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.     

Decreasing stochasticity through enhanced seasonality in measles epidemics

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

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.     

Towards a quantitative understanding of the within-host dynamics of influenza A infections

Although the influenza A virus has been extensively studied, a quantitative understanding of the infection dynamics is still lacking. To make progress in this direction, we designed several mathematical models and compared them with data from influenza A infections of mice. We find that the immune response (IR) plays an important part in the infection dynamics. Both an innate and an adaptive IR are required to provide adequate explanation of the data. In contrast, regrowth of epithelial cells did not seem to be an important mechanism on the time scale of the infection. We also find that different model variants for both innate and adaptive responses fit the data well, indicating the need for additional data to allow further model discrimination.     

Transmission dynamics of methicillin-resistant Staphylococcus aureus in a medical intensive care unit

Intensive care units (ICUs) play an important role in the epidemiology of methicillin-resistant Staphyloccocus aureus (MRSA). Although successful interventions are multi-modal, the relative efficacy of single measures remains unknown. We developed a discrete time, individual-based, stochastic mathematical model calibrated on cross-transmission observed through prospective surveillance to explore the transmission dynamics of MRSA in a medical ICU. Most input parameters were derived from locally acquired data. After fitting the model to the 46 observed cross-transmission events and performing sensitivity analysis, several screening and isolation policies were evaluated by simulating the number of cross-transmissions and isolation-days. The number of all cross-transmission events increased from 54 to 72 if only patients with a past history of MRSA colonization are screened and isolated at admission, to 75 if isolation is put in place only after the results of the admission screening become available, to 82 in the absence of admission screening and with a similar reactive isolation policy, and to 95 when no isolation policy is in place. The method used (culture or polymerase chain reaction) for admission screening had no impact on the number of cross-transmissions. Systematic regular screening during ICU stay provides no added-value, but aggressive admission screening and isolation effectively reduce the number of cross-transmissions. Critically, colonized healthcare workers may play an important role in MRSA transmission and their screening should be reinforced.     

Steady-state invariant genetics: probing the role of morphogen gradient dynamics in developmental patterning

Morphogen-mediated patterning is the predominant mechanism by which positional information is established during animal development. In the classical view, the interpretation of positional signals depends on the equilibrium distribution of a morphogen, regardless of the dynamics of gradient formation. The problem of whether or not morphogen dynamics contribute to developmental patterning has not been explored in detail, partly because genetic experiments, which selectively affect signalling dynamics while maintaining unchanged the steady-state morphogen profile, are difficult to design and interpret. Here, I present a modelling-based approach to identify genetic mutations in developmental patterning that may affect the transient, but leave invariant the steady-state signalling gradient. As a case study, this approach is used to explore the dynamic properties of Hedgehog (Hh) signalling in the developing wing of the fruitfly, Drosophila melanogaster. This analysis provides insights into how different properties of the Hh gradient dynamics, such as the duration of exposure to the signal or the maximum width of the transient gradient, can be genetically perturbed without affecting the steady-state distribution of the Hh concentration profile. I propose that this method can be used as an experimental design tool to investigate the role of transient morphogen gradients in developmental patterning and discuss the generality of these ideas in other problems.     

A deterministic model for the occurrence and dynamics of multiple mutations in hierarchically organized tissues

Cancers are rarely caused by single mutations, but often develop as a result of the combined effects of multiple mutations. For most cells, the number of possible cell divisions is limited because of various biological constraints, such as progressive telomere shortening, cell senescence cascades or a hierarchically organized tissue structure. Thus, the risk of accumulating cells carrying multiple mutations is low. Nonetheless, many diseases are based on the accumulation of such multiple mutations. We model a general, hierarchically organized tissue by a multi-compartment approach, allowing any number of mutations within a cell. We derive closed solutions for the deterministic clonal dynamics and the reproductive capacity of single clones. Our results hold for the average dynamics in a hierarchical tissue characterized by an arbitrary combination of proliferation parameters. We show that hierarchically organized tissues strongly suppress cells carrying multiple mutations and derive closed solutions for the expected size and diversity of clonal populations founded by a single mutant within the hierarchy. We discuss the example of childhood acute lymphoblastic leukaemia in detail and find good agreement between our predicted results and recently observed clonal diversities in patients. This result can contribute to the explanation of very diverse mutation profiles observed by whole genome sequencing of many different cancers.     

Arterial pulse wave propagation across stenoses and aneurysms: assessment of one-dimensional simulations against three-dimensional simulations and in vitro measurements

One-dimensional (1-D) arterial blood flow modelling was tested in a series of idealized vascular geometries representing the abdominal aorta, common carotid and iliac arteries with different sizes of stenoses and/or aneurysms. Three-dimensional (3-D) modelling and in vitro measurements were used as ground truth to assess the accuracy of 1-D model pressure and flow waves. The 1-D and 3-D formulations shared identical boundary conditions and had equivalent vascular geometries and material properties. The parameters of an experimental set-up of the abdominal aorta for different aneurysm sizes were matched in corresponding 1-D models. Results show the ability of 1-D modelling to capture the main features of pressure and flow waves, pressure drop across the stenoses and energy dissipation across aneurysms observed in the 3-D and experimental models. Under physiological Reynolds numbers (Re), root mean square errors were smaller than 5.4% for pressure and 7.3% for the flow, for stenosis and aneurysm sizes of up to 85% and 400%, respectively. Relative errors increased with the increasing stenosis and aneurysm size, aneurysm length and Re, and decreasing stenosis length. All data generated in this study are freely available and provide a valuable resource for future research.     

A dose and time response Markov model for the in-host dynamics of infection with intracellular bacteria following inhalation: with application to Francisella tularensis

In a novel approach, the standard birth–death process is extended to incorporate a fundamental mechanism undergone by intracellular bacteria, phagocytosis. The model accounts for stochastic interaction between bacteria and cells of the immune system and heterogeneity in susceptibility to infection of individual hosts within a population. Model output is the dose–response relation and the dose-dependent distribution of time until response, where response is the onset of symptoms. The model is thereafter parametrized with respect to the highly virulent Schu S4 strain of Francisella tularensis, in the first such study to consider a biologically plausible mathematical model for early human infection with this bacterium. Results indicate a median infectious dose of about 23 organisms, which is higher than previously thought, and an average incubation period of between 3 and 7 days depending on dose. The distribution of incubation periods is right-skewed up to about 100 organisms and symmetric for larger doses. Moreover, there are some interesting parallels to the hypotheses of some of the classical dose–response models, such as independent action (single-hit model) and individual effective dose (probit model). The findings of this study support experimental evidence and postulations from other investigations that response is, in fact, influenced by both in-host and between-host variability.