Seasonality and the persistence and invasion of measles

An immuno-epidemiological model for transient immune protection: A case study for viral respiratory infections

The dynamics of infectious disease in a population critically involves both within-host pathogen replication and between host pathogen transmission. While modeling efforts have recently explored how within-host dynamics contribute to shaping population transmission, fewer have explored how ongoing circulation of an epidemic infectious disease can impact within-host immunological dynamics. We present a simple, influenza-inspired model that explores the potential for re-exposure during a single, ongoing outbreak to shape individual immune response and epidemiological potential in non-trivial ways. We show how even a simplified system can exhibit complex ongoing dynamics and sensitive thresholds in behavior. We also find epidemiological stochasticity likely plays a critical role in reinfection or in the maintenance of individual immunological protection over time.

Mumps epidemic dynamics in the United States before vaccination (1923-1932)

Mumps is a vaccine-preventable, reemerging, and highly transmissible infectious disease. Widespread vaccination dramatically reduced cases; however, case counts have been increasing over the past 20 years. To provide a quantitative overview of historical mumps dynamics that can act as baseline information to help identify causes of mumps reemergence, we analyzed timeseries of cases reported from 1923 to 1932 in the United States. During that time, 239,230 mumps cases were reported in 70 cities. Larger cities reported annual epidemics and smaller cities reported intermittent, sporadic outbreaks. The critical community size above which transmission continuously occurred was likely between 365,583 and 781,188 individuals but could range as high as 3,376,438 individuals. Mumps cases increased as city size increased, suggesting density-dependent transmission. Using a density-dependent SEIR model, we calculated a mean effective reproductive number (Re) of 1.2. Re varied by city and over time, with periodic high values that could characterize short periods of very high transmission known as superspreading events. Case counts most often peaked in March, with higher-than-average transmission from December through April and showed a correlation with weekly births. While certain city pairs in Midwestern states had synchronous outbreaks, most outbreaks were less synchronous and not driven by distance between cities. This work demonstrates the importance of long-term infectious disease surveillance data and will inform future studies on mumps reemergence and control.

Measles outbreak investigation in Kasoshekumer kebele, Sinana district, South-Eastern Oromia, Ethiopia: A case-control study

Objective

This study aimed at verifying the outbreak, preventing further transmission, and identifying the risk factors for the outbreak.

Method

A case-control study design was conducted from March 4, 2020 to April 30, 2020, in Kasoshekumer kebele of Sinana district. Thirty-seven carer-patient pairs and 74 controls were recruited in a 1:2 case-control ratio. Patients were selected using a national standard case definition of the Ministry of Health. Controls were selected from neighbors of patients and interviewed after the second incubation period from the last patient of the outbreak by a structured questionnaire similar to patients. All cases were included while controls were selected purposively for their exposure status. Vaccine efficacy, attack rate, and case fatality rate were conducted as a descriptive epidemiology. Multivariate analysis was used to identify associated factors.

Result

Out of 109 study participants, there were 37 cases and 72 controls. The mean age was 58.8 months with ±54 standard deviation. Thirty-one (83.8%) of the cases had a known contact history and 23 (62.2%) were unvaccinated. The age-specific attack rate was 36/1000 in <5 years and 53/100,000 in >15 years. The estimated vaccine efficacy was 73.7%. Having any type of socialization behavior (adjusted odds ratio = 6.8, confidence interval: 4.25, 11.4), maternal poor knowledge of measles prevention (adjusted odds ratio = 4.152, confidence interval: 1.226, 8.058), and being unvaccinated (adjusted odds ratio = 7.79, confidence interval: 2.281, 12.63) were associated factors for measles infection.

Conclusion

Poor knowledge of the parents on vaccination and the less effort taken to deliver good attitude toward vaccination are resulting in measles outbreak. However, herd immunity can prevent the outbreak; if not, the community will continue to suffer from mortality and morbidity. A resilient immunization program with sustainable and equitable supply, delivery, monitoring, and evaluation is a fundamental action in prevention of measles outbreak. Increased awareness of vaccination and other prevention measures are also crucial to end the outbreaks.

Urban Scaling of Health Outcomes: a Scoping Review

Urban scaling is a framework that describes how city-level characteristics scale with variations in city size. This scoping review mapped the existing evidence on the urban scaling of health outcomes to identify gaps and inform future research. Using a structured search strategy, we identified and reviewed a total of 102 studies, a majority set in high-income countries using diverse city definitions. We found several historical studies that examined the dynamic relationships between city size and mortality occurring during the nineteenth and early twentieth centuries. In more recent years, we documented heterogeneity in the relation between city size and health. Measles and influenza are influenced by city size in conjunction with other factors like geographic proximity, while STIs, HIV, and dengue tend to occur more frequently in larger cities. NCDs showed a heterogeneous pattern that depends on the specific outcome and context. Homicides and other crimes are more common in larger cities, suicides are more common in smaller cities, and traffic-related injuries show a less clear pattern that differs by context and type of injury. Future research should aim to understand the consequences of urban growth on health outcomes in low- and middle-income countries, capitalize on longitudinal designs, systematically adjust for covariates, and examine the implications of using different city definitions.

Comparative study on molecular epidemiology of measles H1 outbreak and sporadic cases in Shandong Province, 2013–2019

Background

Measles caused by measles virus (MeV) is a highly contagious viral disease which has also been associated with complications including pneumonia, myocarditis, encephalitis, and subacute sclerosing panencephalitis. The current study isolated 33 strains belonging to 2 groups, outbreak and sporadic strains, in 13 cities of Shandong province, China from 2013 to 2019. Comparison of genetic characterization among 15 outbreak strains and 18 sporadic strains was performed by analyzing nucleotide sequences of the C-terminal region of N protein gene (N-450).

Results

All 33 stains belonged to genotype H1. The outbreak strains and sporadic strains distributed crossly in phylogenetic tree. Sequences alignment revealed some interesting G to A transversion which changed the amino acids on genomic sites 1317, 1422, and 1543. The nucleotide and amino acid similarities among outbreak isolates were 98–100% (0–10 nucleotide variations) and 97.7–100%, respectively; They were 97.3–100% and 96.6–100%, respectively for sporadic isolates. Evolutionary genetics analysis revealed that the mean evolution rates of outbreak and sporadic isolates were 1.26 N 10− 3 and 1.48 N 10− 3 substitutions per site per year separately, which were similar with corresponding data before 2012. Local transmission analysis suggested that there were three transmission chains in this study, two of them originated from Japan. Outbreak cases and sporadic cases emerged alternatively and were reciprocal causation on the transmission chains.

Conclusions

Our study investigated the phylogeny and evolutional genetics of MeV during a 7-year surveillance, and compared epidemic and genetic characteristics of outbreak strains and sporadic strains. These results underscore the importance of evolutionary study alongside with sporadic cases in discovering and tracing possible outbreaks, especially in the stage of measles elimination.

The seasonal behaviour of COVID-19 and its galectin-like culprit of the viral spike

Seasonal behaviour is an attribute of many viral diseases. Like other 'winter' RNA viruses, infections caused by the causative agent of COVID-19, SARS-CoV-2, appear to exhibit significant seasonal changes. Here we discuss the seasonal behaviour of COVID-19, emerging viral phenotypes, viral evolution, and how the mutational landscape of the virus affects the seasonal attributes of the disease. We propose that the multiple seasonal drivers behind infectious disease spread (and the spread of COVID-19 specifically) are in 'trade-off' relationships and can be better described within a framework of a 'triangle of viral persistence' modulated by the environment, physiology, and behaviour. This 'trade-off' exists as one trait cannot increase without a decrease in another. We also propose that molecular components of the virus can act as sensors of environment and physiology, and could represent molecular culprits of seasonality. We searched for flexible protein structures capable of being modulated by the environment and identified a galectin-like fold within the N-terminal domain of the spike protein of SARS-CoV-2 as a potential candidate. Tracking the prevalence of mutations in this structure resulted in the identification of a hemisphere-dependent seasonal pattern driven by mutational bursts. We propose that the galectin-like structure is a frequent target of mutations because it helps the virus evade or modulate the physiological responses of the host to further its spread and survival. The flexible regions of the N-terminal domain should now become a focus for mitigation through vaccines and therapeutics and for prediction and informed public health decision making.

Herd immunity drives the epidemic fadeout of avian cholera in Arctic-nesting seabirds

Avian cholera, caused by the bacterium Pasteurella multocida, is a common and important infectious disease of wild birds in North America. Between 2005 and 2012, avian cholera caused annual mortality of widely varying magnitudes in Northern common eiders (Somateria mollissima borealis) breeding at the largest colony in the Canadian Arctic, Mitivik Island, Nunavut. Although herd immunity, in which a large proportion of the population acquires immunity to the disease, has been suggested to play a role in epidemic fadeout, immunological studies exploring this hypothesis have been missing. We investigated the role of three potential drivers of fadeout of avian cholera in eiders, including immunity, prevalence of infection, and colony size. Each potential driver was examined in relation to the annual real-time reproductive number (R_t) of P. multocida, previously calculated for eiders at Mitivik Island. Each year, colony size was estimated and eiders were closely monitored, and evaluated for infection and serological status. We demonstrate that acquired immunity approximated using antibody titers to P. multocida in both sexes was likely a key driver for the epidemic fadeout. This study exemplifies the importance of herd immunity in influencing the dynamics and fadeout of epidemics in a wildlife population.

COVID-19 cycles and rapidly evaluating lockdown strategies using spectral analysis

Spectral analysis characterises oscillatory time series behaviours such as cycles, but accurate estimation requires reasonable numbers of observations. At the time of writing, COVID-19 time series for many countries are short: pre- and post-lockdown series are shorter still. Accurate estimation of potentially interesting cycles seems beyond reach with such short series. We solve the problem of obtaining accurate estimates from short series by using recent Bayesian spectral fusion methods. We show that transformed daily COVID-19 cases for many countries generally contain three cycles operating at wavelengths of around 2.7, 4.1 and 6.7 days (weekly) and that shorter wavelength cycles are suppressed after lockdown. The pre- and post-lockdown differences suggest that the weekly effect is at least partly due to non-epidemic factors. Unconstrained, new cases grow exponentially, but the internal cyclic structure causes periodic declines. This suggests that lockdown success might only be indicated by four or more daily falls. Spectral learning for epidemic time series contributes to the understanding of the epidemic process and can help evaluate interventions. Spectral fusion is a general technique that can fuse spectra recorded at different sampling rates, which can be applied to a wide range of time series from many disciplines.

What drives disease flows between locations?

Communicable diseases 'flow' between locations. These flows dictate where and when certain communities will be affected. While the prediction of disease flows is essential for the timely intervention of epidemics, few studies have addressed this critical issue. This study predicts disease flows during an epidemic by considering the epidemiological, network, and temporal contextual factors using a deep learning approach. A series of scenario analyses helps identify the effects of these contextual factors on disease flows. Results show that the extended spatial-temporal effect of the epidemiological factors stimulates disease flows. The compound effects of the network factors enhance the transmission efficiency of these flows. Lastly, the temporal effect accelerates the combined effects of epidemiological and network factors on the flows. Findings of this study reveal the intricate nature of disease flows and lay a solid foundation for real-time surveillance of epidemics and pandemics to inform timely interventions for a broad range of communicable diseases.

Long-range movements coupled with heterogeneous incubation period sustain dog rabies at the national scale in Africa

Dog-transmitted rabies is responsible for more than 98% of human cases worldwide, remaining a persistent problem in developing countries. Mass vaccination targets predominantly major cities, often compromising disease control due to re-introductions. Previous work suggested that areas neighboring cities may behave as the source of these re-introductions. To evaluate this hypothesis, we introduce a spatially explicit metapopulation model for rabies diffusion in Central African Republic. Calibrated on epidemiological data for the capital city, Bangui, the model predicts that long-range movements are essential for continuous re-introductions of rabies-exposed dogs across settlements, eased by the large fluctuations of the incubation period. Bangui's neighborhood, instead, would not be enough to self-sustain the epidemic, contrary to previous expectations. Our findings suggest that restricting long-range travels may be very efficient in limiting rabies persistence in a large and fragmented dog population. Our framework can be applied to other geographical contexts where dog rabies is endemic.

Global trends in measles publications

INTRODUCTION

Beginning with the 1960s, this review analyzes trends in publications on measles indexed by the National Library of Medicine from January 1960 to mid-2018. It notes both the growth in numbers of published papers, and the increasing number and proportion of publications, in the current century, of articles on such items as costing, measles elimination, and determinants of coverage.

METHODS

A two-person team extracted from the National Library of Medicine (NLM) homepage all citations on measles beginning in 1960 and continuing through mid-2018. These were then classified both by overall number and by subject matter, with tabular summaries of both by decade and by subject matter. The tabular presentation forms the basis for a discussion of the ten most frequently cited subjects, and publication trends, with a special emphasis on the current century.

RESULTS

As in the past, the most often currently published items have been on coverage and its determinants, measles elimination, outbreak reports, SSPE, and SIAs. The putative relationship between vaccination and autism saw a spurt of articles in the 1990s, rapidly declining after the IOM report rejecting the causative hypothesis.

CONCLUSION

There is a discussion on the sequencing of polio and measles eradication, the former unlikely before 2022, and an examination of likely research priorities as the world moves from measles control to measles eradication. There is a key role for social science in combatting vaccination reticence. The role of technical innovations, such as micropatch vaccination, is discussed.

Phylogenetic and Epidemiological Analysis of Measles Viruses in Shenzhen, China from January 2015 to July 2019

Background

Measles morbidity and mortality were significantly reduced after the measles vaccine was introduced in China in 1965. However, measles outbreaks easily occur in densely populated areas, especially where there is no universal vaccination. The outbreak that occurred in Shenzhen, the Chinese city with the largest internal immigration, provides a lesson in measles virus mutation and measles prevention. The present study is a phylogenetic analysis of measles viruses and comparison of clinical signs between individuals with and without vaccination.

Material/Methods

We performed phylogenetic analysis of the nucleoprotein (N) genes of measles virus from 129 measles patients in Shenzhen from January 2015 to July 2019. Phylogenetic trees were constructed using the neighbor-joining method.

Results

The phylogenetic analysis showed all viruses were classified into genotype H1. In addition, there is often a seasonal measles outbreak in July each year. The clinical data showed that patients who were unvaccinated were more likely to have coughing, chronic bronchitis, conjunctivitis, catarrh, Koplik spots, and diarrhea. Children of migrant workers and those living in mountainous and rural districts accounted for most measles cases.

Conclusions

Our results showed there was a seasonal measles outbreak in Shenzhen Children’s Hospital. All the measles virus from 129 measles patients were H1 viruses. The clinical signs also showed a difference between unvaccinated and vaccinated patients. Moreover, most of the unvaccinated patients came from migrant worker families. We suggest there is a need for increased health promotion and vaccination programs for migrant workers and people living in remote villages.

Measles outbreak response decision-making under uncertainty: a retrospective analysis

Resurgent outbreaks of vaccine-preventable diseases that have previously been controlled or eliminated have been observed in many settings. Reactive vaccination campaigns may successfully control outbreaks but must necessarily be implemented in the face of considerable uncertainty. Real-time surveillance may provide critical information about at-risk population and optimal vaccination targets, but may itself be limited by the specificity of disease confirmation. We propose an integrated modelling approach that synthesizes historical demographic and vaccination data with real-time outbreak surveillance via a dynamic transmission model and an age-specific disease confirmation model. We apply this framework to data from the 1996-1997 measles outbreak in São Paulo, Brazil. To simulate the information available to decision-makers, we truncated the surveillance data to what would have been available at 1 or 2 months prior to the realized interventions. We use the model, fitted to real-time observations, to evaluate the likelihood that candidate age-targeted interventions could control the outbreak. Using only data available prior to the interventions, we estimate that a significant excess of susceptible adults would prevent child-targeted campaigns from controlling the outbreak and that failing to account for age-specific confirmation rates would underestimate the importance of adult-targeted vaccination.

Endemic Disease Models

In this chapter, we consider models for disease that may be endemic. In the preceding chapter we studied SIS models with and without demographics and SIR models with demographics. In each model, 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} $$\mathcal {R}_0$$ \end{document}R0 determined a threshold. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {R}_0 < 1$$ \end{document}R0<1 the disease dies out, while if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {R}_0 > 1$$ \end{document}R0>1 the disease becomes endemic. The analysis in each case involves determination of equilibria and determining the asymptotic stability of each equilibrium by linearization about the equilibrium. In each of the cases studied in the preceding chapter the disease-free equilibrium was asymptotically stable if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {R}_0 < 1$$ \end{document}R0<1 and if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {R}_0 > 1$$ \end{document}R0>1 there was a unique endemic equilibrium that was asymptotically stable. In this chapter, we will see that these properties continue to hold for many more general models, but there are situations in which there may be an asymptotically stable endemic equilibrium when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {R}_0 < 1$$ \end{document}R0<1, and other situations in which there is an endemic equilibrium that is unstable for some values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {R}_0 > 1$$ \end{document}R0>1.

A stochastic SIR model on a graph with epidemiological and population dynamics occurring over the same time scale

We define and study an open stochastic SIR (Susceptible-Infected-Removed) model on a graph in order to describe the spread of an epidemic on a cattle trade network with epidemiological and demographic dynamics occurring over the same time scale. Population transition intensities are assumed to be density-dependent with a constant component, the amplitude of which determines the overall scale of the population process. Standard branching approximation results for the epidemic process are first given, along with a numerical computation method for the probability of a major epidemic outbreak. This procedure is illustrated using real data on trade-related cattle movements from a densely populated livestock farming region in western France (Finistère) and epidemiological parameters corresponding to an infectious epizootic disease. Then we exhibit an exponential lower bound for the extinction time and the total size of the epidemic in the stable endemic case as a scaling parameter goes to infinity using results inspired by the Freidlin-Wentzell theory of large deviations from a dynamical system.

Association of Previous Measles Infection With Markers of Acute Infectious Disease Among 9- to 59-Month-Old Children in the Democratic Republic of the Congo

BACKGROUND

Transient immunosuppression and increased susceptibility to other infections after measles infection is well known, but recent studies have suggested the occurrence of an "immune amnesia" that could have long-term immunosuppressive effects.

METHODS

We examined the association between past measles infection and acute episodes of fever, cough, and diarrhea among 2350 children aged 9 to 59 months whose mothers were selected for interview in the 2013-2014 Democratic Republic of the Congo (DRC) Demographic and Health Survey (DHS). Classification of children who had had measles was completed using maternal recall and measles immunoglobulin G serostatus obtained via dried-blood-spot analysis with a multiplex immunoassay. The association with time since measles infection and fever, cough, and diarrhea outcomes was also examined.

RESULTS

The odds of fever in the previous 2 weeks were 1.80 (95% confidence interval [CI], 1.25-2.60) among children for whom measles was reported compared to children with no history of measles. Measles vaccination demonstrated a protective association against selected clinical markers of acute infectious diseases.

CONCLUSION

Our results suggest that measles might have a long-term effect on selected clinical markers of acute infectious diseases among children aged 9 to 59 months in the DRC. These findings support the immune-amnesia hypothesis suggested by others and underscore the need for continued evaluation and improvement of the DRC's measles vaccination program.

Seasonal transmission dynamics of measles in China

Measles, a highly contagious infection caused by the measles virus, is a major public health problem in China. The reported measles cases decreased dramatically from 2004 to 2012 due to the mandatory measles vaccine program started in 2005 and the goal of eliminating measles by 2012. However, after reaching its lowest level in 2012, measles has resurged again since 2013. Since the monthly data of measles cases exhibit a seasonally fluctuating pattern, based on the measles model in Earn et al. (Science 287:667-670, 2000), we propose a susceptible, exposed, infectious, and recovered model with periodic transmission rate to investigate the seasonal measles epidemics and the effect of vaccination. We calculate the basic reproduction number [Formula: see text], analyze the dynamical behavior of the model, and use the model to simulate the monthly data of measles cases reported in China. We also carry out some sensitivity analysis of [Formula: see text] in the terms of various model parameters which shows that measles can be controlled and eventually eradicated by increasing the immunization rate, improving the effective vaccine management, and enhancing the awareness of people about measles.

Evidence of cryptic incidence in childhood diseases

Persistence and extinction are key processes in infectious disease dynamics that, owing to incomplete reporting, are seldom directly observable. For fully immunizing diseases, reporting probabilities can be readily estimated from demographic records and case reports. Yet reporting probabilities are not sufficient to unambiguously reconstruct disease incidence from case reports. Here, we focus on disease presence (i.e. marginal probability of non-zero incidence), which provides an upper bound on the marginal probability of disease extinction. We examine measles and pertussis in pre-vaccine era United States (US) cities, and describe a conserved scaling relationship between population size, reporting probability and observed presence (i.e. non-zero case reports). We use this relationship to estimate disease presence given perfect reporting, and define cryptic presence as the difference between estimated and observed presence. We estimate that, in early twentieth century US cities, pertussis presence was higher than measles presence across a range of population sizes, and that cryptic presence was common in small cities with imperfect reporting. While the methods employed here are specific to fully immunizing diseases, our results suggest that cryptic incidence deserves careful attention, particularly in diseases with low case counts, poor reporting and longer infectious periods.

An ecological and digital epidemiology analysis on the role of human behavior on the 2014 Chikungunya outbreak in Martinique

Understanding the spatio-temporal dynamics of endemic infections is of critical importance for a deeper understanding of pathogen transmission, and for the design of more efficient public health strategies. However, very few studies in this domain have focused on emerging infections, generating a gap of knowledge that hampers epidemiological response planning. Here, we analyze the case of a Chikungunya outbreak that occurred in Martinique in 2014. Using time series estimates from a network of sentinel practitioners covering the entire island, we first analyze the spatio-temporal dynamics and show that the largest city has served as the epicenter of this epidemic. We further show that the epidemic spread from there through two different propagation waves moving northwards and southwards, probably by individuals moving along the road network. We then develop a mathematical model to explore the drivers of the temporal dynamics of this mosquito-borne virus. Finally, we show that human behavior, inferred by a textual analysis of messages published on the social network Twitter, is required to explain the epidemiological dynamics over time. Overall, our results suggest that human behavior has been a key component of the outbreak propagation, and we argue that such results can lead to more efficient public health strategies specifically targeting the propagation process.

Measles elimination and immunisation: national surveillance trends in Japan, 2008-2015

Measles elimination relies on vaccination programmes. In Japan, a major outbreak started in 2007. In response, 5-year two-dose catch-up vaccination programme was initiated in April 2008 for children 13-16-years-old. In this study, we analysed the epidemic curves, incidence rates for each age group, virus genotype, vaccination coverage and ratio of measles gelatin particle agglutination (PA) antibody using surveillance data for 2008-2015. Monthly case counts markedly decreased as vaccination coverage increased. D5, which is the endemic virus type, disappeared after 2011, with the following epidemic caused by imported viruses. Most cases were confirmed to have a no-dose or single-dose vaccination status. Although the incidence rate among all age groups ⩾5-years-old decreased during the study period, for children <5-years-old, the incidence rate remained relatively high and increased in 2014. The ratio of PA antibody (⩾1:128 titres) increased for the majority of age groups, but with a decrease for specific age groups: the 0-5 months and the 2-4, 14, 19 and most of the 26-55- and the 60-year-old groups (-1 to -9%). This seems to be the result of higher vaccination coverage, which would result in decreasing natural immunity booster along with decreasing passive immunity in infants whose mothers did not have the natural immunity booster. The 20-29- and 30-39-year-old age groups had higher number of cases, suggesting that vaccination within these age groups might be important for eliminating imported viruses.

Geospatial characteristics of measles transmission in China during 2005-2014

Measles is a highly contagious and severe disease. Despite mass vaccination, it remains a leading cause of death in children in developing regions, killing 114,900 globally in 2014. In 2006, China committed to eliminating measles by 2012; to this end, the country enhanced its mandatory vaccination programs and achieved vaccination rates reported above 95% by 2008. However, in spite of these efforts, during the last 3 years (2013-2015) China documented 27,695, 52,656, and 42,874 confirmed measles cases. How measles manages to spread in China-the world's largest population-in the mass vaccination era remains poorly understood. To address this conundrum and provide insights for future public health efforts, we analyze the geospatial pattern of measles transmission across China during 2005-2014. We map measles incidence and incidence rates for each of the 344 cities in mainland China, identify the key socioeconomic and demographic features associated with high disease burden, and identify transmission clusters based on the synchrony of outbreak cycles. Using hierarchical cluster analysis, we identify 21 epidemic clusters, of which 12 were cross-regional. The cross-regional clusters included more underdeveloped cities with large numbers of emigrants than would be expected by chance (p = 0.011; bootstrap sampling), indicating that cities in these clusters were likely linked by internal worker migration in response to uneven economic development. In contrast, cities in regional clusters were more likely to have high rates of minorities and high natural growth rates than would be expected by chance (p = 0.074; bootstrap sampling). Our findings suggest that multiple highly connected foci of measles transmission coexist in China and that migrant workers likely facilitate the transmission of measles across regions. This complex connection renders eradication of measles challenging in China despite its high overall vaccination coverage. Future immunization programs should therefore target these transmission foci simultaneously.

Environmental, dietary and case-control study of Nodding Syndrome in Uganda: A post-measles brain disorder triggered by malnutrition?

Nodding Syndrome (NS) is an epileptic encephalopathy characterized by involuntary vertical head nodding, other types of seizures, and progressive neurological deficits. The etiology of the east African NS epidemic is unknown. In March 2014, we conducted a case-control study of medical, nutritional and other risk factors associated with NS among children (aged 5-18years) of Kitgum District, northern Uganda (Acholiland). Data on food availability, rainfall, and prevalent disease temporally related to the NS epidemic were also analyzed. In NS Cases, the mean age of reported head nodding onset was 7.6years (range 1-17years). The epidemiologic curve of NS incidence spanned 2000-2013, with peaks in 2003 and 2008. Month of onset of head nodding was non-uniform, with all-year-aggregated peaks in April and June when food availability was low. Families with one or more NS Cases had been significantly more dependent on emergency food and, immediately prior to head nodding onset in the child, subsistence on moldy plant materials, specifically moldy maize. Medical history revealed a single significant association with NS, namely prior measles infection. NS is compared with the post-measles disorder subacute sclerosing panencephalitis, with clinical expression triggered by factors associated with poor nutrition.

Evolution and Use of Dynamic Transmission Models for Measles and Rubella Risk and Policy Analysis

The devastation caused by periodic measles outbreaks motivated efforts over more than a century to mathematically model measles disease and transmission. Following the identification of rubella, which similarly presents with fever and rash and causes congenital rubella syndrome (CRS) in infants born to women first infected with rubella early in pregnancy, modelers also began to characterize rubella disease and transmission. Despite the relatively large literature, no comprehensive review to date provides an overview of dynamic transmission models for measles and rubella developed to support risk and policy analysis. This systematic review of the literature identifies quantitative measles and/or rubella dynamic transmission models and characterizes key insights relevant for prospective modeling efforts. Overall, measles and rubella represent some of the relatively simplest viruses to model due to their ability to impact only humans and the apparent life-long immunity that follows survival of infection and/or protection by vaccination, although complexities arise due to maternal antibodies and heterogeneity in mixing and some models considered potential waning immunity and reinfection. This review finds significant underreporting of measles and rubella infections and widespread recognition of the importance of achieving and maintaining high population immunity to stop and prevent measles and rubella transmission. The significantly lower transmissibility of rubella compared to measles implies that all countries could eliminate rubella and CRS by using combination of measles- and rubella-containing vaccines (MRCVs) as they strive to meet regional measles elimination goals, which leads to the recommendation of changing the formulation of national measles-containing vaccines from measles only to MRCV as the standard of care.

Hand, Foot, and Mouth Disease in China: Critical Community Size and Spatial Vaccination Strategies

Hand Foot and Mouth Disease (HFMD) constitutes a considerable burden for health care systems across China. Yet this burden displays important geographic heterogeneity that directly affects the local persistence and the dynamics of the disease, and thus the ability to control it through vaccination campaigns. Here, we use detailed geographic surveillance data and epidemic models to estimate the critical community size (CCS) of HFMD associated enterovirus serotypes CV-A16 and EV-A71 and we explore what spatial vaccination strategies may best reduce the burden of HFMD. We found CCS ranging from 336,979 (±225,866) to 722,372 (±150,562) with the lowest estimates associated with EV-A71 in the southern region of China where multiple transmission seasons have previously been identified. Our results suggest the existence of a regional immigration-recolonization dynamic driven by urban centers. If EV-A71 vaccines doses are limited, these would be optimally deployed in highly populated urban centers and in high-prevalence areas. If HFMD vaccines are included in China's National Immunization Program in order to achieve high coverage rates (>85%), routine vaccination of newborns largely outperforms strategies in which the equivalent number of doses is equally divided between routine vaccination of newborns and pulse vaccination of the community at large.

Controlling measles using supplemental immunization activities: a mathematical model to inform optimal policy

BACKGROUND

The Measles & Rubella Initiative, a broad consortium of global health agencies, has provided support to measles-burdened countries, focusing on sustaining high coverage of routine immunization of children and supplementing it with a second dose opportunity for measles vaccine through supplemental immunization activities (SIAs). We estimate optimal scheduling of SIAs in countries with the highest measles burden.

METHODS

We develop an age-stratified dynamic compartmental model of measles transmission. We explore the frequency of SIAs in order to achieve measles control in selected countries and two Indian states with high measles burden. Specifically, we compute the maximum allowable time period between two consecutive SIAs to achieve measles control.

RESULTS

Our analysis indicates that a single SIA will not control measles transmission in any of the countries with high measles burden. However, regular SIAs at high coverage levels are a viable strategy to prevent measles outbreaks. The periodicity of SIAs differs between countries and even within a single country, and is determined by population demographics and existing routine immunization coverage.

CONCLUSIONS

Our analysis can guide country policymakers deciding on the optimal scheduling of SIA campaigns and the best combination of routine and SIA vaccination to control measles.

A Short Introduction to Disease Emergence

The purpose of this introductory chapter is to provide the reader with a brief overview of the factors that drive disease emergence in order to set the scene for the more detailed chapters that follow. The chapter is divided into three parts. The first deals with the activities of humans that drive disease emergence. This ranges from historical factors such as animal domestication to the impact of new technologies such as air travel and agricultural intensification in response to population growth. The second section deals with virus properties that enable them to adapt to new hosts, particularly jumping between species. The final section deals with the activities of animals themselves that contribute to disease emergence.

Characterizing the dynamics of rubella relative to measles: the role of stochasticity

Rubella is a completely immunizing and mild infection in children. Understanding its behaviour is of considerable public health importance because of congenital rubella syndrome, which results from infection with rubella during early pregnancy and may entail a variety of birth defects. The recurrent dynamics of rubella are relatively poorly resolved, and appear to show considerable diversity globally. Here, we investigate the behaviour of a stochastic seasonally forced susceptible-infected-recovered model to characterize the determinants of these dynamics and illustrate patterns by comparison with measles. We perform a systematic analysis of spectra of stochastic fluctuations around stable attractors of the corresponding deterministic model and compare them with spectra from full stochastic simulations in large populations. This approach allows us to quantify the effects of demographic stochasticity and to give a coherent picture of measles and rubella dynamics, explaining essential differences in the recurrent patterns exhibited by these diseases. We discuss the implications of our findings in the context of vaccination and changing birth rates as well as the persistence of these two childhood infections.

Contribution of seasonality in transmission of Mycobacterium tuberculosis to seasonality in tuberculosis disease: a simulation study

A seasonal rise in tuberculosis (TB) notifications has been confirmed in several studies. Here, we examined one hypothesis for its cause: increased transmission of TB during wintertime due to crowding. Seasonality analysis was performed on actual and simulated notifications of clustered TB cases, which are considered to be representative of recent transmission, diagnosed from 1993 to 2004 in the Netherlands (n = 4,746). To test the hypothesis of winter crowding, notifications were simulated by adding patient delay and incubation period to an infection date randomly taken to be in winter in 80% of cases. The incubation periods were derived from frequency distributions for different TB disease localizations drawn from the literature. Seasonality analysis was performed using autocorrelation function plots and spectral analysis. Actual notifications showed strong seasonality in clustered TB and clustered extrapulmonary TB cases but not in clustered pulmonary TB cases. Analysis of simulated notifications revealed barely significant seasonality only in extrapulmonary TB cases. Our results suggest that increased transmission of TB during wintertime is unlikely to be the only cause of the seasonal peak in TB notifications. A factor closer to the notification date probably contributes to the seasonality observed in TB notifications.

Persistence in epidemic metapopulations: quantifying the rescue effects for measles, mumps, rubella and whooping cough

Metapopulation rescue effects are thought to be key to the persistence of many acute immunizing infections. Yet the enhancement of persistence through spatial coupling has not been previously quantified. Here we estimate the metapopulation rescue effects for four childhood infections using global WHO reported incidence data by comparing persistence on island countries vs all other countries, while controlling for key variables such as vaccine cover, birth rates and economic development. The relative risk of extinction on islands is significantly higher, and approximately double the risk of extinction in mainland countries. Furthermore, as may be expected, infections with longer infectious periods tend to have the strongest metapopulation rescue effects. Our results quantitate the notion that demography and local community size controls disease persistence.

Think globally, act locally: the role of local demographics and vaccination coverage in the dynamic response of measles infection to control

The global reduction of the burden of morbidity and mortality owing to measles has been a major triumph of public health. However, the continued persistence of measles infection probably not only reflects local variation in progress towards vaccination target goals, but may also reflect local variation in dynamic processes of transmission, susceptible replenishment through births and stochastic local extinction. Dynamic models predict that vaccination should increase the mean age of infection and increase inter-annual variability in incidence. Through a comparative approach, we assess national-level patterns in the mean age of infection and measles persistence. We find that while the classic predictions do hold in general, the impact of vaccination on the age distribution of cases and stochastic fadeout are mediated by local birth rate. Thus, broad-scale vaccine coverage goals are unlikely to have the same impact on the interruption of measles transmission in all demographic settings. Indeed, these results suggest that the achievement of further measles reduction or elimination goals is likely to require programmatic and vaccine coverage goals that are tailored to local demographic conditions.

Probabilistic measures of persistence and extinction in measles (meta)populations

Persistence and extinction are fundamental processes in ecological systems that are difficult to accurately measure due to stochasticity and incomplete observation. Moreover, these processes operate on multiple scales, from individual populations to metapopulations. Here, we examine an extensive new data set of measles case reports and associated demographics in pre-vaccine era US cities, alongside a classic England & Wales data set. We first infer the per-population quasi-continuous distribution of log incidence. We then use stochastic, spatially implicit metapopulation models to explore the frequency of rescue events and apparent extinctions. We show that, unlike critical community size, the inferred distributions account for observational processes, allowing direct comparisons between metapopulations. The inferred distributions scale with population size. We use these scalings to estimate extinction boundary probabilities. We compare these predictions with measurements in individual populations and random aggregates of populations, highlighting the importance of medium-sized populations in metapopulation persistence.

Modeling the Spread and Outbreak Dynamics of Avian Influenza (H5N1) Virus and Its Possible Control

Avian Influenza, commonly known as Bird Flu, is an epidemic caused by H5N1 Virus, that primarily affects birds such as chickens, wild water birds, ducks, and swans etc. On rare occasions, pigs and humans will also be affected with this virus In recent years this epidemic has emerged as a major global health concern. The present chapter is aimed at developing mathematical models that predict the spread and outbreak diversity of low pathogenic avian influenza virus. Essentially, we present (1) a deterministic mathematical model which deals with the dynamics of human infection by avian influenza both in birds and in human, (2) a discrete dynamical model for the spread of H5N1, and (3) the statistical transmission model of bird flu taking into account the factors that affect the epidemic transmission such as source of infection and social and natural factors.

Measured dynamic social contact patterns explain the spread of H1N1v influenza

Patterns of social mixing are key determinants of epidemic spread. Here we present the results of an internet-based social contact survey completed by a cohort of participants over 9,000 times between July 2009 and March 2010, during the 2009 H1N1v influenza epidemic. We quantify the changes in social contact patterns over time, finding that school children make 40% fewer contacts during holiday periods than during term time. We use these dynamically varying contact patterns to parameterise an age-structured model of influenza spread, capturing well the observed patterns of incidence; the changing contact patterns resulted in a fall of approximately 35% in the reproduction number of influenza during the holidays. This work illustrates the importance of including changing mixing patterns in epidemic models. We conclude that changes in contact patterns explain changes in disease incidence, and that the timing of school terms drove the 2009 H1N1v epidemic in the UK. Changes in social mixing patterns can be usefully measured through simple internet-based surveys.

Changes in patterns of social mixing can result in changes in epidemic behaviour; this was observed during the 2009 influenza pandemic, in which the epidemic declined during school holidays and grew during term time. Until now, little information has been available to quantify how people's mixing patterns change over time. Here, we present the results of an internet-based survey of social mixing patterns that was carried out in the UK throughout the 2009 pandemic. We show that school holidays resulted in a substantial drop in the number of social contacts made each day, particularly between children. To test whether these measured patterns of social mixing could explain the observed epidemic, we used our mixing data in a simple mathematical model of influenza spread. We found that changing social contact behaviour could explain levels of infection in the community, and conclude that the timing of school terms was responsible for the shape of the influenza epidemic.

Agricultural intensification, priming for persistence and the emergence of Nipah virus: a lethal bat-borne zoonosis

Emerging zoonoses threaten global health, yet the processes by which they emerge are complex and poorly understood. Nipah virus (NiV) is an important threat owing to its broad host and geographical range, high case fatality, potential for human-to-human transmission and lack of effective prevention or therapies. Here, we investigate the origin of the first identified outbreak of NiV encephalitis in Malaysia and Singapore. We analyse data on livestock production from the index site (a commercial pig farm in Malaysia) prior to and during the outbreak, on Malaysian agricultural production, and from surveys of NiV's wildlife reservoir (flying foxes). Our analyses suggest that repeated introduction of NiV from wildlife changed infection dynamics in pigs. Initial viral introduction produced an explosive epizootic that drove itself to extinction but primed the population for enzootic persistence upon reintroduction of the virus. The resultant within-farm persistence permitted regional spread and increased the number of human infections. This study refutes an earlier hypothesis that anomalous El Niño Southern Oscillation-related climatic conditions drove emergence and suggests that priming for persistence drove the emergence of a novel zoonotic pathogen. Thus, we provide empirical evidence for a causative mechanism previously proposed as a precursor to widespread infection with H5N1 avian influenza and other emerging pathogens.

Converging towards the optimal path to extinction

Extinction appears ubiquitously in many fields, including chemical reactions, population biology, evolution and epidemiology. Even though extinction as a random process is a rare event, its occurrence is observed in large finite populations. Extinction occurs when fluctuations owing to random transitions act as an effective force that drives one or more components or species to vanish. Although there are many random paths to an extinct state, there is an optimal path that maximizes the probability to extinction. In this paper, we show that the optimal path is associated with the dynamical systems idea of having maximum sensitive dependence to initial conditions. Using the equivalence between the sensitive dependence and the path to extinction, we show that the dynamical systems picture of extinction evolves naturally towards the optimal path in several stochastic models of epidemics.

Maximal sensitive dependence and the optimal path to epidemic extinction

Extinction of an epidemic or a species is a rare event that occurs due to a large, rare stochastic fluctuation. Although the extinction process is dynamically unstable, it follows an optimal path that maximizes the probability of extinction. We show that the optimal path is also directly related to the finite-time Lyapunov exponents of the underlying dynamical system in that the optimal path displays maximum sensitivity to initial conditions. We consider several stochastic epidemic models, and examine the extinction process in a dynamical systems framework. Using the dynamics of the finite-time Lyapunov exponents as a constructive tool, we demonstrate that the dynamical systems viewpoint of extinction evolves naturally toward the optimal path.

Outbreak properties of epidemic models: the roles of temporal forcing and stochasticity on pathogen invasion dynamics

Despite temporally forced transmission driving many infectious diseases, analytical insight into its role when combined with stochastic disease processes and non-linear transmission has received little attention. During disease outbreaks, however, the absence of saturation effects early on in well-mixed populations mean that epidemic models may be linearised and we can calculate outbreak properties, including the effects of temporal forcing on fade-out, disease emergence and system dynamics, via analysis of the associated master equations. The approach is illustrated for the unforced and forced SIR and SEIR epidemic models. We demonstrate that in unforced models, initial conditions (and any uncertainty therein) play a stronger role in driving outbreak properties than the basic reproduction number R0, while the same properties are highly sensitive to small amplitude temporal forcing, particularly when R0 is small. Although illustrated for the SIR and SEIR models, the master equation framework may be applied to more realistic models, although analytical intractability scales rapidly with increasing system dimensionality. One application of these methods is obtaining a better understanding of the rate at which vector-borne and waterborne infectious diseases invade new regions given variability in environmental drivers, a particularly important question when addressing potential shifts in the global distribution and intensity of infectious diseases under climate change.

Persistence of canine distemper virus in the Greater Yellowstone ecosystem's carnivore community

Canine distemper virus (CDV) is an acute, highly immunizing pathogen that should require high densities and large populations of hosts for long-term persistence, yet CDV persists among terrestrial carnivores with small, patchily distributed groups. We used CDV in the Greater Yellowstone ecosystem's (GYE) wolves (Canis lupus) and coyotes (Canis latrans) as a case study for exploring how metapopulation structure, host demographics, and multi-host transmission affect the critical community size and spatial scale required for CDV persistence. We illustrate how host spatial connectivity and demographic turnover interact to affect both local epidemic dynamics, such as the length and variation in inter-epidemic periods, and pathogen persistence using stochastic, spatially explicit susceptible-exposed-infectious-recovered simulation models. Given the apparent absence of other known persistence mechanisms (e.g., a carrier or environmental state, densely populated host, chronic infection, or a vector), we suggest that CDV requires either large spatial scales or multi-host transmission for persistence. Current GYE wolf populations are probably too small to support endemic CDV. Coyotes are a plausible reservoir host, but CDV would still require 50000-100000 individuals for moderate persistence (> 50% over 10 years), which would equate to an area of 1-3 times the size of the GYE (60000-200000 km2). Coyotes, and carnivores in general, are not uniformly distributed; therefore, this is probably a gross underestimate of the spatial scale of CDV persistence. However, the presence of a second competent host species can greatly increase the probability of long-term CDV persistence at much smaller spatial scales. Although no management of CDV is currently recommended for the GYE, wolf managers in the region should expect periodic but unpredictable CDV-related population declines as often as every 2-5 years. Awareness and monitoring of such outbreaks will allow corresponding adjustments in management activities such as regulated public harvest, creating a smooth transition to state wolf management and conservation after > 30 years of being protected by the Endangered Species Act.

Decreasing stochasticity through enhanced seasonality in measles epidemics

Seasonal changes in the environment are known to be important drivers of population dynamics, giving rise to sustained population cycles. However, it is often difficult to measure the strength and shape of seasonal forces affecting populations. In recent years, statistical time-series methods have been applied to the incidence records of childhood infectious diseases in an attempt to estimate seasonal variation in transmission rates, as driven by the pattern of school terms. In turn, school-term forcing was used to show how susceptible influx rates affect the interepidemic period. In this paper, we document the response of measles dynamics to distinct shifts in the parameter regime using previously unexplored records of measles mortality from the early decades of the twentieth century. We describe temporal patterns of measles epidemics using spectral analysis techniques, and point out a marked decrease in birth rates over time. Changes in host demography alone do not, however, suffice to explain epidemiological transitions. By fitting the time-series susceptible-infected-recovered model to measles mortality data, we obtain estimates of seasonal transmission in different eras, and find that seasonality increased over time. This analysis supports theoretical work linking complex population dynamics and the balance between stochastic and deterministic forces as determined by the strength of seasonality.

Rural-urban gradient in seasonal forcing of measles transmission in Niger

Seasonally driven cycles of incidence have been consistently observed for a range of directly transmitted pathogens. Though frequently observed, the mechanism of seasonality for directly transmitted human pathogens is rarely well understood. Despite significant annual variation in magnitude, measles outbreaks in Niger consistently begin in the dry season and decline at the onset of the seasonal rains. We estimate the seasonal fluctuation in measles transmission rates for the 38 districts and urban centres of Niger, from 11 years of weekly incidence reports. We show that transmission rates are consistently in anti-phase to the rainfall patterns across the country. The strength of the seasonal forcing of transmission is not correlated with the latitudinal rainfall gradient, as would be expected if transmission rates were determined purely by environmental conditions. Rather, seasonal forcing is correlated with the population size, with larger seasonal fluctuation in more populous, urban areas. This pattern is consistent with seasonal variation in human density and contact rates due to agricultural cycles. The stronger seasonality in large cities drives deep inter-epidemic troughs and results in frequent local extinction of measles, which contrasts starkly to the conventional observation that large cities, by virtue of their size, act as reservoirs of measles.

Stochasticity in staged models of epidemics: quantifying the dynamics of whooping cough

Although many stochastic models can accurately capture the qualitative epidemic patterns of many childhood diseases, there is still considerable discussion concerning the basic mechanisms generating these patterns; much of this stems from the use of deterministic models to try to understand stochastic simulations. We argue that a systematic method of analysing models of the spread of childhood diseases is required in order to consistently separate out the effects of demographic stochasticity, external forcing and modelling choices. Such a technique is provided by formulating the models as master equations and using the van Kampen system-size expansion to provide analytical expressions for quantities of interest. We apply this method to the susceptible-exposed-infected-recovered (SEIR) model with distributed exposed and infectious periods and calculate the form that stochastic oscillations take on in terms of the model parameters. With the use of a suitable approximation, we apply the formalism to analyse a model of whooping cough which includes seasonal forcing. This allows us to more accurately interpret the results of simulations and to make a more quantitative assessment of the predictions of the model. We show that the observed dynamics are a result of a macroscopic limit cycle induced by the external forcing and resonant stochastic oscillations about this cycle.

Stochastic epidemic models: a survey

This paper is a survey paper on stochastic epidemic models. A simple stochastic epidemic model is defined and exact and asymptotic (relying on a large community) properties are presented. The purpose of modelling is illustrated by studying effects of vaccination and also in terms of inference procedures for important parameters, such as the basic reproduction number and the critical vaccination coverage. Several generalizations towards realism, e.g. multitype and household epidemic models, are also presented, as is a model for endemic diseases.

Influenza A gradual and epochal evolution: insights from simple models

The recurrence of influenza A epidemics has originally been explained by a “continuous antigenic drift” scenario. Recently, it has been shown that if genetic drift is gradual, the evolution of influenza A main antigen, the haemagglutinin, is punctuated. As a consequence, it has been suggested that influenza A dynamics at the population level should be approximated by a serial model. Here, simple models are used to test whether a serial model requires gradual antigenic drift within groups of strains with the same antigenic properties (antigenic clusters). We compare the effect of status based and history based frameworks and the influence of reduced susceptibility and infectivity assumptions on the transient dynamics of antigenic clusters. Our results reveal that the replacement of a resident antigenic cluster by a mutant cluster, as observed in data, is reproduced only by the status based model integrating the reduced infectivity assumption. This combination of assumptions is useful to overcome the otherwise extremely high model dimensionality of models incorporating many strains, but relies on a biological hypothesis not obviously satisfied. Our findings finally suggest the dynamical importance of gradual antigenic drift even in the presence of punctuated immune escape. A more regular renewal of susceptible pool than the one implemented in a serial model should be part of a minimal theory for influenza at the population level.

Resolving the impact of waiting time distributions on the persistence of measles

Measles epidemics in human populations exhibit what is perhaps the best empirically characterized, and certainly the most studied, stochastic persistence threshold in population biology. A critical community size (CCS) of around 250,000-500,000 separates populations where measles is predominantly persistent from smaller communities where there are frequent extinctions of measles between major epidemics. The fundamental mechanisms contributing to this pattern of persistence, which are long-lasting immunity to re-infection, recruitment of susceptibles, seasonality in transmission, age dependence of transmission and the spatial coupling between communities, have all been quantified and, to a greater or lesser level of success, captured by theoretical models. Despite these successes there has not been a consensus over whether simple models can successfully predict the value of the CCS, or indeed which mechanisms determine the persistence of measles over a broader range of population sizes. Specifically, the level of the CCS has been thought to be particularly sensitive to the detailed stochastic dynamics generated by the waiting time distribution (WTD) in the infectious and latent periods. We show that the relative patterns of persistence between models with different WTDs are highly sensitive to the criterion of comparison-in particular, the statistical measure of persistence that is employed. To this end, we introduce two new statistical measures of persistence-fade-outs post epidemic and fade-outs post invasion. Contrary to previous reports, we demonstrate that, no matter the choice of persistence measure, appropriately parametrized models of measles demonstrate similar predictions for the level of the CCS.