F. M. G. Magpantay

Avoidable errors in the modelling of outbreaks of emerging pathogens, with special reference to Ebola

As an emergent infectious disease outbreak unfolds, public health response is reliant on information on key epidemiological quantities, such as transmission potential and serial interval. Increasingly, transmission models fit to incidence data are used to estimate these parameters and guide policy. Some widely used modelling practices lead to potentially large errors in parameter estimates and, consequently, errors in model-based forecasts. Even more worryingly, in such situations, confidence in parameter estimates and forecasts can itself be far overestimated, leading to the potential for large errors that mask their own presence. Fortunately, straightforward and computationally inexpensive alternatives exist that avoid these problems. Here, we first use a simulation study to demonstrate potential pitfalls of the standard practice of fitting deterministic models to cumulative incidence data. Next, we demonstrate an alternative based on stochastic models fit to raw data from an early phase of 2014 West Africa Ebola virus disease outbreak. We show not only that bias is thereby reduced, but that uncertainty in estimates and forecasts is better quantified and that, critically, lack of model fit is more readily diagnosed. We conclude with a short list of principles to guide the modelling response to future infectious disease outbreaks.

The pertussis enigma: reconciling epidemiology, immunology and evolution.

Pertussis, a highly contagious respiratory infection, remains a public health priority despite the availability of vaccines for 70 years. Still a leading cause of mortality in developing countries, pertussis has re-emerged in several developed countries with high vaccination coverage. Resurgence of pertussis in these countries has routinely been attributed to increased awareness of the disease, imperfect vaccinal protection or high infection rates in adults. In this review, we first present 1980–2012 incidence data from 63 countries and show that pertussis resurgence is not universal. We further argue that the large geographical variation in trends probably precludes a simple explanation, such as the transition from whole-cell to acellular pertussis vaccines. Reviewing available evidence, we then propose that prevailing views on pertussis epidemiology are inconsistent with both historical and contemporary data. Indeed, we summarize epidemiological evidence showing that natural infection and vaccination both appear to provide long-term protection against transmission and disease, so that previously infected or vaccinated adults contribute little to overall transmission at a population level. Finally, we identify several promising avenues that may lead to a consistent explanation of global pertussis epidemiology and to more effective control strategies.

Epidemiological evidence for herd immunity induced by acellular pertussis vaccines

In a series of elegant experiments on baboons, Warfel et al. conclude that acellular pertussis vaccines (aP) prevent disease but fail to protect against transmissible infection (1). The authors speculate that this fact may explain the resurgence of pertussis in some countries (2). Although the animal model of Warfel et al. is a true breakthrough, we question the soundness of their extrapolation to transmission in human populations. Indeed, much available epidemiological evidence argues against it.

Wave fronts in neuronal fields with nonlocal post-synaptic axonal connections and delayed nonlocal feedback connections.

We consider a neuronal network model with both axonal connections (in the form of synaptic coupling) and delayed non-local feedback connections. The kernel in the feedback channel is assumed to be a standard non-local one, while for the kernel in the synaptic coupling, four types are considered. The main concern is the existence of travelling wave front. By employing the speed index function, we are able to obtain the existence of a travelling wave front for each of these four types within certain range of model parameters. We are also able to describe how the feedback coupling strength and the magnitude of the delay affect the wave speed. Some particular kernel functions for these four cases are chosen to numerically demonstrate the theoretical results.

Pertussis immunity and epidemiology: mode and duration of vaccine-induced immunity.

The resurgence of pertussis in some countries that maintain high vaccination coverage has drawn attention to gaps in our understanding of the epidemiological effects of pertussis vaccines. In particular, major questions surround the nature, degree and durability of vaccine protection. To address these questions, we used mechanistic transmission models to examine regional time series incidence data from Italy in the period immediately following the introduction of acellular pertussis (aP) vaccine. Our results concur with recent animal-challenge experiments wherein infections in aP-vaccinated individuals proved as transmissible as those in naive individuals but much less symptomatic. On the other hand, the data provide evidence for vaccine-driven reduction in susceptibility, which we quantify via a synthetic measure of vaccine impact. As to the precise nature of vaccine failure, the data do not allow us to distinguish between leakiness and waning of vaccine immunity, or some combination of these. Across the range of well-supported models, the nature and duration of vaccine protection, the age profile of incidence and the range of projected epidemiological futures differ substantially, underscoring the importance of the remaining unknowns. We identify key data gaps: sources of data that can supply the information needed to eliminate these remaining uncertainties.

Dynamics of Pertussis Transmission in the United States

Past patterns of infectious disease transmission set the stage on which modern epidemiologic dynamics are played out. Here, we present a comprehensive account of pertussis (whooping cough) transmission in the United States during the early vaccine era. We analyzed recently digitized weekly incidence records from Morbidity and Mortality Weekly Reports from 1938 to 1955, when the whole-cell pertussis vaccine was rolled out, and related them to contemporary patterns of transmission and resurgence documented in monthly incidence data from the National Notifiable Diseases Surveillance System. We found that, during the early vaccine era, pertussis epidemics in US states could be categorized as 1) annual, 2) initially annual and later multiennial, or 3) multiennial. States with predominantly annual cycles tended to have higher per capita birth rates, more household crowding, more children per family, and lower rates of school attendance than the states with multiennial cycles. Additionally, states that exhibited annual epidemics during 1938-1955 have had the highest recent (2001-2010) incidence, while those states that transitioned from annual cycles to multiennial cycles have had relatively low recent incidence. Our study provides an extensive picture of pertussis epidemiology in the United States dating back to the onset of vaccination, a back-story that could aid epidemiologists in understanding contemporary transmission patterns.

AN AGE-STRUCTURED POPULATION MODEL WITH STATE-DEPENDENT DELAY: DERIVATION AND NUMERICAL INTEGRATION ∗

C ‡ , AND J. WU ‡ Abstract. We present an age-structured population model that accounts for complex life cycles and competition for resources limiting the transition to maturity. Taking all of these into account leads to a new mathematical model with a state-dependent delay that cannot be analyzed using the established framework of functional differential equations or directly simulated by standard numerical schemes for age-structured populations. In this paper we present the derivation of the model and a numerical scheme to integrate the equations. Convergence of the method is proven by verifying first consistency and then stability. The difficulties involved in this proof, as well as in the implementation of the scheme, stem from the state-dependent delay term. The numerical scheme is shown to be of order 2 in the given norm, and at least of order 3 in the supremum norm over the mesh point values.

Vaccine impact in homogeneous and age-structured models

A general model of an imperfect vaccine for a childhood disease is presented and the effects of different types of vaccine failure on transmission were investigated using models that consider both homogeneous and age-specific mixing. The models are extensions of the standard SEIR equations with an additional vaccinated component that allows for five different vaccine parameters: three types of vaccine failure in decreasing susceptibility to infection via failure in degree (“leakiness”), take (“all-or-nothingness”) and duration (waning of vaccine-derived immunity); one parameter reflecting the relative reduction in infectiousness of vaccinated individuals who get infected; and one parameter that reflects the relative reduction in reporting probability of vaccinated individuals due to a possible reduction in severity of symptoms. Only the first four parameters affect disease transmission (as measured by the basic reproduction number). The reduction in transmission due to vaccination is different for age-structured models than for homogeneous models. Notably, if the vaccine exhibits waning protection this could be larger for an age-structured model with high contact rates between young children who are still protected by the vaccine and lower contact rates between adults for whom protection might have already waned. Analytic expressions for age-specific “vaccine impacts” were also derived. The overall vaccine impact is bounded between the age-specific impact for the oldest age class and that of the youngest age class.

The impact of past vaccination coverage and immunity on pertussis resurgence

Pertussis reemergence is underpinned by slowly waning immunity to both whole-cell and acellular vaccines together with incomplete past vaccination coverage. The problem of pertussis The recent rise of pertussis in developed countries has generated controversy as to its cause. Domenech de Cellès et al. modeled pertussis transmission using incidence data from Massachusetts, United States. They found little evidence that the switch to the acellular vaccine contributed to the Massachusetts outbreaks. Instead, waning vaccine-conferred immunity, as opposed to vaccine failure to mount a full or even partial immune response, best explained the local rise in pertussis cases along with a historical gap in vaccination coverage. Simulations suggested that administering existing boosters to children may be an effective strategy to halt pertussis transmission. The resurgence of pertussis over the past decades has resulted in incidence levels not witnessed in the United States since the 1950s. The underlying causes have been the subject of much speculation, with particular attention paid to the shortcomings of the latest generation of vaccines. We formulated transmission models comprising competing hypotheses regarding vaccine failure and challenged them to explain 16 years of highly resolved incidence data from Massachusetts, United States. Our results suggest that the resurgence of pertussis is a predictable consequence of incomplete historical coverage with an imperfect vaccine that confers slowly waning immunity. We found evidence that the vaccine itself is effective at reducing overall transmission, yet that routine vaccination alone would be insufficient for elimination of the disease. Our results indicated that the core transmission group is schoolchildren. Therefore, efforts aimed at curtailing transmission in the population at large, and especially in vulnerable infants, are more likely to succeed if targeted at schoolchildren, rather than adults.