Program on Mathematical Modeling of Infectious Diseases: Dynamics and Control - IMS

Mathematical Modeling of Infectious Diseases:
Dynamics and Control
(15 Aug - 9 Oct 2005)

Jointly organized by Institute for Mathematical Sciences, National University of Singapore
and Regional Emerging Diseases Intervention (REDI) Centre, Singapore

~ Abstracts ~

Conducting genomic research on SARS during and after the outbreak
Lawrence Walter Stanton, Genome Institute of Singapore

I will describe our application of genomics research tools to understand the SARS corona virus (SARS-CoV) and the response of host cells to infection. Complete genome sequences of 22 isolates of SARS-CoV from Singapore provided an opportunity to evaluate the genetic diversity and mutational dynamics of the virus as it spread from patient to patient. We have looked at the mutational dynamics of the virus propagated in cultured cells and in a rare case of lab to human transmission. In addition, we have now adopted chip-based resequencing methods and genotyping techniques that avoid culturing of the virus and, thus, provide a means to assess the genetic diversity of emerging SARS-CoV quickly and comprehensively. These methods will aid in contact tracing of SARS transmissions in the population and for monitoring the appearance of new and potentially more virulent strains should there be another outbreak.

I will also present gene expression data derived from cells that were infected by SARS-CoV. We have found that human peripheral blood monocytes can become infected and support replication of SARS-CoV. Gene expression profiles of the infected monocytes offer a revealing look at the response of these cells to SARS infection, which sheds some light on the underlying pathology of the disease and insight into possible treatment.

The critical community size of the classic endemic model
Ingemar Nåsell, Royal Institute of Technology, Sweden

A stochastic version of the classic endemic model is formulated as an absorbing bivariate Markov chain, with discrete state space and continuous time. The absorbing states of this Markov chain correspond to absence of infection. The phenomena of recurrence and extinction of the infection in the classic endemic model are studied mathematically by consideration of the quasistationary distribution of the Markov chain. This distribution can in turn be used to derive information about the time to extinction.

The so called critical community size for measles was defined by Bartlett in 1960 as “the size of the community for which measles is as likely as not to fade out after a major epidemic”. This definition describes a threshold behavior for the stochastic model, with any community larger than the critical community size lying above threshold.

We study the quasi-stationary distribution for typical measles parameters and for community sizes of the same order as the critical community size. We show that the marginal distribution of infected individuals in quasi-stationarity is then well approximated by a geometric distribution, and that this result leads to a simple explicit expression for the critical community size in terms of model parameters.

Stochastic and deterministic models are related in the important sense that the deterministic model can be derived from the stochastic one by a rescaling followed by letting the population size approach infinity. The stochastic model threshold behavior inherent in the critical community size concept is consistent with the well-known deterministic threshold phenomenon.

We show that the deterministic model is an unacceptably poor approximation of the stochastic model for typical pre-vaccination measles parameters. Future work on the classic endemic model, as well as on extensions of it, will require work to be done on both deterministic and stochastic model formulations.

Back-calculation and environmental modeling for the 2003 SARS epidemic and developing effective engineering control methods
Yuguo Li, The University of Hong Kong, China

The 2003 severe acute respiratory syndrome (SARS) epidemic was characterized by the occurrence of large clusterings (i.e. super-spreading events) of infections. Environmental factors were suspected to play significant roles in some of these large outbreaks. For determining the environmental factors, it is important to estimate the time of infection. Estimating the time of infection, an unobservable event per se is also important in determining the other key transmission parameters and epidemiological characteristics of emerging infectious diseases. We first discuss a back-calculation method for estimating time of infection and the daily number of infected cases. We compare the estimated solutions from our proposed method to those obtained by the widely used expectation maximization method, for a simulated data set and two empirical SARS data sets from the 2003 Hong Kong SARS epidemic.

We then present a review of literature on the use of experimental and mathematical modelling of fluid mechanics in investigating the roles of building ventilation and air flow in some airborne diseases as well as developing new engineering control methods. The pros and cons of mathematical modelling in providing environmental investigations for large outbreaks of suspected airborne infectious diseases are discussed using the two example studies, i.e. the Amoy Gardens outbreak and the Prince of Wales Hospital 8A ward outbreak, and in studying ventilation systems in SARS isolation wards. A study of this nature needs an inter-disciplinary study with involvement of medical professionals, infection control experts, microbiologists, epidemiologists and engineers.

Timely identification of control strategies for emerging infectious diseases
Zhilan Feng, Purdue University, USA

This is a continuation of a lecture by John Glasser. The first part, we introduce a mathematical model that includes typical features of directly transmitted infectious disease such as SARS and various possible control measures including quarantine, timely diagnosis and effective isolation. Now, we present mathematical analyses to identify the most effective control strategy. We compute reproductive numbers associated with disease-transmitting stages as well as the overall reproductive number. A sensitivity and uncertainty analysis allows us to determine how these numbers depend on the uncertainty of various parameters. We illustrate the process by which some time-dependent parameters can be estimated by fitting data to the dynamical system. Then we assess the effectiveness of various control strategies by evaluating the reproductive number's partial derivatives with respect to each and comparing them.

The basic epidemiology models I: models and expressions for Ro
Herbert W Hethcote, University of Iowa, USA

This introductory lecture develops the basic types of epidemiological models for infectious diseases, formulates them as systems of nonlinear differential equations, identifies the thresholds and equilibrium points, and describes their dynamical behavior that is found by analyzing local and global stability of these models. The goal is to introduce notation, terminology, and results that will be generalized in later lectures on more advanced models. Intuitive interpretations will be given for the basic reproduction number, the contact number, and the infective replacement number. The model types considered are SI, SIS, SEI, SEIS, SIR, SIRS, SEIR, SEIRS, MSEIR, MSEIRS.

The basic epidemiology models II: parameter estimation and applications
Herbert W Hethcote, University of Iowa, USA

This continuation of the previous introductory lecture looks at applications of the basic epidemiology models. The basic reproduction number is estimated for many directly transmitted diseases, herd immunity is defined, and the fraction that must be vaccinated to obtain herd immunity is determined for diseases such as measles, rubella, mumps, chickenpox, influenza, polio, and smallpox. These diseases are compared with each other and the modeling results are compared with national and international experience with these diseases.

Epidemiology models with variable population size
Herbert W Hethcote, University of Iowa, USA

Basic epidemiology models often assume that births balance deaths, so that the total population size is constant. But populations may be growing or decreasing significantly due to differences in the natural birth and death rates, an excess disease-related mortality, or disease-related decreases in reproduction. In models with a variable total population size, the persistence of the infectious disease may slow the growth rate of a naturally growing population, lead to a lower equilibrium population size, or even reverse the population growth to a decay to extinction. Variable population size models of SIS and SIR type are formulated and analyzed in order to demonstrate the effects of the disease on the population size and the effects of the population structure on the disease dynamics.

Age structured epidemiology models and expressions for Ro
Herbert W Hethcote, University of Iowa, USA

When modeling a disease in which vaccinations are given at different ages, it is necessary to include both age and time as independent variables. SEIR and MSEIR models are formulated with either continuous age or discrete age groups. Expressions for the basic reproduction number are derived. Values of the basic reproduction number and contact number are estimated for various diseases including measles in Niger, Africa and pertussis in the United States.

Simulations of rubella vaccination strategies in China
Herbert W Hethcote, University of Iowa, USA

Many infants whose mothers have rubella infections during their first trimester of pregnancy have birth defects called congenital rubella syndrome (CRS). China does not routinely vaccinate against rubella in the public sector, but may need to start as its “one child per couple” policy changes the population age distribution and the dynamics of rubella epidemiology, so that the incidence of rubella in pregnant women increases. Computer simulations with demographic transitions and rubella transmission dynamics predict that, with no or limited rubella vaccination, CRS incidence in China in the 30 years after 2020 will be more than twice the level in 2005. Comparisons of rubella vaccination strategies using computer simulations show that routine vaccination of over 80% of one-year-old children would be effective in reducing total CRS cases in 2005-2051 and eliminating rubella in China by 2051. Routine immunizations at higher levels and the addition of early mass vaccinations of 2-14 year old children and women of childbearing ages would further reduce total CRS cases and speed up the elimination of rubella.

Phylodynamics: unifying the evolutionary and epidemiological dynamics of viral pathogens
Oliver George Pybus, University of Oxford, UK

A key priority for infectious disease research is to understand the evolutionary and epidemiological processes that govern pathogen behavior. This is particularly true for rapidly-evolving pathogens, such as RNA viruses, whose evolutionary and epidemiological dynamics occur on a similar timescale and therefore interact. This connection is central to many applied issues, from the evolution of drug resistance to vaccine design and the emergence of new diseases. A qualitative ‘phylodynamic’ framework can be used to classify the evolutionary and dynamical behavior of pathogens. By combining this framework with pathogen genetic data and newly-developed computational methods, it is becoming increasingly possible to investigate the combined evolution, epidemic history and molecular adaptation of infectious pathogens.

Evolutionary analysis of viral genomes 1: quantifying genetic diversity
Oliver George Pybus, University of Oxford, UK

Many infectious diseases, particularly viruses, exhibit considerable genetic diversity as a result of their large population sizes and high rates of mutation. Although pathogen diversity contains much information about evolutionary and epidemiological dynamics, this information is often scrambled, fragmentary or hidden. In these lectures I outline the mathematical and statistical models that are used to recover and interpret the information contained in the genome sequences of sampled pathogens. In the first lecture, I introduce the statistical models of molecular sequence evolution that are used to quantify viral genetic diversity and genetic distance. The problems of ‘multiple hits’ and rate variation among sites will be considered.

Evolutionary analysis of viral genomes 2: phylogenies for epidemiologists
Oliver George Pybus, University of Oxford, UK

In the second lecture I will show how measures of genetic distance among sampled strains can be represented as an evolutionary tree, or phylogeny. Methods of phylogenetic inference will be introduced, with specific reference to model-based approaches such as maximum likelihood. I will explain how the relationship between genetic distance and time can be estimated from sequences sampled at different times using molecular clock models. These models can be used to estimate the rate of viral evolution and to date past epidemiological events.

Evolutionary analysis of viral genomes 3: coalescent models
Oliver George Pybus, University of Oxford, UK

Coalescent models are population genetic models that describe the relationships among observed genetic diversity, phylogenetic tree shape, and population-level dynamic processes (such as population size change, migration, or recombination). This lecture will introduce coalescent models and their application to viral epidemiology, concentrating on the inference of past epidemic dynamics from sampled genome sequences.

Evolutionary analysis of viral genomes 4: natural selection & viral adaptation
Oliver George Pybus, University of Oxford, UK

The rapid adaptation of many pathogens by natural selection greatly complicates the development of anti-viral drugs and vaccines, and may contribute to the emergence of new infectious diseases. In this final lecture I will discuss methods for detecting natural selection from sampled pathogen genomes, including the estimation of synonymous to non-synonymous evolutionary rates, and the identification of specific codons that have been selected. The difficulties inherent in correctly interpreting such results will be discussed.

Modeling of infectious diseases and its implications for public health policy-making
Ying-Hen Hsieh, National Chung Hsing University, Taiwan

Mathematical modeling of infectious diseases has become an important tool for public health authorities all over the world to understand a particular infectious disease outbreak or epidemic event in their efforts to confront the threat of newly emerging and reemerging infectious diseases. We briefly describe the mathematical modeling approaches commonly employed with focus on public health implications. We will give illustrative examples of modeling of infectious diseases pertaining to the SARS outbreak in 2003, ranging from simple single-equation model to complicated compartmental systems.

Epidemics in heterogeneous communities and their vaccination policies
Tom Britton, Stockholm University, Sweden

Real life epidemics are affected by various heterogeneities, for example caused by varying susceptibility and infectivity among individuals, and social "clusters" such as households. In the talk we present models acknowledging such heterogeneities and see how the heterogeneities affect vaccination policies aimed at preventing future outbreaks. This is done both for the case that model parameters are known and for the case that parameters are estimated from a previous outbreak.

How does mass immunisation affect disease incidence?
Niels G Becker, National Centre for Epidemiology and Population Health, Australian National University

In this introductory lecture we use simple SIR transmission models to understand the consequences of mass immunisation, including the effects of introducing routine immunisation, enhancing the immunisation coverage and importation of the infection. We also discuss elimination of the infection and monitoring elimination.

Immunisation strategies for a community of households
Niels G Becker, National Centre for Epidemiology and Population Health, Australian National University

Transmission of infection depends on the structure of the community. In modeling, heterogeneity is usually limited to acknowledging an age-structure. Here we illustrate the importance of the household structure. A discussion of the concept of reproduction number in this setting is included. We also allow for different types of individual. While quite general results can be obtained, this tutorial introduction presents the ideas and methods in the simplest settings.

Immunisation with a partially effective vaccine
Niels G Becker, National Centre for Epidemiology and Population Health, Australian National University

Ideally a vaccine will render every vaccinated individual completely immune. In practice, vaccinated individuals do get infected, making it necessary to determine the effect of vaccination on susceptibility and infectivity of individuals and the effect of using a partially effective vaccine on transmission in the community. Here we present methods for such assessments. It is necessary to base such methods on transmission models, otherwise we can be seriously misled.

Preparedness for an emerging infection
Niels G Becker, National Centre for Epidemiology and Population Health, Australian National University

Emerging infections have always been a threat, but the emergence of SARS, the serious threat of an influenza pandemic from avian influenza and the threat of bioterrorism has generated more interest in preparedness for an emerging infection than ever before. Here we demonstrate how models can be used to assess the relative effectiveness of control measures such as reducing the contact rate, early diagnosis of cases with isolation, quarantining members of households with a diagnosed case, quarantining all traced contacts of diagnosed cases and closing schools.

Vaccination strategies when infection is dose dependent
Niels G Becker, National Centre for Epidemiology and Population Health, Australian National University

The effectiveness of a vaccination strategy to control transmission of an infectious disease depends on the way vaccine doses are distributed to individuals in a community of households. Here we show that this dependence is exacerbated when acquisition and severity of illness are determined by the size of the infecting dose, as is thought to be the case for measles and varicella. Two alternative formulations for the way vaccination changes an individual's susceptibility and infectivity show that vaccination coverage, the nature of the vaccine response and the distribution of household size also have a big impact on which strategy is more effective. These judgements are made by comparing the post-vaccination reproduction numbers corresponding to different vaccination strategies.

Model based estimates of long-terms persistence of induced hepatitis A antibodies
Ziv Shkedy, Limburgs Universitair Centrum (LUC), Belgium

In many situations, vaccination programs which incorporate in mathematical models assume life long Immunity after vaccination. In this study we investigate the change of Hepatitis A antibody level after vaccination. Linear mixed models are used to model both the population and the subject-specific evolution of the antibody over time. Empirical based estimates for the subject-specific random effects are used to in order predict the antibody level for each subject and to evaluate the proportion individuals remain protected over time. The method is illustrated using data from a vaccination clinical trial in which subjects were followed up for 10 years after vaccination.

Interventions in epidemics of influenza-like diseases - insights from computer simulations
Hans-Peter Duerr, University of Tuebingen, Germany

The SARS epidemic in 2003 has shown that preventing the spread of infectious diseases is a major challenge for public health systems, not only on a national but also on an international level. Countries around the globe were faced with the problem to initiate as quickly as possible prevention and intervention programs and choosing the right options from a variety of available control measures, starting from isolation of individuals up to cutting possible routes of transmission via international transportation. A future influenza pandemic will pose similar problems, albeit its control can be supported by vaccination and administration of antivirals.

Designing the best intervention strategy may involve the implementation of several options at the same time, implying a complex optimization problem, particularly when country-specific potentials or restrictions must be considered. By means of stochastic, individual-based simulations we investigate the effects of such combinations of interventions and particularly pay attention to different contact structures within the population. This network-based approach is also intended to identify uncertainties which can threaten the success of certain prevention and intervention strategies.

Controlling international spread and monitoring incidence of a newly-emerged infectious disease
Kathryn Anne Glass, Australian National University, Australia

When faced with an outbreak of a newly-emerged infectious disease, it is necessary to make use of basic control measures that limit opportunities for infectious contact. In addition to local measures, border control measures are introduced to try to reduce spread of disease between countries. We describe methods for quantifying the effect of these measures, and compare these to the effects of local measures. In order to quantify the risks of international spread, it is necessary to have good estimates of the numbers of infected individuals. In the event of a newly-emerged infection, there may be considerable under-reporting due to lack of knowledge about the disease, especially if some individuals experience a mild, atypical or asymptomatic infection. We describe a Bayesian approach to estimating the number of hidden cases, and show how this can also be used to predict future case numbers. Towards the end of the outbreak, these predictions provide valuable guidance on when control measures may safely be lifted.

Estimating the force of infection from serological data
Ziv Shkedy, Limburgs Universitair Centrum (LUC), Belgium

The force of infection is one of the primary epidemiological parameters of infectious diseases. For many infectious diseases it is assumed that the force of infection is age dependent. Although the force of infection can be estimated directly from a follow up study, it is much more common to have cross-sectional seroprevalence data from which the prevalence and the force of infection can be estimated. In this talk we review several parametric and nonparametric examples from the literature and discuss new approaches to estimate the force of infection including local polynomials, factional polynomials and hierarchical Bayesian models. The methods are illustrated on five seroprevalence samples, two of Hepatitis A, and one of Rubella, Mumps and Varicella.

Timely identification of control strategies for emerging infectious diseases
John W Glasser, US CDC Atlanta, USA

Within weeks of a traveler from Guangdong Province, China, with Severe Acute Respiratory Syndrome (SARS) infecting others in their Hong Kong hotel, several case-series were published and the responsible pathogen was identified. Public health responses to the ensuing outbreaks in Hong Kong and elsewhere were equally swift, but not equally effective. To mitigate future outbreaks of new diseases, we modeled a generic infectious disease apparently transmitted by close contact, but about which little else is known. We intended to 1) elucidate social phenomena affecting disease transmission, and 2) determine if timely control of this new disease could have been achieved solely by influencing them. But our preliminary assessment of factors contributing to control of the outbreak in Singapore motivated us to 3) compare official communiqués and social responses elsewhere and 4) derive analytical expressions for the impact of all possible interventions, including such communications, that could be evaluated quickly in future. Our model is a system of non-autonomous differential equations (DE) in which proportions seeking medical care during the prodrome and being diagnosed and effectively isolated may evolve, and their contacts be quarantined soon after exposure, whereas contacts of those misdiagnosed or who present with acute respiratory symptoms may be identified too late. We are estimating stage-specific infection rates, conditional on clinical observations and these social phenomena, by minimizing disparities between predicted and observed hospitalizations following the 2003 importations of SARS to Singapore and Taiwan. We have derived an expression for the average number of infections per infectious person, which must be <1 for control, from the autonomous DE underlying our new disease model, and described its relationship to R0, the reproductive number in a wholly susceptible population absent intervention. To evaluate various social phenomena Ministries of Health could influence, as well as their quarantine of possible contacts, we took partial derivatives with respect to each intervention. During the outbreak in Singapore, people with compatible symptoms sought medical care earlier (proportion hospitalized within 4 days of onset increased from 0.3 to 0.9) and clinicians became proficient at diagnosing them (proportion isolated on admission increased similarly). We do not yet know which patients had been quarantined, but use the proportion isolated within one day of symptom onset, which increased from 0.05 to 0.6, as a surrogate. Conditional on other relevant social phenomena, our preliminary assessment of the impact of quarantine is modest. Because 7,863 people were quarantined, only 11 of whom became ill, the societal cost of this intervention was enormous. We show that, for biologically reasonable pathogen, and fixed but otherwise reasonable host response parameters, small increments in the proportion of cases seeking care during their prodrome versus acute illnesses are equivalent to quarantining an order of magnitude more contacts than officials managed in either Singapore or Taiwan (where only about 5% were quarantined). We have not yet modeled the outbreak in Taiwan, but believe SARS could have been controlled in Singapore solely by ensuring that people with compatible symptoms sought medical care during the prodrome, especially ones who might have been exposed, and effectively isolating those diagnosed. Given biological parameters estimable from early case-series or experience with related pathogens, we believe our analytical expressions would permit identification of the most promising response to emerging infectious diseases. By refining initial estimates as information accumulated and monitoring intervention effects, we believe modelers could ensure timelier and otherwise more advantageous allocation of resources in future outbreaks of new diseases. We are comparing parameters estimated from observations available after successive two-week periods to evaluate these beliefs.

En route to reliable policymaking tools: models as hypotheses
John W Glasser, US CDC Atlanta, USA

Public health policymakers cannot identify optimal strategies by experimenting with human populations. Thus, at the Centers for Disease Control and Prevention, models are assisting increasingly in the design, evaluation and improvement of health policy. En route to reliable policymaking tools, mathematical epidemiologists must capture pathophysiology, estimate parameters, and replicate historical observations, ideally in disparate settings. But problematic diseases often are poorly understood, with published results and informed opinions inconsistent, if not contradictory. Crucial observations may be unavailable, and rarely can well-designed studies be executed before decisions must be reached. Moreover, existing observations generally are required for parameter estimation, leaving a paucity of information for model validation. In such circumstances, scientists learn to frame hypotheses and seek natural or design artificial experiments capable of disproving them. Hypotheses found wanting are revised and reevaluated or abandoned in favor of alternatives, which are evaluated in turn; others are scrutinized more closely. But mathematics is more explicit than other languages available for theorizing, and myriad quantitative methods are available for analyzing and simulating equations. These attributes of mathematical models, in turn, transform the ethical constraints that prevent medical epidemiologists from experimenting with human populations into assets insofar as they motivate realistic modeling instead. The arbitrary precision of quantitative results facilitates evaluation, and causes of disparate predictions are relatively easily diagnosed and remedied, whereupon improved models can be reevaluated at rates that would leave traditional experimentalists breathless. But, to increase our understanding of natural phenomena, model states and functional relations must correspond to the elements and processes responsible.

1997 Measles outbreak in São Paulo, Brazil: evaluation of vaccination strategies via mathematical modeling
John W Glasser, US CDC Atlanta, USA

Despite a routine two-dose measles vaccination program, mass campaigns in 1987 and 1992 and low subsequent incidence, São Paulo experienced an outbreak between May and October of 1997 with over 42,000 confirmed cases, mostly young adults, and 42 measles-associated deaths, mostly infants. To eliminate measles, the Pan American Health Organization recommends supplementing routine childhood vaccination with mass campaigns, initially to reduce (catch-up) and periodically to maintain (follow-up) susceptible numbers below the herd immunity threshold. To a) determine if a follow-up campaign during 1996, when due in São Paulo State, might have prevented or mitigated this measles outbreak, b) evaluate the impact of outbreak-control efforts and c) explore the potential effectiveness of supplementary adolescent or adult vaccination, we modeled measles in metropolitan São Paulo. Because disease dynamics reflect interactions between demographic and epidemiological processes, our model is unusually realistic, comprising 5 immune states, 10 age classes and all relevant demographic as well as epidemiological processes. Simulations suggest that a 1996 follow-up campaign would have mitigated, but could not have prevented, this outbreak and that the impact of control efforts, relatively late in the outbreak and directed predominately at children, was modest. But targeted vaccination of adolescents and young adults, including long-time residents neither vaccinated as children nor infected subsequently, as well as recent immigrants from rural areas whose agricultural economies fluctuate with the weather, might have prevented this outbreak. Susceptible adolescents and young adults must be immunized in metropolitan São Paulo, especially those living in crowded favellas and traveling to and from limited workplaces for unskilled and semi-skilled laborers via crowded public conveyances, possibly by vaccinating them where employed. And, because measles remains a significant source of morbidity and mortality in the increasingly urbanized developing world, we must consider differences among and movement between rural and urban sub-populations in devising control, elimination and eradication strategies.

Mathematical modeling of Rubella and Congenital Rubella Syndrome
John W Glasser, US CDC Atlanta, USA

Congenital rubella syndrome (CRS) comprises a panoply of lesions, some fatal and others diagnosed too long after birth to be readily associated with first trimester maternal infections. Consequently, this syndrome is under-ascertained except by searching medical records for compatible conditions and confirming them serologically. Mathematical modeling not only obviates the need for routine studies of this sort, permitting countries and the international health community to estimate the burden of this disease, but also enables them to consider means by which it might be mitigated. Our estimates of CRS from disease or sero-surveillance and demographic information compare favorably to those from retrospective chart reviews in children’s hospitals and clinics specializing on afflictions that may be attributable to infection in utero. The burden can be mitigated by reducing either susceptibility among women of childbearing age (WCBA) or their risk of infection. The first approach might involve vaccinating adolescent girls where they attend school or mothers where they deliver in hospital, neither of which however is universal. The second necessarily involves childhood vaccination, via routine age-appropriate doses, mass campaigns, or both. But if sufficient coverage is not sustained, the immunization of some children only temporarily protects others, whose risk of infection diminishes until they have children of their own. Unless vaccinated, those children may be infected while playing or attending school, and infect family members, including their susceptible mothers, who may be pregnant. The potential for this perverse outcome is well known, but we will describe a 30-year childhood vaccination program in Costa Rica that attained 80% coverage, transforming rubella from an annual disease largely afflicting children into irregular epidemics that also engulfed adolescents and young adults. Insofar as childhood vaccination increased susceptibility among WCBA, its impact on CRS is equivocal. We will also describe an abrupt change in disease dynamics followed by a dramatic increase in CRS attributable to falling birth rates in Romania, and models with which we have evaluated possible strategies for mitigating this burden in these and other developing countries. Given estimated costs of feasible tactics, policymakers could consider whether to devote their scarce resources to CRS or other health problems.

Implications of an explanation for secular patterns in reported Pertussis in the United States
John W Glasser, US CDC Atlanta, USA

In the United States, pertussis seems to be increasing among adolescents, middle-aged adults, and infants <4 months of age, for whom it may be fatal. A multi-state study implicates parents, and to a lesser extent grandparents and siblings, as sources of infant infection. The increase among adolescents and middle-aged adults coincides with a decrease among children in Massachusetts (MA). We modeled pertussis via realistic systems of differential equations to evaluate one hypothetical cause of these secular patterns, diminished boosting of shorter-duration, artificially induced immunity as vaccination has reduced childhood disease, and its remedy by re-vaccination. We estimated initial conditions from IgG titers to pertussis toxin (PT) in sera from southeasterners who participated in an unrelated vaccine trial. And we adjusted age-specific infection rates, obtained from pre-vaccination disease histories, to minimize disparities between predictions and reports to the National Notifiable Diseases Surveillance System (NNDSS). Because disease peaks among adolescents in fall and other age groups in summer, and policymakers are considering adolescent re-vaccination, we simultaneously estimated coefficients of age-specific seasonal forcing. In the US, risks per susceptible (forces of infection, FOI) calculated from these adjusted rates peak at successively lower levels during childhood, adolescence and middle age, as does IgG to PT from sera obtained in conjunction with the 1991-’94 National Health and Nutrition Examination Survey (NHANES III). In MA, the FOI on adolescents apparently exceeds that on children. And, compared with reports to the NNDSS, our MA model predicts proportionately fewer cases among persons 15-19 years old, and more aged 20-24, while our US model’s predictions are concordant (p<0.01 via Kolmogorov-Smirnov). Increasing diagnostic suspicion, evolution of Bordetella pertussis in response to vaccination or deterioration of vaccine potency are not required to explain observed secular patterns, but neither are these alternative hypothesis mutually-exclusive. Nonetheless, the discrepant MA model predictions, disparate age-specific FOI and regional analyses of sera from NHANES III corroborate our explanation: relatively few children in MA are susceptible by virtue of sustained vaccine coverage at higher levels than yet attained throughout most of the US, which is why childhood disease is scarce. And relatively more adolescents and middle-aged adults are susceptible by virtue of the waning of artificially induced immunity, which is why adolescent and middle-aged adult pertussis is increasing. Re-vaccination of adolescents, and possibly selected middle-aged adults, could not only reduce disease in MA, but also ensure that adults caring for infants (i.e., 20-24 and 30-49 year olds contribute most to their FOI in our US model) were immune or subject to reduced FOI. Simulations confirm these deductions, with late adolescent vaccination optimal. In Sweden, where 3-dose coverage exceeded 98% soon after resumption of vaccination following a 17-year hiatus, boosting via exposure must have declined abruptly as immunity began shifting from longer-duration, naturally acquired to shorter-duration, artificially induced. Absent adolescent re-vaccination, disease among young infants will increase as parental immunity wanes.

Mathematical epidemiology of Varicella and Herpes Zoster
John W Glasser, US CDC Atlanta, USA

Varicella persists in the US despite vaccination coverage approaching 80%, with outbreaks increasingly involving modified disease among vaccinated children who may be less infectious (probability of transmission on contact) than ones with typical disease, but have greater contact rates. Our goals are to determine long-term impacts of 1) childhood vaccination on varicella and herpes zoster (HZ) caused by wild-type and vaccine virus, including the role, if any, of internal versus external boosting in maintaining protection against HZ, and 2) adult vaccination on HZ and varicella. We have modeled the transmission and control of varicella-zoster virus in human populations via a system of differential equations. Our model features realistic population dynamics, by virtue of age-specific birth and death rates, seasonal forcing that accounts for such age-dependent and independent effects as the school calendar and temperature, respectively, and hypothetical host-pathogen relations, including progressively less well controlled reactivations of latent virus that results solely in internal boosting initially, but leads eventually to HZ. We will describe parameters estimated from published observations and by fitting our model with different biological assumptions to time-series from active surveillance since before 1995, when vaccination began, in Antelope Valley, CA, and West Philadelphia, PA. These include estimates of the duration of naturally-acquired and artificially-induced immunity and assessments of the importance of internal and external boosting. Given satisfactory reproduction of historical observations in these rural and urban communities, we will evaluate vaccination policy options.

Re-introduction of smallpox into dynamic, socially-structured and spatially-distributed populations
John W Glasser, US CDC Atlanta, USA

The risk of terrorists re-introducing smallpox has motivated preparations in potential target countries. After reproducing the spatiotemporal pattern of smallpox transmission following the 1972 importation into Yugoslavia via coupled, biologically realistic systems of ordinary differential equations, we developed dynamic models with current US age distributions and typical spatially-distributed social structures. We simulated aerosol exposure of 10-105 people in hypothetical communities varying in size and structural complexity, and evaluated nested public health responses. Smallpox was eradicated by the isolation of cases, vaccination of traced contacts, and surveillance for prodromal symptoms, whereupon patients would be immediately isolated, a strategy now known as surveillance and containment (S&C), following the failure of mass vaccination in densely populated African and Asian countries. Consequently, our simulated primary response was S&C coupled with the vaccination of 95% of hospital-based health-care workers (HCW) within 2 days of first diagnosis, an estimated 18 days post-aerosol release. We assumed that 90% of patients would remain at home or be hospitalized within days of symptom onset and 75% of their contacts (i.e., people whom they had already exposed or whose risk was increased by close proximity) would be vaccinated and monitored, especially those vaccinated more than 4 days after exposure. Conditional on S&C, we also evaluated pre-emptive immunization of 10% or 50% of HCW, alone or together with closing schools for 10 days and vaccinating 40% or 80% of the populace one year and older within a week of first diagnosis. When this last measure is local, it resembles the historical practice of vaccinating most intensely near compounds in which cases were discovered, now known as ring vaccination. Compared with no intervention, S&C reduced simulated cases by 82-99%, depending on setting and scenario. Additional measures had necessarily small marginal benefits. We are analyzing our model to elucidate control via S&C, which differs fundamentally from population immunity.

The role of stochasticity in epidemic models
Valerie Isham, University College London, UK

Mathematical modelling of infectious diseases has a long and distinguished history and is one of the most successful areas of mathematical biology. Understanding the dynamics of the spread of an infectious disease brings possibilities for its control, and a clear understanding of the basic theory of epidemic models is a prerequisite for this.

In this talk, my focus will be on stochastic epidemic models, on which there has been increasing emphasis in recent years. I will briefly review the historical background and discuss some of the effects that allowing for heterogeneity and stochasticity has on model properties. I will use recent work on macroparasite infections to illustrate some of the issues involved.

Heterogeneity in epidemic modeling
Daryl Daley, The Australian National University, Australia

The talk surveys work on how the standard epidemic models with homogeneous mixing of an homogeneous population of individuals can be modified to take account of the population being heterogeneous with respect to mixing, to susceptibility, and infectivity. The evolutionary behaviour of the modified models can be compared with some pseudo-equivalent standard model(s) as benchmark(s).

Sexual networks and the evolution of sexually-transmitted pathogens: N. gonorrhoeae as an example
Azra Ghani, London School of Hygiene & Tropical Medicine, UK

Identification of areas of the sexual ntework in which infection persists is important to the control of STIs. However the epidemiological data required for this task is frequently difficult to obtain. In a study in London we tested the feasibility of using molecular typing methods to identify clusters of related isolates. To aid interpretation of the clusters observed I will present details and results from a network model of gonorrhoea transmission to investigate the patterns of strain structure expected under a variety of network structures. The dynamic network model for gonorrhoea transmission was adapted to incorporate sequence types (ST) defined by their allelic profile at two loci. All ST were assumed to be equally fit with new ST arising through recombination and mutation. The talk will explore the extent of strain diversity that can be expected in the absence of significant mutation, how strain structure might relate to the underlying properties of the sexual network, to what extent we can understand the underlying sexual network from the strain structure observed in a population and under which scenarios we can expect a high concordance in ST in sexual partnerships or short transmission chains.

A chain multinomial model for estimating real time fatality rate of a disease with application to severe acute respiratory syndrome (SARS)
Paul Yip, University of Hong Kong, China

It is well known that statistics using cumulative data directly are insensitive to changes. The World Health Organisation (WHO) estimates of the fatality rates is of the above type which may not be able to reflect latest changes of fatality due to treatment or government policy in a timely fashion. An estimate of a real time fatality rate based on a chain multinomial model with a kernel function is proposed here. It is more accurate than that of the WHO estimate in describing the fatality especially in the earlier period over the course of the epidemic. The estimator provides useful information for public health policy maker to understand the severity of the disease or to evaluate the effects of treatments or policies within a shorter period, which is critical in disease control. Simulation results showed that the performance of the proposed estimator is superior to that of the WHO estimator in terms of sensitiveness to changes and its timeliness to reflect the severity of the disease.

Characterization of dengue viruses from Singapore at the Environmental Health Institute
Tim Hart, The National Environment Agency, Singapore

Disease due to infections with dengue virus have surged in Singapore in recent years to record levels. This is despite a long period of relatively low incidence of dengue fever/dengue haemorrhagic fever (DF/DHF), which has been attributed to successful mosquito control efforts on the island. Although a number of explanations have been proposed regarding the reason for this sudden rise in cases, there is clearly a need for further research to understand the patterns of dengue infection in Singapore. The Environmental Health Institute (EHI) is a research department within the National Environment Agency (NEA), which is responsible for mosquito control in Singapore. As such, EHI takes a particular interest in the epidemiology, diagnostics and molecular characteristics of dengue viruses isolated in Singapore. This talk will give a short summary of the activities of EHI in using basic and field research to evaluate and target the dengue control effort. An update of the current projects taking place in EHI will be accompanied with examples including analysis of the recent rise in DF/DHF.

Determinants and consequences of seasonality in measles dynamics
Natalia B. Mantilla-Beniers, University of Cambridge, UK

Morbidity records of childhood infections have made important contributions to our understanding of host-pathogen population dynamics. Statistical analyses deploying extant data from developed countries have supported field observations that attributed school terms a central role in sustaining recurrent epidemics. In this talk, I will demonstrate how changes in seasonal transmission and host demography have affected temporal patterns of infection. To achieve this, I propose a method to draw a historical contrast of transmission estimates using previously unexplored time series of measles mortality from early 20th century Europe and well-known incidence time series from the 1950s. Descriptions of host demography and the overall patterns of infection through time are used to assess the role of different dynamic forces through alternative models. Host demography is seen to be the primary factor underlying the changing patterns of disease transmission, highlighting key issues of the population dynamics of host-parasite interactions that are relevant to disease control.

Smallpox transmission and control: spatial dynamics in the mainland United Kingdom
Steven Riley, University of Hong Kong, China

Contingency planning for the possible deliberate reintroduction of smallpox has become a priority for many national public health organisations in recent years. We used an individual-based spatial model of smallpox transmission in the UK and census derived journey-to-work data in order to accurately describe the spatiotemporal dynamics of an outbreak of smallpox in the community. A novel Markov chain Monte Carlo algorithm was developed in order to generate spatial networks which were consistent with commuting behaviour. We identified 7 key epidemiological parameters which are unknown at this time for smallpox transmission and tested the sensitivity of model predictions to these parameters before choosing three representative scenarios of smallpox transmission in the UK. We describe the spatiotemporal dynamics for these three illustrative scenarios and assess the efficacy of the following interventions 1) self-imposed rash-motivated quarantine 2) contact tracing with vaccination and identification of at-risk individuals and 3) case-triggered regional vaccination. We agree with some – but not all – recent studies: public information campaigns aimed at reducing symptomatic transmission coupled with contact tracing and vaccination would most likely be sufficient to rapidly halt ongoing transmission. Further, a slight reduction in the expected size and duration of an outbreak could be achieved with regional vaccination. However, additional benefits of regional vaccination are small and do not justify the high numbers of vaccine doses required, with their associated negative side-effects.

Marker selection for population structure in genetic association studies
Yik-Ying Teo Oxford University, UK

Genetic association studies of the future will increasingly progress in the direction of complete genome sequencing of common single nucleotide polymorphisms (SNPs), especially in relation to understanding the allelic variation underlying most common or complex diseases. Improvements in genotyping technology and significantly reduced costs mean that such genome-wide association studies are now possible, heralding an era from the traditional linkage and candidate gene studies. The orders of magnitude increase in the number of genetic markers genotyped introduced new problems with regards to multiple testing and substantial false positive rates, as well as provide vast amount of information for deducing unsuspected structure between the individuals. Typically affected individuals and unrelated controls are sampled from populations defined on the basis of geography, culture and ethnic membership and rarely on inherent genetic structure. Such unaccounted genetic structure within the samples have the potential to produce spurious associations, particularly in studies of complex diseases where the modest signals from multiple disease genes are comparable to the confounding signals from unaccounted population structure. The availability of large number of unlinked markers means that it is possible to statistically infer the population membership of individuals before testing for association for these stratified samples. It was previously suggested that large number of unlinked markers will be required to infer existing population structure to reasonable levels of accuracy. We introduced a measure of information to assess the informativeness of each marker for population structure. We also introduce a novel marker selection scheme for selecting optimal sets of markers. We show that the number of markers required is greatly reduced by selecting optimal sets of informative markers. In this session, we will provide a brief overview of genome-wide association studies as well as the challenges faced. We specifically focus on the area of inferring population structure and the selection of informative markers. This is discussed in the context of a number of simulation studies, and with applications using genome-wide data from the International HapMap Project.

The role of crossimmunity on influenza dynamics
Carlos Castillo-Chavez, Arizona State University, USA

The challenges posed by the variation generated within and between subtypes of influenza type A at the population level will be discussed using mathematical models for homogenously mixing and structured populations.

Models for communicable diseases
Carlos Castillo-Chavez, Arizona State University, USA

An introduction to simple epidemiological models illustrated with applications to influenza, tuberculosis and rotaviruses will be provided in these three lectures.

SituationESP: simulation tools for disease surveillance groups
Chris Skelly, Institute for the Environment, Brunel University, UK

Disease surveillance is a collaborative effort to collect, manage, analyse and disseminate data for the purpose of supporting public health interventions. Surveillance 'systems' are defined by these collaborative 'data sharing' efforts, which continuously strive to improve data quality, timeliness and the efficiency with which it provides 'information for action'. These traditional surveillance systems are 'data centric' and they have a number of limitations that will be difficult to overcome, as SARS and Avian Influenza are demonstrating. However, we are moving closer to a 'model centric' surveillance paradigm that may offer surveillance groups more tools, including disease transmission simulators that attempt to capture the dynamics of disease activity in the communities being monitored.

With surveillance groups and their objectives in mind, we have begun developing simulation tools. Simulations are initialised with available surveillance data and the evaluation of simulations surveillance groups to test their underlying knowledge and assumptions on a regular basis. Surveillance professionals may find that in the future 'model centric' systems allow the goals of 'data centric' systems to be achieved more efficiently, while facilitating an ongoing assessment of disease dynamics at work in their communities. The first SituationESP module developed is a spatial SEIR micro-simulator for person-person diseases.

Modeling healthcare associated infections
Ben Cooper, Health Protection Agency, UK

Healthcare associated infections present three distinct challenges to the modeler that do not apply to typical community pathogens: (i) patient stays in hospitals are typically only a matter of days so the population under investigation changes rapidly over time; (ii) populations of interest are usually very small, so stochastic effects may be expected to dominate; (iii) for most important organisms a large proportion of the transmission is likely to be due to asymptomatically colonized patients. As a consequence, epidemics are usually only partially observed, and swab data may give only a very limited picture of underlying transmission patterns.

This lecture will show how these challenges can be met using mathematical models and modern statistical tools. In particular, fundamental differences between model formulations for studying the transmission of resistant organisms in the community and in hospitals are emphasized, and model predictions are contrasted. The seminar will also introduce the use of models as inferential tools for hospital epidemic data.

From infection cycles to pathogen phylogeny
Jacco Wallinga, National Institutes of Public Health and the Environment, Netherland

Outbreak investigators routinely collect the epidemiological characteristics of infected cases, such as their age, gender, time of symptom onset. The advent of modern molecular typing techniques allow the investigators to collect additional information on the pathogens themaselves, such as their DNA sequences. Unfortunately, it is hard to relate results of molecular typing directly to those of epidemiologic analyses. It is even harder to see how we can relate both of them to predictions of mathematical transmission models that capture our current knowledge of the infection cycle of the pathogen. Here, we attempt to find common ground for analysis of both observed genetic and epidemiologic characteristics. We ask the question: given all the information on two arbitrary cases, what is the likelihood that one case has infected the other? We point out that answering this question requires knowledge on the infection cycle of the pathogen, as captured by the mathematical transmission models. We show that we can use both genetic information or epidemiologic information to reconstruct the likely shape of the infection tree. This opens up perspectives for detecting ancestral relationships among pathogens within and between outbreaks.

Long-term phylogenetic history of dengue from Puerto Rico demonstrates selection and population dynamics in sequence evolution
Shannon N. Bennett, University of Hawaii at Manoa, USA

Dengue fever, caused by four antigenically distinct arboviruses (serotypes DENV-1–DENV-4), has increased dramatically in prevalence worldwide in recent decades, partly due to the spread of the vector and globalization, with concomitant increases in the severe form of the disease, dengue hemorrhagic fever. This dynamic epidemiologic landscape has been accompanied by significant viral evolution. I illustrate several of the topics covered by previous lectures, namely selection, coalescent estimation and population dynamics, using sequence data on dengue isolates from Puerto Rico over the last 25 years.

Experiences of a medical statistician working in a SARS-designated hospital
Arul Earnest, University of Sydney, Australia

In the first part of this presentation, I will share my experience as a medical statistician working in Tan Tock Seng hospital during the recent 2003 SARS outbreak. Starting from the early stage of the outbreak, this talk will guide through the research that was undertaken in the hospital, and highlight the synergy between statistical modelling and clinicians’ experience in dealing with an outbreak of an infectious disease. The second topic will discuss the use of spatial epidemiology as a tool for policy-makers in the study of an infectious disease such as dengue. The application of this relatively new field in Singapore warrants further discussion. Possible uses of this methodology include development and refinement of proactive cluster surveillance systems, models to identify hotspots of areas that show a significant increase of cases over time, models to examine the relationship of population and environmental risk factors with the disease and also as a tool to evaluate effectiveness of intervention measures.

Economic outcomes of influenza pandemic stockpiling strategies in Singapore
Vernon Lee, Tan Tock Seng Hospital, Singapore

Influenza pandemics result in high morbidity, mortality, and economic costs. Treatment and prophylaxis with Neuramidase-inhibitors are of interest because of their effectiveness against influenza; with many nations building stockpiles. We compared the economic outcomes of prophylaxis and treatment with oseltamivir versus the status quo in Singapore.
We used cost-benefit and cost-effectiveness models to determine economic outcomes in a pandemic. Input variables were obtained from various local and international sources. There is much uncertainty surrounding influenza pandemics; and thus sensitivity and Monte Carlo simulation analyses were performed.
If no action is taken during a pandemic, we estimated that there would be 525 - 1775 deaths with an economic cost of $0.9 to $2.2 billion. In severe pandemics, there may be up to 50,000 deaths and a cost of $112 billion. The treatment only strategy, at $31 per person, has the overall optimal mean economic benefit. The optimal oseltamivir treatment stockpile level lie between 40% and 60% of the population. Prophylaxis, at $22 per person per week, is economically beneficial in the age-stratified high risk sub-populations which account for 75% of all deaths. In high case-fatality pandemics, prophylaxis is most beneficial compared with the other options.
The treatment only strategy maximizes the mean economic benefit but prophylaxis maximizes lives saved, and is most economically beneficial in a severe pandemic. The final decision involves risk analyses and trade offs between lives saved and economic costs.

Epidemic dengue: recrudescence of an old enemy
Steven Ooi, Ministry of Health, Singapore

Singapore has not been spared from the regional recrudescence of dengue fever and dengue haemorrhagic fever. In the first 35 epidemiological weeks of 2005, 8,850 cases were reported. Disease notifications exceeded 750 cases a week in Sep 2005. To date, there have been eight dengue deaths this year, giving a case fatality of 0.9 per 1,000 cases. The current epidemic is unprecedented and could not be attributed to more imported infections, decline in population herd immunity, re-emergence of the DEN-3 virus serotype (DEN-1 remains as the predominant strain), or regional climate change. Two epidemiological features, a disproportionate rise in dengue among HDB dwellers and a shift in the disease distribution towards younger age groups, bucked the historical pattern that occurred from 1980-2000 and suggested an increase in infections acquired outside the homes in HDB public housing estates. Mathematical models have shown that the effective force of infection (ie, no of new infections per unit case and time) is a product of four factors: number of mosquitoes per person, mosquito biting rate, vector competence (proportion of infective bites), and infective mosquito prevalence. Our paper outlines the current dengue outbreak management strategies, and the findings of a 2004 MOH-NEA study in a highly dengue endemic area which confirmed that elimination of Aedes mosquitoes is still the key to curb rising incidence.

Some entomological factors in dengue dynamics in Singapore
Christina Liew, Environmental Health Institute, Singapore

Abstract: A brief overview of mosquito vectors in Singapore, with emphasis on Aedes mosquitoes and some entomological factors (i.e. flight range, temperature, parity and survival) that affect dengue dynamics in Singapore.

Recent changes in the local epidemiology of dengue: the view from outside
Mark Chen, Tan Tock Seng Hospital, Singapore

Singapore is experiencing its worst ever recorded outbreak of dengue. As of 17th Sept this year (2005), 9919 cases have been notified, compared to 5145 cases for the same period last year.

In the face of mounting numbers, the press, the public and the professionals have all offered their views as to the possible reasons for the increase in the number of cases. We review the prevailing hypotheses against existing published data, comment on the validity of these assumptions, and suggest how these hypotheses may be verified. In the course of the talk, we will identify the present gaps in our knowledge, and propose ways to fill those gaps through entomological studies, classical epidemiology, spatio-temporal analyses and mathematical models of disease transmission.