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Interventions to mitigate the spread of infectious diseases, while succeeding in their goal, have economic and social costs associated with them. These limit the duration and intensity of the interventions. We study a class of interventions which reduce the reproduction number and find the optimal strength of the intervention which minimises the attack rate of an immunity inducing infection. The intervention works by eliminating the overshoot part of an epidemic, and avoids a second-wave of infections. We extend the framework by considering a heterogeneous population and find that the optimal intervention can pose an ethical dilemma for decision and policy makers. This ethical dilemma is shown to be analogous to the trolley problem and we discuss how the dilemma can be avoided.

Infectious disease control measures often require collective compliance of large numbers of individuals to benefit public health. This raises ethical questions regarding the value of the public health benefit created by individual and collective compliance. Answering these requires estimating the extent to which individual actions prevent infection of others. We develop mathematical techniques enabling quantification of the impacts of individuals or groups complying with three public health measures: border quarantine, isolation of infected individuals, and prevention via vaccination/prophylaxis. The results suggest that (i) these interventions exhibit synergy: they become more effective on a per-individual basis as compliance increases and (ii) There is often significant “overdetermination” of transmission: if a susceptible person contacts multiple infectious individuals, an intervention preventing one transmission may not change the ultimate outcome (thus risk imposed by some individuals may erode the benefits of others’ compliance). These results have implications for public health policy during epidemics.

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 $$1-1/{\mathcal {R}}_0$$ 1 - 1 / R 0 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.

Deterministic approximations to stochastic Susceptible–Infectious–Susceptible models typically predict a stable endemic steady-state when above threshold. This can be hard to relate to the underlying stochastic dynamics, which has no endemic steady-state but can exhibit approximately stable behaviour. Here, we relate the approximate models to the stochastic dynamics via the definition of the quasi-stationary distribution (QSD), which captures this approximately stable behaviour. We develop a system of ordinary differential equations that approximate the number of infected individuals in the QSD for arbitrary contact networks and parameter values. When the epidemic level is high, these QSD approximations coincide with the existing approximation methods. However, as we approach the epidemic threshold, the models deviate, with these models following the QSD and the existing methods approaching the all susceptible state. Through consistently approximating the QSD, the proposed methods provide a more robust link to the stochastic models.

For a recently derived pairwise model of network epidemics with non-Markovian recovery, we prove that under some mild technical conditions on the distribution of the infectious periods, smaller variance in the recovery time leads to higher reproduction number, and consequently to a larger epidemic outbreak, when the mean infectious period is fixed. We discuss how this result is related to various stochastic orderings of the distributions of infectious periods. The results are illustrated by a number of explicit stochastic simulations, suggesting that their validity goes beyond regular networks.

SummaryBackgroundSince its onset, the COVID-19 pandemic has caused significant morbidity and mortality worldwide, with particularly severe outcomes in healthcare institutions and congregate settings. To mitigate spread, healthcare systems have been cohorting patients to limit contacts between uninfected patients and potentially infected patients or healthcare workers (HCWs). A major challenge in managing the pandemic is the presence of currently asymptomatic individuals capable of transmitting the virus, who could introduce COVID-19 into uninfected cohorts. The optimal combination of personal protective equipment (PPE) and testing approaches to prevent these events is unclear, especially in light of ongoing limitations in access to both.MethodsUsing stochastic simulations with an SEIR model we quantified and compared the impacts of PPE use, patient and HCWs testing, and cohorting.FindingsIn the base case without testing or PPE, the healthcare system was rapidly overwhelmed, and became a net contributor to the force of infection. We found that effective use of PPE by both HCWs and patients could prevent this scenario, while random testing of apparently asymptomatic individuals on a weekly basis was less effective. We also found that even imperfect use of PPE could provide substantial protection by decreasing the force of infection, and that creation of smaller patient/HCW subcohorts can provide additional resilience to outbreak development.InterpretationThese findings reinforce the importance of ensuring adequate PPE supplies even in the absence of testing, and provide support for strict subcohorting regimens to reduce outbreak potential in healthcare institutions.FundingNational Institute of General Medical Sciences, National Institutes of Health.Research in contextEvidence beforePreserving healthcare from outbreaks of respiratory viruses is a longstanding concern, brought into sharp relief by the covid-19 pandemic. Early case series and numerous anecdotal reports suggest that health care workers (HCWs) and patients receiving treatment for conditions other than SARS-CoV-2 infection are at elevated risk of becoming infected, and the consequences of infections in long term care facilities are well known. In addition, the early stages of the pandemic have been marked by shortages of personal protective equipment (PPE) and diagnostic testing, but the most effective strategies for their use given the specific characteristics of SARS-CoV-2 transmission are unclear.Value addedOur research plainly shows the importance of presymptomatic transmission. Given reasonable estimates of this, random testing of currently asymptomatic staff and patients once a week is not able to prevent large outbreaks. We show that PPE is, as expected, the most effective intervention and moreover even suboptimal PPE use is highly beneficial. To further limit transmission, we show the benefit of sub-cohorting into smaller groups of HCWs and patients. When the force of infection in the community is low, this can entirely prevent the establishment of infection in a large fraction of healthcare.ImplicationsPPE should be used throughout healthcare, on the assumption that any patient or HCWs is potentially infected. Further work should determine the most effective means of PPE for the non-COVID cohort. If PPE resources are limited, whether in general or due to a second surge, we recommend subcohorting to limit the impact of introductions from the community.

Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2), the etiological agent of the Coronavirus Disease 2019 (COVID-19) disease, has moved rapidly around the globe, infecting millions and killing hundreds of thousands. The basic reproduction number, which has been widely used—appropriately and less appropriately—to characterize the transmissibility of the virus, hides the fact that transmission is stochastic, often dominated by a small number of individuals, and heavily influenced by superspreading events (SSEs). The distinct transmission features of SARS-CoV-2, e.g., high stochasticity under low prevalence (as compared to other pathogens, such as influenza), and the central role played by SSEs on transmission dynamics cannot be overlooked. Many explosive SSEs have occurred in indoor settings, stoking the pandemic and shaping its spread, such as long-term care facilities, prisons, meat-packing plants, produce processing facilities, fish factories, cruise ships, family gatherings, parties, and nightclubs. These SSEs demonstrate the urgent need to understand routes of transmission, while posing an opportunity to effectively contain outbreaks with targeted interventions to eliminate SSEs. Here, we describe the different types of SSEs, how they influence transmission, empirical evidence for their role in the COVID-19 pandemic, and give recommendations for control of SARS-CoV-2., This Essay discusses how the transmission of COVID-19 is dominated by superspreading events, proposing that finding ways to cut the long tail of secondary infections is important for controlling onward transmission.

As COVID-19 continues to spread, public health interventions are crucial to minimize its impact. The most desirable goal is to drive the pathogen quickly to extinction. This generally involves applying interventions as strongly as possible, which worked for SARS, but so far has failed for COVID-19. If fast eradication is not achievable, the next best goal is to delay the spread and minimize cases and burden on the health care system until suitable drugs or vaccines are available. This suppression approach also calls for strong interventions, potentially applied for a long time.

The interventions and outcomes in the ongoing SARS-CoV-2 pandemic are highly varied. The disease and the interventions both impose costs and harm on society. Some interventions with particularly high costs may only be implemented briefly. The design of optimal policy requires consideration of many intervention scenarios. In this paper we investigate the optimal timing of interventions that are not sustainable for a long period. Specifically, we look at at the impact of a single short-term non-repeated intervention (a “one-shot intervention”) on an epidemic and consider the impact of the intervention’s timing. To minimize the total number infected, the intervention should start close to the peak so that there is minimal rebound once the intervention is stopped. To minimise the peak prevalence, it should start earlier, leading to initial reduction and then having a rebound to the same prevalence as the pre-intervention peak rather than one very large peak. To delay infections as much as possible (as might be appropriate if we expect improved interventions or treatments to be developed), earlier interventions have clear benefit. In populations with distinct subgroups, synchronized interventions are less effective than targeting the interventions in each subcommunity separately. Author Summary Some interventions which help control a spreading epidemic have significant adverse effects on the population, and cannot be maintained long-term. The optimal timing of such an intervention will depend on the ultimate goal. Interventions to delay the epidemic while new treatments or interventions are developed are best implemented as soon as possible. Interventions to minimize the peak prevalence are best implemented partway through the growth phase allowing immunity to build up so that the eventual rebound is not larger than the initial peak. Interventions to minimize the total number of infections are best implemented late in the growth phase to minimize the amount of rebound. For a population with subcommunities which would have asynchronous outbreaks, similar results hold. Additionally, we find that it is best to target the intervention asynchronously to each subcommunity rather than synchronously across the population.

We provide a description of the Epidemics on Networks (EoN) python package designed for studying disease spread in static networks. The package consists of over $100$ methods available for users to perform stochastic simulation of a range of different processes including SIS and SIR disease, and generic simple or comlex contagions., Comment: This paper is available in a shorter form without examples at https://joss.theoj.org/papers/10.21105/joss.01731.pdf

This paper introduces a novel extension of the edge-based compartmental model to epidemics where the transmission and recovery processes are driven by general independent probability distributions. Edge-based compartmental modelling is just one of many different approaches used to model the spread of an infectious disease on a network; the major result of this paper is the rigorous proof that the edge-based compartmental model and the message passing models are equivalent for general independent transmission and recovery processes. This implies that the new model is exact on the ensemble of configuration model networks of infinite size. For the case of Markovian transmission the message passing model is re-parametrised into a pairwise-like model which is then used to derive many well-known pairwise models for regular networks, or when the infectious period is exponentially distributed or is of a fixed length.

There is an emerging consensus that achieving global tuberculosis control targets will require more proactive case finding approaches than are currently used in high-incidence settings. Household contact tracing (HHCT), for which households of newly diagnosed cases are actively screened for additional infected individuals is a potentially efficient approach to finding new cases of tuberculosis, however randomized trials assessing the population-level effects of such interventions in settings with sustained community transmission have shown mixed results. One potential explanation for this is that household transmission is responsible for a variable proportion of population-level tuberculosis burden between settings. For example, transmission is more likely to occur in households in settings with a lower tuberculosis burden and where individuals mix preferentially in local areas, compared with settings with higher disease burden and more dispersed mixing. To better understand the relationship between endemic incidence levels, social mixing, and the impact of HHCT, we developed a spatially explicit model of coupled household and community transmission. We found that the impact of HHCT was robust across settings of varied incidence and community contact patterns. In contrast, we found that the effects of community contact tracing interventions were sensitive to community contact patterns. Our results suggest that the protective benefits of HHCT are robust and the benefits of this intervention are likely to be maintained across epidemiological settings., Author summary Screening household members of newly detected tuberculosis cases is an efficient method for finding previously undiagnosed cases in high-burden settings. Despite the intuitive appeal of this approach, randomized trials examining the population-level effects of these interventions in settings with sustained community transmission have shown mixed results. One explanation for these inconclusive findings is that household transmission is responsible for a varying proportion of overall tuberculosis burden between locations, with the impact of household transmission being a function of both the overall incidence and the relative intensity of disease-transmitting contacts in the community and the household. In this manuscript, we use an individual-based network model to explore how local incidence levels and patterns of community contact impact the effectiveness of household-based approaches for interrupting tuberculosis transmission. Our analyses suggest that protective benefits of household-based interventions are maintained across a wide range of epidemiological settings. Our findings provide evidence for the robustness of household-based interventions and suggest that variable results from trials may be primarily due to implementation challenges rather than inherent limitations of these interventions.

The ongoing Coronavirus Disease 2019 (COVID-19) pandemic threatens the health of humans and causes great economic losses. Predictive modelling and forecasting the epidemic trends are essential for developing countermeasures to mitigate this pandemic. We develop a network model, where each node represents an individual and the edges represent contacts between individuals where the infection can spread. The individuals are classified based on the number of contacts they have each day (their node degrees) and their infection status. The transmission network model was respectively fitted to the reported data for the COVID-19 epidemic in Wuhan (China), Toronto (Canada), and the Italian Republic using a Markov Chain Monte Carlo (MCMC) optimization algorithm. Our model fits all three regions well with narrow confidence intervals and could be adapted to simulate other megacities or regions. The model projections on the role of containment strategies can help inform public health authorities to plan control measures., Highlights • Predictive modelling and forecasting the epidemic trends for COVID-19 are essential for developing countermeasures to mitigate this pandemic. • We develop a network model, taking into account underlying social contact heterogeneity in the real world. • We derived explicit expressions of the basic reproduction number and final epidemic size. • Our model fitted the outbreaks for the three selected places very well with narrow confidence intervals and could be adapted to other megacities or other regions. • The model projections on the role of containment strategies can help inform public health authorities to plan control measures.

Individual-based models provide modularity and structural flexibility necessary for modeling of infectious diseases at the within-host and population levels, but are challenging to implement. Levels of complexity can exceed the capacity and timescales for students and trainees in most academic institutions. Here we describe the process and advantages of a multi-disease framework approach developed with formal software support. The epidemiological modeling software, EMOD, has undergone a decade of software development. It is structured so that a majority of code is shared across disease modeling including malaria, HIV, tuberculosis, dengue, polio and typhoid. In additional to implementation efficiency, the sharing increases code usage and testing. The freely available codebase also includes hundreds of regression tests, scientific feature tests and component tests to help verify functionality and avoid inadvertent changes to functionality during future development. Here we describe the levels of detail, flexible configurability and modularity enabled by EMOD and the role of software development principles and processes in its development., To address multi-scale questions in infectious disease transmission, a multi-disease modeling platform EMOD has been developed with modularity, code sharing across diseases, and formal software development and testing processes.

A complex contagion is an infectious process in which individuals may require multiple transmissions before changing state. These are used to model behaviours if an individual only adopts a particular behaviour after perceiving a consensus among others. We may think of individuals as beginning inactive and becoming active once they are contacted by a sufficient number of active partners. These have been studied in a number of cases, but analytic models for the dynamic spread of complex contagions are typically complex. Here we study the dynamics of the Watts threshold model (WTM) assuming that transmission occurs in continuous time as a Poisson process, or in discrete time where individuals transmit to all partners in the time step following their infection. We adapt techniques developed for infectious disease modelling to develop an analyse analytic models for the dynamics of the WTM in configuration model networks and a class of random clustered (triangle-based) networks. The resulting model is relatively simple and compact. We use it to gain insights into the dynamics of the contagion. Taking the infinite population limit, we derive conditions under which cascades happen with an arbitrarily small initial proportion active, confirming a hypothesis of Watts for this case. We also observe hybrid phase transitions when cascades are not possible for small initial conditions, but occur for large enough initial conditions. We derive sufficient conditions for this hybrid phase transition to occur. We show that in many cases, if the hybrid phase transition occurs, then all individuals eventually become active. Finally, we discuss the role clustering plays in facilitating or impeding the spread and find that the hypothesis of Watts that was confirmed in configuration model networks does not hold in general. This approach allows us to unify many existing disparate observations and derive new results.

Influenza pandemics can emerge unexpectedly and wreak global devastation. However, each of the six pandemics since 1889 emerged in the Northern Hemisphere just after the flu season, suggesting that pandemic timing may be predictable. Using a stochastic model fit to seasonal flu surveillance data from the United States, we find that seasonal flu leaves a transient wake of heterosubtypic immunity that impedes the emergence of novel flu viruses. This refractory period provides a simple explanation for not only the spring-summer timing of historical pandemics, but also early increases in pandemic severity and multiple waves of transmission. Thus, pandemic risk may be seasonal and predictable, with the accuracy of pre-pandemic and real-time risk assessments hinging on reliable seasonal influenza surveillance and precise estimates of the breadth and duration of heterosubtypic immunity., Author summary Influenza pandemics emerge via genomic reassortment between circulating human and animal strains. The risk of pandemic emergence should therefore be high during the flu season, when viruses are abundant and conditions favor transmission. However, the six pandemics on record since 1889 all emerged in the Northern Hemisphere following the flu season, suggesting that other forces may predictably constrain pandemic risk. We find that seasonal influenza epidemics leave a wake of immunity that impedes pandemic emergence. This transient refractory period is consistent with the spring-summer emergence, multiple wave dynamics of recent pandemics, and may cause initial underestimation of the viral transmission rate. These findings may improve pre-pandemic risk assessments and real-time situational awareness, particularly as we gain greater insight into the extent of immunity.

An important feature of many real-world networks is the transient nature of some interactions. Thus far, our models have explicitly assumed that the network is static. That is, we assume that the rate of partner turnover is so slow that we can ignore its impact on epidemic dynamics.Over the past decade, there has been tremendous progress in modelling and analysing disease spread in non-static networks, i.e. networks whose structure changes due to endogenous or exogenous factors, or because of the disease dynamics unfolding on the network. The terminology used to describe such networks is not standardised. We summarise some common terminology and link it to the relation between the time scales of the dynamics on the network and the dynamics of the network. Ordered from static to networks that change quickly, we have

Mathematical models are caricatures of real systems that aim to capture the fundamental mechanisms of some process in order to explain observations or predict outcomes. No model — no matter how complicated — is perfect, or in the words of George Box [46]: “All models are wrong; some models are useful”. A useful model can provide valuable insights which improve our understanding of a system, and ultimately informs our decision-making.

In this chapter, we present a different approach to deriving exact models. In Chapter 2, we began with equations for every possible state of the system and then aggregated them into a simpler form. Here, we begin by deriving separate equations for the status of each node. These typically depend on the states of pairs of nodes, so we introduce equations for the pairs, which in turn depend on triples. We build up equations at each level. For a typical network, the number of equations we obtain is too large to be tractable. Consequently, we introduce “closures”, whereby terms corresponding to larger structures are represented in terms of smaller structures, in order to create a closed system of equations. In most cases, this representation involves an approximation, but in the case of SIR dynamics on trees or networks with cut-vertices, it is possible to reduce the number of equations considerably while keeping the model exact.

This chapter focuses on the relationships between the continuous-time SIR models we have previously derived and identifying conditions under which they are appropriate. Unless otherwise noted, the models discussed in this chapter are SIR models. Each of these models involves some assumptions, and to understand their limitations, we need to understand whether the true spread “respects” these assumptions.

Early studies of non-Markovian epidemics focused on SIR dynamics on fully connected networks, or homogeneously mixing populations, with the infection process being Markovian but with the infectious period taken from a general distribution [8, 278, 292, 293]. These approaches use probability theory arguments and typically focus on characterising the distribution of final epidemic sizes for finite populations, or on the average size in the infinite population limit. Similarly, the quasi-stationary distribution in a stochastic SIS model, again in a fully connected network, has been the subject of many studies [66, 230, 231]. More recently, it has been shown that one can readily apply results from queueing [19] or branching process [233] theory, or use martingales [65] to cast the same questions within a different framework and obtain results more readily.

This book has developed analytic models of disease spread on networks. All of our tractable models require closure assumptions. The closure process assumes that we can explain the dynamics at the network scale by understanding the dynamics locally at the level of small units with random connections between these units.

In previous chapters, it was shown that dynamics on networks can be described by continuous-time Markov chains, where probabilities of states are determined by master equations. While limiting mean-field ODE models can provide a good approximation of the expected value of certain random variables, a PDE-based approach is needed if information on the probability distribution of these is desired. The aim of this chapter is to develop and study mathematical methods to estimate the accuracy of mean-field ODE and PDE approximations. This will be carried out for continuous-time Markov chains with state space {0, 1, …, N}, for which transition from state k is possible only to states k − 1 and k + 1. These processes are called birth–death or one-step processes.

We consider the susceptible – infected – susceptible (SIS) epidemic on a dynamic network model with addition and deletion of links depending on node status. We analyse the resulting pairwise model using classical bifurcation theory to map out the spectrum of all possible epidemic behaviours. However, the major focus of the chapter is on the evolution and possible equilibria of the resulting networks. Whereas most studies are driven by determining system-level outcomes, e.g., whether the epidemic dies out or becomes endemic, with little regard for the emerging network structure, here, we want to buck this trend by augmenting the system-level results with mapping out of the structure and properties of the resulting networks. We find that depending on parameter values the network can become disconnected and show bistable-like behaviour whereas the endemic steady state sees the emergence of networks with qualitatively different degree distributions. In particular, we observe de-phased oscillations of both prevalence and network degree during which there is role reversal between the level and nature of the connectivity of susceptible and infected nodes. We conclude with an attempt at describing what a potential bifurcation theory for networks would look like.

The methods introduced thus far are applicable to both SIS and SIR diseases. This chapter focuses primarily on SIR disease. Once a node u becomes infected with an SIR disease, no other node affects the timing of any other action of u. This permits simplifications leading to a more complete theory of SIR disease.

The existence of sexual partnerships that overlap in time (concurrent relationships) is believed by some to be a significant contributing factor to the spread of HIV, although this is controversial. We derive an analytic model which allows us to investigate and compare disease spread in populations with and without concurrency. We can identify regions of parameter space in which its impact is negligible, and other regions in which it plays a major role. We also see that the impact of concurrency on the initial growth phase can be much larger than its impact on the equilibrium size. We see that the effect of concurrency saturates, which leads to the perhaps surprising conclusion that interventions targeting concurrency may be most effective in populations with low to moderate levels of concurrency.Author SummaryWe consider the spread of an infectious disease through a population modeled by a dynamic network with demographic turnover. We develop a stochastic model of the disease and derive governing equations that exactly predict the large population (deterministic) limit of the stochastic model. We use this to investigate the role of concurrency and find that interventions targeting concurrency may be most effective in populations with lower levels of concurrency.Our model is not intended to be an accurate representation of any single population. Rather it is intended to give general insights for intervention design and to provide a framework which can be further specialized to particular populations.This model is the first model to allow for analytic investigation of the impact of concurrent partnerships in a population exhibiting demographic turnover. Thus it will be useful for investigating the “concurrency hypothesis.”