Extracting the time-dependent transmission rate from infection data via solution of an inverse ODE problem

The transmission rate of many acute infectious diseases varies significantly in time, but the underlying mechanisms are usually uncertain. They may include seasonal changes in the environment, contact rate, immune system response, etc. The transmission rate has been thought difficult to measure directly. We present a new algorithm to compute the time-dependent transmission rate directly from prevalence data, which makes no assumptions about the number of susceptible or vital rates. The algorithm follows our complete and explicit solution of a mathematical inverse problem for SIR-type transmission models. We prove that almost any infection profile can be perfectly fitted by an SIR model with variable transmission rate. This clearly shows a serious danger of overfitting such transmission models. We illustrate the algorithm with historic UK measles data and our observations support the common belief that measles transmission was predominantly driven by school contacts.     

Cities and villages: infection hierarchies in a measles metapopulation

An important issue in the dynamics of directly transmitted microparasites is the relationship between infection probability and host density. We use models and extensive spatio-temporal data for the incidence of measles to examine evidence for spatial heterogeneity in transmission probability, in terms of urban–rural hierarchies in infection rate. Pre-vaccination measles data for England and Wales show strong evidence for urban–rural heterogeneities in infection rate – the proportion of urban cases rises significantly before major epidemics. The model shows that this effect is consistent with a higher infection rate in large cities, though small towns have epidemic characteristics intermediate between town and country. Surprisingly, urban and rural areas of the same population size have a similar propensity for local extinction of infection. A spatial map of urban–rural correlations reveals complex regional patterns of synchronization of towns and cities. The hierarchical heterogeneities in infection persist into the vaccine era; their implications for disease persistence and control are discussed.     

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

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

A Network-based Analysis of the 1861 Hagelloch Measles Data

Summary In this article, we demonstrate a statistical method for fitting the parameters of a sophisticated network and epidemic model to disease data. The pattern of contacts between hosts is described by a class of dyadic independence exponential-family random graph models (ERGMs), whereas the transmission process that runs over the network is modeled as a stochastic susceptible-exposed-infectious-removed (SEIR) epidemic. We fit these models to very detailed data from the 1861 measles outbreak in Hagelloch, Germany. The network models include parameters for all recorded host covariates including age, sex, household, and classroom membership and household location whereas the SEIR epidemic model has exponentially distributed transmission times with gamma-distributed latent and infective periods. This approach allows us to make meaningful statements about the structure of the population-separate from the transmission process-as well as to provide estimates of various biological quantities of interest, such as the effective reproductive number, R. Using reversible jump Markov chain Monte Carlo, we produce samples from the joint posterior distribution of all the parameters of this model-the network, transmission tree, network parameters, and SEIR parameters-and perform Bayesian model selection to find the best-fitting network model. We compare our results with those of previous analyses and show that the ERGM network model better fits the data than a Bernoulli network model previously used. We also provide a software package, written in R, that performs this type of analysis.     

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

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

A review of data needed to parameterize a dynamic model of measles in developing countries

Background

Dynamic models of infection transmission can project future disease burden within a population. Few dynamic measles models have been developed for low-income countries, where measles disease burden is highest. Our objective was to review the literature on measles epidemiology in low-income countries, with a particular focus on data that are needed to parameterize dynamic models.

Methods

We included age-stratified case reporting and seroprevalence studies with fair to good sample sizes for mostly urban African and Indian populations. We emphasized studies conducted before widespread immunization. We summarized age-stratified attack rates and seroprevalence profiles across these populations. Using the study data, we fitted a "representative" seroprevalence profile for African and Indian settings. We also used a catalytic model to estimate the age-dependent force of infection for individual African and Indian studies where seroprevalence was surveyed. We used these data to quantify the effects of population density on the basic reproductive number R ₀.

Results

The peak attack rate usually occurred at age 1 year in Africa, and 1 to 2 years in India, which is earlier than in developed countries before mass vaccination. Approximately 60% of children were seropositive for measles antibody by age 2 in Africa and India, according to the representative seroprevalence profiles. A statistically significant decline in the force of infection with age was found in 4 of 6 Indian seroprevalence studies, but not in 2 African studies. This implies that the classic threshold result describing the critical proportion immune (p _c) required to eradicate an infectious disease, p _c = 1-1/R ₀, may overestimate the required proportion immune to eradicate measles in some developing country populations. A possible, though not statistically significant, positive relation between population density and R ₀ for various Indian and African populations was also found. These populations also showed a similar pattern of waning of maternal antibodies. Attack rates in rural Indian populations show little dependence on vaccine coverage or population density compared to urban Indian populations. Estimated R ₀ values varied widely across populations which has further implications for measles elimination.

Conclusions

It is possible to develop a broadly informative dynamic model of measles transmission in low-income country settings based on existing literature, though it may be difficult to develop a model that is closely tailored to any given country. Greater efforts to collect data specific to low-income countries would aid in control efforts by allowing highly population-specific models to be developed.     

An in-host model of acute infection: measles as a case study

The epidemiology of acute infections is strongly influenced by the immune status of individuals. In-host models can provide quantitative predictions of immune status and can thus offer valuable insights into the factors that influence transmission between individuals and the effectiveness of vaccination protocols with respect to individual immunity. Here we develop an in-host model of measles infection. The model explicitly considers the effects of immune system memory and CD8 T-cells, which are key to measles clearance. The model is used to determine the effects of waning immunity through vaccination and infection, the effects of booster exposures or vaccines on the level of immunity, and the immune system characteristics that result in measles transmission (R(0)>1) even if an individual has no apparent clinical symptoms. We find that the level of immune system CD8 T-cells at the time of exposure to measles determines whether an individual will experience a measles infection or simply a boost in immunity. We also find that the infected cell dynamics are a good indicator of measles transmission and the degree of symptoms that will be experienced. Our results indicate that the degree of immunity in adults is independent of the source of exposure in early childhood, be it vaccine or natural infection.     

Seasonal transmission dynamics of measles in China

Measles, a highly contagious infection caused by the measles virus, is a major public health problem in China. The reported measles cases decreased dramatically from 2004 to 2012 due to the mandatory measles vaccine program started in 2005 and the goal of eliminating measles by 2012. However, after reaching its lowest level in 2012, measles has resurged again since 2013. Since the monthly data of measles cases exhibit a seasonally fluctuating pattern, based on the measles model in Earn et al. (Science 287:667–670, 2000), we propose a susceptible, exposed, infectious, and recovered model with periodic transmission rate to investigate the seasonal measles epidemics and the effect of vaccination. We calculate the basic reproduction number \({\mathcal {R}}_{0}\), analyze the dynamical behavior of the model, and use the model to simulate the monthly data of measles cases reported in China. We also carry out some sensitivity analysis of \({\mathcal {R}}_{0}\) in the terms of various model parameters which shows that measles can be controlled and eventually eradicated by increasing the immunization rate, improving the effective vaccine management, and enhancing the awareness of people about measles.     

Seasonality and comparative dynamics of six childhood infections in pre-vaccination Copenhagen

Seasonal variation in infection transmission is a key determinant of epidemic dynamics of acute infections. For measles, the best-understood strongly immunizing directly transmitted childhood infection, the perception is that term-time forcing is the main driver of seasonality in developed countries. The degree to which this holds true across other acute immunizing childhood infections is not clear. Here, we identify seasonal transmission patterns using a unique long-term dataset with weekly incidence of six infections including measles. Data on age–incidence allow us to quantify the mean age of infection. Results indicate correspondence between dips in transmission and school holidays for some infections, but there are puzzling discrepancies, despite close correspondence between average age of infection and age of schooling. Theoretical predictions of the relationship between amplitude of seasonality and basic reproductive rate of infections that should result from term-time forcing are also not upheld. We conclude that where yearly trajectories of susceptible numbers are perturbed, e.g. via waning of immunity, seasonality is unlikely to be entirely driven by term-time forcing. For the three bacterial infections, pertussis, scarlet fever and diphtheria, there is additionally a strong increase in transmission during the late summer before the end of school vacations.     

Generalized character process models: estimating the genetic basis of traits that cannot be observed and that change with age or environmental conditions

The genetic analysis of characters that change as a function of some independent and continuous variable has received increasing attention in the biological and statistical literature. Previous work in this area has focused on the analysis of normally distributed characters that are directly observed. We propose a framework for the development and specification of models for a quantitative genetic analysis of function-valued characters that are not directly observed, such as genetic variation in age-specific mortality rates or complex threshold characters. We employ a hybrid Markov chain Monte Carlo algorithm involving a Monte Carlo EM algorithm coupled with a Markov chain approximation to the likelihood, which is quite robust and provides accurate estimates of the parameters in our models. The methods are investigated using simulated data and are applied to a large data set measuring mortality rates in the fruit fly, Drosophila melanogaster.     

Integrating data mining and transmission theory in the ecology of infectious diseases

Our understanding of ecological processes is built on patterns inferred from data. Applying modern analytical tools such as machine learning to increasingly high dimensional data offers the potential to expand our perspectives on these processes, shedding new light on complex ecological phenomena such as pathogen transmission in wild populations. Here, we propose a novel approach that combines data mining with theoretical models of disease dynamics. Using rodents as an example, we incorporate statistical differences in the life history features of zoonotic reservoir hosts into pathogen transmission models, enabling us to bound the range of dynamical phenomena associated with hosts, based on their traits. We then test for associations between equilibrium prevalence, a key epidemiological metric and data on human outbreaks of rodent‐borne zoonoses, identifying matches between empirical evidence and theoretical predictions of transmission dynamics. We show how this framework can be generalized to other systems through a rubric of disease models and parameters that can be derived from empirical data. By linking life history components directly to their effects on disease dynamics, our mining‐modelling approach integrates machine learning and theoretical models to explore mechanisms in the macroecology of pathogen transmission and their consequences for spillover infection to humans.     

Inferring the effective reproductive number from deterministic and semi-deterministic compartmental models using incidence and mobility data

The effective reproduction number (ℜ_t) is a theoretical indicator of the course of an infectious disease that allows policymakers to evaluate whether current or previous control efforts have been successful or whether additional interventions are necessary. This metric, however, cannot be directly observed and must be inferred from available data. One approach to obtaining such estimates is fitting compartmental models to incidence data. We can envision these dynamic models as the ensemble of structures that describe the disease’s natural history and individuals’ behavioural patterns. In the context of the response to the COVID-19 pandemic, the assumption of a constant transmission rate is rendered unrealistic, and it is critical to identify a mathematical formulation that accounts for changes in contact patterns. In this work, we leverage existing approaches to propose three complementary formulations that yield similar estimates for ℜ_t based on data from Ireland’s first COVID-19 wave. We describe these Data Generating Processes (DGP) in terms of State-Space models. Two (DGP1 and DGP2) correspond to stochastic process models whose transmission rate is modelled as Brownian motion processes (Geometric and Cox-Ingersoll-Ross). These DGPs share a measurement model that accounts for incidence and transmission rates, where mobility data is assumed as a proxy of the transmission rate. We perform inference on these structures using Iterated Filtering and the Particle Filter. The final DGP (DGP3) is built from a pool of deterministic models that describe the transmission rate as information delays. We calibrate this pool of models to incidence reports using Hamiltonian Monte Carlo. By following this complementary approach, we assess the tradeoffs associated with each formulation and reflect on the benefits/risks of incorporating proxy data into the inference process. We anticipate this work will help evaluate the implications of choosing a particular formulation for the dynamics and observation of the time-varying transmission rate.     

Multivariate Markov process models for the transmission of methicillin-resistant Staphylococcus aureus in a hospital ward

Summary .   Methicillin-resistant Staphylococcus Aureus (MRSA) is a pathogen that continues to be of major concern in hospitals. We develop models and computational schemes based on observed weekly incidence data to estimate MRSA transmission parameters. We extend the deterministic model of McBryde, Pettitt, and McElwain (2007, Journal of Theoretical Biology 245, 470–481) involving an underlying population of MRSA colonized patients and health-care workers that describes, among other processes, transmission between uncolonized patients and colonized health-care workers and vice versa. We develop new bivariate and trivariate Markov models to include incidence so that estimated transmission rates can be based directly on new colonizations rather than indirectly on prevalence. Imperfect sensitivity of pathogen detection is modeled using a hidden Markov process. The advantages of our approach include (i) a discrete valued assumption for the number of colonized health-care workers, (ii) two transmission parameters can be incorporated into the likelihood, (iii) the likelihood depends on the number of new cases to improve precision of inference, (iv) individual patient records are not required, and (v) the possibility of imperfect detection of colonization is incorporated. We compare our approach with that used by McBryde et al. (2007) based on an approximation that eliminates the health-care workers from the model, uses Markov chain Monte Carlo and individual patient data. We apply these models to MRSA colonization data collected in a small intensive care unit at the Princess Alexandra Hospital, Brisbane, Australia.     

Cell-associated immunity to measles (rubeola): The demonstration of in vitro lymphocyte tritiated thymidine incorporation in response to measles complement fixation antigen

Lack of a reliable in vitro assay for lymphocyte responsiveness to measles (rubeola) has hampered our understanding of the cell-associated response in diseases caused by, or related to, the measles virus. We report a reliable and reproducible system for demonstrating specific lymphocyte incorporation of ³H-thymidine in response to measles complement fixation antigen (CFA). Seventeen patients with positive histories of measles as children demonstrated a dose-response curve that varied between individuals but was constant for each individual. Kinetic data disclosed maximal responsiveness on day 7, and viral inactivation experiments disclosed that live virus was neither necessary for nor inhibitory to the reaction. The implications of this assay in terms of our understanding of the cell-associated response to measles virus in clinical measles and SSPE are discussed. The concept is explored that membrane-associated antigen is crucial in demonstrating the host's cellular immune response to viruses that can grow by cell-to-contiguous cell spread.     

Deciphering the relative weights of demographic transition and vaccination in the decrease of measles incidence in Italy

In Italy, during the course of the past century to the present-day, measles incidence underwent a remarkable decreasing trend that started well before the introduction of the national immunization programme. In this work, we aim at examining to what extent both the demographic transition, characterized by declining mortality and fertility rates over time, and the vaccination programme are responsible for the observed epidemiological pattern. Making use of a non-stationary, age-structured disease transmission model, we show that in the pre-vaccination era, from 1901 to 1982, the decline in birth rates has resulted in a drastic decrease in the effective transmission rate, which in turn has determined a declining trend of measles incidence (from 25.2 to 10.3 infections per 1000 individuals). However, since 1983, vaccination appears to have become the major contributing factor in the decrease of measles incidence, which otherwise would have remained stable as a consequence of the nearly constant birth rates. This led to a remarkable decrease in the effective transmission rate, to a level well below the critical threshold for disease persistence. These findings call for the adoption of epidemiological models, which deviate the age structure from stationary equilibrium solutions, to better understand the biology of infectious diseases and evaluate immunization programmes.     

Persistent Chaos of Measles Epidemics in the Prevaccination United States Caused by a Small Change in Seasonal Transmission Patterns

Epidemics of infectious diseases often occur in predictable limit cycles. Theory suggests these cycles can be disrupted by high amplitude seasonal fluctuations in transmission rates, resulting in deterministic chaos. However, persistent deterministic chaos has never been observed, in part because sufficiently large oscillations in transmission rates are uncommon. Where they do occur, the resulting deep epidemic troughs break the chain of transmission, leading to epidemic extinction, even in large cities. Here we demonstrate a new path to locally persistent chaotic epidemics via subtle shifts in seasonal patterns of transmission, rather than through high-amplitude fluctuations in transmission rates. We base our analysis on a comparison of measles incidence in 80 major cities in the prevaccination era United States and United Kingdom. Unlike the regular limit cycles seen in the UK, measles cycles in US cities consistently exhibit spontaneous shifts in epidemic periodicity resulting in chaotic patterns. We show that these patterns were driven by small systematic differences between countries in the duration of the summer period of low transmission. This example demonstrates empirically that small perturbations in disease transmission patterns can fundamentally alter the regularity and spatiotemporal coherence of epidemics.     

On parameter estimation in population models III: time-inhomogeneous processes and observation error

Essential to applying a mathematical model to a real-world application is calibrating the model to data. Methods for calibrating population models often become computationally infeasible when the population size (more generally the size of the state space) becomes large, or other complexities such as time-dependent transition rates, or sampling error, are present. Continuing previous work in this series on the use of diffusion approximations for efficient calibration of continuous-time Markov chains, I present efficient techniques for time-inhomogeneous chains and accounting for observation error. Observation error (partial observability) is accounted for by joint estimation using a scaled unscented Kalman filter for state-space models. The methodology will be illustrated with respect to models of disease dynamics incorporating seasonal transmission rate and in the presence of observation error, including application to two influenza outbreaks and measles in London in the pre-vaccination era.     

Effect of thermal environment on the temporal,spatial and seasonal occurrence of measles in Ondo state,Nigeria

We investigated the temporal and spatial dynamics, as well as the seasonal occurrence of measles in Ondo state, Nigeria, to better understand the role of the thermal environment in the occurrence of the childhood killer disease measles, which ranks among the top ten leading causes of child deaths worldwide. The linkages between measles and atmospheric environmental factors were examined by correlating human-biometeorological parameters in the study area with reported clinical cases of measles for the period 1998–2008. We also applied stepwise regression analysis in order to determine the human-biometeorological parameters that lead to statistical changes in reported clinical cases of measles. We found that high reported cases of measles are associated with the least populated areas, where rearing and cohabitation of livestock/domestic animals within human communities are common. There was a significant correlation (P < 0.01) between monthly cases of measles and human-biometeorological parameters except wind speed and vapour pressure. High transmission of measles occurred in the months of January to May during the dry season when human thermal comfort indices are very high. This highlights the importance of the thermal environment in disease demographics since it accounted for more than 40% variation in measles transmission within the study period.     

Molecular epidemiology of measles virus

Genetic analysis of viruses associated with recent outbreaks of measles in the United States indicated that at least four genotypes were present during 1994 and 1995. None of these more recent genotypes were related to the genotype responsible for the resurgence of measles cases in the United States between 1989 and 1992. The sequence data confirmed that the majority of measles cases that occurred in the United States between 1994 and 1995 were the result of international importation of virus. The data also suggested that transmission of the genotype associated with the resurgence had been interrupted by aggressive control measures. Therefore, molecular epidemiologic studies will provide a powerful means to measure the success of measles control strategies.