Estimability and interpretation of vaccine efficacy using frailty mixing models

Quantifying the Individual Variation in Susceptibility to Endemic Coronavirus and SARS-CoV-2 with Human Challenge Trials

Human challenge trials reveal how the infection risk depends on a given infectious dose. We propose a mathematical framework to analyze and interpret the outcomes of human challenge trials by incorporating the variability between individuals in susceptibility to infection. We illustrate the framework for two distinctive diseases; endemic diseases where a fraction of the study population has been exposed to the target pathogen previously and is thus immune, and novel diseases where the study population is fully susceptible. Based on available data from published trials, we estimate the immune proportion and the variation in susceptibility to endemic HCoV-229E and present plausible infection risks with SARS-CoV-2 over multiple orders of magnitude of the infectious dose. The results show that the proposed method captures heterogeneous background susceptibility in the study population, and we suggest ways to improve the design of future trials and to translate their outcomes to the general population.

Competing Heterogeneities in Vaccine Effectiveness Estimation

Understanding the waning of vaccine-induced protection is important for both immunology and public health. Population heterogeneities in underlying (pre-vaccination) susceptibility and vaccine response can cause measured vaccine effectiveness (mVE) to change over time, even in the absence of pathogen evolution and any actual waning of immune responses. We use multi-scale agent-based models parameterized using epidemiological and immunological data, to investigate the effect of these heterogeneities on mVE as measured by the hazard ratio. Based on our previous work, we consider the waning of antibodies according to a power law and link it to protection in two ways: (1) motivated by correlates of risk data and (2) using a within-host model of stochastic viral extinction. The effect of the heterogeneities is given by concise and understandable formulas, one of which is essentially a generalization of Fisher's fundamental theorem of natural selection to include higher derivatives. Heterogeneity in underlying susceptibility accelerates apparent waning, whereas heterogeneity in vaccine response slows down apparent waning. Our models suggest that heterogeneity in underlying susceptibility is likely to dominate. However, heterogeneity in vaccine response offsets <10% to >100% (median of 29%) of this effect in our simulations. Our study suggests heterogeneity is more likely to 'bias' mVE downwards towards the faster waning of immunity but a subtle bias in the opposite direction is also plausible.

Identification of Vaccine Effects When Exposure Status Is Unknown

Results from randomized controlled trials (RCTs) help determine vaccination strategies and related public health policies. However, defining and identifying estimands that can guide policies in infectious disease settings is difficult, even in an RCT. The effects of vaccination critically depend on characteristics of the population of interest, such as the prevalence of infection, the number of vaccinated, and social behaviors. To mitigate the dependence on such characteristics, estimands, and study designs, that require conditioning or intervening on exposure to the infectious agent have been advocated. But a fundamental problem for both RCTs and observational studies is that exposure status is often unavailable or difficult to measure, which has made it impossible to apply existing methodology to study vaccine effects that account for exposure status. In this study, we present new results on this type of vaccine effects. Under plausible conditions, we show that point identification of certain relative effects is possible even when the exposure status is unknown. Furthermore, we derive sharp bounds on the corresponding absolute effects. We apply these results to estimate the effects of the ChAdOx1 nCoV-19 vaccine on SARS-CoV-2 disease (COVID-19) conditional on postvaccine exposure to the virus, using data from a large RCT.

A Systematic Review on COVID-19 Vaccine Strategies, Their Effectiveness, and Issues

COVID-19 vaccines are indispensable, with the number of cases and mortality still rising, and currently no medicines are routinely available for reducing morbidity and mortality, apart from dexamethasone, although others are being trialed and launched. To date, only a limited number of vaccines have been given emergency use authorization by the US Food and Drug Administration and the European Medicines Agency. There is a need to systematically review the existing vaccine candidates and investigate their safety, efficacy, immunogenicity, unwanted events, and limitations. The review was undertaken by searching online databases, i.e., Google Scholar, PubMed, and ScienceDirect, with finally 59 studies selected. Our findings showed several types of vaccine candidates with different strategies against SARS-CoV-2, including inactivated, mRNA-based, recombinant, and nanoparticle-based vaccines, are being developed and launched. We have compared these vaccines in terms of their efficacy, side effects, and seroconversion based on data reported in the literature. We found mRNA vaccines appeared to have better efficacy, and inactivated ones had fewer side effects and similar seroconversion in all types of vaccines. Overall, global variant surveillance and systematic tweaking of vaccines, coupled with the evaluation and administering vaccines with the same or different technology in successive doses along with homologous and heterologous prime-booster strategy, have become essential to impede the pandemic. Their effectiveness appreciably outweighs any concerns with any adverse events.

Spotlight influenza: Estimation of influenza vaccine effectiveness in elderly people with assessment of residual confounding by negative control outcomes, Finland, 2012/13 to 2019/20

Background

Cohort studies on vaccine effectiveness are prone to confounding bias if the distribution of risk factors is unbalanced between vaccinated and unvaccinated study subjects.

Aim

We aimed to estimate influenza vaccine effectiveness in the elderly population in Finland by controlling for a sufficient set of confounders based on routinely available register data.

Methods

For each of the eight consecutive influenza seasons from 2012/13 through 2019/20, we conducted a cohort study comparing the hazards of laboratory-confirmed influenza in vaccinated and unvaccinated people aged 65–100 years using individual-level medical and demographic data. Vaccine effectiveness was estimated as 1 minus the hazard ratio adjusted for the confounders age, sex, vaccination history, nights hospitalised in the past and presence of underlying chronic conditions. To assess the adequacy of the selected set of confounders, we estimated hazard ratios of off-season hospitalisation for acute respiratory infection as a negative control outcome.

Results

Each analysed cohort comprised around 1 million subjects, of whom 37% to 49% were vaccinated. Vaccine effectiveness against laboratory-confirmed influenza ranged from 16% (95% confidence interval (CI): 12–19) to 48% (95% CI: 41–54). More than 80% of the laboratory-confirmed cases were hospitalised. The adjusted off-season hazard ratio estimates varied between 1.00 (95% CI: 0.94–1.05) and 1.08 (95% CI: 1.01–1.15), indicating that residual confounding was absent or negligible.

Conclusion

Seasonal influenza vaccination reduces the hazard of severe influenza disease in vaccinated elderly people. Data about age, sex, vaccination history and utilisation of hospital care proved sufficient to control confounding.

Heterogeneity in susceptibility dictates the order of epidemic models

The fundamental models of epidemiology describe the progression of an infectious disease through a population using compartmentalized differential equations, but typically do not incorporate population-level heterogeneity in infection susceptibility. Here we combine a generalized analytical framework of contagion with computational models of epidemic dynamics to show that variation strongly influences the rate of infection, while the infection process simultaneously sculpts the susceptibility distribution. These joint dynamics influence the force of infection and are, in turn, influenced by the shape of the initial variability. We find that certain susceptibility distributions (the exponential and the gamma) are unchanged through the course of the outbreak, and lead naturally to power-law behavior in the force of infection; other distributions are often sculpted towards these "eigen-distributions" through the process of contagion. The power-law behavior fundamentally alters predictions of the long-term infection rate, and suggests that first-order epidemic models that are parameterized in the exponential-like phase may systematically and significantly over-estimate the final severity of the outbreak. In summary, our study suggests the need to examine the shape of susceptibility in natural populations as part of efforts to improve prediction models and to prioritize interventions that leverage heterogeneity to mitigate against spread.

Interpreting vaccine efficacy trial results for infection and transmission

Randomized controlled trials (RCTs) have shown high efficacy of multiple vaccines against SARS-CoV-2 disease (COVID-19), and recent studies have shown the vaccines are also effective against infection. Evidence for the effect of each of these vaccines on ability to transmit the virus is also beginning to emerge. We describe an approach to estimate these vaccines' effects on viral positivity, a prevalence measure which under the reasonable assumption that vaccinated individuals who become infected are no more infectious than unvaccinated individuals forms a lower bound on efficacy against transmission. Specifically, we recommend separate analysis of positive tests triggered by symptoms (usually the primary RCT outcome) and cross-sectional prevalence of positive tests obtained regardless of symptoms. The odds ratio of carriage for vaccine vs. placebo provides an unbiased estimate of vaccine effectiveness against viral positivity, under certain assumptions, and we show through simulations that likely departures from these assumptions will only modestly bias this estimate. Applying this approach to published data from the RCT of the Moderna vaccine, we estimate that one dose of vaccine reduces the potential for transmission by at least 61%, possibly considerably more. We describe how these approaches can be translated into observational studies of vaccine effectiveness.

Estimating vaccine efficacy over time after a randomized study is unblinded

The COVID‐19 pandemic due to the novel coronavirus SARS CoV‐2 has inspired remarkable breakthroughs in the development of vaccines against the virus and the launch of several phase 3 vaccine trials in Summer 2020 to evaluate vaccine efficacy (VE). Trials of vaccine candidates using mRNA delivery systems developed by Pfizer‐BioNTech and Moderna have shown substantial VEs of 94–95%, leading the US Food and Drug Administration to issue Emergency Use Authorizations and subsequent widespread administration of the vaccines. As the trials continue, a key issue is the possibility that VE may wane over time. Ethical considerations dictate that trial participants be unblinded and those randomized to placebo be offered study vaccine, leading to trial protocol amendments specifying unblinding strategies. Crossover of placebo subjects to vaccine complicates inference on waning of VE. We focus on the particular features of the Moderna trial and propose a statistical framework based on a potential outcomes formulation within which we develop methods for inference on potential waning of VE over time and estimation of VE at any postvaccination time. The framework clarifies assumptions made regarding individual‐ and population‐level phenomena and acknowledges the possibility that subjects who are more or less likely to become infected may be crossed over to vaccine differentially over time. The principles of the framework can be adapted straightforwardly to other trials.

Interpreting vaccine efficacy trial results for infection and transmission

Randomized controlled trials (RCTs) have shown high efficacy of multiple vaccines against SARS-CoV-2 disease (COVID-19), and recent studies have shown the vaccines are also effective against infection. Evidence for the effect of each of these vaccines on ability to transmit the virus is also beginning to emerge. We describe an approach to estimate these vaccines' effects on viral positivity, a prevalence measure which under the reasonable assumption that vaccinated individuals who become infected are no more infectious than unvaccinated individuals forms a lower bound on efficacy against transmission. Specifically, we recommend separate analysis of positive tests triggered by symptoms (usually the primary outcome) and cross-sectional prevalence of positive tests obtained regardless of symptoms. The odds ratio of carriage for vaccine vs. placebo provides an unbiased estimate of vaccine effectiveness against viral positivity, under certain assumptions, and we show through simulations that likely departures from these assumptions will only modestly bias this estimate. Applying this approach to published data from the RCT of the Moderna vaccine, we estimate that one dose of vaccine reduces the potential for transmission by at least 61%, possibly considerably more. We describe how these approaches can be translated into observational studies of vaccine effectiveness.

Mitigation of biases in estimating hazard ratios under non-sensitive and non-specific observation of outcomes-applications to influenza vaccine effectiveness

Background

Non-sensitive and non-specific observation of outcomes in time-to-event data affects event counts as well as the risk sets, thus, biasing the estimation of hazard ratios. We investigate how imperfect observation of incident events affects the estimation of vaccine effectiveness based on hazard ratios.

Methods

Imperfect time-to-event data contain two classes of events: a portion of the true events of interest; and false-positive events mistakenly recorded as events of interest. We develop an estimation method utilising a weighted partial likelihood and probabilistic deletion of false-positive events and assuming the sensitivity and the false-positive rate are known. The performance of the method is evaluated using simulated and Finnish register data.

Results

The novel method enables unbiased semiparametric estimation of hazard ratios from imperfect time-to-event data. False-positive rates that are small can be approximated to be zero without inducing bias. The method is robust to misspecification of the sensitivity as long as the ratio of the sensitivity in the vaccinated and the unvaccinated is specified correctly and the cumulative risk of the true event is small.

Conclusions

The weighted partial likelihood can be used to adjust for outcome measurement errors in the estimation of hazard ratios and effectiveness but requires specifying the sensitivity and the false-positive rate. In absence of exact information about these parameters, the method works as a tool for assessing the potential magnitude of bias given a range of likely parameter values.

Disentangling non-specific and specific transgenerational immune priming components in host-parasite interactions

Exposure to a pathogen primes many organisms to respond faster or more efficiently to subsequent exposures. Such priming can be non-specific or specific, and has been found to extend across generations. Disentangling and quantifying specific and non-specific effects is essential for understanding the genetic epidemiology of a system. By combining a large infection experiment and mathematical modelling, we disentangle different transgenerational effects in the crustacean model Daphnia magna exposed to different strains of the bacterial parasite Pasteuria ramosa. In the experiment, we exposed hosts to a high dose of one of three parasite strains, and subsequently challenged their offspring with multiple doses of the same (homologous) or a different (heterologous) strain. We find that exposure of Daphnia to Pasteuria decreases the susceptibility of their offspring by approximately 50%. This transgenerational protection is not larger for homologous than for heterologous parasite challenges. Methodologically, our work represents an important contribution not only to the analysis of immune priming in ecological systems but also to the experimental assessment of vaccines. We present, for the first time, an inference framework to investigate specific and non-specific effects of immune priming on the susceptibility distribution of hosts—effects that are central to understanding immunity and the effect of vaccines.

Depletion-of-susceptibles bias in influenza vaccine waning studies: how to ensure robust results

Vaccine effectiveness studies are subject to biases due to depletion-of-persons at risk of infection, or at especially high risk of infection, at different rates from different groups (depletion-of-susceptibles bias), a problem that can also lead to biased estimates of waning effectiveness, including spurious inference of waning when none exists. An alternative study design to identify waning is to study only vaccinated persons, and compare for each day the incidence in persons with earlier or later dates of vaccination to assess waning in vaccine protection as a function of vaccination time (namely whether earlier vaccination would result in lower subsequent protection compared to later vaccination). Prior studies suggested under what conditions this alternative would yield correct estimates of waning. Here we define the depletion-of-susceptibles process formally and show mathematically that for influenza vaccine waning studies, a randomised trial or corresponding observational study that compares incidence at a specific calendar time among individuals vaccinated at different times before the influenza season begins will not be vulnerable to depletion-of-susceptibles bias in its inference of waning as a function of vaccination time under the null hypothesis that none exists, and will - if waning does actually occur - underestimate the extent of waning. Such a design is thus robust in the sense that a finding of waning in that inference framework reflects actual waning of vaccine-induced immunity. We recommend such a design for future studies of waning, whether observational or randomised.

Heterogeneous susceptibility to rotavirus infection and gastroenteritis in two birth cohort studies: Parameter estimation and epidemiological implications

Cohort studies, randomized trials, and post-licensure studies have reported reduced natural and vaccine-derived protection against rotavirus gastroenteritis (RVGE) in low- and middle-income countries. While susceptibility of children to rotavirus is known to vary within and between settings, implications for estimation of immune protection are not well understood. We sought to re-estimate naturally-acquired protection against rotavirus infection and RVGE, and to understand how differences in susceptibility among children impacted estimates. We re-analyzed data from studies conducted in Mexico City, Mexico and Vellore, India. Cumulatively, 573 rotavirus-unvaccinated children experienced 1418 rotavirus infections and 371 episodes of RVGE over 17,636 child-months. We developed a model that characterized susceptibility to rotavirus infection and RVGE among children, accounting for aspects of the natural history of rotavirus and differences in transmission rates between settings. We tested whether model-generated susceptibility measurements were associated with demographic and anthropometric factors, and with the severity of RVGE symptoms. We identified greater variation in susceptibility to rotavirus infection and RVGE in Vellore than in Mexico City. In both cohorts, susceptibility to rotavirus infection and RVGE were associated with male sex, lower birth weight, lower maternal education, and having fewer siblings; within Vellore, susceptibility was also associated with lower socioeconomic status. Children who were more susceptible to rotavirus also experienced higher rates of rotavirus-negative diarrhea, and higher risk of moderate-to-severe symptoms when experiencing RVGE. Simulations suggested that discrepant estimates of naturally-acquired immunity against RVGE can be attributed, in part, to between-setting differences in susceptibility of children, but result primarily from the interaction of transmission rates with age-dependent risk for infections to cause RVGE. We found that more children in Vellore than in Mexico City belong to a high-risk group for rotavirus infection and RVGE, and demonstrate that unmeasured individual- and age-dependent susceptibility may influence estimates of naturally-acquired immune protection against RVGE.

Differences in susceptibility can help explain why some individuals, and not others, acquire infection and exhibit symptoms when exposed to infectious disease agents. However, it is difficult to distinguish between differences in susceptibility versus exposure in epidemiological studies. We developed a modeling approach to distinguish transmission intensity and susceptibility in data from cohort studies of rotavirus infection among children in Mexico City, Mexico, and Vellore, India, and evaluated how these factors may have contributed to differences in estimates of naturally-acquired immune protection between the studies. Given the same exposure, more children were at high risk of acquiring rotavirus infection, and of experiencing gastroenteritis when infected, in Vellore than in Mexico City. The probability of belonging to this high-risk stratum was associated with well-known individual factors such as lower socioeconomic status, lower birth weight, and incidence of diarrhea due to other causes. We also found the risk for rotavirus infections to cause symptoms declined with age, independent of acquired immunity. These findings can, in part, account for estimates of lower protective efficacy of acquired immunity against rotavirus gastroenteritis in high-incidence settings, mirroring estimates of reduced effectiveness of live oral rotavirus vaccines in low- and middle-income countries.

Biased efficacy estimates in phase-III dengue vaccine trials due to heterogeneous exposure and differential detectability of primary infections across trial arms

Vaccine efficacy (VE) estimates are crucial for assessing the suitability of dengue vaccine candidates for public health implementation, but efficacy trials are subject to a known bias to estimate VE toward the null if heterogeneous exposure is not accounted for in the analysis of trial data. In light of many well-characterized sources of heterogeneity in dengue virus (DENV) transmission, our goal was to estimate the potential magnitude of this bias in VE estimates for a hypothetical dengue vaccine. To ensure that we realistically modeled heterogeneous exposure, we simulated city-wide DENV transmission and vaccine trial protocols using an agent-based model calibrated with entomological and epidemiological data from long-term field studies in Iquitos, Peru. By simulating a vaccine with a true VE of 0.8 in 1,000 replicate trials each designed to attain 90% power, we found that conventional methods underestimated VE by as much as 21% due to heterogeneous exposure. Accounting for the number of exposures in the vaccine and placebo arms eliminated this bias completely, and the more realistic option of including a frailty term to model exposure as a random effect reduced this bias partially. We also discovered a distinct bias in VE estimates away from the null due to lower detectability of primary DENV infections among seronegative individuals in the vaccinated group. This difference in detectability resulted from our assumption that primary infections in vaccinees who are seronegative at baseline resemble secondary infections, which experience a shorter window of detectable viremia due to a quicker immune response. This resulted in an artefactual finding that VE estimates for the seronegative group were approximately 1% greater than for the seropositive group. Simulation models of vaccine trials that account for these factors can be used to anticipate the extent of bias in field trials and to aid in their interpretation.

Measurement of Vaccine Direct Effects Under the Test-Negative Design

Test-negative designs are commonplace in assessments of influenza vaccination effectiveness, estimating this value from the exposure odds ratio of vaccination among individuals treated for acute respiratory illness who test positive for influenza virus infection. This approach is widely believed to recover the vaccine direct effect by correcting for differential health-care-seeking behavior among vaccinated and unvaccinated persons. However, the relationship of the measured odds ratio to true vaccine effectiveness is poorly understood. We derived the odds ratio under circumstances of real-world test-negative studies. The odds ratio recovers the vaccine direct effect when 2 conditions are met: 1) Individuals' vaccination decisions are uncorrelated with exposure or susceptibility to the test-positive or test-negative conditions, and 2) vaccination confers "all-or-nothing" protection (whereby certain individuals have no protection while others are perfectly protected). Biased effect-size estimates arise if either condition is unmet. Such bias might suggest misleading associations of vaccine effectiveness with time since vaccination or the force of infection of influenza. The test-negative design could also fail to correct for differential health-care-seeking behavior among vaccinated and unvaccinated persons without stringent criteria for enrollment and testing. Our findings demonstrate a need to reassess how data from test-negative studies can inform policy decisions.

Temporally Varying Relative Risks for Infectious Diseases: Implications for Infectious Disease Control

Risks for disease in some population groups relative to others (relative risks) are usually considered to be consistent over time, although they are often modified by other, nontemporal factors. For infectious diseases, in which overall incidence often varies substantially over time, the patterns of temporal changes in relative risks can inform our understanding of basic epidemiologic questions. For example, recent studies suggest that temporal changes in relative risks of infection over the course of an epidemic cycle can both be used to identify population groups that drive infectious disease outbreaks, and help elucidate differences in the effect of vaccination against infection (that is relevant to transmission control) compared with its effect against disease episodes (that reflects individual protection). Patterns of change in the age groups affected over the course of seasonal outbreaks can provide clues to the types of pathogens that could be responsible for diseases for which an infectious cause is suspected. Changing apparent efficacy of vaccines during trials may provide clues to the vaccine's mode of action and/or indicate risk heterogeneity in the trial population. Declining importance of unusual behavioral risk factors may be a signal of increased local transmission of an infection. We review these developments and the related public health implications.

Impact of stochastically generated heterogeneity in hazard rates on individually randomized vaccine efficacy trials

Background/aims Network structure and individuals' level of exposure to a pathogen can impact results from efficacy evaluation studies of interventions against infectious diseases. Heterogeneity in infection risk can cause randomized groups to increasingly differ as a trial progresses and as more high-risk individuals become infected (described in prior work as the "frailty" phenomenon). Here, we show the impact this phenomenon can have on an individually randomized trial of a leaky vaccine in which all participants are exchangeable a priori. Methods We model a vaccine trial by generating a network of individuals grouped into communities, which are connected to a larger main population. We then simulate an epidemic, deterministically and with time-varying transmission rates in the main population and stochastically in the communities. The disease natural history follows a susceptible-exposed-infectious-recovered model. Simulation results are used to estimate vaccine efficacy [Formula: see text] with a Cox proportional hazards model. Results We find downward bias in [Formula: see text] associated with low connectivity between communities in the study population and high force of infection, even when all participants in the trial are exchangeable at the time of randomization. This phenomenon arises because the stochastic dynamics in such a setting randomly lead to community-level variation in the force of infection. Stratifying a Cox model by community alleviates this bias with no loss of power. Conclusion Understanding and accounting for the impact of heterogeneous hazard rates can allow for more accurate estimates of [Formula: see text] in epidemic settings.

Vaccine Effects on Heterogeneity in Susceptibility and Implications for Population Health Management

Heterogeneity in host susceptibility is a key determinant of infectious disease dynamics but is rarely accounted for in assessment of disease control measures. Understanding how susceptibility is distributed in populations, and how control measures change this distribution, is integral to predicting the course of epidemics with and without interventions. Using multiple experimental and modeling approaches, we show that rainbow trout have relatively homogeneous susceptibility to infection with infectious hematopoietic necrosis virus and that vaccination increases heterogeneity in susceptibility in a nearly all-or-nothing fashion. In a simple transmission model with an R₀ of 2, the highly heterogeneous vaccine protection would cause a 35 percentage-point reduction in outbreak size over an intervention inducing homogenous protection at the same mean level. More broadly, these findings provide validation of methodology that can help to reduce biases in predictions of vaccine impact in natural settings and provide insight into how vaccination shapes population susceptibility.

Differences among individuals influence transmission and spread of infectious diseases as well as the effectiveness of control measures. Control measures, such as vaccines, may provide leaky protection, protecting all hosts to an identical degree, or all-or-nothing protection, protecting some hosts completely while leaving others completely unprotected. This distinction can have a dramatic influence on disease dynamics, yet this distribution of protection is frequently unaccounted for in epidemiological models and estimates of vaccine efficacy. Here, we apply new methodology to experimentally examine host heterogeneity in susceptibility and mode of vaccine action as distinct components influencing disease outcome. Through multiple experiments and new modeling approaches, we show that the distribution of vaccine effects can be robustly estimated. These results offer new experimental and inferential methodology that can improve predictions of vaccine effectiveness and have broad applicability to human, wildlife, and ecosystem health.

Displacement of sexual partnerships in trials of sexual behavior interventions: A model-based assessment of consequences

We investigated the impact of the displacement of sexual activity from adherent recipients of an intervention to others within or outside a trial population on the results from hypothetical trials of different sexual behavior interventions. A short-term model of HIV-prevention interventions that lead to female rejection of male partnership requests showed the impact of displacement expected at the start of a trial. An agent-based model, with sexual mixing and other South African specific demographics, evaluated consequences of displacement for sexual behavior interventions targeting young females in South Africa. This model measured the cumulative incidence among adherent, non-adherent, control and non-enrolled females in a hypothetical trial of HIV prevention. When males made more than one attempt to seek a partnership, interventions reduced short-term HIV infection risk among adherent females, but increased it among non-adherent females as well as controls, non-enrolled (females eligible for the trial but not chosen to participate) and ineligible females (females that did not qualify for the trial due to age). The impact of displacement depends on the intervention and the adherence. In both models, the risk to individuals who are not members of the adherent intervention group will increase with displacement leading to a biased calculation for the effect estimates for the trial. Likewise, intent-to-treat effect estimates become nonlinear functions of the proportion adherent.

Transmission dynamics of Ebola virus disease and intervention effectiveness in Sierra Leone

Sierra Leone is the most severely affected country by an unprecedented outbreak of Ebola virus disease (EVD) in West Africa. Although successfully contained, the transmission dynamics of EVD and the impact of interventions in the country remain unclear. We established a database of confirmed and suspected EVD cases from May 2014 to September 2015 in Sierra Leone and mapped the spatiotemporal distribution of cases at the chiefdom level. A Poisson transmission model revealed that the transmissibility at the chiefdom level, estimated as the average number of secondary infections caused by a patient per week, was reduced by 43% [95% confidence interval (CI): 30%, 52%] after October 2014, when the strategic plan of the United Nations Mission for Emergency Ebola Response was initiated, and by 65% (95% CI: 57%, 71%) after the end of December 2014, when 100% case isolation and safe burials were essentially achieved, both compared with before October 2014. Population density, proximity to Ebola treatment centers, cropland coverage, and atmospheric temperature were associated with EVD transmission. The household secondary attack rate (SAR) was estimated to be 0.059 (95% CI: 0.050, 0.070) for the overall outbreak. The household SAR was reduced by 82%, from 0.093 to 0.017, after the nationwide campaign to achieve 100% case isolation and safe burials had been conducted. This study provides a complete overview of the transmission dynamics of the 2014-2015 EVD outbreak in Sierra Leone at both chiefdom and household levels. The interventions implemented in Sierra Leone seem effective in containing the epidemic, particularly in interrupting household transmission.

Estimation and interpretation of heterogeneous vaccine efficacy against recurrent infections

Vaccine-induced protection may not be homogeneous across individuals. It is possible that a vaccine gives complete protection for a portion of individuals, while the rest acquire only incomplete (leaky) protection of varying magnitude. If vaccine efficacy is estimated under wrong assumptions about such individual level heterogeneity, the resulting estimates may be difficult to interpret. For instance, population-level predictions based on such estimates may be biased. We consider the problem of estimating heterogeneous vaccine efficacy against an infection that can be acquired multiple times (susceptible-infected-susceptible model). The estimation is based on a limited number of repeated measurements of the current status of each individual, a situation commonly encountered in practice. We investigate how the placement of consecutive samples affects the estimability and efficiency of vaccine efficacy parameters. The same sampling frequency may not be optimal for efficient estimation of all components of heterogeneous vaccine protection. However, we suggest practical guidelines allowing estimation of all components. For situations in which the estimability of individual components fails, we suggest to use summary measures of vaccine efficacy.

Vaccination Programs for Endemic Infections: Modelling Real versus Apparent Impacts of Vaccine and Infection Characteristics

Vaccine effect, as measured in clinical trials, may not accurately reflect population-level impact. Furthermore, little is known about how sensitive apparent or real vaccine impacts are to factors such as the risk of re-infection or the mechanism of protection. We present a dynamic compartmental model to simulate vaccination for endemic infections. Several measures of effectiveness are calculated to compare the real and apparent impact of vaccination, and assess the effect of a range of infection and vaccine characteristics on these measures. Although broadly correlated, measures of real and apparent vaccine effectiveness can differ widely. Vaccine impact is markedly underestimated when primary infection provides partial natural immunity, when coverage is high and when post-vaccination infectiousness is reduced. Despite equivalent efficacy, 'all or nothing' vaccines are more effective than 'leaky' vaccines, particularly in settings with high risk of re-infection and transmissibility. Latent periods result in greater real impacts when risk of re-infection is high, but this effect diminishes if partial natural immunity is assumed. Assessments of population-level vaccine effects against endemic infections from clinical trials may be significantly biased, and vaccine and infection characteristics should be considered when modelling outcomes of vaccination programs, as their impact may be dramatic.

Influenza vaccine efficacy trials: a simulation approach to understand failures from the past

The success of a seasonal influenza vaccine efficacy trial depends not only upon the design but also upon the annual epidemic characteristics. In this context, simulation methods are an essential tool in evaluating the performances of study designs under various circumstances. However, traditional methods for simulating time-to-event data are not suitable for the simulation of influenza vaccine efficacy trials because of the seasonality and heterogeneity of influenza epidemics. Instead, we propose a mathematical model parameterized with historical surveillance data, heterogeneous frailty among the subjects, survey-based heterogeneous number of daily contact, and a mixed vaccine protection mechanism. We illustrate our methodology by generating multiple-trial data similar to a large phase III trial that failed to show additional relative vaccine efficacy of an experimental adjuvanted vaccine compared with the reference vaccine. We show that small departures from the designing assumptions, such as a smaller range of strain protection for the experimental vaccine or the chosen endpoint, could lead to smaller probabilities of success in showing significant relative vaccine efficacy.

The theoretical impact and cost-effectiveness of vaccines that protect against sexually transmitted infections and disease

Sexually transmitted diseases, a source of widespread morbidity and sometimes mortality, are caused by a diverse group of infections with a common route of transmission. Existing vaccines against hepatitis B virus (HBV) and human papilloma virus 16, 18, 6 and 11 are highly efficacious and cost effective. In reviewing the potential role for other vaccines against sexually transmitted infections (STIs) a series of questions needs to be addressed about the burden of disease, the potential characteristics of a new vaccine, and the impact of other interventions. These questions can be viewed in the light of the population dynamics of sexually transmitted infections as a group and how a vaccine can impact these dynamics. Mathematical models show the potential for substantial impact, especially if vaccines are widely used. To better make the case for sexually transmitted infection vaccines we need better data and analyses of the burden of disease, especially severe disease. However, cost effectiveness analyses using a wide range of assumptions show that STI vaccines would be cost effective and their development a worthwhile investment.

Unveiling time in dose-response models to infer host susceptibility to pathogens

The biological effects of interventions to control infectious diseases typically depend on the intensity of pathogen challenge. As much as the levels of natural pathogen circulation vary over time and geographical location, the development of invariant efficacy measures is of major importance, even if only indirectly inferrable. Here a method is introduced to assess host susceptibility to pathogens, and applied to a detailed dataset generated by challenging groups of insect hosts (Drosophila melanogaster) with a range of pathogen (Drosophila C Virus) doses and recording survival over time. The experiment was replicated for flies carrying the Wolbachia symbiont, which is known to reduce host susceptibility to viral infections. The entire dataset is fitted by a novel quantitative framework that significantly extends classical methods for microbial risk assessment and provides accurate distributions of symbiont-induced protection. More generally, our data-driven modeling procedure provides novel insights for study design and analyses to assess interventions.

While control options for plant, animal, and human pathogens are emerging rapidly, reliable assessment of the effect of interventions in biological systems presents many challenges. A major question is how to connect laboratory experiments and measurements with the relevant process in natural settings, where hosts are subject to pathogen exposures that vary in time and geographical location. With this aim, measures of protection that are invariant under varying exposure intensity need to be developed and integrated with mathematical models. In this article, we introduce a method to assess host susceptibility to pathogens, and apply it to survival of Drosophila melanogaster challenged with different doses of Drosophila C virus. By replicating the procedure in groups of flies that carry the symbiont Wolbachia, we are able to estimate how the viral protection induced by this intracellular bacterium is distributed in the host population. Our results disentangle host infection status from observed mortality, accounting naturally for time since exposure. The multiple-dose design proposed challenges traditional study designs to assess interventions.

Estimating the per-exposure effect of infectious disease interventions

The average effect of an infectious disease intervention (eg, a vaccine) varies across populations with different degrees of exposure to the pathogen. As a result, many investigators favor a per-exposure effect measure that is considered independent of the population level of exposure and that can be used in simulations to estimate the total disease burden averted by an intervention across different populations. However, while per-exposure effects are frequently estimated, the quantity of interest is often poorly defined, and assumptions in its calculation are typically left implicit. In this article, we build upon work by Halloran and Struchiner (Epidemiology. 1995;6:142-151) to develop a formal definition of the per-exposure effect and discuss conditions necessary for its unbiased estimation. With greater care paid to the parameterization of transmission models, their results can be better understood and can thereby be of greater value to decision-makers.

Analysis of partial and complete protection in malaria cohort studies

Background

Malaria transmission is highly heterogeneous and analysis of incidence data must account for this for correct statistical inference. Less widely appreciated is the occurrence of a large number of zero counts (children without a malaria episode) in malaria cohort studies. Zero-inflated regression methods provide one means of addressing this issue, and also allow risk factors providing complete and partial protection to be disentangled.

Methods

Poisson, negative binomial (NB), zero-inflated Poisson (ZIP) and zero-inflated negative binomial (ZINB) regression models were fitted to data from two cohort studies of malaria in children in Ghana. Multivariate models were used to understand risk factors for elevated incidence of malaria and for remaining malaria-free, and to estimate the fraction of the population not at risk of malaria.

Results

ZINB models, which account for both heterogeneity in individual risk and an unexposed sub-group within the population, provided the best fit to data in both cohorts. These approaches gave additional insight into the mechanism of factors influencing the incidence of malaria compared to simpler approaches, such as NB regression. For example, compared to urban areas, rural residence was found to both increase the incidence rate of malaria among exposed children, and increase the probability of being exposed. In Navrongo, 34% of urban residents were estimated to be at no risk, compared to 3% of rural residents. In Kintampo, 47% of urban residents and 13% of rural residents were estimated to be at no risk.

Conclusion

These results illustrate the utility of zero-inflated regression methods for analysis of malaria cohort data that include a large number of zero counts. Specifically, these results suggest that interventions that reach mainly urban residents will have limited overall impact, since some urban residents are essentially at no risk, even in areas of high endemicity, such as in Ghana.

Estimating vaccine efficacy under the heterogeneity of vaccine action in a nonrandomly mixing population

Vaccines often have heterogeneous actions because of possible variation in the immune systems of hosts. One must consider such heterogeneity of vaccine action when developing a vaccine efficacy parameter. Addressing this issue the summary model of vaccine action has been proposed in the literature to estimate vaccine efficacy in a randomly mixing population. However, nonrandom mixing is common, particularly in a small-group-mixing population. This article extends the summary model of vaccine action to such a nonrandomly mixing population. The interpretation and estimation of the summary vaccine efficacy were discussed in light of other two models of vaccine action: the leaky and all-or-nothing model. Vaccine efficacy under all models is defined as the relative reduction in transmission probability due to vaccine. Estimation of the transmission probabilities is described based on a deterministic epidemic model of an acute transmitted disease. This article further discusses, based on the above vaccine models, the estimation of vaccination coverage required to control epidemic. Methods are illustrated using data simulated by considering different patterns of mixing and vaccine action. Results confirm that the summary model performs better than other two models when vaccine action is heterogeneous.

Synergism from combinations of infection-blocking malaria vaccines

Plasmodium falciparum infections present novel challenges for vaccine development, including parasite replication dynamics not previously encountered for viral pathogens, and enormous diversity in target antigens. These challenges are illustrated by using a mathematical model to describe the association between the proportion of pre-erythrocytic or blood-stage parasites eliminated by vaccine-induced immune responses and the proportion of infections prevented. It is hypothesized that due to the requirement for all sporozoites to be eliminated to prevent infection, combining infection-blocking vaccines that confer protection through different biological mechanisms could lead to synergistic combinations of efficacy. Vaccines targeting blood-stage parasites may also combine synergistically if they combine to reduce the parasite multiplication rate to below the threshold of 1.

Design questions for Streptococcus pneumoniae vaccine trials with a colonisation endpoint

Evaluation of vaccine efficacy for protection against colonisation (VEcol) with Streptococcus pneumoniae and other bacterial pathogens is often based on a cross-sectional study design, in which only one nasopharyngeal sample is obtained per study subject. Here we investigate the feasibility of this study design by investigating a number of practical design problems. Specific questions are related to the timing of colonisation measurement with respect to the time of vaccination, the adjustment for the within-host replacement of vaccine-type colonisation by the non-vaccine type pneumococci, and the impact of multiple serotype colonisation on VEcol estimation. We also discuss the issue of choosing the control vaccine, including comparison of two active pneumococcal vaccines, as well as the sample size and the statistical power of colonisation endpoint trials. In addition, the statistical design with the specific aim to include information about VEcol in the licensure process of new pneumococcal vaccine products is discussed.

The role of exposure history on HIV acquisition: insights from repeated low-dose challenge studies

To assess the efficacy of HIV vaccine candidates or preventive treatment, many research groups have started to challenge monkeys repeatedly with low doses of the virus. Such challenge data provide a unique opportunity to assess the importance of exposure history for the acquisition of the infection. I developed stochastic models to analyze previously published challenge data. In the mathematical models, I allowed for variation of the animals' susceptibility to infection across challenge repeats, or across animals. In none of the studies I analyzed, I found evidence for an immunizing effect of non-infecting challenges, and in most studies, there is no evidence for variation in the susceptibilities to the challenges across animals. A notable exception was a challenge experiment by Letvin et al. Sci Translat Med (2011) conducted with the strain SIVsmE660. The challenge data of this experiment showed significant susceptibility variation from animal-to-animal, which is consistent with previously established genetic differences between the involved animals. For the studies which did not show significant immunizing effects and susceptibility differences, I conducted a power analysis and could thus exclude a very strong immunization effect for some of the studies. These findings validate the assumption that non-infecting challenges do not immunize an animal — an assumption that is central in the argument that repeated low-dose challenge experiments increase the statistical power of preclinical HIV vaccine trials. They are also relevant for our understanding of the role of exposure history for HIV acquisition and forecasting the epidemiological spread of HIV.

Individuals are exposed to Human Immunodeficiency Virus (HIV) many times before they contract the virus. It is not known what an instance of exposure, which does not result in infection, does to the host. Frequent exposures to the virus are hypothesized to immunize an individual, and result in resistance to infection with HIV. This hypothesis may explain the resistance observed in some individuals despite frequent exposure to the virus. Since it is very difficult to monitor the HIV exposure and infection status of humans, this question is easier to address in animal models. I took data from previously published infection experiments of monkeys with Simian Immunodeficiency Virus (SIV) and analyzed them with newly developed mathematical models. I found that there is no evidence that challenging monkeys with the virus reduces their susceptibility to infection. These findings have important repercussions for the testing of HIV vaccines in monkeys, and also for our understanding of the role of exposure history for the acquisition of HIV.

Estimating individual exposure to malaria using local prevalence of malaria infection in the field

Background

Heterogeneity in malaria exposure complicates survival analyses of vaccine efficacy trials and confounds the association between immune correlates of protection and malaria infection in longitudinal studies. Analysis may be facilitated by taking into account the variability in individual exposure levels, but it is unclear how exposure can be estimated at an individual level.

Method and Findings

We studied three cohorts (Chonyi, Junju and Ngerenya) in Kilifi District, Kenya to assess measures of malaria exposure. Prospective data were available on malaria episodes, geospatial coordinates, proximity to infected and uninfected individuals and residence in predefined malaria hotspots for 2,425 individuals. Antibody levels to the malaria antigens AMA1 and MSP1₁₄₂ were available for 291 children from Junju. We calculated distance-weighted local prevalence of malaria infection within 1 km radius as a marker of individual's malaria exposure. We used multivariable modified Poisson regression model to assess the discriminatory power of these markers for malaria infection (i.e. asymptomatic parasitaemia or clinical malaria). The area under the receiver operating characteristic (ROC) curve was used to assess the discriminatory power of the models. Local malaria prevalence within 1 km radius and AMA1 and MSP1₁₄₂ antibodies levels were independently associated with malaria infection. Weighted local malaria prevalence had an area under ROC curve of 0.72 (95%CI: 0.66–0.73), 0.71 (95%CI: 0.69–0.73) and 0.82 (95%CI: 0.80–0.83) among cohorts in Chonyi, Junju and Ngerenya respectively. In a small subset of children from Junju, a model incorporating weighted local malaria prevalence with AMA1 and MSP1₁₄₂ antibody levels provided an AUC of 0.83 (95%CI: 0.79–0.88).

Conclusion

We have proposed an approach to estimating the intensity of an individual's malaria exposure in the field. The weighted local malaria prevalence can be used as individual marker of malaria exposure in malaria vaccine trials and longitudinal studies of natural immunity to malaria.

Spectrum bias and loss of statistical power in discordant couple studies of sexually transmitted infections

BACKGROUND

Discordant couple studies are frequently used to evaluate preventive interventions for sexually transmitted infections (STI). This study design may be vulnerable to spectrum bias when transmission risk is heterogeneous.

METHODS

We used Markov models to assess the effect of heterogeneous transmission risk on the ability to detect effective interventions using a discordant couple study design. We also evaluated the implications that such bias may have for statistical power. Models incorporated potential health states in a population of initially infection-discordant couples, according to infection status with a hypothetical STI and participation in a hypothetical clinical research study. We evaluated the effect of length of discordant relationship at time of study enrollment, the shape of distribution describing transmission risk among couples, and the effect of sex-specific differential transmission probabilities, on model outcomes.

RESULTS

The results demonstrate that discordant couple studies are prone to spectrum bias, the degree of which is affected by the shape of the underlying transmission probability density function.

CONCLUSIONS

Such bias could lead to unexpected study findings, including gender-specific vaccine effects, and loss of statistical power, making this an important and underrecognized consideration in the design and interpretation of discordant couple studies.

Incorporating frailty in a multi-state model: application to disease natural history modelling of adenoma-carcinoma in the large bowel

Homogeneous multi-state models of disease progression have been widely used for designing and evaluating cancer screening programs. However, in screening for premalignant conditions of the cervix or large bowel, it is unlikely that all premalignant lesions have the same underlying propensity for progression. Incorporating frailty into multi-state models raises practical difficulties as it precludes the derivation of finite transition probabilities by matrix solution of the Kolmogorov equations. We address this problem by formulating a heterogeneous process as a series of homogeneous processes linked by transitions which are subject to heterogeneity (frailty). Continuous frailty and discrete mover—stayer models were developed. We applied these to the example of progression of adenoma to colorectal cancer in a three-state model and to a five-state model including consideration of adenoma size. Results were compared with those of purely homogeneous models in a previous study in terms of cumulative risk of malignant transformation from adenoma to invasive colorectal cancer.

Heterogeneity in malaria exposure and vaccine response: implications for the interpretation of vaccine efficacy trials

Background

Phase III trials of the malaria vaccine, RTS, S, are now underway across multiple sites of varying transmission intensity in Africa. Heterogeneity in exposure, vaccine response and waning of efficacy may bias estimates of vaccine efficacy.

Methods

Theoretical arguments are used to identify the expected effects of a) heterogeneity in exposure to infectious bites; b) heterogeneity in individual's response to the vaccine; and c) waning efficacy on measures of vaccine efficacy from clinical trials for an infection-blocking vaccine.

Results

Heterogeneity in exposure and vaccine response leads to a smaller proportion of trial participants becoming infected than one would expect in a homogeneous setting. This causes estimates of vaccine efficacy from clinical trials to be underestimated if transmission heterogeneity is ignored, and overestimated if heterogeneity in vaccine response is ignored. Waning of vaccine efficacy can bias estimates of vaccine efficacy in both directions.

Conclusions

Failure to account for heterogeneities in exposure and response, and waning of efficacy in clinical trials can lead to biased estimates of malaria vaccine efficacy. Appropriate methods to reduce these biases need to be used to ensure accurate interpretation and comparability between trial sites of results from the upcoming Phase III clinical trials of RTS, S.

Pathogen dose infectivity curves as a method to analyze the distribution of host susceptibility: a quantitative assessment of maternal effects after food stress and pathogen exposure

Stress conditions have been found to change the susceptibility of hosts or their offspring to infection. The usual method of testing at just one parasite dose level does not allow conclusions on the distribution of susceptibility. To better understand the epidemiology and evolution of host-parasite systems, however, knowledge about the distribution of host susceptibility, the parameters that characterize it, and how it changes in response to environmental conditions is required. We investigated transgenerational effects of different stress factors by exposing Daphnia magna to standard conditions, to low food levels, or to a high dose of the bacterial pathogen Pasteuria ramosa and then measuring the susceptibility of the offspring to different spore doses of the parasite. For the analysis we used a mathematical model that predicts the fraction of infected hosts at different parasite doses, allowing us to estimate the mean and variance of host susceptibility. We find that low food levels reduce both the mean and the variance of offspring susceptibility. Parasite exposure, on the other hand, widens the offspring's susceptibility distribution without affecting its mean. Our analysis uncovered previously unknown transgenerational effects on the distribution of susceptibilities. The finding of an alteration in the variance of susceptibility to infection has implications for host and parasite dynamics and can contribute to our understanding of the stability of host-parasite interactions.

Malaria vaccine trial endpoints - bridging the gaps between trial design, public health and the next generation of vaccines

A recent working group convened by the World Health Organization recommended that time to first or only episode of clinical malaria should be used to evaluate vaccine efficacy in phase III trials. However, calculating vaccine efficacy based on this endpoint misses important aspects of malaria disease and transmission. Here, we discuss the gaps that this approach leaves in predicting the potential public health impact of a vaccine and the challenges faced by vaccine trial designers. We examine the implications of current vaccine trial design on effectiveness studies and the next generation of malaria vaccines.

What should vaccine developers ask? Simulation of the effectiveness of malaria vaccines

Background

A number of different malaria vaccine candidates are currently in pre-clinical or clinical development. Even though they vary greatly in their characteristics, it is unlikely that any of them will provide long-lasting sterilizing immunity against the malaria parasite. There is great uncertainty about what the minimal vaccine profile should be before registration is worthwhile; how to allocate resources between different candidates with different profiles; which candidates to consider combining; and what deployment strategies to consider.

Methods and Findings

We use previously published stochastic simulation models, calibrated against extensive epidemiological data, to make quantitative predictions of the population effects of malaria vaccines on malaria transmission, morbidity and mortality. The models are fitted and simulations obtained via volunteer computing. We consider a range of endemic malaria settings with deployment of vaccines via the Expanded program on immunization (EPI), with and without additional booster doses, and also via 5-yearly mass campaigns for a range of coverages. The simulation scenarios account for the dynamic effects of natural and vaccine induced immunity, for treatment of clinical episodes, and for births, ageing and deaths in the cohort. Simulated pre-erythrocytic vaccines have greatest benefits in low endemic settings (<EIR of 10.5) where between 12% and 14% of all deaths are averted when initial efficacy is 50%. In some high transmission scenarios (>EIR of 84) PEV may lead to increased incidence of severe disease in the long term, if efficacy is moderate to low (<70%). Blood stage vaccines (BSV) are most useful in high transmission settings, and are comparable to PEV for low transmission settings. Combinations of PEV and BSV generally perform little better than the best of the contributing components. A minimum half-life of protection of 2–3 years appears to be a precondition for substantial epidemiological effects. Herd immunity effects can be achieved with even moderately effective (>20%) malaria vaccines (either PEV or BSV) when deployed through mass campaigns targeting all age-groups as well as EPI, and especially if combined with highly efficacious transmission-blocking components.

Conclusions

We present for the first time a stochastic simulation approach to compare likely effects on morbidity, mortality and transmission of a range of malaria vaccines and vaccine combinations in realistic epidemiological and health systems settings. The results raise several issues for vaccine clinical development, in particular appropriateness of vaccine types for different transmission settings; the need to assess transmission to the vector and duration of protection; and the importance of deployment additional to the EPI, which again may make the issue of number of doses required more critical. To test the validity and robustness of our conclusions there is a need for further modeling (and, of course, field research) using alternative formulations for both natural and vaccine induced immunity. Evaluation of alternative deployment strategies outside EPI needs to consider the operational implications of different approaches to mass vaccination.

Estimation of vaccine efficacy in a repeated measures study under heterogeneity of exposure or susceptibility to infection

Vaccine efficacy (VE) is commonly estimated through proportional hazards modelling of the time to first infection or disease, even when the event of interest can recur. These methods can result in biased estimates when VE is heterogeneous across levels of exposure and susceptibility in subjects. These two factors are important sources of unmeasured heterogeneity, since they vary within and across areas, and often cannot be individually quantified. We propose an estimator of VE per exposure that accounts for heterogeneous susceptibility and exposure for a repeated measures study with binary recurrent outcomes. The estimator requires only information about the probability distribution of environmental exposures. Through simulation studies, we compare the properties of this estimator with proportional hazards estimation under the heterogeneity of exposure. The methods are applied to a reanalysis of a malaria vaccine trial in Brazil.

A new SEIR epidemic model with applications to the theory of eradication and control of diseases, and to the calculation of R0

We present a novel SEIR (susceptible-exposure-infective-recovered) model that is suitable for modeling the eradication of diseases by mass vaccination or control of diseases by case isolation combined with contact tracing, incorporating the vaccine efficacy or the control efficacy into the model. Moreover, relying on this novel SEIR model and some probabilistic arguments, we have found four formulas that are suitable for estimating the basic reproductive numbers R(0) in terms of the ratio of the mean infectious period to the mean latent period of a disease. The ranges of R(0) for most known diseases, that are calculated by our formulas, coincide very well with the values of R(0) estimated by the usual method of fitting the models to observed data.

A quantitative test of the relationship between parasite dose and infection probability across different host-parasite combinations

Epidemiological models generally assume that the number of susceptible individuals that become infected within a unit of time depends on the density of the hosts and the concentration of parasites (i.e. mass-action principle). However, empirical studies have found significant deviations from this assumption due to biotic and abiotic factors, such as seasonality, the spatial structure of the host population and host heterogeneity with respect to immunity and susceptibility. In this paper, we examine the effect of the dose level of the bacterial endoparasite Pasteuria ramosa on the infection rate of its host, the water flea Daphnia magna. Using seven host clones and two parasite isolates, we measure the fraction of infected hosts after exposure to eight different parasite doses to determine whether there is variation in the infection process across different host clone-parasite isolate combinations. In five combinations, a pronounced dose-dependent infection pattern was found. Using a likelihood approach, we compare the infection data of these five combinations to the fit of three mathematical models: a mass-action model, a parasite antagonism model (i.e. an increase in the parasite dose leads to an under-proportionate increase in the infection rate per host) and a heterogeneous host model. We found that the host heterogeneity model, in which we assumed the existence of non-inherited phenotypic differences in host susceptibilities to the parasite, provides the best fit. Our analysis suggests that among 5 out of the 14 host clone-parasite isolate combinations that resulted in appreciable infections, non-genetic host heterogeneity plays an important role.

Attributable fractions with survival data

Attributable fraction (AF) is an important concept in clinical and epidemiological studies. The concept has mainly been discussed in relation to case-control studies, cross-sectional studies, and follow-up studies of fixed length. Here, we propose and discuss several ways of defining and estimating AFs with right-censored survival data, and thus with varying lengths of follow-up. In particular, we define the attributable hazard fraction, the AF before time t, and the AF within study. These measures have different interpretations and may give different numerical values, as illustrated in an application to real data on time to the first receiving of cash benefits for hearing impairment in children. The results underline the need for careful selection of the type of measure and interpretation when reporting AFs for survival data.

Accounting for variability in sample size estimation with applications to nonadherence and estimation of variance and effect size

We consider sample size calculations for testing differences in means between two samples and allowing for different variances in the two groups. Typically, the power functions depend on the sample size and a set of parameters assumed known, and the sample size needed to obtain a prespecified power is calculated. Here, we account for two sources of variability: we allow the sample size in the power function to be a stochastic variable, and we consider estimating the parameters from preliminary data. An example of the first source of variability is nonadherence (noncompliance). We assume that the proportion of subjects who will adhere to their treatment regimen is not known before the study, but that the proportion is a stochastic variable with a known distribution. Under this assumption, we develop simple closed form sample size calculations based on asymptotic normality. The second source of variability is in parameter estimates that are estimated from prior data. For example, we account for variability in estimating the variance of the normal response from existing data which are assumed to have the same variance as the study for which we are calculating the sample size. We show that we can account for the variability of the variance estimate by simply using a slightly larger nominal power in the usual sample size calculation, which we call the calibrated power. We show that the calculation of the calibrated power depends only on the sample size of the existing data, and we give a table of calibrated power by sample size. Further, we consider the calculation of the sample size in the rarer situation where we account for the variability in estimating the standardized effect size from some existing data. This latter situation, as well as several of the previous ones, is motivated by sample size calculations for a Phase II trial of a malaria vaccine candidate.

Randomization and baseline transmission in vaccine field trials

In randomized trials, the treatment assignment mechanism is independent of the outcome of interest and other covariates thought to be relevant in determining this outcome. It also allows, on average, for a balanced distribution of these covariates in the vaccine and placebo groups. Randomization, however, does not guarantee that the estimated effect is an unbiased estimate of the biological effect of interest. We show how exposure to infection can be a confounder even in randomized vaccine field trials. Based on a simple model of the biological efficacy of interest, we extend the arguments on comparability and collapsibility to examine the limits of randomization to control for unmeasured covariates. Estimates from randomized, placebo-controlled Phase III vaccine field trials that differ in baseline transmission are not comparable unless explicit control for baseline transmission is taken into account.