Effects of heterogeneous and clustered contact patterns on infectious disease dynamics

The complex interplay between risk tolerance and the spread of infectious diseases

Risk-driven behaviour provides a feedback mechanism through which individuals both shape and are collectively affected by an epidemic. We introduce a general and flexible compartmental model to study the effect of heterogeneity in the population with regard to risk tolerance. The interplay between behaviour and epidemiology leads to a rich set of possible epidemic dynamics. Depending on the behavioural composition of the population, we find that increasing heterogeneity in risk tolerance can either increase or decrease the epidemic size. We find that multiple waves of infection can arise due to the interplay between transmission and behaviour, even without the replenishment of susceptibles. We find that increasing protective mechanisms such as the effectiveness of interventions, the fraction of risk-averse people in the population and the duration of intervention usage reduce the epidemic overshoot. When the protection is pushed past a critical threshold, the epidemic dynamics enter an underdamped regime where the epidemic size exactly equals the herd immunity threshold and overshoot is eliminated. Finally, we can find regimes where epidemic size does not monotonically decrease with a population that becomes increasingly risk-averse.

Preserving friendships in school contacts: An algorithm to construct synthetic temporal networks for epidemic modelling

High-resolution temporal data on contacts between hosts provide crucial information on the mixing patterns underlying infectious disease transmission. Publicly available data sets of contact data are however typically recorded over short time windows with respect to the duration of an epidemic. To inform models of disease transmission, data are thus often repeated several times, yielding synthetic data covering long enough timescales. Looping over short term data to approximate contact patterns on longer timescales can lead to unrealistic transmission chains because of the deterministic repetition of all contacts, without any renewal of the contact partners of each individual between successive periods. Real contacts indeed include a combination of regularly repeated contacts (e.g., due to friendship relations) and of more casual ones. In this paper, we propose an algorithm to longitudinally extend contact data recorded in a school setting, taking into account this dual aspect of contacts and in particular the presence of repeated contacts due to friendships. To illustrate the interest of such an algorithm, we then simulate the spread of SARS-CoV-2 on our synthetic contacts using an agent-based model specific to the school setting. We compare the results with simulations performed on synthetic data extended with simpler algorithms to determine the impact of preserving friendships in the data extension method. Notably, the preservation of friendships does not strongly affect transmission routes between classes in the school but leads to different infection pathways between individual students. Our results moreover indicate that gathering contact data during two days in a population is sufficient to generate realistic synthetic contact sequences between individuals in that population on longer timescales. The proposed tool will allow modellers to leverage existing contact data, and contributes to the design of optimal future field data collection.

Effects of clustering heterogeneity on the spectral density of sparse networks

We derive exact equations for the spectral density of sparse networks with an arbitrary distribution of the number of single edges and triangles per node. These equations enable a systematic investigation of the effects of clustering on the spectral properties of the network adjacency matrix. In the case of heterogeneous networks, we demonstrate that the spectral density becomes more symmetric as the fluctuations in the triangle-degree sequence increase. This phenomenon is explained by the small clustering coefficient of networks with a large variance of the triangle-degree distribution. In the homogeneous case of regular clustered networks, we find that both perturbative and nonperturbative approximations fail to predict the spectral density in the high-connectivity limit. This suggests that traditional large-degree approximations may be ineffective in studying the spectral properties of networks with more complex motifs. Our theoretical results are fully confirmed by numerical diagonalizations of finite adjacency matrices.

Intervention effect of targeted workplace closures may be approximated by single-layered networks in an individual-based model of COVID-19 control

Individual-based models of infectious disease dynamics commonly use network structures to represent human interactions. Network structures can vary in complexity, from single-layered with homogeneous mixing to multi-layered with clustering and layer-specific contact weights. Here we assessed policy-relevant consequences of network choice by simulating different network structures within an established individual-based model of SARS-CoV-2 dynamics. We determined the clustering coefficient of each network structure and compared this to several epidemiological outcomes, such as cumulative and peak infections. High-clustered networks estimate fewer cumulative infections and peak infections than less-clustered networks when transmission probabilities are equal. However, by altering transmission probabilities, we find that high-clustered networks can essentially recover the dynamics of low-clustered networks. We further assessed the effect of workplace closures as a layer-targeted intervention on epidemiological outcomes and found in this scenario a single-layered network provides a sufficient approximation of intervention effect relative to a multi-layered network when layer-specific contact weightings are equal. Overall, network structure choice within models should consider the knowledge of contact weights in different environments and pathogen mode of transmission to avoid over- or under-estimating disease burden and impact of interventions.

Population Behavior Changes Underlying Phasic Shifts of SARS-CoV-2 Exposure Settings Across 3 Omicron Epidemic Waves in Hong Kong: Prospective Cohort Study

BACKGROUND

Exposure risk was shown to have affected individual susceptibility and the epidemic spread of COVID-19. The dynamics of risk by and across exposure settings alongside the variations following the implementation of social distancing interventions are understudied.

OBJECTIVE

This study aims to examine the population's trajectory of exposure risk in different settings and its association with SARS-CoV-2 infection across 3 consecutive Omicron epidemic waves in Hong Kong.

METHODS

From March to June 2022, invitation letters were posted to 41,132 randomly selected residential addresses for the recruitment of households into a prospective population cohort. Through web-based monthly surveys coupled with email reminders, a representative from each enrolled household self-reported incidents of SARS-CoV-2 infections, COVID-19 vaccination uptake, their activity pattern in the workplace, and daily and social settings in the preceding month. As a proxy of their exposure risk, the reported activity trend in each setting was differentiated into trajectories based on latent class growth analyses. The associations of different trajectories of SARS-CoV-2 infection overall and by Omicron wave (wave 1: February-April; wave 2: May-September; wave 3: October-December) in 2022 were evaluated by using Cox proportional hazards models and Kaplan-Meier analysis.

RESULTS

In total, 33,501 monthly responses in the observation period of February-December 2022 were collected from 5321 individuals, with 41.7% (2221/5321) being male and a median age of 46 (IQR 34-57) years. Against an expanding COVID-19 vaccination coverage from 81.9% to 95.9% for 2 doses and 20% to 77.7% for 3 doses, the cumulative incidence of SARS-CoV-2 infection escalated from <0.2% to 25.3%, 32.4%, and 43.8% by the end of waves 1, 2, and 3, respectively. Throughout February-December 2022, 52.2% (647/1240) of participants had worked regularly on-site, 28.7% (356/1240) worked remotely, and 19.1% (237/1240) showed an assorted pattern. For daily and social settings, 4 and 5 trajectories were identified, respectively, with 11.5% (142/1240) and 14.6% (181/1240) of the participants gauged to have a high exposure risk. Compared to remote working, working regularly on-site (adjusted hazard ratio [aHR] 1.47, 95% CI 1.19-1.80) and living in a larger household (aHR 1.12, 95% CI 1.06-1.18) were associated with a higher risk of SARS-CoV-2 infection in wave 1. Those from the highest daily exposure risk trajectory (aHR 1.46, 95% CI 1.07-2.00) and the second highest social exposure risk trajectory (aHR 1.52, 95% CI 1.18-1.97) were also at an increased risk of infection in waves 2 and 3, respectively, relative to the lowest risk trajectory.

CONCLUSIONS

In an infection-naive population, SARS-CoV-2 transmission was predominantly initiated at the workplace, accelerated in the household, and perpetuated in the daily and social environments, as stringent restrictions were scaled down. These patterns highlight the phasic shift of exposure settings, which is important for informing the effective calibration of targeted social distancing measures as an alternative to lockdown.

SIR+ models: accounting for interaction-dependent disease susceptibility in the planning of public health interventions

Avoiding physical contact is regarded as one of the safest and most advisable strategies to follow to reduce pathogen spread. The flip side of this approach is that a lack of social interactions may negatively affect other dimensions of health, like induction of immunosuppressive anxiety and depression or preventing interactions of importance with a diversity of microbes, which may be necessary to train our immune system or to maintain its normal levels of activity. These may in turn negatively affect a population's susceptibility to infection and the incidence of severe disease. We suggest that future pandemic modelling may benefit from relying on 'SIR+ models': epidemiological models extended to account for the benefits of social interactions that affect immune resilience. We develop an SIR+ model and discuss which specific interventions may be more effective in balancing the trade-off between minimizing pathogen spread and maximizing other interaction-dependent health benefits. Our SIR+ model reflects the idea that health is not just the mere absence of disease, but rather a state of physical, mental and social well-being that can also be dependent on the same social connections that allow pathogen spread, and the modelling of public health interventions for future pandemics should account for this multidimensionality.

Evaluating Seasonal Variations in Human Contact Patterns and Their Impact on the Transmission of Respiratory Infectious Diseases

BACKGROUND

Human contact patterns are a key determinant driving the spread of respiratory infectious diseases. However, the relationship between contact patterns and seasonality as well as their possible association with the seasonality of respiratory diseases is yet to be clarified.

METHODS

We investigated the association between temperature and human contact patterns using data collected through a cross-sectional diary-based contact survey in Shanghai, China, between December 24, 2017, and May 30, 2018. We then developed a compartmental model of influenza transmission informed by the derived seasonal trends in the number of contacts and validated it against A(H1N1)pdm09 influenza data collected in Shanghai during the same period.

RESULTS

We identified a significant inverse relationship between the number of contacts and the seasonal temperature trend defined as a spline interpolation of temperature data (p = 0.003). We estimated an average of 16.4 (95% PrI: 15.1-17.5) contacts per day in December 2017 that increased to an average of 17.6 contacts (95% PrI: 16.5-19.3) in January 2018 and then declined to an average of 10.3 (95% PrI: 9.4-10.8) in May 2018. Estimates of influenza incidence obtained by the compartmental model comply with the observed epidemiological data. The reproduction number was estimated to increase from 1.24 (95% CI: 1.21-1.27) in December to a peak of 1.34 (95% CI: 1.31-1.37) in January. The estimated median infection attack rate at the end of the season was 27.4% (95% CI: 23.7-30.5%).

CONCLUSIONS

Our findings support a relationship between temperature and contact patterns, which can contribute to deepen the understanding of the relationship between social interactions and the epidemiology of respiratory infectious diseases.

Assessing the impact of different contact patterns on disease transmission: Taking COVID-19 as a case

Human-to-human contact plays a leading role in the transmission of infectious diseases, and the contact pattern between individuals has an important influence on the intensity and trend of disease transmission. In this paper, we define regular contacts and random contacts. Then, taking the COVID-19 outbreak in Yangzhou City, China as an example, we consider age heterogeneity, household structure and two contact patterns to establish discrete dynamic models with switching between daytime and nighttime to depict the transmission mechanism of COVID-19 in population. We studied the changes in the reproduction number with different age groups and household sizes at different stages. The effects of the proportion of two contacts patterns on reproduction number were also studied. Furthermore, taking the final size, the peak value of infected individuals in community and the peak value of quarantine infected individuals and nucleic acid test positive individuals as indicators, we evaluate the impact of the number of random contacts, the duration of the free transmission stage and summer vacation on the spread of the disease. The results show that a series of prevention and control measures taken by the Chinese government in response to the epidemic situation are reasonable and effective, and the young and middle-aged adults (aged 18-59) with household size of 6 have the strongest transmission ability. In addition, the results also indicate that increasing the proportion of random contact is beneficial to the control of the infectious disease in the phase with interventions. This work enriches the content of infectious disease modeling and provides theoretical guidance for the prevention and control of follow-up major infectious diseases.

Modelling the impact of human behavior using a two-layer Watts-Strogatz network for transmission and control of Mpox

PURPOSE

This study aims to evaluate the effectiveness of mitigation strategies and analyze the impact of human behavior on the transmission of Mpox. The results can provide guidance to public health authorities on comprehensive prevention and control for the new Mpox virus strain in the Democratic Republic of Congo as of December 2023.

METHODS

We develop a two-layer Watts-Strogatz network model. The basic reproduction number is calculated using the next-generation matrix approach. Markov chain Monte Carlo (MCMC) optimization algorithm is used to fit Mpox cases in Canada into the network model. Numerical simulations are used to assess the impact of mitigation strategies and human behavior on the final epidemic size.

RESULTS

Our results show that the contact transmission rate of low-risk groups and susceptible humans increases when the contact transmission rate of high-risk groups and susceptible humans is controlled as the Mpox epidemic spreads. The contact transmission rate of high-risk groups after May 18, 2022, is approximately 20% lower than that before May 18, 2022. Our findings indicate a positive correlation between the basic reproduction number and the level of heterogeneity in human contacts, with the basic reproduction number estimated at 2.3475 (95% CI: 0.0749-6.9084). Reducing the average number of sexual contacts to two per week effectively reduces the reproduction number to below one.

CONCLUSION

We need to pay attention to the re-emergence of the epidemics caused by low-risk groups when an outbreak dominated by high-risk groups is under control. Numerical simulations show that reducing the average number of sexual contacts to two per week is effective in slowing down the rapid spread of the epidemic. Our findings offer guidance for the public health authorities of the Democratic Republic of Congo in developing effective mitigation strategies.

Opinion disparity in hypergraphs with community structure

The division of a social group into subgroups with opposing opinions, which we refer to as opinion disparity, is a prevalent phenomenon in society. This phenomenon has been modeled by including mechanisms such as opinion homophily, bounded confidence interactions, and social reinforcement mechanisms. In this paper, we study a complementary mechanism for the formation of opinion disparity based on higher-order interactions, i.e., simultaneous interactions between multiple agents. We present an extension of the planted partition model for uniform hypergraphs as a simple model of community structure, and we consider the hypergraph Susceptible-Infected-Susceptible (SIS) model on a hypergraph with two communities where the binary ideology can spread via links (pairwise interactions) and triangles (three-way interactions). We approximate this contagion process with a mean-field model and find that for strong enough community structure, the two communities can hold very different average opinions. We determine the regimes of structural and infectious parameters for which this opinion disparity can exist, and we find that the existence of these disparities is much more sensitive to the triangle community structure than to the link community structure. We show that the existence and type of opinion disparities are extremely sensitive to differences in the sizes of the two communities.

The use of networks in spatial and temporal computational models for outbreak spread in epidemiology: A systematic review

OBJECTIVES

To examine recent literature in order to present a comprehensive overview of the current trends as regards the computational models used to represent the propagation of an infectious outbreak in a population, paying particular attention to those that represent network-based transmission.

METHODS

a systematic review was conducted following the PRISMA (Preferred Reporting Items for Systematic Reviews and Meta-Analyses) guidelines. Papers published in English between 2010 and September 2021 were sought in the ACM Digital Library, IEEE Xplore, PubMed and Scopus databases.

RESULTS

Upon considering their titles and abstracts, 832 papers were obtained, of which 192 were selected for a full content-body check. Of these, 112 studies were eventually deemed suitable for quantitative and qualitative analysis. Emphasis was placed on the spatial and temporal scales studied, the use of networks or graphs, and the granularity of the data used to evaluate the models. The models principally used to represent the spreading of outbreaks have been stochastic (55.36%), while the type of networks most frequently used are relationship networks (32.14%). The most common spatial dimension used is a region (19.64%) and the most used unit of time is a day (28.57%). Synthetic data as opposed to an external source were used in 51.79% of the papers. With regard to the granularity of the data sources, aggregated data such as censuses or transportation surveys are the most common.

CONCLUSION

We identified a growing interest in the use of networks to represent disease transmission. We detected that research is focused on only certain combinations of the computational model, type of network (in both the expressive and the structural sense) and spatial scale, while the search for other interesting combinations has been left for the future.

When might host heterogeneity drive the evolution of asymptomatic, pandemic coronaviruses?

Controlling many infectious diseases, including SARS-Coronavirus-2 (SARS-CoV-2), requires surveillance followed by isolation, contact-tracing and quarantining. These interventions often begin by identifying symptomatic individuals. However, actively removing pathogen strains causing symptomatic infections may inadvertently select for strains less likely to cause symptomatic infections. Moreover, a pathogen's fitness landscape is structured around a heterogeneous host pool; uneven surveillance efforts and distinct transmission risks across host classes can meaningfully alter selection pressures. Here, we explore this interplay between evolution caused by disease control efforts and the evolutionary consequences of host heterogeneity. Using an evolutionary epidemiology model parameterized for coronaviruses, we show that intense symptoms-driven disease control selects for asymptomatic strains, particularly when these efforts are applied unevenly across host groups. Under these conditions, increasing quarantine efforts have diverging effects. If isolation alone cannot eradicate, intensive quarantine efforts combined with uneven detections of asymptomatic infections (e.g., via neglect of some host classes) can favor the evolution of asymptomatic strains. We further show how, when intervention intensity depends on the prevalence of symptomatic infections, higher removal efforts (and isolating symptomatic cases in particular) more readily select for asymptomatic strains than when these efforts do not depend on prevalence. The selection pressures on pathogens caused by isolation and quarantining likely lie between the extremes of no intervention and thoroughly successful eradication. Thus, analyzing how different public health responses can select for asymptomatic pathogen strains is critical for identifying disease suppression efforts that can effectively manage emerging infectious diseases.

Supplementary Information

The online version contains supplementary material available at 10.1007/s11071-022-07548-7.

Network topological determinants of pathogen spread

How do we best constrain social interactions to decrease transmission of communicable diseases? Indiscriminate suppression is unsustainable long term and presupposes that all interactions carry equal importance. Instead, transmission within a social network has been shown to be determined by its topology. In this paper, we deploy simulations to understand and quantify the impact on disease transmission of a set of topological network features, building a dataset of 9000 interaction graphs using generators of different types of synthetic social networks. Independently of the topology of the network, we maintain constant the total volume of social interactions in our simulations, to show how even with the same social contact some network structures are more or less resilient to the spread. We find a suitable intervention to be specific suppression of unfamiliar and casual interactions that contribute to the network's global efficiency. This is, pathogen spread is significantly reduced by limiting specific kinds of contact rather than their global number. Our numerical studies might inspire further investigation in connection to public health, as an integrative framework to craft and evaluate social interventions in communicable diseases with different social graphs or as a highlight of network metrics that should be captured in social studies.

Individual risk perception and empirical social structures shape the dynamics of infectious disease outbreaks

The dynamics of a spreading disease and individual behavioral changes are entangled processes that have to be addressed together in order to effectively manage an outbreak. Here, we relate individual risk perception to the adoption of a specific set of control measures, as obtained from an extensive large-scale survey performed via Facebook-involving more than 500,000 respondents from 64 countries-showing that there is a "one-to-one" relationship between perceived epidemic risk and compliance with a set of mitigation rules. We then develop a mathematical model for the spreading of a disease-sharing epidemiological features with COVID-19-that explicitly takes into account non-compliant individual behaviors and evaluates the impact of a population fraction of infectious risk-deniers on the epidemic dynamics. Our modeling study grounds on a wide set of structures, including both synthetic and more than 180 real-world contact patterns, to evaluate, in realistic scenarios, how network features typical of human interaction patterns impact the spread of a disease. In both synthetic and real contact patterns we find that epidemic spreading is hindered for decreasing population fractions of risk-denier individuals. From empirical contact patterns we demonstrate that connectivity heterogeneity and group structure significantly affect the peak of hospitalized population: higher modularity and heterogeneity of social contacts are linked to lower peaks at a fixed fraction of risk-denier individuals while, at the same time, such features increase the relative impact on hospitalizations with respect to the case where everyone correctly perceive the risks.

The Impact of Contact Structure and Mixing on Control Measures and Disease-Induced Herd Immunity in Epidemic Models: A Mean-Field Model Perspective

The contact structure of a population plays an important role in transmission of infection. Many 'structured models' capture aspects of the contact pattern through an underlying network or a mixing matrix. An important observation in unstructured models of a disease that confers immunity is that once a fraction [Formula: see text] has been infected, the residual susceptible population can no longer sustain an epidemic. A recent observation of some structured models is that this threshold can be crossed with a smaller fraction of infected individuals, because the disease acts like a targeted vaccine, preferentially immunising higher-risk individuals who play a greater role in transmission. Therefore, a limited 'first wave' may leave behind a residual population that cannot support a second wave once interventions are lifted. In this paper, we set out to investigate this more systematically. While networks offer a flexible framework to model contact patterns explicitly, they suffer from several shortcomings: (i) high-fidelity network models require a large amount of data which can be difficult to harvest, and (ii) very few, if any, theoretical contact network models offer the flexibility to tune different contact network properties within the same framework. Therefore, we opt to systematically analyse a number of well-known mean-field models. These are computationally efficient and provide good flexibility in varying contact network properties such as heterogeneity in the number contacts, clustering and household structure or differentiating between local and global contacts. In particular, we consider the question of herd immunity under several scenarios. When modelling interventions as changes in transmission rates, we confirm that in networks with significant degree heterogeneity, the first wave of the epidemic confers herd immunity with significantly fewer infections than equivalent models with less or no degree heterogeneity. However, if modelling the intervention as a change in the contact network, then this effect may become much more subtle. Indeed, modifying the structure disproportionately can shield highly connected nodes from becoming infected during the first wave and therefore make the second wave more substantial. We strengthen this finding by using an age-structured compartmental model parameterised with real data and comparing lockdown periods implemented either as a global scaling of the mixing matrix or age-specific structural changes. Overall, we find that results regarding (disease-induced) herd immunity levels are strongly dependent on the model, the duration of the lockdown and how the lockdown is implemented in the model.

Concurrency measures in the era of temporal network epidemiology: a review

Diseases spread over temporal networks of interaction events between individuals. Structures of these temporal networks hold the keys to understanding epidemic propagation. One early concept of the literature to aid in discussing these structures is concurrency-quantifying individuals' tendency to form time-overlapping 'partnerships'. Although conflicting evaluations and an overabundance of operational definitions have marred the history of concurrency, it remains important, especially in the area of sexually transmitted infections. Today, much of theoretical epidemiology uses more direct models of contact patterns, and there is an emerging body of literature trying to connect methods to the concurrency literature. In this review, we will cover the development of the concept of concurrency and these new approaches.

Disease dynamics and mean field models for clustered networks

Social networks are clustered networks with short mean path length. In this work we analyze the disease dynamics in a class of this type of small-world networks composed of set of households and a set of workplaces. Individuals from each household are randomly assigned to workplaces. In both environments we assumed complete mixing and therefore we obtain highly clustered networks with short mean path lengths. Basic reproduction numbers were computed numerically and we show that at endemic equilibrium the average susceptible proportion <S/N> is different from the inverse of the basic reproduction number (R₀^-1). Therefore exist an exponent p≠1 for which <S/N>^p=R₀^-1. Using this exponent we developed a mean field model which closely capture the disease dynamics in the network. Finally we outline how this model could be use to model vector-borne diseases in social networks.

Effectiveness of isolation policies in schools: evidence from a mathematical model of influenza and COVID-19

Background

Non-pharmaceutical interventions such as social distancing, school closures and travel restrictions are often implemented to control outbreaks of infectious diseases. For influenza in schools, the Center of Disease Control (CDC) recommends that febrile students remain isolated at home until they have been fever-free for at least one day and a related policy is recommended for SARS-CoV-2 (COVID-19). Other authors proposed using a school week of four or fewer days of in-person instruction for all students to reduce transmission. However, there is limited evidence supporting the effectiveness of these interventions.

Methods

We introduced a mathematical model of school outbreaks that considers both intervention methods. Our model accounts for the school structure and schedule, as well as the time-progression of fever symptoms and viral shedding. The model was validated on outbreaks of seasonal and pandemic influenza and COVID-19 in schools. It was then used to estimate the outbreak curves and the proportion of the population infected (attack rate) under the proposed interventions.

Results

For influenza, the CDC-recommended one day of post-fever isolation can reduce the attack rate by a median (interquartile range) of 29 (13–59)%. With 2 days of post-fever isolation the attack rate could be reduced by 70 (55–85)%. Alternatively, shortening the school week to 4 and 3 days reduces the attack rate by 73 (64–88)% and 93 (91–97)%, respectively. For COVID-19, application of post-fever isolation policy was found to be less effective and reduced the attack rate by 10 (5–17)% for a 2-day isolation policy and by 14 (5–26)% for 14 days. A 4-day school week would reduce the median attack rate in a COVID-19 outbreak by 57 (52–64)%, while a 3-day school week would reduce it by 81 (79–83)%. In both infections, shortening the school week significantly reduced the duration of outbreaks.

Conclusions

Shortening the school week could be an important tool for controlling influenza and COVID-19 in schools and similar settings. Additionally, the CDC-recommended post-fever isolation policy for influenza could be enhanced by requiring two days of isolation instead of one.

COVID-19 Superspreading Suggests Mitigation by Social Network Modulation

Although COVID-19 has caused severe suffering globally, the efficacy of nonpharmaceutical interventions has been greater than typical models have predicted. Meanwhile, evidence is mounting that the pandemic is characterized by superspreading. Capturing this phenomenon theoretically requires modeling at the scale of individuals. Using a mathematical model, we show that superspreading drastically enhances mitigations which reduce the overall personal contact number and that social clustering increases this effect.

Disease-driven reduction in human mobility influences human-mosquito contacts and dengue transmission dynamics

Heterogeneous exposure to mosquitoes determines an individual’s contribution to vector-borne pathogen transmission. Particularly for dengue virus (DENV), there is a major difficulty in quantifying human-vector contacts due to the unknown coupled effect of key heterogeneities. To test the hypothesis that the reduction of human out-of-home mobility due to dengue illness will significantly influence population-level dynamics and the structure of DENV transmission chains, we extended an existing modeling framework to include social structure, disease-driven mobility reductions, and heterogeneous transmissibility from different infectious groups. Compared to a baseline model, naïve to human pre-symptomatic infectiousness and disease-driven mobility changes, a model including both parameters predicted an increase of 37% in the probability of a DENV outbreak occurring; a model including mobility change alone predicted a 15.5% increase compared to the baseline model. At the individual level, models including mobility change led to a reduction of the importance of out-of-home onward transmission (R, the fraction of secondary cases predicted to be generated by an individual) by symptomatic individuals (up to -62%) at the expense of an increase in the relevance of their home (up to +40%). An individual’s positive contribution to R could be predicted by a GAM including a non-linear interaction between an individual’s biting suitability and the number of mosquitoes in their home (>10 mosquitoes and 0.6 individual attractiveness significantly increased R). We conclude that the complex fabric of social relationships and differential behavioral response to dengue illness cause the fraction of symptomatic DENV infections to concentrate transmission in specific locations, whereas asymptomatic carriers (including individuals in their pre-symptomatic period) move the virus throughout the landscape. Our findings point to the difficulty of focusing vector control interventions reactively on the home of symptomatic individuals, as this approach will fail to contain virus propagation by visitors to their house and asymptomatic carriers.

Human mobility patterns can play an integral role in vector-borne disease dynamics by characterizing an individual’s potential contacts with disease-transmitting vectors. Dengue virus is transmitted by a sedentary vector, but human mobility allows individuals to have contact with mosquitoes at their home and other houses they frequent (their activity space). When accounting for the decreased mobility of symptomatic dengue cases in an agent-based simulation model, however, we found a severely diminished role of the activity space in onward transmission. Those who received the majority of their mosquito contacts outside their home experienced decreases in expected bites and onward transmission when mobility changes were accounted for. Onward transmission was driven by a synergistic relationship between the number of mosquitoes in an individual’s home and their biting suitability, where even those with the highest biting suitability would have limited contribution to transmission given a low number of household mosquitoes. Reactive vector control, which often targets symptomatic cases, could be effective for slowing onward transmission from these cases, but will fail to control virus transmission due to the disproportionate contribution of asymptomatic infections.

28 Models Later: Model Competition and the Zombie Apocalypse

Between Fall 2011 and Fall 2012 students at Utah State University played several rounds of Humans versus Zombies (HvZ), a role-playing variant of tag popular on college campuses. The goal of the game is for the zombies to tag humans, converting them into more zombies. Based on portrayals of 'zombieism' in popular culture, one might treat HvZ as a disease system. However, a traditional SIR model with mass-action dynamics does a poor job of modeling HvZ, leading to the natural question: What mechanisms drive the dynamics of the HvZ system? We use model competition, with Bayesian Information Criterion as arbiter, to answer this question. First, we develop a suite of models with a variety of transmission mechanisms and fit to data from fall 2011. We use model competition to determine which model(s) have the most support from the data, thereby offering insight into driving mechanisms for HvZ. Bootstrapping is used to both assess the significance of individual mechanisms and to determine confidence in the performance of our models. Finally, we test predictions of the best models with data from fall 2012. Results indicate that through both years of the game humans tend to cluster defensively, zombies tend to hunt in groups, some zombies are more proficient hunters, and some humans leave the game.

Transmission heterogeneities, kinetics, and controllability of SARS-CoV-2

A long-standing question in infectious disease dynamics concerns the role of transmission heterogeneities, which are driven by demography, behavior, and interventions. On the basis of detailed patient and contact-tracing data in Hunan, China, we find that 80% of secondary infections traced back to 15% of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) primary infections, which indicates substantial transmission heterogeneities. Transmission risk scales positively with the duration of exposure and the closeness of social interactions and is modulated by demographic and clinical factors. The lockdown period increases transmission risk in the family and households, whereas isolation and quarantine reduce risks across all types of contacts. The reconstructed infectiousness profile of a typical SARS-CoV-2 patient peaks just before symptom presentation. Modeling indicates that SARS-CoV-2 control requires the synergistic efforts of case isolation, contact quarantine, and population-level interventions because of the specific transmission kinetics of this virus.

The role of social structure and dynamics in the maintenance of endemic disease

Social interactions are required for the direct transmission of infectious diseases. Consequently, the social network structure of populations plays a key role in shaping infectious disease dynamics. A huge research effort has examined how specific social network structures make populations more (or less) vulnerable to damaging epidemics. However, it can be just as important to understand how social networks can contribute to endemic disease dynamics, in which pathogens are maintained at stable levels for prolonged periods of time. Hosts that can maintain endemic disease may serve as keystone hosts for multi-host pathogens within an ecological community, and also have greater potential to act as key wildlife reservoirs of agricultural and zoonotic diseases. Here, we examine combinations of social and demographic processes that can foster endemic disease in hosts. We synthesise theoretical and empirical work to demonstrate the importance of both social structure and social dynamics in maintaining endemic disease. We also highlight the importance of distinguishing between the local and global persistence of infection and reveal how different social processes drive variation in the scale at which infectious diseases appear endemic. Our synthesis provides a framework by which to understand how sociality contributes to the long-term maintenance of infectious disease in wildlife hosts and provides a set of tools to unpick the social and demographic mechanisms involved in any given host-pathogen system.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s00265-021-03055-8.

Accounting for cross-immunity can improve forecast accuracy during influenza epidemics

Previous exposure to influenza viruses confers cross-immunity against future infections with related strains. However, this is not always accounted for explicitly in mathematical models used for forecasting during influenza outbreaks. We show that, if an influenza outbreak is due to a strain that is similar to one that has emerged previously, then accounting for cross-immunity explicitly can improve the accuracy of real-time forecasts. To do this, we consider two infectious disease outbreak forecasting models. In the first (the "1-group model"), all individuals are assumed to be identical and cross-immunity is not accounted for. In the second (the "2-group model"), individuals who have previously been infected by a related strain are assumed to be less likely to experience severe disease, and therefore recover more quickly, than immunologically naive individuals. We fit both models to estimated case notification data (including symptomatic individuals as well as laboratory-confirmed cases) from Japan from the 2009 H1N1 influenza pandemic, and then generate synthetic data for a future outbreak by assuming that the 2-group model represents the epidemiology of influenza infections more accurately. We use the 1-group model (as well as the 2-group model for comparison) to generate forecasts that would be obtained in real-time as the future outbreak is ongoing, using parameter values estimated from the 2009 epidemic as informative priors, motivated by the fact that without using prior information from 2009, the forecasts are highly uncertain. In the scenario that we consider, the 1-group model only produces accurate outbreak forecasts once the peak of the epidemic has passed, even when the values of important epidemiological parameters such as the lengths of the mean incubation and infectious periods are known exactly. As a result, it is necessary to use the more epidemiologically realistic 2-group model to generate accurate forecasts. Accounting for cross-immunity driven by exposures in previous outbreaks explicitly is expected to improve the accuracy of epidemiological modelling forecasts during influenza outbreaks.

Effectiveness of Isolation Policies in Schools: Evidence from a Mathematical Model of Influenza and COVID-19

BACKGROUND

Non-pharmaceutical interventions such as social distancing, school closures and travel restrictions are often implemented to control outbreaks of infectious diseases. For influenza in schools, the Center of Disease Control (CDC) recommends that febrile students remain isolated at home until they have been fever-free for at least one day and a related policy is recommended for SARS-CoV2 (COVID-19). Other authors proposed using a school week of four or fewer days of in-person instruction for all students to reduce transmission. However, there is limited evidence supporting the effectiveness of these interventions.

METHODS

We introduced a mathematical model of school outbreaks that considers both intervention methods. Our model accounts for the school structure and schedule, as well as the time-progression of fever symptoms and viral shedding. The model was validated on outbreaks of seasonal and pandemic influenza and COVID-19 in schools. It was then used to estimate the outbreak curves and the proportion of the population infected (attack rate) under the proposed interventions.

RESULTS

For influenza, the CDC-recommended one day of post-fever isolation can reduce the attack rate by a median (interquartile range) of 29 (13 - 59)%. With two days of post-fever isolation the attack rate could be reduced by 70 (55 - 85)%. Alternatively, shortening the school week to four and three days reduces the attack rate by 73 (64 - 88)% and 93 (91 - 97)%, respectively. For COVID-19, application of post-fever isolation policy was found to be less effective and reduced the attack rate by 10 (5 - 17)% for a two-day isolation policy and by 14 (5 - 26)% for 14 days. A four-day school week would reduce the median attack rate in a COVID-19 outbreak by 57 (52 - 64)%, while a three-day school week would reduce it by 81 (79 - 83)%. In both infections, shortening the school week significantly reduced the duration of outbreaks.

CONCLUSIONS

Shortening the school week could be an important tool for controlling influenza and COVID-19 in schools and similar settings. Additionally, the CDC-recommended post-fever isolation policy for influenza could be enhanced by requiring two days of isolation instead of one.

Transmission heterogeneities, kinetics, and controllability of SARS-CoV-2

A long-standing question in infectious disease dynamics concerns the role of transmission heterogeneities, driven by demography, behavior and interventions. Based on detailed patient and contact tracing data in Hunan, China we find 80% of secondary infections traced back to 15% of SARS-CoV-2 primary infections, indicating substantial transmission heterogeneities. Transmission risk scales positively with the duration of exposure and the closeness of social interactions and is modulated by demographic and clinical factors. The lockdown period increases transmission risk in the family and households, while isolation and quarantine reduce risks across all types of contacts. The reconstructed infectiousness profile of a typical SARS-CoV-2 patient peaks just before symptom presentation. Modeling indicates SARS-CoV-2 control requires the synergistic efforts of case isolation, contact quarantine, and population-level interventions, owing to the specific transmission kinetics of this virus.

One Sentence Summary

Public health measures to control SARS-CoV-2 could be designed to block the specific transmission characteristics of the virus.

A model-based evaluation of the efficacy of COVID-19 social distancing, testing and hospital triage policies

A stochastic compartmental network model of SARS-CoV-2 spread explores the simultaneous effects of policy choices in three domains: social distancing, hospital triaging, and testing. Considering policy domains together provides insight into how different policy decisions interact. The model incorporates important characteristics of COVID-19, the disease caused by SARS-CoV-2, such as heterogeneous risk factors and asymptomatic transmission, and enables a reliable qualitative comparison of policy choices despite the current uncertainty in key virus and disease parameters. Results suggest possible refinements to current policies, including emphasizing the need to reduce random encounters more than personal contacts, and testing low-risk symptomatic individuals before high-risk symptomatic individuals. The strength of social distancing of symptomatic individuals affects the degree to which asymptomatic cases drive the epidemic as well as the level of population-wide contact reduction needed to keep hospitals below capacity. The relative importance of testing and triaging also depends on the overall level of social distancing.

Impact of Social Distancing Measures on Coronavirus Disease Healthcare Demand, Central Texas, USA

Social distancing orders have been enacted worldwide to slow the coronavirus disease (COVID-19) pandemic, reduce strain on healthcare systems, and prevent deaths. To estimate the impact of the timing and intensity of such measures, we built a mathematical model of COVID-19 transmission that incorporates age-stratified risks and contact patterns and projects numbers of hospitalizations, patients in intensive care units, ventilator needs, and deaths within US cities. Focusing on the Austin metropolitan area of Texas, we found that immediate and extensive social distancing measures were required to ensure that COVID-19 cases did not exceed local hospital capacity by early May 2020. School closures alone hardly changed the epidemic curve. A 2-week delay in implementation was projected to accelerate the timing of peak healthcare needs by 4 weeks and cause a bed shortage in intensive care units. This analysis informed the Stay Home-Work Safe order enacted by Austin on March 24, 2020.

Contact Mixing Patterns and Population Movement among Migrant Workers in an Urban Setting in Thailand

Data relating to contact mixing patterns among humans are essential for the accurate modeling of infectious disease transmission dynamics. Here, we describe contact mixing patterns among migrant workers in urban settings in Thailand, based on a survey of 369 migrant workers of three nationalities. Respondents recorded their demographic data, including age, sex, nationality, workplace, income, and education. Each respondent chose a single day to record their contacts; this resulted in a total of more than 8300 contacts. The characteristics of contacts were recorded, including their age, sex, nationality, location of contact, and occurrence of physical contact. More than 75% of all contacts occurred among migrants aged 15 to 39 years. The contacts were highly clustered in this age group among migrant workers of all three nationalities. There were far fewer contacts between migrant workers with younger and older age groups. The pattern varied slightly among different nationalities, which was mostly dependent upon the types of jobs taken. Half of migrant workers always returned to their home country at most once a year and on a seasonal basis. The present study has helped us gain a better understanding of contact mixing patterns among migrant workers in urban settings. This information is useful both when simulating disease epidemics and for guiding optimal disease control strategies among this vulnerable section of the population.

Conscientious vaccination exemptions in kindergarten to eighth-grade children across Texas schools from 2012 to 2018: A regression analysis

BACKGROUND

As conscientious vaccination exemption (CVE) percentages rise across the United States, so does the risk and occurrence of outbreaks of vaccine-preventable diseases such as measles. In the state of Texas, the median CVE percentage across school systems more than doubled between 2012 and 2018. During this period, the proportion of schools surpassing a CVE percentage of 3% rose from 2% to 6% for public schools, 20% to 26% for private schools, and 17% to 22% for charter schools. The aim of this study was to investigate this phenomenon at a fine scale.

METHODS AND FINDINGS

Here, we use beta regression models to study the socioeconomic and geographic drivers of CVE trends in Texas. Using annual counts of CVEs at the school system level from the 2012-2013 to the 2017-2018 school year, we identified county-level predictors of median CVE percentage among public, private, and charter schools, the proportion of schools below a high-risk threshold for vaccination coverage, and five-year trends in CVEs. Since the 2012-2013 school year, CVE percentages have increased in 41 out of 46 counties in the top 10 metropolitan areas of Texas. We find that 77.6% of the variation in CVE percentages across metropolitan counties is explained by median income, the proportion of the population that holds a bachelor's degree, the proportion of the population that self-reports as ethnically white, the proportion of the population that is English speaking, and the proportion of the population that is under the age of five years old. Across the 10 top metropolitan areas in Texas, counties vary considerably in the proportion of school systems reporting CVE percentages above 3%. Sixty-six percent of that variation is explained by the proportion of the population that holds a bachelor's degree and the proportion of the population affiliated with a religious congregation. Three of the largest metropolitan areas-Austin, Dallas-Fort Worth, and Houston-are potential vaccination exemption "hotspots," with over 13% of local school systems above this risk threshold. The major limitations of this study are inconsistent school-system-level CVE reporting during the study period and a lack of geographic and socioeconomic data for individual private schools.

CONCLUSIONS

In this study, we have identified high-risk communities that are typically obscured in county-level risk assessments and found that public schools, like private schools, are exhibiting predictable increases in vaccination exemption percentages. As public health agencies confront the reemerging threat of measles and other vaccine-preventable diseases, findings such as ours can guide targeted interventions and surveillance within schools, cities, counties, and sociodemographic subgroups.

Suppressing epidemic spreading by optimizing the allocation of resources between prevention and treatment

The rational allocation of resources is crucial to suppress the outbreak of epidemics. Here, we propose an epidemic spreading model in which resources are used simultaneously to prevent and treat disease. Based on the model, we study the impacts of different resource allocation strategies on epidemic spreading. First, we analytically obtain the epidemic threshold of disease using the recurrent dynamical message passing method. Then, we simulate the spreading of epidemics on the Erdős-Rényi (ER) network and the scale-free network and investigate the infection density of disease as a function of the disease infection rate. We find hysteresis loops in the phase transition of the infection density on both types of networks. Intriguingly, when different resource allocation schemes are adopted, the phase transition on the ER network is always a first-order phase transition, while the phase transition on the scale-free network transforms from a hybrid phase transition to a first-order phase transition. Particularly, through extensive numerical simulations, we find that there is an optimal resource allocation scheme, which can best suppress epidemic spreading. In addition, we find that the degree heterogeneity of the network promotes the spreading of disease. Finally, by comparing theoretical and numerical results on a real-world network, we find that our method can accurately predict the spreading of disease on the real-world network.

Dynamics of beneficial epidemics

Pathogens can spread epidemically through populations. Beneficial contagions, such as viruses that enhance host survival or technological innovations that improve quality of life, also have the potential to spread epidemically. How do the dynamics of beneficial biological and social epidemics differ from those of detrimental epidemics? We investigate this question using a breadth-first modeling approach involving three distinct theoretical models. First, in the context of population genetics, we show that a horizontally-transmissible element that increases fitness, such as viral DNA, spreads superexponentially through a population, more quickly than a beneficial mutation. Second, in the context of behavioral epidemiology, we show that infections that cause increased connectivity lead to superexponential fixation in the population. Third, in the context of dynamic social networks, we find that preferences for increased global infection accelerate spread and produce superexponential fixation, but preferences for local assortativity halt epidemics by disconnecting the infected from the susceptible. We conclude that the dynamics of beneficial biological and social epidemics are characterized by the rapid spread of beneficial elements, which is facilitated in biological systems by horizontal transmission and in social systems by active spreading behavior of infected individuals.

Contrasting the value of targeted versus area-wide mosquito control scenarios to limit arbovirus transmission with human mobility patterns based on different tropical urban population centers

Vector control is still our primary intervention for both prevention and mitigation of epidemics of many vector-borne diseases. Efficiently targeting control measures is important since control can involve substantial economic costs. Targeting is not always straightforward, as transmission of vector-borne diseases is affected by various types of host movement. Here we assess how taking daily commuting patterns into consideration can help improve vector control efforts. We examine three tropical urban centers (San Juan, Recife, and Jakarta) that have recently been exposed to Zika and/or dengue infections and consider whether the distribution of human populations and resulting commuting flows affects the optimal scale at which control interventions should be implemented. We developed a stochastic, spatial model and investigated four control scenarios. The scenarios differed in the spatial extent of their implementation and were: 1) a response at the level of an individual neighborhood; 2) a response targeted at a neighborhood in which infected humans were detected and the one with which it was most strongly connected by human movement; 3) a limited area-wide response where all neighborhoods within a certain radius of the focal area were included; and 4) a collective response where all participating neighborhoods implemented control. The relative effectiveness of the scenarios varied only slightly between different settings, with the number of infections averted over time increasing with the scale of implementation. This difference depended on the efficacy of control at the neighborhood level. At low levels of efficacy, the scenarios mirrored each other in infections averted. At high levels of efficacy, impact increased with the scale of the intervention. As a result, the choice between scenarios will not only be a function of the amount of effort decision-makers are willing to invest, but largely epend on the overall effectiveness of vector control approaches.

Control and prevention of Aedes-transmitted viruses, such as dengue, chikungunya, or Zika relies heavily on vector control approaches. Given the effort and cost involved in implementation of vector control, targeting of control measures is highly desirable. However, it is unclear to what extent the effectiveness of highly focal and reactive control measures depends on the commuting and movement patterns of humans. To investigate this question, we developed a model and four control scenarios that ranged from highly focal to area-wide larval control. The distribution of humans and their commuting patterns were modelled after three major tropical urban centers, San Juan, Recife, and Jakarta. We show that as implementation is applied across a wider area, a greater number of infections is averted. Critically, this only occurs if the efficacy of control at the neighborhood level is sufficiently high. A consistent outcome across the three settings was that the focal strategy was most likely to provide the best outcome at lower levels of effort, and when the efficacy of control was low. These outcomes suggest that optimal control strategies will likely have to be tailored to individual settings by decision makers and would benefit from localized cost-effectiveness modelling studies.

Epidemic threshold in pairwise models for clustered networks: closures and fast correlations

The epidemic threshold is probably the most studied quantity in the modelling of epidemics on networks. For a large class of networks and dynamics, it is well studied and understood. However, it is less so for clustered networks where theoretical results are mostly limited to idealised networks. In this paper we focus on a class of models known as pairwise models where, to our knowledge, no analytical result for the epidemic threshold exists. We show that by exploiting the presence of fast variables and using some standard techniques from perturbation theory we are able to obtain the epidemic threshold analytically. We validate this new threshold by comparing it to the threshold based on the numerical solution of the full system. The agreement is found to be excellent over a wide range of values of the clustering coefficient, transmission rate and average degree of the network. Interestingly, we find that the analytical form of the threshold depends on the choice of closure, highlighting the importance of model selection when dealing with real-world epidemics. Nevertheless, we expect that our method will extend to other systems in which fast variables are present.

SIR epidemics and vaccination on random graphs with clustering

In this paper we consider Susceptible \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rightarrow $$\end{document}→ Infectious \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rightarrow $$\end{document}→ Recovered (SIR) epidemics on random graphs with clustering. To incorporate group structure of the underlying social network, we use a generalized version of the configuration model in which each node is a member of a specified number of triangles. SIR epidemics on this type of graph have earlier been investigated under the assumption of homogeneous infectivity and also under the assumption of Poisson transmission and recovery rates. We extend known results from literature by relaxing the assumption of homogeneous infectivity both in individual infectivity and between different kinds of neighbours. An important special case of the epidemic model analysed in this paper is epidemics in continuous time with arbitrary infectious period distribution. We use branching process approximations of the spread of the disease to provide expressions for the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document}R0, the probability of a major outbreak and the expected final size. In addition, the impact of random vaccination with a perfect vaccine on the final outcome of the epidemic is investigated. We find that, for this particular model, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document}R0 equals the perfect vaccine-associated reproduction number. Generalizations to groups larger than three are discussed briefly.

Edge-Based Compartmental Modelling of an SIR Epidemic on a Dual-Layer Static-Dynamic Multiplex Network with Tunable Clustering

The duration, type and structure of connections between individuals in real-world populations play a crucial role in how diseases invade and spread. Here, we incorporate the aforementioned heterogeneities into a model by considering a dual-layer static–dynamic multiplex network. The static network layer affords tunable clustering and describes an individual’s permanent community structure. The dynamic network layer describes the transient connections an individual makes with members of the wider population by imposing constant edge rewiring. We follow the edge-based compartmental modelling approach to derive equations describing the evolution of a susceptible–infected–recovered epidemic spreading through this multiplex network of individuals. We derive the basic reproduction number, measuring the expected number of new infectious cases caused by a single infectious individual in an otherwise susceptible population. We validate model equations by showing convergence to pre-existing edge-based compartmental model equations in limiting cases and by comparison with stochastically simulated epidemics. We explore the effects of altering model parameters and multiplex network attributes on resultant epidemic dynamics. We validate the basic reproduction number by plotting its value against associated final epidemic sizes measured from simulation and predicted by model equations for a number of set-ups. Further, we explore the effect of varying individual model parameters on the basic reproduction number. We conclude with a discussion of the significance and interpretation of the model and its relation to existing research literature. We highlight intrinsic limitations and potential extensions of the present model and outline future research considerations, both experimental and theoretical.

Edge-based epidemic spreading in degree-correlated complex networks

Networks that grow through the addition of new nodes or edges may acquire degree-degree correlations. When one considers a short epidemic on a slowly growing network, such as the spread of a strain of influenza in a population for one season, it is reasonable to assume that the degree-correlated network is static during the course of an epidemic. In this case using only information about the network degree distribution is not enough to capture the exponential growth phase, the epidemic peak or the final epidemic size. Hence, in this paper we formulate an edge-based SIR epidemic model on degree-correlated networks, which includes the Miller model on configuration networks as a special case. The model is relatively low-dimensional; in particular, considering the fact that it captures degree correlations. Moreover, we derive rate equations to compute two node degree correlations in a growing network. Predictions of our model agree well with the corresponding stochastic SIR process on degree-correlated networks, such as the exponential growth phase, the epidemic peak and the final epidemic size. The basic reproduction number R₀ and the final epidemic size are theoretically derived, which are equivalent to those based on the percolation theory. However, our model has the advantage that it can trace the dynamic spread of an epidemic on degree-correlated networks. This provides us with more accurate information to predict and control the spread of diseases in growing populations with biased-mixing. Finally, our model is tested on degree-correlated networks with clustering, and it is shown that our model is robust to degree-correlated networks with small clustering.

Impact of degree truncation on the spread of a contagious process on networks

Understanding how person-to-person contagious processes spread through a population requires accurate information on connections between population members. However, such connectivity data, when collected via interview, is often incomplete due to partial recall, respondent fatigue or study design, e.g., fixed choice designs (FCD) truncate out-degree by limiting the number of contacts each respondent can report. Past research has shown how FCD truncation affects network properties, but its implications for predicted speed and size of spreading processes remain largely unexplored. To study the impact of degree truncation on predictions of spreading process outcomes, we generated collections of synthetic networks containing specific properties (degree distribution, degree-assortativity, clustering), and also used empirical social network data from 75 villages in Karnataka, India. We simulated FCD using various truncation thresholds and ran a susceptible-infectious-recovered (SIR) process on each network. We found that spreading processes propagated on truncated networks resulted in slower and smaller epidemics, with a sudden decrease in prediction accuracy at a level of truncation that varied by network type. Our results have implications beyond FCD to truncation due to any limited sampling from a larger network. We conclude that knowledge of network structure is important for understanding the accuracy of predictions of process spread on degree truncated networks.

Pairwise approximation for SIR-type network epidemics with non-Markovian recovery

We present the generalized mean-field and pairwise models for non-Markovian epidemics on networks with arbitrary recovery time distributions. First we consider a hyperbolic partial differential equation (PDE) system, where the population of infective nodes and links are structured by age since infection. We show that the PDE system can be reduced to a system of integro-differential equations, which is analysed analytically and numerically. We investigate the asymptotic behaviour of the generalized model and provide an implicit analytical expression involving the final epidemic size and pairwise reproduction number. As an illustration of the applicability of the general model, we recover known results for the exponentially distributed and fixed recovery time cases. For gamma- and uniformly distributed infectious periods, new pairwise models are derived. Theoretical findings are confirmed by comparing results from the new pairwise model and explicit stochastic network simulation. A major benefit of the generalized pairwise model lies in approximating the time evolution of the epidemic.

Contact networks structured by sex underpin sex-specific epidemiology of infection

Contact networks are fundamental to the transmission of infection and host sex often affects the acquisition and progression of infection. However, the epidemiological impacts of sex-related variation in animal contact networks have rarely been investigated. We test the hypothesis that sex-biases in infection are related to variation in multilayer contact networks structured by sex in a population of European badgers Meles meles naturally infected with Mycobacterium bovis. Our key results are that male-male and between-sex networks are structured at broader spatial scales than female-female networks and that in male-male and between-sex contact networks, but not female-female networks, there is a significant relationship between infection and contacts with individuals in other groups. These sex differences in social behaviour may underpin male-biased acquisition of infection and may result in males being responsible for more between-group transmission. This highlights the importance of sex-related variation in host behaviour when managing animal diseases.

Multiple peaks patterns of epidemic spreading in multi-layer networks

The study of epidemic spreading on populations of networked individuals has seen recently a great deal of significant progresses. A common point in many of past studies is, however, that there is only one peak of infected density in each single epidemic spreading episode. At variance, real data from different cities over the world suggest that, besides a major single peak trait of infected density, a finite probability exists for a pattern made of two (or multiple) peaks. We show that such a latter feature is distinctive of a multilayered network of interactions, and reveal that a two peaks pattern may emerge from different time delays at which the epidemic spreads in between the two layers. Further, we show that the essential ingredient is a weak coupling condition between the layers themselves, while different degree distributions in the two layers are also helpful. Moreover, an edge-based theory is developed which fully explains all numerical results. Our findings may therefore be of significance for protecting secondary disasters of epidemics, which are definitely undesired in real life.

The epidemic model based on the approximation for third-order motifs on networks

The spread of an infectious disease may depend on the structure of the network. To study the influence of the structure parameters of the network on the spread of the epidemic, we need to put these parameters into the epidemic model. The method of moment closure introduces structure parameters into the epidemic model. In this paper, we present a new moment closure epidemic model based on the approximation of third-order motifs in networks. The order of a motif defined in this paper is determined by the number of the edges in the motif, rather than by the number of nodes in the motif as defined in the literature. We provide a general approach to deriving a set of ordinary differential equations that describes, to a high degree of accuracy, the spread of an infectious disease. Using this method, we establish a susceptible-infected-recovered (SIR) model. We then calculate the basic reproduction number of the SIR model, and find that it decreases as the clustering coefficient increases. Finally, we perform some simulations using the proposed model to study the influence of the clustering coefficient on the final epidemic size, the maximum number of infected, and the peak time of the disease. The numerical simulations based on the SIR model in this paper fit the stochastic simulations based on the Monte Carlo method well at different levels of clustering. Our results show that the clustering coefficient poses impediments to the spread of disease under an SIR model.

Identification of Homophily and Preferential Recruitment in Respondent-Driven Sampling

Respondent-driven sampling (RDS) is a link-tracing procedure used in epidemiologic research on hidden or hard-to-reach populations in which subjects recruit others via their social networks. Estimates from RDS studies may have poor statistical properties due to statistical dependence in sampled subjects' traits. Two distinct mechanisms account for dependence in an RDS study: homophily, the tendency for individuals to share social ties with others exhibiting similar characteristics, and preferential recruitment, in which recruiters do not recruit uniformly at random from their network alters. The different effects of network homophily and preferential recruitment in RDS studies have been a source of confusion and controversy in methodological and empirical research in epidemiology. In this work, we gave formal definitions of homophily and preferential recruitment and showed that neither is identified in typical RDS studies. We derived nonparametric identification regions for homophily and preferential recruitment and showed that these parameters were not identified unless the network took a degenerate form. The results indicated that claims of homophily or recruitment bias measured from empirical RDS studies may not be credible. We applied our identification results to a study involving both a network census and RDS on a population of injection drug users in Hartford, Connecticut (2012-2013).

Determining whether a class of random graphs is consistent with an observed contact network

We demonstrate a general method to analyze the sensitivity of attack rate in a network model of infectious disease epidemiology to the structure of the network. We use Moore and Shannon’s “network reliability” statistic to measure the epidemic potential of a network. A number of networks are generated using exponential random graph models based on the properties of the contact network structure of one of the Add Health surveys. The expected number of infections on the original Add Health network is significantly different from that on any of the models derived from it. Because individual-level transmissibility and network structure are not separately identifiable parameters given population-level attack rate data it is possible to re-calibrate the transmissibility to fix this difference. However, the temporal behavior of the outbreak remains significantly different. Hence any estimates of the effectiveness of time dependent interventions on one network are unlikely to generalize to the other. Moreover, we show that in one case even a small perturbation to the network spoils the re-calibration. Unfortunately, the set of sufficient statistics for specifying a contact network model is not yet known. Until it is, estimates of the outcome of a dynamical process on a particular network obtained from simulations on a different network are not reliable.

Social contagions on weighted networks

We investigate critical behaviors of a social contagion model on weighted networks. An edge-weight compartmental approach is applied to analyze the weighted social contagion on strongly heterogenous networks with skewed degree and weight distributions. We find that degree heterogeneity cannot only alter the nature of contagion transition from discontinuous to continuous but also can enhance or hamper the size of adoption, depending on the unit transmission probability. We also show that the heterogeneity of weight distribution always hinders social contagions, and does not alter the transition type.

Spread of competing viruses on heterogeneous networks

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

Phylodynamics on local sexual contact networks

Phylodynamic models are widely used in infectious disease epidemiology to infer the dynamics and structure of pathogen populations. However, these models generally assume that individual hosts contact one another at random, ignoring the fact that many pathogens spread through highly structured contact networks. We present a new framework for phylodynamics on local contact networks based on pairwise epidemiological models that track the status of pairs of nodes in the network rather than just individuals. Shifting our focus from individuals to pairs leads naturally to coalescent models that describe how lineages move through networks and the rate at which lineages coalesce. These pairwise coalescent models not only consider how network structure directly shapes pathogen phylogenies, but also how the relationship between phylogenies and contact networks changes depending on epidemic dynamics and the fraction of infected hosts sampled. By considering pathogen phylogenies in a probabilistic framework, these coalescent models can also be used to estimate the statistical properties of contact networks directly from phylogenies using likelihood-based inference. We use this framework to explore how much information phylogenies retain about the underlying structure of contact networks and to infer the structure of a sexual contact network underlying a large HIV-1 sub-epidemic in Switzerland.

Phylodynamic models relate the branching pattern of a pathogen’s phylogenetic tree to the tree-like growth of an epidemic as it spreads through a host population. Such models are increasingly used to learn about the epidemiology of different pathogens. We extend current models to consider the structure of host contact networks—the web of physical interactions through which pathogens spread. By considering how local interactions among hosts shape the phylogeny of a pathogen, our models offer a “pathogen’s eye view” of these networks. Our models also provide a statistical framework that can be used to infer network structure directly from phylogenies, which we use to estimate the properties of a sexual contact network in Switzerland from a HIV phylogeny.

Disease dynamics in a stochastic network game: a little empathy goes a long way in averting outbreaks

Individuals change their behavior during an epidemic in response to whether they and/or those they interact with are healthy or sick. Healthy individuals may utilize protective measures to avoid contracting a disease. Sick individuals may utilize preemptive measures to avoid spreading a disease. Yet, in practice both protective and preemptive changes in behavior come with costs. This paper proposes a stochastic network disease game model that captures the self-interests of individuals during the spread of a susceptible-infected-susceptible disease. In this model, individuals strategically modify their behavior based on current disease conditions. These reactions influence disease spread. We show that there is a critical level of concern, i.e., empathy, by the sick individuals above which disease is eradicated rapidly. Furthermore, we find that risk averse behavior by the healthy individuals cannot eradicate the disease without the preemptive measures of the sick individuals. Empathy is more effective than risk-aversion because when infectious individuals change behavior, they reduce all of their potential infections, whereas when healthy individuals change behavior, they reduce only a small portion of potential infections. This imbalance in the role played by the response of the infected versus the susceptible individuals on disease eradication affords critical policy insights.

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

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