A nonlinear programming approach for estimation of transmission parameters in childhood infectious disease using a continuous time model

Mathematical models can enhance our understanding of childhood infectious disease dynamics, but these models depend on appropriate parameter values that are often unknown and must be estimated from disease case data. In this paper, we develop a framework for efficient estimation of childhood infectious disease models with seasonal transmission parameters using continuous differential equations containing model and measurement noise. The problem is formulated using the simultaneous approach where all state variables are discretized, and the discretized differential equations are included as constraints, giving a large-scale algebraic nonlinear programming problem that is solved using a nonlinear primal–dual interior-point solver. The technique is demonstrated using measles case data from three different locations having different school holiday schedules, and our estimates of the seasonality of the transmission parameter show strong correlation to school term holidays. Our approach gives dramatic efficiency gains, showing a 40–400-fold reduction in solution time over other published methods. While our approach has an increased susceptibility to bias over techniques that integrate over the entire unknown state-space, a detailed simulation study shows no evidence of bias. Furthermore, the computational efficiency of our approach allows for investigation of a large model space compared with more computationally intensive approaches.     

Effects of the infectious period distribution on predicted transitions in childhood disease dynamics

The population dynamics of infectious diseases occasionally undergo rapid qualitative changes, such as transitions from annual to biennial cycles or to irregular dynamics. Previous work, based on the standard seasonally forced ‘susceptible–exposed–infectious–removed’ (SEIR) model has found that transitions in the dynamics of many childhood diseases result from bifurcations induced by slow changes in birth and vaccination rates. However, the standard SEIR formulation assumes that the stage durations (latent and infectious periods) are exponentially distributed, whereas real distributions are narrower and centred around the mean. Much recent work has indicated that realistically distributed stage durations strongly affect the dynamical structure of seasonally forced epidemic models. We investigate whether inferences drawn from previous analyses of transitions in patterns of measles dynamics are robust to the shapes of the stage duration distributions. As an illustrative example, we analyse measles dynamics in New York City from 1928 to 1972. We find that with a fixed mean infectious period in the susceptible–infectious–removed (SIR) model, the dynamical structure and predicted transitions vary substantially as a function of the shape of the infectious period distribution. By contrast, with fixed mean latent and infectious periods in the SEIR model, the shapes of the stage duration distributions have a less dramatic effect on model dynamical structure and predicted transitions. All these results can be understood more easily by considering the distribution of the disease generation time as opposed to the distributions of individual disease stages. Numerical bifurcation analysis reveals that for a given mean generation time the dynamics of the SIR and SEIR models for measles are nearly equivalent and are insensitive to the shapes of the disease stage distributions.     

A dynamic dose–response model to account for exposure patterns in risk assessment: a case study in inhalation anthrax

The most commonly used dose–response models implicitly assume that accumulation of dose is a time-independent process where each pathogen has a fixed risk of initiating infection. Immune particle neutralization of pathogens, however, may create strong time dependence; i.e. temporally clustered pathogens have a better chance of overwhelming the immune particles than pathogen exposures that occur at lower levels for longer periods of time. In environmental transmission systems, we expect different routes of transmission to elicit different dose–timing patterns and thus potentially different realizations of risk. We present a dose–response model that captures time dependence in a manner that incorporates the dynamics of initial immune response. We then demonstrate the parameter estimation of our model in a dose–response survival analysis using empirical time-series data of inhalational anthrax in monkeys in which we find slight dose–timing effects. Future dose–response experiments should include varying the time pattern of exposure in addition to varying the total doses delivered. Ultimately, the dynamic dose–response paradigm presented here will improve modelling of environmental transmission systems where different systems have different time patterns of exposure.     

Influence of birth rates and transmission rates on the global seasonality of rotavirus incidence

Rotavirus is a major cause of mortality in developing countries, and yet the dynamics of rotavirus in such settings are poorly understood. Rotavirus is typically less seasonal in the tropics, although recent observational studies have challenged the universality of this pattern. While numerous studies have examined the association between environmental factors and rotavirus incidence, here we explore the role of intrinsic factors. By fitting a mathematical model of rotavirus transmission dynamics to published age distributions of cases from 15 countries, we obtain estimates of local transmission rates. Model-predicted patterns of seasonal incidence based solely on differences in birth rates and transmission rates are significantly correlated with those observed (Spearman''s ρ = 0.65, p < 0.05). We then examine seasonal patterns of rotavirus predicted across a range of different birth rates and transmission rates and explore how vaccination may impact these patterns. Our results suggest that the relative lack of rotavirus seasonality observed in many tropical countries may be due to the high birth rates and transmission rates typical of developing countries rather than being driven primarily by environmental conditions. While vaccination is expected to decrease the overall burden of disease, it may increase the degree of seasonal variation in the incidence of rotavirus in some settings.     

Essential epidemiological mechanisms underpinning the transmission dynamics of seasonal influenza

Seasonal influenza has considerable impact around the world, both economically and in mortality among risk groups, but there is considerable uncertainty as to the essential mechanisms and their parametrization. In this paper, we identify a number of characteristic features of influenza incidence time series in temperate regions, including ranges of annual attack rates and outbreak durations. By constraining the output of simple models to match these characteristic features, we investigate the role played by population heterogeneity, multiple strains, cross-immunity and the rate of strain evolution in the generation of incidence time series. Results indicate that an age-structured model with non-random mixing and co-circulating strains are both required to match observed time-series data. Our work gives estimates of the seasonal peak basic reproduction number, R₀, in the range 1.6–3. Estimates of R₀ are strongly correlated with the timescale for waning of immunity to current circulating seasonal influenza strain, which we estimate is between 3 and 8 years. Seasonal variation in transmissibility is largely confined to 15–30% of its mean value. While population heterogeneity and cross-immunity are required mechanisms, the degree of heterogeneity and cross-immunity is not tightly constrained. We discuss our findings in the context of other work fitting to seasonal influenza data.     

Predictability in a highly stochastic system: final size of measles epidemics in small populations

A standard assumption in the modelling of epidemic dynamics is that the population of interest is well mixed, and that no clusters of metapopulations exist. The well-known and oft-used SIR model, arguably the most important compartmental model in theoretical epidemiology, assumes that the disease being modelled is strongly immunizing, directly transmitted and has a well-defined period of infection, in addition to these population mixing assumptions. Childhood infections, such as measles, are prime examples of diseases that fit the SIR-like mechanism. These infections have been well studied for many systems with large, well-mixed populations with endemic infection. Here, we consider a setting where populations are small and isolated. The dynamics of infection are driven by stochastic extinction–recolonization events, producing large, sudden and short-lived epidemics before rapidly dying out from a lack of susceptible hosts. Using a TSIR model, we fit prevaccination measles incidence and demographic data in Bornholm, the Faroe Islands and four districts of Iceland, between 1901 and 1965. The datasets for each of these countries suffer from different levels of data heterogeneity and sparsity. We explore the potential for prediction of this model: given historical incidence data and up-to-date demographic information, and knowing that a new epidemic has just begun, can we predict how large it will be? We show that, despite a lack of significant seasonality in the incidence of measles cases, and potentially severe heterogeneity at the population level, we are able to estimate the size of upcoming epidemics, conditioned on the first time step, to within reasonable confidence. Our results have potential implications for possible control measures for the early stages of new epidemics in small populations.     

The interaction of seasonal forcing and immunity and the resonance dynamics of malaria

Theory has emphasized the importance of both intrinsic factors such as host immunity and extrinsic drivers such as climate in determining disease dynamics. In particular, seasonality may lead to multi-annual cycles in prevalence, but the likelihood of this depends on the role of acquired immunity. Some diseases including malaria have immunity that falls between the classic susceptible–infectious–removed and susceptible–infectious–susceptible models. Here, we investigate the general conditions promoting the subharmonic resonance behaviour that may lead to multi-annual cycles in a general malaria dynamical model. Utilizing two complementary approaches to bifurcation analyses, we show that resonance is promoted by processes shortening the length of the infectious period and that subharmonic cycles are favoured in situations with strong seasonality in transmission but at intermediate levels of endemicity. We discuss the implications of our results for understanding prevalence patterns in long-term malaria datasets from Kenya that show multi-annual cycles and one from Thailand that does not and discuss the possible implications of treatment.     

A century of transitions in New York City's measles dynamics

Infectious diseases spreading in a human population occasionally exhibit sudden transitions in their qualitative dynamics. Previous work has successfully predicted such transitions in New York City''s historical measles incidence using the seasonally forced susceptible–infectious–recovered (SIR) model. This work relied on a dataset spanning 45 years (1928–1973), which we have extended to 93 years (1891–1984). We identify additional dynamical transitions in the longer dataset and successfully explain them by analysing attractors and transients of the same mechanistic epidemiological model.     

Estimating epidemic coupling between populations from the time to invasion

Identifying the mechanisms by which diseases spread among populations is important for understanding and forecasting patterns of epidemics and pandemics. Estimating transmission coupling among populations is challenging because transmission events are difficult to observe in practice, and connectivity among populations is often obscured by local disease dynamics. We consider the common situation in which an epidemic is seeded in one population and later spreads to a second population. We present a method for estimating transmission coupling between the two populations, assuming they can be modelled as susceptible–infected–removed (SIR) systems. We show that the strength of coupling between the two populations can be estimated from the time taken for the disease to invade the second population. Confidence in the estimate is low if only a single invasion event has been observed, but is substantially improved if numerous independent invasion events are observed. Our analysis of this simplest, idealized scenario represents a first step toward developing and verifying methods for estimating epidemic coupling among populations in an ever-more-connected global human population.     

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

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

Distribution of vaccine/antivirals and the ‘least spread line’ in a stratified population

We describe a prioritization scheme for an allocation of a sizeable quantity of vaccine or antivirals in a stratified population. The scheme builds on an optimal strategy for reducing the epidemic''s initial growth rate in a stratified mass-action model. The strategy is tested on the EpiSims network describing interactions and influenza dynamics in the population of Utah, where the stratification we have chosen is by age (0–6, 7–13, 14–18, adults). No prior immunity information is available, thus everyone is assumed to be susceptible—this may be relevant, possibly with the exception of persons over 50, to the 2009 H1N1 influenza outbreak. We have found that the top priority in an allocation of a sizeable quantity of seasonal influenza vaccinations goes to young children (0–6), followed by teens (14–18), then children (7–13), with the adult share being quite low. These results, which rely on the structure of the EpiSims network, are compared with the current influenza vaccination coverage levels in the US population.     

Parameterizing state–space models for infectious disease dynamics by generalized profiling: measles in Ontario

Parameter estimation for infectious disease models is important for basic understanding (e.g. to identify major transmission pathways), for forecasting emerging epidemics, and for designing control measures. Differential equation models are often used, but statistical inference for differential equations suffers from numerical challenges and poor agreement between observational data and deterministic models. Accounting for these departures via stochastic model terms requires full specification of the probabilistic dynamics, and computationally demanding estimation methods. Here, we demonstrate the utility of an alternative approach, generalized profiling, which provides robustness to violations of a deterministic model without needing to specify a complete probabilistic model. We introduce novel means for estimating the robustness parameters and for statistical inference in this framework. The methods are applied to a model for pre-vaccination measles incidence in Ontario, and we demonstrate the statistical validity of our inference through extensive simulation. The results confirm that school term versus summer drives seasonality of transmission, but we find no effects of short school breaks and the estimated basic reproductive ratio ℛ₀ greatly exceeds previous estimates. The approach applies naturally to any system for which candidate differential equations are available, and avoids many challenges that have limited Monte Carlo inference for state–space models.     

Age-dependent partnering and the HIV transmission chain: a microsimulation analysis

Efficient planning and evaluation of human immunodeficiency virus (HIV) prevention programmes requires an understanding of what sustains the epidemic, including the mechanism by which HIV transmission keeps pace with the ageing of the infected population. Recently, more detailed population models have been developed which represent the epidemic with sufficient detail to characterize the dynamics of ongoing transmission. Here, we describe the structure and parameters of such a model, called EMOD-HIV v. 0.7. We analyse the chains of transmission that allow the HIV epidemic to propagate across age groups in this model. In order to prevent the epidemic from dying out, the virus must find younger victims faster than its extant victims age and die. The individuals who enable such transmission events in EMOD-HIV v. 0.7 are higher concurrency, co-infected males aged 26–29 and females aged 23–24. Prevention programmes that target these populations could efficiently interrupt the mechanisms that allow HIV to transmit at a pace that is faster than the progress of time.     

Rheological and micro-Raman time-series characterization of enzyme sol–gel solution toward morphological control of electrospun fibers

Rheological and micro-Raman time-series characterizations were used to investigate the chemical evolutionary changes of silica sol–gel mixtures for electrospinning fibers to immobilize an enzyme (tyrosinase). Results of dynamic rheological measurements agreed with the expected structural transitions associated with reacting sol–gel systems. The electrospinning sols exhibited shear-thinning behavior typical of a power law model. Ultrafine (200–300 nm diameter) fibers were produced at early and late times within the reaction window of approximately one hour from initial mixing of sol solutions with and without enzyme; diameter distributions of these fibers showed much smaller deviations than expected. The enzyme markedly increased magnitudes of both elastic and viscous moduli but had no significant impact on final fiber diameters, suggesting that the shear-thinning behavior of both sol–gel mixtures is dominant in the fiber elongation process. The time course and scale for the electrospinning batch fabrication show strong correlations between the magnitudes in rheological property changes over time and the chemical functional group evolution obtained from micro-Raman time-series analysis of the reacting sol–gel systems.     

Noise, nonlinearity and seasonality: the epidemics of whooping cough revisited

Understanding the mechanisms that generate oscillations in the incidence of childhood infectious diseases has preoccupied epidemiologists and population ecologists for nearly two centuries. This body of work has generated simple yet powerful explanations for the epidemics of measles and chickenpox, while the dynamics of other infectious diseases, such as whooping cough, have proved more challenging to decipher. A number of authors have, in recent years, proposed that the noisy and somewhat irregular epidemics of whooping cough may arise due to stochasticity and its interaction with nonlinearity in transmission and seasonal variation in contact rates. The reason underlying the susceptibility of whooping cough dynamics to noise and the precise nature of its transient dynamics remain poorly understood. Here we use household data on the incubation period in order to parametrize more realistic distributions of the latent and infectious periods. We demonstrate that previously reported phenomena result from transients following the interaction between the stable annual attractor and unstable multiennial solutions.     

Modelling the long-term dynamics of pre-vaccination pertussis

The dynamics of strongly immunizing childhood infections is still not well understood. Although reports of successful modelling of several data records can be found in the previous literature, the key determinants of the observed temporal patterns have not yet been clearly identified. In particular, different models of immunity waning and degree of protection applied to disease- and vaccine-induced immunity have been debated in the previous literature on pertussis. Here, we study the effect of disease-acquired immunity on the long-term patterns of pertussis prevalence. We compare five minimal models, all of which are stochastic, seasonally forced, well-mixed models of infection, based on susceptible–infective–recovered dynamics in a closed population. These models reflect different assumptions about the immune response of naive hosts, namely total permanent immunity, immunity waning, immunity waning together with immunity boosting, reinfection of recovered and repeat infection after partial immunity waning. The power spectra of the output prevalence time series characterize the long-term dynamics of the models. For epidemiological parameters consistent with published data, the power spectra show quantitative and even qualitative differences, which can be used to test their assumptions by comparison with ensembles of several-decades-long pre-vaccination data records. We illustrate this strategy on two publicly available historical datasets.     

Joint modelling of serological and hospitalization data reveals that high levels of pre-existing immunity and school holidays shaped the influenza A pandemic of 2009 in The Netherlands

Obtaining a quantitative understanding of the transmission dynamics of influenza A is important for predicting healthcare demand and assessing the likely impact of intervention measures. The pandemic of 2009 provides an ideal platform for developing integrative analyses as it has been studied intensively, and a wealth of data sources is available. Here, we analyse two complementary datasets in a disease transmission framework: cross-sectional serological surveys providing data on infection attack rates, and hospitalization data that convey information on the timing and duration of the pandemic. We estimate key epidemic determinants such as infection and hospitalization rates, and the impact of a school holiday. In contrast to previous approaches, our novel modelling of serological data with mixture distributions provides a probabilistic classification of individual samples (susceptible, immune and infected), propagating classification uncertainties to the transmission model and enabling serological classifications to be informed by hospitalization data. The analyses show that high levels of immunity among persons 20 years and older provide a consistent explanation of the skewed attack rates observed during the pandemic and yield precise estimates of the probability of hospitalization per infection (1–4 years: 0.00096 (95%CrI: 0.00078–0.0012); 5–19 years: 0.00036 (0.00031–0.0044); 20–64 years: 0.0015 (0.00091–0.0020); 65+ years: 0.0084 (0.0028–0.016)). The analyses suggest that in The Netherlands, the school holiday period reduced the number of infectious contacts between 5- and 9-year-old children substantially (estimated reduction: 54%; 95%CrI: 29–82%), thereby delaying the unfolding of the pandemic in The Netherlands by approximately a week.     

Disease control across urban–rural gradients

Controlling the regional re-emergence of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) after its initial spread in ever-changing personal contact networks and disease landscapes is a challenging task. In a landscape context, contact opportunities within and between populations are changing rapidly as lockdown measures are relaxed and a number of social activities re-activated. Using an individual-based metapopulation model, we explored the efficacy of different control strategies across an urban–rural gradient in Wales, UK. Our model shows that isolation of symptomatic cases or regional lockdowns in response to local outbreaks have limited efficacy unless the overall transmission rate is kept persistently low. Additional isolation of non-symptomatic infected individuals, who may be detected by effective test-and-trace strategies, is pivotal to reducing the overall epidemic size over a wider range of transmission scenarios. We define an ‘urban–rural gradient in epidemic size'' as a correlation between regional epidemic size and connectivity within the region, with more highly connected urban populations experiencing relatively larger outbreaks. For interventions focused on regional lockdowns, the strength of such gradients in epidemic size increased with higher travel frequencies, indicating a reduced efficacy of the control measure in the urban regions under these conditions. When both non-symptomatic and symptomatic individuals are isolated or regional lockdown strategies are enforced, we further found the strongest urban–rural epidemic gradients at high transmission rates. This effect was reversed for strategies targeted at symptomatic individuals only. Our results emphasize the importance of test-and-trace strategies and maintaining low transmission rates for efficiently controlling SARS-CoV-2 spread, both at landscape scale and in urban areas.     

Partially observed epidemics in wildlife hosts: modelling an outbreak of dolphin morbillivirus in the northwestern Atlantic,June 2013–2014

Morbilliviruses cause major mortality in marine mammals, but the dynamics of transmission and persistence are ill understood compared to terrestrial counterparts such as measles; this is especially true for epidemics in cetaceans. However, the recent outbreak of dolphin morbillivirus in the northwestern Atlantic Ocean can provide new insights into the epidemiology and spatio-temporal spread of this pathogen. To deal with uncertainties surrounding the ecology of this system (only stranded animals were observed), we develop a statistical framework that can extract key information about the underlying transmission process given only sparse data. Our self-exciting Poisson process model suggests that individuals are infectious for at most 24 days and can transfer infection up to two latitude degrees (220 km) within this time. In addition, the effective reproduction number is generally below one, but reaches 2.6 during a period of heightened stranding numbers near Virginia Beach, Virginia, in summer 2013. Network analysis suggests local movements dominate spatial spread, with seasonal migration facilitating wider dissemination along the coast. Finally, a low virus transmission rate or high levels of pre-existing immunity can explain the lack of viral spread into the Gulf of Mexico. More generally, our approach illustrates novel methodologies for analysing very indirectly observed epidemics.     

Strong seasonality produces spatial asynchrony in the outbreak of infectious diseases

Models for infectious diseases usually assume a fixed demographic structure. Yet, a disease can spread over a region encountering different local demographic variations that may significantly alter local dynamics. Spatial heterogeneity in the resulting dynamics can lead to important differences in the design of surveillance and control strategies. We illustrate this by exploring the north–south gradient in the seasonal demography of raccoon rabies over the eastern USA. We find that the greater variance in the timing of spring births characteristic of southern populations can lead to the spatial synchronization of southern epidemics, while the narrow birth-pulse associated with northern populations can lead to an irregular patchwork of epidemics. These results indicate that surveillance in the southern states can be reduced relative to northern locations without loss of detection ability. This approach could yield significant savings in vaccination programmes. The importance of seasonality in many widely distributed diseases indicates that our findings will find applications beyond raccoon rabies.