach of infected individuals and daily and weekly routines fragmenting the contact network at night and over weekends. For Table 8 we have also calculated the t-statistic of the difference, using the sample standard deviation of the realizations of the ABM used to calculate the p = 0.00035, respectively, for a total resident population of 626,240, together with the t-statistic average, emphasizing that the output of the ABM is indeed significantly different from that of the SEIR model. ...
Article
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Coronavirus disease 2019 (COVID-19) is an infectious disease of humans caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Since the first case was identified in China in December 2019 the disease has spread worldwide, leading to an ongoing pandemic. In this article, we present an agent-based model of COVID-19 in Luxembourg, and use it to estimate the impact, on cases and deaths, of interventions including testing, contact tracing, lockdown, curfew and vaccination. Our model is based on collation, with agents performing activities and moving between locations accordingly. The model is highly heterogeneous, featuring spatial clustering, over 2000 behavioural types and a 10 minute time resolution. The model is validated against COVID-19 clinical monitoring data collected in Luxembourg in 2020. Our model predicts far fewer cases and deaths than the equivalent equation-based SEIR model. In particular, with R 0 = 2.45, the SEIR model infects 87% of the resident population while our agent-based model infects only around 23% of the resident population. Our simulations suggest that testing and contract tracing reduce cases substantially, but are less effective at reducing deaths. Lockdowns are very effective although costly, while the impact of an 11pm-6am curfew is relatively small. When vaccinating against a future outbreak, our results suggest that herd immunity can be achieved at relatively low coverage, with substantial levels of protection achieved with only 30% of the population fully immune. When vaccinating in the midst of an outbreak, the challenge is more difficult. In this context, we investigate the impact of vaccine efficacy, capacity, hesitancy and strategy. We conclude that, short of a permanent lockdown, vaccination is by far the most effective way to suppress and ultimately control the spread of COVID-19.
... In order to assess public health actions, it is important to gain an understanding of the spread of COVID-19 in closed environments as well as behavioral aspects of susceptible individuals. Significant literature exists on simulating the spread of COVID-19 through compartment models or social-network analysis [10][11][12]. Agent-based models (ABMs) have also been widely used to incorporate the propagation of COVID-19 at the interpersonal level based on individual actions [13]. ...
Article
Many schools and universities have seen a significant increase in the spread of COVID-19. As such, a number of non-pharmaceutical interventions have been proposed including distancing requirements, surveillance testing, and updating ventilation systems. Unfortunately, there is limited guidance for which policy or set of policies are most effective for a specific school system. We develop a novel approach to model the spread of SARS-CoV-2 quanta in a closed classroom environment that extends traditional transmission models that assume uniform mixing through air recirculation by including the local spread of quanta from a contagious source. In addition, the behavior of students with respect to guideline compliance was modeled through an agent-based simulation. Estimated infection rates were on average lower using traditional transmission models compared to our approach. Further, we found that although ventilation changes were effective at reducing mean transmission risk, it had much less impact than distancing practices. Duration of the class was an important factor in determining the transmission risk. For the same total number of semester hours for a class, delivering lectures more frequently for shorter durations was preferable to less frequently with longer durations. Finally, as expected, as the contact tracing level increased, more infectious students were identified and removed from the environment and the spread slowed, though there were diminishing returns. These findings can help provide guidance as to which school-based policies would be most effective at reducing risk and can be used in a cost/comparative effectiveness estimation study given local costs and constraints.
... Grossmann et al. [39] propose a stochastic network-based COVID-19 spreading model and compare its results with those obtained through an ordinary differential equations (ODE) model. In their network-based model, they use random graph models to represent interaction structures and human connections. ...
Preprint
As the COVID-19 outbreak evolves around the world, the World Health Organization (WHO) and its Member States have been heavily relying on staying at home and lock down measures to control the spread of the virus. In the last months, various signs showed that the COVID-19 curve was flattening, but even the partial lifting of some containment measures (e.g., school closures and telecommuting) appear to favor a second wave of the disease. The accurate evaluation of possible countermeasures and their well-timed revocation are therefore crucial to avoid future waves or reduce their duration. In this paper, we analyze patient and route data of infected patients from January 20, 2020, to May 31, 2020, collected by the Korean Center for Disease Control & Prevention (KCDC). This data analysis helps us to characterize patient mobility patterns and then use this characterization to parameterize simulations to evaluate different what-if scenarios. Although this is not a definitive model of how COVID-19 spreads in a population, its usefulness and flexibility are illustrated using real-world data for exploring virus spread under a variety of circumstances.
... Specifically, nodes (eventually) become immune after an infection (or die) and do not transmit the pathogen further. However, our framework is easily adaptable to epidemic models with more disease stages, such as COVID-19 models [5]. We consider as input an (undirected, unweighted) contact network with n nodes and a budget k (number of vaccines). ...
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We address the problem of reducing the spread of an epidemic over a contact network by vaccinating a limited number of nodes that represent individuals or agents. We propose a Simulation based vaccine allocation method (Simba), a combination of (i) numerous repetitions of an efficient Monte-Carlo simulation , (ii) a PageRank-type influence analysis on an empirical transmission graph which is learned from the simulations, and (iii) discrete stochastic optimization. Our method scales very well with the size of the network and is suitable for networks with millions of nodes. Moreover, in contrast to most approaches that are model-agnostic approaches and solely perform graph-analysis on the contact graph, the stochastic simulations explicitly take the exact diffusion dynamics of the epidemic into account. Thereby, we make our vaccination strategy sensitive to the specific clinical and transmission parameters of the epidemic.
Preprint
Full-text available
In the recent COVID-19 pandemic, computer simulations are used to predict the evolution of the virus propagation and to evaluate the prospective effectiveness of non-pharmaceutical interventions. As such, the corresponding mathematical models and their simulations are central tools to guide political decision-making. Typically, ODE-based models are considered, in which fractions of infected and healthy individuals change deterministically and continuously over time. In this work, we translate an ODE-based COVID-19 spreading model from literature to a stochastic multi-agent system and use a contact network to mimic complex interaction structures. We observe a large dependency of the epidemic's dynamics on the structure of the underlying contact graph, which is not adequately captured by existing ODE-models. For instance, existence of super-spreaders leads to a higher infection peak but a lower death toll compared to interaction structures without super-spreaders. Overall, we observe that the interaction structure has a crucial impact on the spreading dynamics, which exceeds the effects of other parameters such as the basic reproduction number R0. We conclude that deterministic models fitted to COVID-19 outbreak data have limited predictive power or may even lead to wrong conclusions while stochastic models taking interaction structure into account offer different and probably more realistic epidemiological insights.
Preprint
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The COVID-19 pandemic is straining public health systems worldwide and major non-pharmaceutical interventions have been implemented to slow its spread. During the initial phase of the outbreak the spread was primarily determined by human mobility. Yet empirical evidence on the effect of key geographic factors on local epidemic spread is lacking. We analyse highly-resolved spatial variables for cities in China together with case count data in order to investigate the role of climate, urbanization, and variation in interventions across China. Here we show that the epidemic intensity of COVID-19 is strongly shaped by crowding, such that epidemics in dense cities are more spread out through time, and denser cities have larger total incidence. Observed differences in epidemic intensity are well captured by a metapopulation model of COVID-19 that explicitly accounts for spatial hierarchies. Densely-populated cities worldwide may experience more prolonged epidemics. Whilst stringent interventions can shorten the time length of these local epidemics, although these may be difficult to implement in many affected settings.
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The spread of COVID-19 is posing an unprecedented threat to health systems worldwide[1]. The fast propagation of the disease combined with the existence of covert contagions by asymptomatic individuals make the controlling of this disease particularly challenging. The key parameter to track the progression of the epidemics is the effective reproduction number R, defined as the number of secondary infections generated by an infected individual[2]. The suppression of the epidemics is directly related to this value, and is attained when R<1.Here, we find an analytical expression for R as a function of mobility restrictions and confinement measures, using an epidemic model tailored for COVID-19. This expression for R is an extremely useful tool to design containment policies that are able to suppress the epidemics. We applied our epidemic model for the case of Spain, successfully forecasting both the observed incidence in each region and the overload of the health system. The expression for R allowed us to determine the precise reduction of mobility kappa_0 needed to bend the curve of epidemic incidence, which turned out to be kappa_0 ≈ 0.7. This value, for the case of Spain, translates to a total lockdown with the exception of the mobility associated to essential services, a policy that was finally enforced on March 28.
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The novel Coronavirus SARS-CoV-2 (CoV) has induced a world-wide pandemic and subsequent non-pharmaceutical interventions (NPI) in order to control the spreading of the virus. NPIs are considered to be critical in order to at least delay the peak number of infected individuals and to prevent the health care system becoming overwhelmed by the number of patients to treat in hospitals or in intensive care units (ICUs). However, there is also increasing concern that the NPIs in place would increase mortality because of other diseases, increase the frequency of suicide and increase the risk of an economic recession with unforeseeable implications. It is therefore instrumental to evaluate the necessity of NPIs and to monitor the progress of containment of the virus spreading. We used a data-driven estimation of the evolution of the reproduction number for viral spreading in Germany as well as in all its federal states. Based on an extended infection-epidemic model, parameterized with data from the Robert Koch-Institute and, alternatively, with parameters stemming from a fit to the initial phase of CoV spreading in different regions of Italy, we consistently found that the reproduction number was turned down to a range near 1 in all federal states. We used the latest reproduction number as a starting point for the simulation of epidemic progression and varied the reproduction number, mimicking either release or strengthening of NPIs. Germany is currently, April 3rd, 2020, at the border line of a reproduction number between the scenarios of major immunisation of the population or eradication of the virus. We strongly recommend to keep all NPIs in place and suggest to even strengthen the measures in order to accelerate reaching the state of full control, thus, also limiting collateral damage of the NPIs in time.
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The spread of a novel pathogenic infectious agent eliciting protective immunity is typically characterised by three distinct phases: (I) an initial phase of slow accumulation of new infections (often undetectable), (II) a second phase of rapid growth in cases of infection, disease and death, and (III) an eventual slow down of transmission due to the depletion of susceptible individuals, typically leading to the termination of the (first) epidemic wave. Before the implementation of control measures (e.g. social distancing, travel bans, etc) and under the assumption that infection elicits protective immunity, epidemiological theory indicates that the ongoing epidemic of SARS-CoV-2 will conform to this pattern. Here, we calibrate a susceptible-infected-recovered (SIR) model to data on cumulative reported SARS-CoV-2 associated deaths from the United Kingdom (UK) and Italy under the assumption that such deaths are well reported events that occur only in a vulnerable fraction of the population. We focus on model solutions which take into consideration previous estimates of critical epidemiological parameters such as the basic reproduction number (R0), probability of death in the vulnerable fraction of the population, infectious period and time from infection to death, with the intention of exploring the sensitivity of the system to the actual fraction of the population vulnerable to severe disease and death. Our simulations are in agreement with other studies that the current epidemic wave in the UK and Italy in the absence of interventions should have an approximate duration of 2-3 months, with numbers of deaths lagging behind in time relative to overall infections. Importantly, the results we present here suggest the ongoing epidemics in the UK and Italy started at least a month before the first reported death and have already led to the accumulation of significant levels of herd immunity in both countries. There is an inverse relationship between the proportion currently immune and the fraction of the population vulnerable to severe disease. This relationship can be used to determine how many people will require hospitalisation (and possibly die) in the coming weeks if we are able to accurately determine current levels of herd immunity. There is thus an urgent need for investment in technologies such as virus (or viral pseudotype) neutralization assays and other robust assays which provide reliable read-outs of protective immunity, and for the provision of open access to valuable data sources such as blood banks and paired samples of acute and convalescent sera from confirmed cases of SARS-CoV-2 to validate these. Urgent development and assessment of such tests should be followed by rapid implementation at scale to provide real-time data. These data will be critical to the proper assessment of the effects of social distancing and other measures currently being adopted to slow down the case incidence and for informing future policy direction.
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Background In December, 2019, severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), a novel coronavirus, emerged in Wuhan, China. Since then, the city of Wuhan has taken unprecedented measures in response to the outbreak, including extended school and workplace closures. We aimed to estimate the effects of physical distancing measures on the progression of the COVID-19 epidemic, hoping to provide some insights for the rest of the world. Methods To examine how changes in population mixing have affected outbreak progression in Wuhan, we used synthetic location-specific contact patterns in Wuhan and adapted these in the presence of school closures, extended workplace closures, and a reduction in mixing in the general community. Using these matrices and the latest estimates of the epidemiological parameters of the Wuhan outbreak, we simulated the ongoing trajectory of an outbreak in Wuhan using an age-structured susceptible-exposed-infected-removed (SEIR) model for several physical distancing measures. We fitted the latest estimates of epidemic parameters from a transmission model to data on local and internationally exported cases from Wuhan in an age-structured epidemic framework and investigated the age distribution of cases. We also simulated lifting of the control measures by allowing people to return to work in a phased-in way and looked at the effects of returning to work at different stages of the underlying outbreak (at the beginning of March or April). Findings Our projections show that physical distancing measures were most effective if the staggered return to work was at the beginning of April; this reduced the median number of infections by more than 92% (IQR 66–97) and 24% (13–90) in mid-2020 and end-2020, respectively. There are benefits to sustaining these measures until April in terms of delaying and reducing the height of the peak, median epidemic size at end-2020, and affording health-care systems more time to expand and respond. However, the modelled effects of physical distancing measures vary by the duration of infectiousness and the role school children have in the epidemic. Interpretation Restrictions on activities in Wuhan, if maintained until April, would probably help to delay the epidemic peak. Our projections suggest that premature and sudden lifting of interventions could lead to an earlier secondary peak, which could be flattened by relaxing the interventions gradually. However, there are limitations to our analysis, including large uncertainties around estimates of R0 and the duration of infectiousness. Funding Bill & Melinda Gates Foundation, National Institute for Health Research, Wellcome Trust, and Health Data Research UK.
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On the basis of a semi-realistic SIR microsimulation for Germany and Poland, we show that the R 0 parameter interval for which the COVID-19 epidemic stays overcritical but below the capacity limit of the health care system to reach herd immunity is so narrow that a successful implementation of this strategy is likely to fail. Our microsimulation is based on official census data and involves household composition and age distribution as the main population structure variables. Outside household contacts are characterised by an out-reproduction number R* which is the only free parameter of the model. For a subcritical domain we compute the time till extinction and prevalence as a function of the initial number of infected individuals and R*. For the Polish city of Wroc law we also discuss the combined impact of testing coverage and contact reduction. For both countries we estimate R* for disease progression until 20th of March 2020.
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A SEIR simulation model for the COVID-19 pandemic was developed (http://covidsim.eu) and applied to a hypothetical European country of 10 million population. Our results show which interventions potentially push the epidemic peak into the subsequent year (when vaccinations may be available) or which fail. Different levels of control (via contact reduction) resulted in 22% to 63% of the population sick, 0.2% to 0.6% hospitalised, and 0.07% to 0.28% dead (n=6,450 to 28,228).
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Outbreak to pandemic In response to global dispersion of severe acute respiratory syndrome–coronavirus 2 (SARS-CoV-2), quarantine measures have been implemented around the world. To understand how travel and quarantine influence the dynamics of the spread of this novel human virus, Chinazzi et al. applied a global metapopulation disease transmission model to epidemiological data from China. They concluded that the travel quarantine introduced in Wuhan on 23 January 2020 only delayed epidemic progression by 3 to 5 days within China, but international travel restrictions did help to slow spread elsewhere in the world until mid-February. Their results suggest that early detection, hand washing, self-isolation, and household quarantine will likely be more effective than travel restrictions at mitigating this pandemic. Science , this issue p. 395
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There is an urgent need to project how transmission of the novel betacoronavirus SARS-CoV-2 will unfold in coming years. These dynamics will depend on seasonality, the duration of immunity, and the strength of cross-immunity to/from the other human coronaviruses. Using data from the United States, we measured how these factors affect transmission of human betacoronaviruses HCoV-OC43 and HCoV-HKU1. We then built a mathematical model to simulate transmission of SARS-CoV-2 through the year 2025. We project that recurrent wintertime outbreaks of SARS-CoV-2 will probably occur after an initial pandemic wave. We summarize the full range of plausible transmission scenarios and identify key data still needed to distinguish between them, most importantly longitudinal serological studies to determine the duration of immunity to SARS-CoV-2.
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Background Rapid spread of SARS-CoV-2 in Wuhan prompted heightened surveillance in Shenzhen and elsewhere in China. The resulting data provide a rare opportunity to measure key metrics of disease course, transmission, and the impact of control. Methods The Shenzhen CDC identified 391 SARS-CoV-2 cases from January 14 to February 12, 2020 and 1286 close contacts. We compare cases identified through symptomatic surveillance and contact tracing, and estimate the time from symptom onset to confirmation, isolation, and hospitalization. We estimate metrics of disease transmission and analyze factors influencing transmission risk. Findings Cases were older than the general population (mean age 45) and balanced between males (187) and females (204). Ninety-one percent had mild or moderate clinical severity at initial assessment. Three have died, 225 have recovered (median time to recovery is 32 days). Cases were isolated on average 4.6 days after developing symptoms; contact tracing reduced this by 1.9 days. Household contacts and those travelling with a case where at higher risk of infection (ORs 6 and 7) than other close contacts. The household secondary attack rate was 15%, and children were as likely to be infected as adults. The observed reproductive number was 0.4, with a mean serial interval of 6.3 days. Interpretation Our data on cases as well as their infected and uninfected close contacts provide key insights into SARS-CoV-2 epidemiology. This work shows that heightened surveillance and isolation, particularly contact tracing, reduces the time cases are infectious in the community, thereby reducing R. Its overall impact, however, is uncertain and highly dependent on the number of asymptomatic cases. We further show that children are at similar risk of infection as the general population, though less likely to have severe symptoms; hence should be considered in analyses of transmission and control.
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Background Isolation of cases and contact tracing is used to control outbreaks of infectious diseases, and has been used for coronavirus disease 2019 (COVID-19). Whether this strategy will achieve control depends on characteristics of both the pathogen and the response. Here we use a mathematical model to assess if isolation and contact tracing are able to control onwards transmission from imported cases of COVID-19. Methods We developed a stochastic transmission model, parameterised to the COVID-19 outbreak. We used the model to quantify the potential effectiveness of contact tracing and isolation of cases at controlling a severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2)-like pathogen. We considered scenarios that varied in the number of initial cases, the basic reproduction number (R0), the delay from symptom onset to isolation, the probability that contacts were traced, the proportion of transmission that occurred before symptom onset, and the proportion of subclinical infections. We assumed isolation prevented all further transmission in the model. Outbreaks were deemed controlled if transmission ended within 12 weeks or before 5000 cases in total. We measured the success of controlling outbreaks using isolation and contact tracing, and quantified the weekly maximum number of cases traced to measure feasibility of public health effort. Findings Simulated outbreaks starting with five initial cases, an R0 of 1·5, and 0% transmission before symptom onset could be controlled even with low contact tracing probability; however, the probability of controlling an outbreak decreased with the number of initial cases, when R0 was 2·5 or 3·5 and with more transmission before symptom onset. Across different initial numbers of cases, the majority of scenarios with an R0 of 1·5 were controllable with less than 50% of contacts successfully traced. To control the majority of outbreaks, for R0 of 2·5 more than 70% of contacts had to be traced, and for an R0 of 3·5 more than 90% of contacts had to be traced. The delay between symptom onset and isolation had the largest role in determining whether an outbreak was controllable when R0 was 1·5. For R0 values of 2·5 or 3·5, if there were 40 initial cases, contact tracing and isolation were only potentially feasible when less than 1% of transmission occurred before symptom onset. Interpretation In most scenarios, highly effective contact tracing and case isolation is enough to control a new outbreak of COVID-19 within 3 months. The probability of control decreases with long delays from symptom onset to isolation, fewer cases ascertained by contact tracing, and increasing transmission before symptoms. This model can be modified to reflect updated transmission characteristics and more specific definitions of outbreak control to assess the potential success of local response efforts. Funding Wellcome Trust, Global Challenges Research Fund, and Health Data Research UK.
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Social mixing patterns are crucial in driving transmission of infectious diseases and informing public health interventions to contain their spread. Age-specific social mixing is often inferred from surveys of self-recorded contacts which by design often have a very limited number of participants. In addition, such surveys are rare, so public health interventions are often evaluated by considering only one such study. Here we report detailed population contact patterns for United Kingdom based self-reported contact data from over 36,000 volunteers that participated in the massive citizen science project BBC Pandemic. The amount of data collected allows us generate fine-scale age-specific population contact matrices by context (home, work, school, other) and type (conversational or physical) of contact that took place. These matrices are highly relevant for informing prevention and control of new outbreaks, and evaluating strategies that reduce the amount of mixing in the population (such as school closures, social distancing, or working from home). In addition, they finally provide the possibility to use multiple sources of social mixing data to evaluate the uncertainty that stems from social mixing when designing public health interventions.
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Since December 2019, China has been experiencing a large outbreak of a novel coronavirus (2019-nCoV) which can cause respiratory disease and severe pneumonia. We estimated the basic reproduction number R0 of 2019-nCoV to be around 2.2 (90% high density interval: 1.4–3.8), indicating the potential for sustained human-to-human transmission. Transmission characteristics appear to be of similar magnitude to severe acute respiratory syndrome-related coronavirus (SARS-CoV) and pandemic influenza, indicating a risk of global spread.
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In this study, we present representative human contact networks among Chinese college students. Unlike schools in the US, human contacts within Chinese colleges are extremely clustered, partly due to the highly organized lifestyle of Chinese college students. Simulations of influenza spreading across real contact networks are in good accordance with real influenza records; however, epidemic simulations across idealized scale-free or small-world networks show considerable overestimation of disease prevalence, thus challenging the widely-applied idealized human contact models in epidemiology. Furthermore, the special contact pattern within Chinese colleges results in disease spreading patterns distinct from those of the US schools. Remarkably, class cancelation, though simple, shows a mitigating power equal to quarantine/vaccination applied on ~25% of college students, which quantitatively explains its success in Chinese colleges during the SARS period. Our findings greatly facilitate reliable prediction of epidemic prevalence, and thus should help establishing effective strategies for respiratory infectious diseases control.
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Several systems can be modeled as sets of interconnected networks or networks with multiple types of connections, here generally called multilayer networks. Spreading processes such as information propagation among users of online social networks, or the diffusion of pathogens among individuals through their contact network, are fundamental phenomena occurring in these networks. However, while information diffusion in single networks has received considerable attention from various disciplines for over a decade, spreading processes in multilayer networks is still a young research area presenting many challenging research issues. In this paper, we review the main models, results and applications of multilayer spreading processes and discuss some promising research directions.
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This article reviews and presents various solved and open problems in the development, analysis, and control of epidemic models. The proper modeling and analysis of spreading processes has been a long-standing area of research among many different fields, including mathematical biology, physics, computer science, engineering, economics, and the social sciences. One of the earliest epidemic models conceived was by Daniel Bernoulli in 1760, which was motivated by studying the spread of smallpox [1]. In addition to Bernoulli, there were many different researchers also working on mathematical epidemic models around this time [2]. These initial models were quite simplistic, and the further development and study of such models dates back to the 1900s [3]-[6], where still-simple models were studied to provide insight into how various diseases can spread through a population. In recent years, there has been a resurgence of interest in these problems as the concept of "networks" becomes increasingly prevalent in modeling many different aspects of the world today. A more comprehensive review of the history of mathematical epidemiology can be found in [7] and [8].
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We consider the problem of controlling the propagation of an epidemic outbreak in an arbitrary contact network by distributing vaccination resources throughout the network. We analyze a networked version of the Susceptible-Infected-Susceptible (SIS) epidemic model when individuals in the network present different levels of susceptibility to the epidemic. In this context, controlling the spread of an epidemic outbreak can be written as a spectral condition involving the eigenvalues of a matrix that depends on the network structure and the parameters of the model. We study the problem of finding the optimal distribution of vaccines throughout the network to control the spread of an epidemic outbreak. We propose a convex framework to find cost-optimal distribution of vaccination resources when different levels of vaccination are allowed. We also propose a greedy approach with quality guarantees for the case of all-or-nothing vaccination. We illustrate our approaches with numerical simulations in a real social network.
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NetworkX is a Python language package for exploration and analysis of networks and network algorithms. The core package provides data structures for representing many types of networks, or graphs, including simple graphs, directed graphs, and graphs with parallel edges and self loops. The nodes in NetworkX graphs can be any (hashable) Python object and edges can contain arbitrary data; this flexibility mades NetworkX ideal for representing networks found in many different scientific fields. In addition to the basic data structures many graph algorithms are implemented for calculating network properties and structure measures: shortest paths, betweenness centrality, clustering, and degree distribution and many more. NetworkX can read and write various graph formats for eash exchange with existing data, and provides generators for many classic graphs and popular graph models, such as the Erdoes-Renyi, Small World, and Barabasi-Albert models, are included. The ease-of-use and flexibility of the Python programming language together with connection to the SciPy tools make NetworkX a powerful tool for scientific computations. We discuss some of our recent work studying synchronization of coupled oscillators to demonstrate how NetworkX enables research in the field of computational networks.
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Background: The integration of empirical data in computational frameworks designed to model the spread of infectious diseases poses a number of challenges that are becoming more pressing with the increasing availability of high-resolution information on human mobility and contacts. This deluge of data has the potential to revolutionize the computational efforts aimed at simulating scenarios, designing containment strategies, and evaluating outcomes. However, the integration of highly detailed data sources yields models that are less transparent and general in their applicability. Hence, given a specific disease model, it is crucial to assess which representations of the raw data work best to inform the model, striking a balance between simplicity and detail. Methods: We consider high-resolution data on the face-to-face interactions of individuals in a pediatric hospital ward, obtained by using wearable proximity sensors. We simulate the spread of a disease in this community by using an SEIR model on top of different mathematical representations of the empirical contact patterns. At the most detailed level, we take into account all contacts between individuals and their exact timing and order. Then, we build a hierarchy of coarse-grained representations of the contact patterns that preserve only partially the temporal and structural information available in the data. We compare the dynamics of the SEIR model across these representations. Results: We show that a contact matrix that only contains average contact durations between role classes fails to reproduce the size of the epidemic obtained using the high-resolution contact data and also fails to identify the most at-risk classes. We introduce a contact matrix of probability distributions that takes into account the heterogeneity of contact durations between (and within) classes of individuals, and we show that, in the case study presented, this representation yields a good approximation of the epidemic spreading properties obtained by using the high-resolution data. Conclusions: Our results mark a first step towards the definition of synopses of high-resolution dynamic contact networks, providing a compact representation of contact patterns that can correctly inform computational models designed to discover risk groups and evaluate containment policies. We show in a typical case of a structured population that this novel kind of representation can preserve in simulation quantitative features of the epidemics that are crucial for their study and management.
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A wide class of binary-state dynamics on networks---including, for example, the voter model, the Bass diffusion model, and threshold models---can be described in terms of transition rates (spin-flip probabilities) that depend on the number of nearest neighbors in each of the two possible states. High-accuracy approximations for the emergent dynamics of such models on uncorrelated, infinite networks are given by recently-developed compartmental models or approximate master equations (AME). Pair approximations (PA) and mean-field theories can be systematically derived from the AME. We show that PA and AME solutions can coincide under certain circumstances, and numerical simulations confirm that PA is highly accurate in these cases. For monotone dynamics (where transitions out of one nodal state are impossible, e.g., SI disease-spread or Bass diffusion), PA and AME give identical results for the fraction of nodes in the infected (active) state for all time, provided the rate of infection depends linearly on the number of infected neighbors. In the more general non-monotone case, we derive a condition---that proves equivalent to a detailed balance condition on the dynamics---for PA and AME solutions to coincide in the limit $t \to \infty$. This permits bifurcation analysis, yielding explicit expressions for the critical (ferromagnetic/paramagnetic transition) point of such dynamics, closely analogous to the critical temperature of the Ising spin model. Finally, the AME for threshold models of propagation is shown to reduce to just two differential equations, and to give excellent agreement with numerical simulations. As part of this work, Octave/Matlab code for implementing and solving the differential equation systems is made available for download.
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The most frequent infectious diseases in humans--and those with the highest potential for rapid pandemic spread--are usually transmitted via droplets during close proximity interactions (CPIs). Despite the importance of this transmission route, very little is known about the dynamic patterns of CPIs. Using wireless sensor network technology, we obtained high-resolution data of CPIs during a typical day at an American high school, permitting the reconstruction of the social network relevant for infectious disease transmission. At 94% coverage, we collected 762,868 CPIs at a maximal distance of 3 m among 788 individuals. The data revealed a high-density network with typical small-world properties and a relatively homogeneous distribution of both interaction time and interaction partners among subjects. Computer simulations of the spread of an influenza-like disease on the weighted contact graph are in good agreement with absentee data during the most recent influenza season. Analysis of targeted immunization strategies suggested that contact network data are required to design strategies that are significantly more effective than random immunization. Immunization strategies based on contact network data were most effective at high vaccination coverage.
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Mathematical models of infection that consider targeted interventions are exquisitely dependent on the assumed mixing patterns of the population. We report on a pilot study designed to assess three different methods (one retrospective, two prospective) for obtaining contact data relevant to the determination of these mixing patterns. 65 adults were asked to record their social encounters in each location visited during 6 study days using a novel method whereby a change in physical location of the study participant triggered data entry. Using a cross-over design, all participants recorded encounters on 3 days in a paper diary and 3 days using an electronic recording device (PDA). Participants were randomised to first prospective recording method. Both methods captured more contacts than a pre-study questionnaire, but ascertainment using the paper diary was superior to the PDA (mean difference: 4.52 (95% CI 0.28, 8.77). Paper diaries were found more acceptable to the participants compared with the PDA. Statistical analysis confirms that our results are broadly consistent with those reported from large-scale European based surveys. An association between household size (trend 0.14, 95% CI (0.06, 0.22), P < 0.001) and composition (presence of child 0.37, 95% CI (0.17, 0.56), P < 0.001) and the total number of reported contacts was observed, highlighting the importance of sampling study populations based on household characteristics as well as age. New contacts were still being recorded on the third study day, but compliance had declined, indicating that the optimal number of sample days represents a trade-off between completeness and quality of data for an individual. The study's location-based reporting design allows greater scope compared to other methods for examining differences in the characteristics of encounters over a range of environments. Improved parameterisation of dynamic transmission models gained from work of this type will aid in the development of more robust decision support tools to assist health policy makers and planners.
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The propagation of communicable diseases through a population is an inherent spatial and temporal process of great importance for modern society. For this reason a spatially explicit epidemiologic model of infectious disease is proposed for a greater understanding of the disease's spatial diffusion through a network of human contacts. The objective of this study is to develop an agent-based modelling approach the integrates geographic information systems (GIS) to simulate the spread of a communicable disease in an urban environment, as a result of individuals' interactions in a geospatial context. The methodology for simulating spatiotemporal dynamics of communicable disease propagation is presented and the model is implemented using measles outbreak in an urban environment as a case study. Individuals in a closed population are explicitly represented by agents associated to places where they interact with other agents. They are endowed with mobility, through a transportation network allowing them to move between places within the urban environment, in order to represent the spatial heterogeneity and the complexity involved in infectious diseases diffusion. The model is implemented on georeferenced land use dataset from Metro Vancouver and makes use of census data sets from Statistics Canada for the municipality of Burnaby, BC, Canada study site. The results provide insights into the application of the model to calculate ratios of susceptible/infected in specific time frames and urban environments, due to its ability to depict the disease progression based on individuals' interactions. It is demonstrated that the dynamic spatial interactions within the population lead to high numbers of exposed individuals who perform stationary activities in areas after they have finished commuting. As a result, the sick individuals are concentrated in geographical locations like schools and universities. The GIS-agent based model designed for this study can be easily customized to study the disease spread dynamics of any other communicable disease by simply adjusting the modeled disease timeline and/or the infection model and modifying the transmission process. This type of simulations can help to improve comprehension of disease spread dynamics and to take better steps towards the prevention and control of an epidemic outbreak.
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In the absence of other evidence, modelling has been used extensively to help policy makers plan for a potential future influenza pandemic. We have constructed an individual based model of a small community in the developed world with detail down to exact household structure obtained from census collection datasets and precise simulation of household demographics, movement within the community and individual contact patterns. We modelled the spread of pandemic influenza in this community and the effect on daily and final attack rates of four social distancing measures: school closure, increased case isolation, workplace non-attendance and community contact reduction. We compared the modelled results of final attack rates in the absence of any interventions and the effect of school closure as a single intervention with other published individual based models of pandemic influenza in the developed world. We showed that published individual based models estimate similar final attack rates over a range of values for R(0) in a pandemic where no interventions have been implemented; that multiple social distancing measures applied early and continuously can be very effective in interrupting transmission of the pandemic virus for R(0) values up to 2.5; and that different conclusions reached on the simulated benefit of school closure in published models appear to result from differences in assumptions about the timing and duration of school closure and flow-on effects on other social contacts resulting from school closure. Models of the spread and control of pandemic influenza have the potential to assist policy makers with decisions about which control strategies to adopt. However, attention needs to be given by policy makers to the assumptions underpinning both the models and the control strategies examined.
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Development of strategies for mitigating the severity of a new influenza pandemic is now a top global public health priority. Influenza prevention and containment strategies can be considered under the broad categories of antiviral, vaccine and non-pharmaceutical (case isolation, household quarantine, school or workplace closure, restrictions on travel) measures. Mathematical models are powerful tools for exploring this complex landscape of intervention strategies and quantifying the potential costs and benefits of different options. Here we use a large-scale epidemic simulation to examine intervention options should initial containment of a novel influenza outbreak fail, using Great Britain and the United States as examples. We find that border restrictions and/or internal travel restrictions are unlikely to delay spread by more than 2-3 weeks unless more than 99% effective. School closure during the peak of a pandemic can reduce peak attack rates by up to 40%, but has little impact on overall attack rates, whereas case isolation or household quarantine could have a significant impact, if feasible. Treatment of clinical cases can reduce transmission, but only if antivirals are given within a day of symptoms starting. Given enough drugs for 50% of the population, household-based prophylaxis coupled with reactive school closure could reduce clinical attack rates by 40-50%. More widespread prophylaxis would be even more logistically challenging but might reduce attack rates by over 75%. Vaccine stockpiled in advance of a pandemic could significantly reduce attack rates even if of low efficacy. Estimates of policy effectiveness will change if the characteristics of a future pandemic strain differ substantially from those seen in past pandemics.
Article
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Planning a response to an outbreak of a pandemic strain of influenza is a high public health priority. Three research groups using different individual-based, stochastic simulation models have examined the consequences of intervention strategies chosen in consultation with U.S. public health workers. The first goal is to simulate the effectiveness of a set of potentially feasible intervention strategies. Combinations called targeted layered containment (TLC) of influenza antiviral treatment and prophylaxis and nonpharmaceutical interventions of quarantine, isolation, school closure, community social distancing, and workplace social distancing are considered. The second goal is to examine the robustness of the results to model assumptions. The comparisons focus on a pandemic outbreak in a population similar to that of Chicago, with ≈8.6 million people. The simulations suggest that at the expected transmissibility of a pandemic strain, timely implementation of a combination of targeted household antiviral prophylaxis, and social distancing measures could substantially lower the illness attack rate before a highly efficacious vaccine could become available. Timely initiation of measures and school closure play important roles. Because of the current lack of data on which to base such models, further field research is recommended to learn more about the sources of transmission and the effectiveness of social distancing measures in reducing influenza transmission. • influenza antiviral agents • mitigation • prophylaxis • social distancing • transmission
Article
SARS-CoV-2, the virus that causes coronavirus disease 2019 (COVID-19), has spread rapidly around the world since it was first recognized in late 2019. Most early reports of person-to-person SARS-CoV-2 transmission have been among household contacts, where the secondary attack rate has been estimated to exceed 10% (1), in health care facilities (2), and in congregate settings (3). However, widespread community transmission, as is currently being observed in the United States, requires more expansive transmission events between nonhousehold contacts. In February and March 2020, the Chicago Department of Public Health (CDPH) investigated a large, multifamily cluster of COVID-19. Patients with confirmed COVID-19 and their close contacts were interviewed to better understand nonhousehold, community transmission of SARS-CoV-2. This report describes the cluster of 16 cases of confirmed or probable COVID-19, including three deaths, likely resulting from transmission of SARS-CoV-2 at two family gatherings (a funeral and a birthday party). These data support current CDC social distancing recommendations intended to reduce SARS-CoV-2 transmission. U.S residents should follow stay-at-home orders when required by state or local authorities.
Preprint
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The evolving coronavirus disease 2019 (COVID‐19) epidemic1 is certainly cause for concern. Proper communication and optimal decision‐making is an ongoing challenge, as data evolve. The challenge is compounded, however, by exaggerated information. This can lead to inappropriate actions. It is important to differentiate promptly the true epidemic from an epidemic of false claims and potentially harmful actions.
Article
Background Three clusters of coronavirus disease 2019 (COVID-19) linked to a tour group from China, a company conference, and a church were identified in Singapore in February, 2020. Methods We gathered epidemiological and clinical data from individuals with confirmed COVID-19, via interviews and inpatient medical records, and we did field investigations to assess interactions and possible modes of transmission of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Open source reports were obtained for overseas cases. We reported the median (IQR) incubation period of SARS-CoV-2. Findings As of Feb 15, 2020, 36 cases of COVID-19 were linked epidemiologically to the first three clusters of circumscribed local transmission in Singapore. 425 close contacts were quarantined. Direct or prolonged close contact was reported among affected individuals, although indirect transmission (eg, via fomites and shared food) could not be excluded. The median incubation period of SARS-CoV-2 was 4 days (IQR 3–6). The serial interval between transmission pairs ranged between 3 days and 8 days. Interpretation SARS-CoV-2 is transmissible in community settings, and local clusters of COVID-19 are expected in countries with high travel volume from China before the lockdown of Wuhan and institution of travel restrictions. Enhanced surveillance and contact tracing is essential to minimise the risk of widespread transmission in the community. Funding None.
Book
This book covers recent developments in epidemic process models and related data on temporally varying networks. It is widely recognized that contact networks are indispensable for describing, understanding, and intervening to stop the spread of infectious diseases in human and animal populations; “network epidemiology” is an umbrella term to describe this research field. More recently, contact networks have been recognized as being highly dynamic. This observation, also supported by an increasing amount of new data, has led to research on temporal networks, a rapidly growing area. Changes in network structure are often informed by epidemic (or other) dynamics, in which case they are referred to as adaptive networks. This volume gathers contributions by prominent authors working in temporal and adaptive network epidemiology, a field essential to understanding infectious diseases in real society.
Book
Article
Models of how outbreaks of infectious disease emerge in a population of humans typically have two components—a so called compartmental model dividing the people into classes with respect to the disease and assigning transition rules between the classes, and a model for the contact pattern between people. Both these aspects affect the propagation of the epidemic outbreak, for example the predicted outbreak sizes. Contact patterns can be represented at different levels of information content: as a temporal network (with temporal information about the contacts and who that is in contact with whom), as a static network (without information bout the time of the contacts) or as a fully connected system (where everyone can be in contact with everyone with equal chance all the time). We compare the predicted final outbreak sizes for these three contact-pattern representations, given the current state of the outbreak. We find that the temporal component has a strong influence on the diversity of outbreak sizes.
Article
In recent years the research community has accumulated overwhelming evidence for the emergence of complex and heterogeneous connectivity patterns in a wide range of biological and socio-technical systems. The complex properties of real world networks have a profound impact on the behavior of equilibrium and non-equilibrium phenomena occurring in various systems, and the study of epidemic spreading is central to our understanding of the unfolding of dynamical processes in complex networks. The theoretical analysis of epidemic spreading in heterogeneous networks requires the development of novel analytical frameworks, and it has produced results of conceptual and practical relevance. Here we present a coherent and comprehensive review of the vast research activity concerning epidemic processes, detailing the successful theoretical approaches as well as making their limits and assumptions clear. Physicists, epidemiologists, computer and social scientists share a common interest in studying epidemic spreading and rely on very similar models for the description of the diffusion of pathogens, knowledge, and innovation. For this reason, while we focus on the main results and the paradigmatic models in infectious disease modeling, we also present the major results concerning generalized social contagion processes. Finally we outline the research activity at the forefront in the study of epidemic spreading in co-evolving and time-varying networks.
Article
Most studies on susceptible-infected-susceptible epidemics in networks implicitly assume Markovian behavior: the time to infect a direct neighbor is exponentially distributed. Much effort so far has been devoted to characterize and precisely compute the epidemic threshold in susceptible-infected-susceptible Markovian epidemics on networks. Here, we report the rather dramatic effect of a nonexponential infection time (while still assuming an exponential curing time) on the epidemic threshold by considering Weibullean infection times with the same mean, but different power exponent α. For three basic classes of graphs, the Erdős-Rényi random graph, scale-free graphs and lattices, the average steady-state fraction of infected nodes is simulated from which the epidemic threshold is deduced. For all graph classes, the epidemic threshold significantly increases with the power exponents α. Hence, real epidemics that violate the exponential or Markovian assumption can behave seriously differently than anticipated based on Markov theory.
Article
This paper is concerned with a stochastic SIR (susceptible-->infective-->removed) model for the spread of an epidemic amongst a population of individuals, with a random network of social contacts, that is also partitioned into households. The behaviour of the model as the population size tends to infinity in an appropriate fashion is investigated. A threshold parameter which determines whether or not an epidemic with few initial infectives can become established and lead to a major outbreak is obtained, as are the probability that a major outbreak occurs and the expected proportion of the population that are ultimately infected by such an outbreak, together with methods for calculating these quantities. Monte Carlo simulations demonstrate that these asymptotic quantities accurately reflect the behaviour of finite populations, even for only moderately sized finite populations. The model is compared and contrasted with related models previously studied in the literature. The effects of the amount of clustering present in the overall population structure and the infectious period distribution on the outcomes of the model are also explored.
Article
Mathematical analysis and modelling is central to infectious disease epidemiology. Here, we provide an intuitive introduction to the process of disease transmission, how this stochastic process can be represented mathematically and how this mathematical representation can be used to analyse the emergent dynamics of observed epidemics. Progress in mathematical analysis and modelling is of fundamental importance to our growing understanding of pathogen evolution and ecology. The fit of mathematical models to surveillance data has informed both scientific research and health policy. This Review is illustrated throughout by such applications and ends with suggestions of open challenges in mathematical epidemiology.
Report 9: impact of non-pharmaceutical interventions (NPIs) to reduce COVID19 mortality and healthcare demand
• N Ferguson
Schätzung der aktuellen entwicklung der sars-cov-2-epidemie in deutschland-nowcasting
• O Hamouda
Estimate of the development of the epidemic reproduction number RT from coronavirus SARS-CoV-2 case data and implications for political measures based on prognostics
• S Khailaie
Derivation of the effective reproduction number R for COVID-19 in relation to mobility restrictions and confinement
• A Arenas