Detecting and distinguishing tipping points using spectral early warning signals.

Theory and observation tell us that many complex systems exhibit tipping points-thresholds involving an abrupt and irreversible transition to a contrasting dynamical regime. Such events are commonly referred to as critical transitions. Current research seeks to develop early warning signals (EWS) of critical transitions that could help prevent undesirable events such as ecosystem collapse. However, conventional EWS do not indicate the type of transition, since they are based on the generic phenomena of critical slowing down. For instance, they may fail to distinguish the onset of oscillations (e.g. Hopf bifurcation) from a transition to a distant attractor (e.g. Fold bifurcation). Moreover, conventional EWS are less reliable in systems with density-dependent noise. Other EWS based on the power spectrum (spectral EWS) have been proposed, but they rely upon spectral reddening, which does not occur prior to critical transitions with an oscillatory component. Here, we use Ornstein-Uhlenbeck theory to derive analytic approximations for EWS prior to each type of local bifurcation, thereby creating new spectral EWS that provide greater sensitivity to transition proximity; higher robustness to density-dependent noise and bifurcation type; and clues to the type of approaching transition. We demonstrate the advantage of applying these spectral EWS in concert with conventional EWS using a population model, and show that they provide a characteristic signal prior to two different Hopf bifurcations in data from a predator-prey chemostat experiment. The ability to better infer and differentiate the nature of upcoming transitions in complex systems will help humanity manage critical transitions in the Anthropocene Era.

Homophilic replicator equations.

Tags are conspicuous attributes of organisms that affect the behaviour of other organisms toward the holder, and have previously been used to explore group formation and altruism. Homophilic imitation, a form of tag-based selection, occurs when organisms imitate those with similar tags. Here we further explore the use of tag-based selection by developing homophilic replicator equations to model homophilic imitation dynamics. We assume that replicators have both tags (sometimes called traits) and strategies. Fitnesses are determined by the strategy profile of the population, and imitation is based upon the strategy profile, fitness differences, and similarity in tag space. We show the characteristics of resulting fixed manifolds and conditions for stability. We discuss the phenomenon of coat-tailing (where tags associated with successful strategies increase in abundance, even though the tags are not inherently beneficial) and its implications for population diversity. We extend our model to incorporate recurrent mutations and invasions to explore their implications upon tag and strategy diversity. We find that homophilic imitation based upon tags significantly affects the diversity of the population, although not the ESS. We classify two different types of invasion scenarios by the strategy and tag compositions of the invaders and invaded. In one scenario, we find that novel tags introduced by invaders become more readily established with homophilic imitation than without it. In the other, diversity decreases. Lastly, we find a negative correlation between homophily and the rate of convergence.

Truncation selection and payoff distributions applied to the replicator equation.

The replicator equation has been frequently used in the theoretical literature to explain a diverse array of biological phenomena. However, it makes several simplifying assumptions, namely complete mixing, an infinite population, asexual reproduction, proportional selection, and mean payoffs. Here, we relax the conditions of mean payoffs and proportional selection by incorporating payoff distributions and truncation selection into extensions of the replicator equation and agent-based models. In truncation selection, replicators with fitnesses above a threshold survive. The reproduction rate is equal for all survivors and is sufficient to replace the replicators that did not survive. We distinguish between two types of truncation: independent and dependent with respect to the fitness threshold. If the payoff variances from all strategy pairing are the same, then we recover the replicator equation from the independent truncation equation. However, if all payoff variances are not equal, then any boundary fixed point can be made stable (or unstable) if only the fitness threshold is chosen appropriately. We observed transient and complex dynamics in our models, which are not observed in replicator equations incorporating the same games. We conclude that the assumptions of mean payoffs and proportional selection in the replicator equation significantly impact replicator dynamics.

Impacts of constrained culling and vaccination on control of foot and mouth disease in near-endemic settings: a pair approximation model.

Many countries have eliminated foot and mouth disease (FMD), but outbreaks remain common in other countries. Rapid development of international trade in animals and animal products has increased the risk of disease introduction to FMD-free countries. Most mathematical models of FMD are tailored to settings that are normally disease-free, and few models have explored the impact of constrained control measures in a 'near-endemic' spatially distributed host population subject to frequent FMD re-introductions from nearby endemic wild populations, as characterizes many low-income, resource-limited countries. Here we construct a pair approximation model of FMD and investigate the impact of constraints on total vaccine supply for prophylactic and ring vaccination, and constraints on culling rates and cumulative culls. We incorporate natural immunity waning and vaccine waning, which are important factors for near-endemic populations. We find that, when vaccine supply is sufficiently limited, the optimal approach for minimizing cumulative infections combines rapid deployment of ring vaccination during outbreaks with a contrasting approach of careful rationing of prophylactic vaccination over the year, such that supplies last as long as possible (and with the bulk of vaccines dedicated toward prophylactic vaccination). Thus, for optimal long-term control of the disease by vaccination in near-endemic settings when vaccine supply is limited, it is best to spread out prophylactic vaccination as much as possible. Regardless of culling constraints, the optimal culling strategy is rapid identification of infected premises and their immediate contacts at the initial stages of an outbreak, and rapid culling of infected premises and farms deemed to be at high risk of infection (as opposed to culling only the infected farms). Optimal culling strategies are similar when social impact is the outcome of interest. We conclude that more FMD transmission models should be developed that are specific to the challenges of FMD control in near-endemic, low-income countries.

Examining Ontario's universal influenza immunization program with a multi-strain dynamic model.

Seasonal influenza imposes a significant worldwide health burden each year. Mathematical models help us to understand how changes in vaccination affect this burden. Here, we develop a new dynamic transmission model which directly tracks the four dominant seasonal influenza strains/lineages, and use it to retrospectively examine the impact of the switch from a targeted to a universal influenza immunization program (UIIP) in the Canadian province of Ontario in 2000. According to our model results, averaged over the first four seasons post-UIIP, the rates of influenza-associated health outcomes in Ontario were reduced to about half of their pre-UIIP values. This is conservative compared to the results of a study estimating the UIIP impact from administrative data, though that study finds age-specific trends similar to those presented here. The strain interaction in our model, together with its flexible parameter calibration scheme, make it readily extensible to studying scenarios beyond the one explored here.

Dynamics and control of foot-and-mouth disease in endemic countries: a pair approximation model.

Previous mathematical models of spatial farm-to-farm transmission of foot and mouth disease (FMD) have explored the impacts of control measures such as culling and vaccination during a single outbreak in a country normally free of FMD. As a result, these models do not include factors that are relevant to countries where FMD is endemic in some regions, like long-term waning natural and vaccine immunity, use of prophylactic vaccination and disease re-importations. These factors may have implications for disease dynamics and control, yet few models have been developed for FMD-endemic settings. Here we develop and study an SEIRV (susceptible-exposed-infectious-recovered-vaccinated) pair approximation model of FMD. We focus on long term dynamics by exploring characteristics of repeated outbreaks of FMD and their dependence on disease re-importation, loss of natural immunity, and vaccine waning. We find that the effectiveness of ring and prophylactic vaccination strongly depends on duration of natural immunity, rate of vaccine waning, and disease re-introduction rate. However, the number and magnitude of FMD outbreaks are generally more sensitive to the duration of natural immunity than the duration of vaccine immunity. If loss of natural immunity and/or vaccine waning happen rapidly, then multiple epidemic outbreaks result, making it difficult to eliminate the disease. Prophylactic vaccination is more effective than ring vaccination, at the same per capita vaccination rate. Finally, more frequent disease re-importation causes a higher cumulative number of infections, although a lower average epidemic peak. Our analysis demonstrates significant differences between dynamics in FMD-free settings versus FMD-endemic settings, and that dynamics in FMD-endemic settings can vary widely depending on factors such as the duration of natural and vaccine immunity and the rate of disease re-importations. We conclude that more mathematical models tailored to FMD-endemic countries should be developed that include these factors.

The impact of personal experiences with infection and vaccination on behaviour-incidence dynamics of seasonal influenza.

Personal experiences with past infection events, or perceived vaccine failures and complications, are known to drive vaccine uptake. We coupled a model of individual vaccinating decisions, influenced by these drivers, with a contact network model of influenza transmission dynamics. The impact of non-influenzal influenza-like illness (niILI) on decision-making was also incorporated: it was possible for individuals to mistake niILI for true influenza. Our objectives were to (1) evaluate the impact of personal experiences on vaccine coverage; (2) understand the impact of niILI on behaviour-incidence dynamics; (3) determine which factors influence vaccine coverage stability; and (4) determine whether vaccination strategies can become correlated on the network in the absence of social influence. We found that certain aspects of personal experience can significantly impact behaviour-incidence dynamics. For instance, longer term memory for past events had a strong stabilising effect on vaccine coverage dynamics, although it could either increase or decrease average vaccine coverage depending on whether memory of past infections or past vaccine failures dominated. When vaccine immunity wanes slowly, vaccine coverage is low and stable, and infection incidence is also very low, unless the effects of niILI are ignored. Strategy correlations can occur in the absence of imitation, on account of the neighbour-neighbour transmission of infection and history-dependent decision making. Finally, niILI weakens the behaviour-incidence coupling and therefore tends to stabilise dynamics, as well as breaking up strategy correlations. Behavioural feedbacks, and the quality of self-diagnosis of niILI, may need to be considered in future programs adopting "universal" flu vaccines conferring long-term immunity. Public health interventions that focus on reminding individuals about their previous influenza infections, as well as communicating facts about vaccine efficacy and the difference between influenza and niILI, may be an effective way to increase vaccine coverage and prevent unexpected drops in coverage.

A game dynamic model for delayer strategies in vaccinating behaviour for pediatric infectious diseases.

Several studies have found that some parents delay the age at which their children receive pediatric vaccines due to perception of higher vaccine risk at the recommended age of vaccination. This has been particularly apparently during the Measles-Mumps-Rubella scare in the United Kingdom. Under a voluntary vaccination policy, vaccine coverage in certain age groups is a potentially complex interplay between vaccinating behaviour, disease dynamics, and age-specific risk factors. Here, we construct an age-structured game dynamic model, where individuals decide whether to vaccinate according to imitation dynamics depending on age-dependent disease prevalence and perceived risk of vaccination. Individuals may be timely vaccinators, delayers, or non-vaccinators. The model exhibits multiple equilibria and a broad range of possible dynamics. For certain parameter regimes, the proportion of timely vaccinators and delayers oscillate in an anti-phase fashion in response to oscillations in infection prevalence. Under an exogenous change to the perceived risk of vaccination as might occur during a vaccine scare, the model can also capture an increase in delayer strategists similar in magnitude to that observed during the Measles-Mumps-Rubella vaccine scare in the United Kingdom. Our model also shows that number of delayers steadily increases with increasing severity of the scare, whereas it saturates to specific value with increases in duration of the scare. Finally, by comparing the model dynamics with and without the option of a delayer strategy, we show that adding a third delayer strategy can have a stabilizing effect on model dynamics. In an era where individual choice--rather than accessibility--is becoming an increasingly important determinant of vaccine uptake, more infectious disease models may need to use game theory or related techniques to determine vaccine uptake.

Scheduling of measles vaccination in low-income countries: projections of a dynamic model.

Large-scale vaccination campaigns (SIAs) and improved routine immunization (RI) have greatly reduced measles incidence in low-income countries. However, the interval between SIAs required to maintain these gains over the long term is not clear. We developed a dynamic model of measles transmission to assess measles vaccination strategies in Cambodia, Ghana, India, Morocco, Nigeria, and Uganda. We projected measles cases from 2008 to 2050 under (a) holding SIAs every 2, 4, 6, or 8 years, (b) improvements in first dose routine measles vaccine (MCV1) coverage of 0%, 1%, 3% annually, and (c) introducing MCV2 once MCV1 coverage reaches 70%, 80%, 90%. If MCV1 continues improving, then India and Nigeria could hold SIAs every 4 years without significant probability of large outbreaks, and the other countries every 6-8 years. If RI remains stagnant, India and Nigeria should hold SIAs every 2 years, and the other countries every 4-6 years.

Cost-utility of universal hepatitis A vaccination in Canada.

Hepatitis A (HA) vaccination in Canada is currently targeted toward high-risk groups. The cost-effectiveness and expected health outcomes of universal vaccination relative to targeted vaccination in low-incidence countries such as Canada are currently unknown. Here, we conducted a cost-utility analysis for this situation, with Canada as the study population. We included vaccine costs, time costs, infection costs, and public health costs. We assessed a range of possible universal vaccination strategies over an 80-year time horizon using multiple cost perspectives. A dynamic model was used to account for herd immunity. Aggregate health gains from switching to universal vaccination are modest (10-30 QALYs per year). However, a "9+9" strategy that replaces two doses of monovalent hepatitis B (HB) vaccine at 9/10 years (universally administered in most provinces) with two doses of bivalent HA/HB vaccine is cost-saving from the societal perspective. At a willingness to pay threshold of $50,000/QALY, mean net benefit is +49.4 QALYs (S.D. 12.6) from the societal perspective and +3.8 QALYS (S.D. 3.0) from the payer perspective for the "9+9" strategy. Net benefit from the payer perspective is sensitive to the marginal cost of HA/HB vaccine relative to HB vaccine. Similar conclusions may apply in other countries with low incidence and a targeted vaccination policy.

A dynamic model for assessing universal Hepatitis A vaccination in Canada.

Vaccination against Hepatitis A virus (HAV) in Canada is currently targeted toward high-risk groups. However, universal vaccination has been adopted in several other countries with a similar disease burden. Here we develop an age-structured compartmental model of HAV transmission and vaccination in Canada to assess potential universal vaccination strategies. The model predicts that universal vaccination at age 1 (respectively 4, 9, 15), with phasing out of targeted vaccination, would reduce reported incidence by 60% (respectively 52, 36, 31%) and mortality attributable to HAV by 56% (respectively 45, 26, 25%), relative to continued targeted vaccination, over 80 years.

A versatile ODE approximation to a network model for the spread of sexually transmitted diseases.

We develop a moment closure approximation (MCA) to a network model of sexually transmitted disease (STD) spread through a steady/casual partnership network. MCA has been used previously to approximate static, regular lattices, whereas application to dynamic, irregular networks is a new endeavour, and application to sociologically-motivated network models has not been attempted. Our goals are 1). to investigate issues relating to the application of moment closure approximations to dynamic and irregular networks, and 2). to understand the impact of concurrent casual partnerships on STD transmission through a population of predominantly steady monogamous partnerships. We are able to derive a moment closure approximation for a dynamic irregular network representing sexual partnership dynamics, however, we are forced to use a triple approximation due to the large error of the standard pair approximation. This example underscores the importance of doing error analysis for moment closure approximations. We also find that a small number of casual partnerships drastically increases the prevalence and rate of spread of the epidemic. Finally, although the approximation is derived for a specific network model, we can recover approximations to a broad range of network models simply by varying model parameters which control the structure of the dynamic network. Thus our moment closure approximation is very flexible in the kinds of network models it can approximate.