Coupled Contagion Dynamics of Fear and Disease: Mathematical and Computational Explorations

Background

In classical mathematical epidemiology, individuals do not adapt their contact behavior during epidemics. They do not endogenously engage, for example, in social distancing based on fear. Yet, adaptive behavior is well-documented in true epidemics. We explore the effect of including such behavior in models of epidemic dynamics.

Methodology/Principal Findings

Using both nonlinear dynamical systems and agent-based computation, we model two interacting contagion processes: one of disease and one of fear of the disease. Individuals can “contract” fear through contact with individuals who are infected with the disease (the sick), infected with fear only (the scared), and infected with both fear and disease (the sick and scared). Scared individuals–whether sick or not–may remove themselves from circulation with some probability, which affects the contact dynamic, and thus the disease epidemic proper. If we allow individuals to recover from fear and return to circulation, the coupled dynamics become quite rich, and can include multiple waves of infection. We also study flight as a behavioral response.

Conclusions/Significance

In a spatially extended setting, even relatively small levels of fear-inspired flight can have a dramatic impact on spatio-temporal epidemic dynamics. Self-isolation and spatial flight are only two of many possible actions that fear-infected individuals may take. Our main point is that behavioral adaptation of some sort must be considered.     

Skip the Trip: Air Travelers' Behavioral Responses to Pandemic Influenza

Theory suggests that human behavior has implications for disease spread. We examine the hypothesis that individuals engage in voluntary defensive behavior during an epidemic. We estimate the number of passengers missing previously purchased flights as a function of concern for swine flu or A/H1N1 influenza using 1.7 million detailed flight records, Google Trends, and the World Health Organization''s FluNet data. We estimate that concern over “swine flu,” as measured by Google Trends, accounted for 0.34% of missed flights during the epidemic. The Google Trends data correlates strongly with media attention, but poorly (at times negatively) with reported cases in FluNet. Passengers show no response to reported cases. Passengers skipping their purchased trips forwent at least $50 M in travel related benefits. Responding to actual cases would have cut this estimate in half. Thus, people appear to respond to an epidemic by voluntarily engaging in self-protection behavior, but this behavior may not be responsive to objective measures of risk. Clearer risk communication could substantially reduce epidemic costs. People undertaking costly risk reduction behavior, for example, forgoing nonrefundable flights, suggests they may also make less costly behavior adjustments to avoid infection. Accounting for defensive behaviors may be important for forecasting epidemics, but linking behavior with epidemics likely requires consideration of risk communication.     

Epidemics in Interconnected Small-World Networks

Networks can be used to describe the interconnections among individuals, which play an important role in the spread of disease. Although the small-world effect has been found to have a significant impact on epidemics in single networks, the small-world effect on epidemics in interconnected networks has rarely been considered. Here, we study the susceptible-infected-susceptible (SIS) model of epidemic spreading in a system comprising two interconnected small-world networks. We find that the epidemic threshold in such networks decreases when the rewiring probability of the component small-world networks increases. When the infection rate is low, the rewiring probability affects the global steady-state infection density, whereas when the infection rate is high, the infection density is insensitive to the rewiring probability. Moreover, epidemics in interconnected small-world networks are found to spread at different velocities that depend on the rewiring probability.     

Validation of the Modes of Transmission Model as a Tool to Prioritize HIV Prevention Targets: A Comparative Modelling Analysis

Background

The static Modes of Transmission (MOT) model predicts the annual fraction of new HIV infections acquired across subgroups (MOT metric), and is used to focus HIV prevention. Using synthetic epidemics via a dynamical model, we assessed the validity of the MOT metric for identifying epidemic drivers (behaviours or subgroups that are sufficient and necessary for HIV to establish and persist), and the potential consequence of MOT-guided policies.

Methods and Findings

To generate benchmark MOT metrics for comparison, we simulated three synthetic epidemics (concentrated, mixed, and generalized) with different epidemic drivers using a dynamical model of heterosexual HIV transmission. MOT metrics from generic and complex MOT models were compared against the benchmark, and to the contribution of epidemic drivers to overall HIV transmission (cumulative population attributable fraction over t years, PAF_t). The complex MOT metric was similar to the benchmark, but the generic MOT underestimated the fraction of infections in epidemic drivers. The benchmark MOT metric identified epidemic drivers early in the epidemics. Over time, the MOT metric did not identify epidemic drivers. This was not due to simplified MOT models or biased parameters but occurred because the MOT metric (irrespective of the model used to generate it) underestimates the contribution of epidemic drivers to HIV transmission over time (PAF_5–30). MOT-directed policies that fail to reach epidemic drivers could undermine long-term impact on HIV incidence, and achieve a similar impact as random allocation of additional resources.

Conclusions

Irrespective of how it is obtained, the MOT metric is not a valid stand-alone tool to identify epidemic drivers, and has limited additional value in guiding the prioritization of HIV prevention targets. Policy-makers should use the MOT model judiciously, in combination with other approaches, to identify epidemic drivers.     

Equilibria of an Epidemic Game with Piecewise Linear Social Distancing Cost

Around the world, infectious disease epidemics continue to threaten people’s health. When epidemics strike, we often respond by changing our behaviors to reduce our risk of infection. This response is sometimes called “social distancing.” Since behavior changes can be costly, we would like to know the optimal social distancing behavior. But the benefits of changes in behavior depend on the course of the epidemic, which itself depends on our behaviors. Differential population game theory provides a method for resolving this circular dependence. Here, I present the analysis of a special case of the differential SIR epidemic population game with social distancing when the relative infection rate is linear, but bounded below by zero. Equilibrium solutions are constructed in closed-form for an open-ended epidemic. Constructions are also provided for epidemics that are stopped by the deployment of a vaccination that becomes available a fixed-time after the start of the epidemic. This can be used to anticipate a window of opportunity during which mass vaccination can significantly reduce the cost of an epidemic.     

Towards a characterization of behavior-disease models

The last decade saw the advent of increasingly realistic epidemic models that leverage on the availability of highly detailed census and human mobility data. Data-driven models aim at a granularity down to the level of households or single individuals. However, relatively little systematic work has been done to provide coupled behavior-disease models able to close the feedback loop between behavioral changes triggered in the population by an individual's perception of the disease spread and the actual disease spread itself. While models lacking this coupling can be extremely successful in mild epidemics, they obviously will be of limited use in situations where social disruption or behavioral alterations are induced in the population by knowledge of the disease. Here we propose a characterization of a set of prototypical mechanisms for self-initiated social distancing induced by local and non-local prevalence-based information available to individuals in the population. We characterize the effects of these mechanisms in the framework of a compartmental scheme that enlarges the basic SIR model by considering separate behavioral classes within the population. The transition of individuals in/out of behavioral classes is coupled with the spreading of the disease and provides a rich phase space with multiple epidemic peaks and tipping points. The class of models presented here can be used in the case of data-driven computational approaches to analyze scenarios of social adaptation and behavioral change.     

Impact of Birth Seasonality on Dynamics of Acute Immunizing Infections in Sub-Saharan Africa

We analyze the impact of birth seasonality (seasonal oscillations in the birth rate) on the dynamics of acute, immunizing childhood infectious diseases. Previous research has explored the effect of human birth seasonality on infectious disease dynamics using parameters appropriate for the developed world. We build on this work by including in our analysis an extended range of baseline birth rates and amplitudes, which correspond to developing world settings. Additionally, our analysis accounts for seasonal forcing both in births and contact rates. We focus in particular on the dynamics of measles. In the absence of seasonal transmission rates or stochastic forcing, for typical measles epidemiological parameters, birth seasonality induces either annual or biennial epidemics. Changes in the magnitude of the birth fluctuations (birth amplitude) can induce significant changes in the size of the epidemic peaks, but have little impact on timing of disease epidemics within the year. In contrast, changes to the birth seasonality phase (location of the peak in birth amplitude within the year) significantly influence the timing of the epidemics. In the presence of seasonality in contact rates, at relatively low birth rates (20 per 1000), birth amplitude has little impact on the dynamics but does have an impact on the magnitude and timing of the epidemics. However, as the mean birth rate increases, both birth amplitude and phase play an important role in driving the dynamics of the epidemic. There are stronger effects at higher birth rates.     

The impact of network clustering and assortativity on epidemic behaviour

Epidemic models have successfully included many aspects of the complex contact structure apparent in real-world populations. However, it is difficult to accommodate variations in the number of contacts, clustering coefficient and assortativity. Investigations of the relationship between these properties and epidemic behaviour have led to inconsistent conclusions and have not accounted for their interrelationship. In this study, simulation is used to estimate the impact of social network structure on the probability of an SIR (susceptible-infective-removed) epidemic occurring and, if it does, the final size. Increases in assortativity and clustering coefficient are associated with smaller epidemics and the impact is cumulative. Derived values of the basic reproduction ratio (R₀) over networks with the highest property values are more than 20% lower than those derived from simulations with zero values of these network properties.     

Behavioral responses to the 2015 MERS epidemic in Korea

Understanding behavioral responses to epidemics is important in evaluating the broad health consequences of emerging infectious diseases. Building on the economic epidemiology literature, this study investigates individual behavioral responses to the 2015 Middle East Respiratory Syndrome Coronavirus (MERS-CoV) epidemic in Korea using a panel of individuals in a nationally representative survey. Results show that exposure to the epidemic led to lasting impacts on smoking and drinking behaviors, indicating that emerging infectious disease outbreaks are motivations for behavioral changes and opportunities for public policy interventions. In particular, individuals in the hardest-hit regions or socially connected persons were more likely to change their risky behaviors, suggesting that intensity of exposure and social interactions are potential mechanisms.     

Epidemic outbreaks on structured populations

Our chances to halt epidemic outbreaks rely on how accurately we represent the population structure underlying the disease spread. When analysing global epidemics this force us to consider metapopulation models taking into account intra- and inter-community interactions. Here I introduce and analyze a metapopulation model which accounts for several features observed in real outbreaks. First, I demonstrate that depending on the intra-community expected outbreak size and the fraction of social bridges the epidemic outbreaks die out or there is a finite probability to observe a global epidemics. Second, I show that the global scenario is characterized by resurgent epidemics, their number increasing with increasing the intra-community average distance between individuals. Finally, I present empirical data for the AIDS epidemics supporting the model predictions.     

Modelling the effect of wheat canopy architecture as affected by sowing density on Septoria tritici epidemics using a coupled epidemic-virtual plant model

Background and Aims

The relationship between Septoria tritici, a splash-dispersed disease, and its host is complex because of the interactions between the dynamic plant architecture and the vertical progress of the disease. The aim of this study was to test the capacity of a coupled virtual wheat–Septoria tritici epidemic model (Septo3D) to simulate disease progress on the different leaf layers for contrasted sowing density treatments.

Methods

A field experiment was performed with winter wheat ‘Soissons’ grown at three contrasted densities. Plant architecture was characterized to parameterize the wheat model, and disease dynamic was monitored to compare with simulations. Three simulation scenarios, differing in the degree of detail with which plant variability of development was represented, were defined.

Key Results

Despite architectural differences between density treatments, few differences were found in disease progress; only the lower-density treatment resulted in a slightly higher rate of lesion development. Model predictions were consistent with field measurements but did not reproduce the higher rate of lesion progress in the low density. The canopy reconstruction scenario in which inter-plant variability was taken into account yielded the best agreement between measured and simulated epidemics. Simulations performed with the canopy represented by a population of the same average plant deviated strongly from the observations.

Conclusions

It was possible to compare the predicted and measured epidemics on detailed variables, supporting the hypothesis that the approach is able to provide new insights into the processes and plant traits that contribute to the epidemics. On the other hand, the complex and dynamic responses to sowing density made it difficult to test the model precisely and to disentangle the various aspects involved. This could be overcome by comparing more contrasted and/or simpler canopy architectures such as those resulting from quasi-isogenic lines differing by single architectural traits.     

Markers of disease evolution: the case of tuberculosis

Abrupt changes in environmental conditions--broadly understood to include demographic and social dynamics--can seriously impact the local or global disease dynamics of a population. These changes in the evolutionary landscape, which may occur over relatively short time-scales, are very likely to play a critical role in disease evolution. The potential impact of demographic, social and epidemiological shifts on the evolution of tuberculosis epidemics in the United States over the past century and a half is the main subject of this article. Evidence is provided to support the hypothesis that the observed substantial decreases in the incidence of active tuberculosis are the result of abrupt reductions in the rates of disease progression.     

Untangling the Interplay between Epidemic Spread and Transmission Network Dynamics

The epidemic spread of infectious diseases is ubiquitous and often has a considerable impact on public health and economic wealth. The large variability in the spatio-temporal patterns of epidemics prohibits simple interventions and requires a detailed analysis of each epidemic with respect to its infectious agent and the corresponding routes of transmission. To facilitate this analysis, we introduce a mathematical framework which links epidemic patterns to the topology and dynamics of the underlying transmission network. The evolution, both in disease prevalence and transmission network topology, is derived from a closed set of partial differential equations for infections without allowing for recovery. The predictions are in excellent agreement with complementarily conducted agent-based simulations. The capacity of this new method is demonstrated in several case studies on HIV epidemics in synthetic populations: it allows us to monitor the evolution of contact behavior among healthy and infected individuals and the contributions of different disease stages to the spreading of the epidemic. This gives both direction to and a test bed for targeted intervention strategies for epidemic control. In conclusion, this mathematical framework provides a capable toolbox for the analysis of epidemics from first principles. This allows for fast, in silico modeling--and manipulation--of epidemics and is especially powerful if complemented with adequate empirical data for parameterization.     

Accuracy of models for the 2001 foot-and-mouth epidemic

Since 2001 models of the spread of foot-and-mouth disease, supported by the data from the UK epidemic, have been expounded as some of the best examples of problem-driven epidemic models. These claims are generally based on a comparison between model results and epidemic data at fairly coarse spatio-temporal resolution. Here, we focus on a comparison between model and data at the individual farm level, assessing the potential of the model to predict the infectious status of farms in both the short and long terms. Although the accuracy with which the model predicts farms reporting infection is between 5 and 15%, these low levels are attributable to the expected level of variation between epidemics, and are comparable to the agreement between two independent model simulations. By contrast, while the accuracy of predicting culls is higher (20-30%), this is lower than expected from the comparison between model epidemics. These results generally support the contention that the type of the model used in 2001 was a reliable representation of the epidemic process, but highlight the difficulties of predicting the complex human response, in terms of control strategies to the perceived epidemic risk.     

Community Size Effects on Epidemic Spreading in Multiplex Social Networks

The dynamical process of epidemic spreading has drawn much attention of the complex network community. In the network paradigm, diseases spread from one person to another through the social ties amongst the population. There are a variety of factors that govern the processes of disease spreading on the networks. A common but not negligible factor is people’s reaction to the outbreak of epidemics. Such reaction can be related information dissemination or self-protection. In this work, we explore the interactions between disease spreading and population response in terms of information diffusion and individuals’ alertness. We model the system by mapping multiplex networks into two-layer networks and incorporating individuals’ risk awareness, on the assumption that their response to the disease spreading depends on the size of the community they belong to. By comparing the final incidence of diseases in multiplex networks, we find that there is considerable mitigation of diseases spreading for full phase of spreading speed when individuals’ protection responses are introduced. Interestingly, the degree of community overlap between the two layers is found to be critical factor that affects the final incidence. We also analyze the consequences of the epidemic incidence in communities with different sizes and the impacts of community overlap between two layers. Specifically, as the diseases information makes individuals alert and take measures to prevent the diseases, the effective protection is more striking in small community. These phenomena can be explained by the multiplexity of the networked system and the competition between two spreading processes.     

Predictive Validation of an Influenza Spread Model

Background

Modeling plays a critical role in mitigating impacts of seasonal influenza epidemics. Complex simulation models are currently at the forefront of evaluating optimal mitigation strategies at multiple scales and levels of organization. Given their evaluative role, these models remain limited in their ability to predict and forecast future epidemics leading some researchers and public-health practitioners to question their usefulness. The objective of this study is to evaluate the predictive ability of an existing complex simulation model of influenza spread.

Methods and Findings

We used extensive data on past epidemics to demonstrate the process of predictive validation. This involved generalizing an individual-based model for influenza spread and fitting it to laboratory-confirmed influenza infection data from a single observed epidemic (1998–1999). Next, we used the fitted model and modified two of its parameters based on data on real-world perturbations (vaccination coverage by age group and strain type). Simulating epidemics under these changes allowed us to estimate the deviation/error between the expected epidemic curve under perturbation and observed epidemics taking place from 1999 to 2006. Our model was able to forecast absolute intensity and epidemic peak week several weeks earlier with reasonable reliability and depended on the method of forecasting-static or dynamic.

Conclusions

Good predictive ability of influenza epidemics is critical for implementing mitigation strategies in an effective and timely manner. Through the process of predictive validation applied to a current complex simulation model of influenza spread, we provided users of the model (e.g. public-health officials and policy-makers) with quantitative metrics and practical recommendations on mitigating impacts of seasonal influenza epidemics. This methodology may be applied to other models of communicable infectious diseases to test and potentially improve their predictive ability.     

扩散对破碎化景观上宿主-寄生种群动态的影响

景观破碎化和扩散是空间种群模型的重要因素,对生物入侵存在着深远的影响。本章将基于偶对近似模型,探讨由局部和全局宿主-寄生相互作用共同决定的扩散模式对破坏性景观上疾病入侵与传播的影响。其中,生境破坏由生境丧失量与生境破碎化程度来描述。模拟结果显示,宿主和病毒的全局扩散对疾病的入侵与种群密度产生不对称效应：病毒的全局扩散对系统产生的影响较宿主的全局扩散更为显著。不同扩散模式下,生境丧失越高或破碎化程度越低,均将越有害于寄生病毒的入侵;同时,生境的破坏程度也显著地影响了入侵阈值对扩散模式的响应机制。本文研究结果暗示,景观破碎化的空间分布格局以及病毒扩散的限制均可作为物种保护与管理中有效的疾病控制策略。该研究结果在一定意义上丰富和发展了寄生感染理论,为物种保护提供了生态学理论依据。     

Nest architecture, activity pattern, worker density and the dynamics of disease transmission in social insects

The role of disease in the organization of insect colonies has become an important focus of research in evolutionary pathobiology, in which the relationship of sociality and disease transmission can be comparatively and experimentally analysed. In this paper we use an individual-based model of disease transmission to assess how an epidemic is influenced by worker density and activity level, the probability of disease transmission, and the structural organization of the nest. First, we observed in our model a nonlinear interaction between worker density and the probability of disease transmission: high levels of both factors interact to enhance the likelihood of an epidemic. Additionally, when we incorporated in our model the empirical observation that only a fraction of the worker population in social insect colonies is active at any given point in time, results suggested that relatively low levels of worker movement can have a significant impact on the spread of disease, slowing its transmission through the colony. Finally, we found that nests having even a simple spatial separation of chambers could delay the spread of infection and diminish the severity of an outbreak. The effect of nest structure in delaying infection spread became more pronounced as nest architecture became increasingly unidimensional, as in the case of simple gallery nests. Therefore, nest architecture and worker activity patterns might indeed exert considerable influence on the dynamics of epidemics in social insects and should be incorporated into models of disease transmission.     

How do toxicants affect epidemiological dynamics?

Populations are formed of their constituent interacting individuals, each with their own respective within‐host biological processes. Infection not only spreads within the host organism but also spreads between individuals. Here we propose and study a multilevel model which links the within‐host statuses of immunity and parasite density to population epidemiology under sublethal and lethal toxicant exposure. We analyse this nested model in order to better understand how toxicants impact the spread of disease within populations. We demonstrate that outbreak of infection within a population is completely determined by the level of toxicant exposure, and that it is maximised by intermediate toxicant dosage. We classify the population epidemiology into five phases of increasing toxicant exposure and calculate the conditions under which disease will spread, showing that there exists a threshold toxicant level under which epidemics will not occur. In general, higher toxicant load results in either extinction of the population or outbreak of infection. The within‐host statuses of the individual host also determine the outcome of the epidemic at the population level. We discuss applications of our model in the context of environmental epidemiology, predicting that increased exposure to toxicants could result in greater risk of epidemics within ecological systems. We predict that reducing sublethal toxicant exposure below our predicted safe threshold could contribute to controlling population level disease and infection.