The dynamical consequences of seasonal forcing, immune boosting and demographic change in a model of disease transmission.

The impact of seasonal effects on the time course of an infectious disease can be dramatic. Seasonal fluctuations in the transmission rate for an infectious disease are known mathematically to induce cyclical behaviour and drive the onset of multistable and chaotic dynamics. These properties of forced dynamical systems have previously been used to explain observed changes in the period of outbreaks of infections such as measles, varicella (chickenpox), rubella and pertussis (whooping cough). Here, we examine in detail the dynamical properties of a seasonally forced extension of a model of infection previously used to study pertussis. The model is novel in that it includes a non-linear feedback term capturing the interaction between exposure and the duration of protection against re-infection. We show that the presence of limit cycles and multistability in the unforced system give rise to complex and intricate behaviour as seasonal forcing is introduced. Through a mixture of numerical simulation and bifurcation analysis, we identify and explain the origins of chaotic regions of parameter space. Furthermore, we identify regions where saddle node lines and period-doubling cascades of different orbital periods overlap, suggesting that the system is particularly sensitive to small perturbations in its parameters and prone to multistable behaviour. From a public health point of view - framed through the 'demographic transition' whereby a population׳s birth rate drops over time (and life-expectancy commensurately increases) - we argue that even weak levels of seasonal-forcing and immune boosting may contribute to the myriad of complex and unexpected epidemiological behaviours observed for diseases such as pertussis. Our approach helps to contextualise these epidemiological observations and provides guidance on how to consider the potential impact of vaccination programs.

Dynamical crises, multistability and the influence of the duration of immunity in a seasonally-forced model of disease transmission.

BACKGROUND: Highly successful strategies to make populations more resilient to infectious diseases, such as childhood vaccinations programs, may nonetheless lead to unpredictable outcomes due to the interplay between seasonal variations in transmission and a population's immune status. METHODS: Motivated by the study of diseases such as pertussis we introduce a seasonally-forced susceptible-infectious-recovered model of disease transmission with waning and boosting of immunity. We study the system's dynamical properties using a combination of numerical simulations and bifurcation techniques, paying particular attention to the properties of the initial condition space. RESULTS: We find that highly unpredictable behaviour can be triggered by changes in biologically relevant model parameters such as the duration of immunity. In the particular system we analyse--used in the literature to study pertussis dynamics--we identify the presence of an initial-condition landscape containing three coexisting attractors. The system's response to interventions which perturb population immunity (e.g. vaccination "catch-up" campaigns) is therefore difficult to predict. CONCLUSION: Given the increasing use of models to inform policy decisions regarding vaccine introduction and scheduling and infectious diseases intervention policy more generally, our findings highlight the importance of thoroughly investigating the dynamical properties of those models to identify key areas of uncertainty. Our findings suggest that the often stated tension between capturing biological complexity and utilising mathematically simple models is perhaps more nuanced than generally suggested. Simple dynamical models, particularly those which include forcing terms, can give rise to incredibly complex behaviour.

Four dimensional chaos and intermittency in a mesoscopic model of the electroencephalogram.

The occurrence of so-called four dimensional chaos in dynamical systems represented by coupled, nonlinear, ordinary differential equations is rarely reported in the literature. In this paper, we present evidence that Liley's mesoscopic theory of the electroencephalogram (EEG), which has been used to describe brain activity in a variety of clinically relevant contexts, possesses a chaotic attractor with a Kaplan-Yorke dimension significantly larger than three. This accounts for simple, high order chaos for a physiologically admissible parameter set. Whilst the Lyapunov spectrum of the attractor has only one positive exponent, the contracting dimensions are such that the integer part of the Kaplan-Yorke dimension is three, thus giving rise to four dimensional chaos. A one-parameter bifurcation analysis with respect to the parameter corresponding to extracortical input is conducted, with results indicating that the origin of chaos is due to an inverse period doubling cascade. Hence, in the vicinity of the high order, strange attractor, the model is shown to display intermittent behavior, with random alternations between oscillatory and chaotic regimes. This phenomenon represents a possible dynamical justification of some of the typical features of clinically established EEG traces, which can arise in the case of burst suppression in anesthesia and epileptic encephalopathies in early infancy.

Seasonality as a driver of pH1N12009 influenza vaccination campaign impact.

Although the most recent respiratory virus pandemic was triggered by a Coronavirus, sustained and elevated prevalence of highly pathogenic avian influenza viruses able to infect mammalian hosts highlight the continued threat of pandemics of influenza A virus (IAV) to global health. Retrospective analysis of pandemic outcomes, including comparative investigation of intervention efficacy in different regions, provide important contributions to the evidence base for future pandemic planning. The swine-origin IAV pandemic of 2009 exhibited regional variation in onset, infection dynamics and annual infection attack rates (IARs). For example, the UK experienced three severe peaks of infection over two influenza seasons, whilst Australia experienced a single severe wave. We adopt a seasonally forced 2-subtype model for the transmission of pH1N12009 and seasonal H3N2 to examine the role vaccination campaigns may play in explaining differences in pandemic trajectories in temperate regions. Our model differentiates between the nature of vaccine- and infection-acquired immunity. In particular, we assume that immunity triggered by infection elicits heterologous cross-protection against viral shedding in addition to long-lasting neutralising antibody, whereas vaccination induces imperfect reduction in susceptibility. We employ an Approximate Bayesian Computation (ABC) framework to calibrate the model using data for pH1N12009 seroprevalence, relative subtype dominance, and annual IARs for Australia and the UK. Heterologous cross-protection substantially suppressed the pandemic IAR over the posterior, with the strength of protection against onward transmission inversely correlated with the initial reproduction number. We show that IAV pandemic timing relative to the usual seasonal influenza cycle influenced the size of the initial waves of pH1N12009 in temperate regions and the impact of vaccination campaigns.

Extensive Four-Dimensional Chaos in a Mesoscopic Model of the Electroencephalogram.

BACKGROUND: In a previous work (Dafilis et al. in Chaos 23(2):023111, 2013), evidence was presented for four-dimensional chaos in Liley's mesoscopic model of the electroencephalogram. The study was limited to one parameter set of the model equations. FINDINGS: In this report we expand that result by presenting evidence for the extension of four-dimensional chaotic behavior to a large area of the biologically admissible parameter space. A two-parameter bifurcation analysis highlights the complexity of the dynamical landscape involved in the creation of such chaos. CONCLUSIONS: The extensive presence of high-order chaos in a well-established physiological model of electrorhythmogenesis further emphasizes the applicability and relevance of mean field mesoscopic models in the description of brain activity at theoretical, experimental, and clinical levels.

Robust chaos in a model of the electroencephalogram: Implications for brain dynamics.

Various techniques designed to extract nonlinear characteristics from experimental time series have provided no clear evidence as to whether the electroencephalogram (EEG) is chaotic. Compounding the lack of firm experimental evidence is the paucity of physiologically plausible theories of EEG that are capable of supporting nonlinear and chaotic dynamics. Here we provide evidence for the existence of chaotic dynamics in a neurophysiologically plausible continuum theory of electrocortical activity and show that the set of parameter values supporting chaos within parameter space has positive measure and exhibits fat fractal scaling. (c) 2001 American Institute of Physics.

Drivers and consequences of influenza antiviral resistant-strain emergence in a capacity-constrained pandemic response.

Antiviral agents remain a key component of most pandemic influenza preparedness plans, but there is considerable uncertainty regarding their optimal use. In particular, concerns exist regarding the likelihood of wide-scale distribution to select for drug-resistant variants. We used a model that considers the influence of logistical constraints on diagnosis and drug delivery to consider achievable 'reach' of alternative antiviral intervention strategies targeted at cases of varying severity, with or without pre-exposure prophylaxis of contacts. To identify key drivers of epidemic mitigation and resistance emergence, we used Latin hypercube sampling to explore plausible ranges of parameters describing characteristics of wild type and resistant viruses, along with intervention efficacy, target coverage and distribution capacity. Within our model framework, 'real world' constraints substantially reduced achievable drug coverage below stated targets as the epidemic progressed. In consequence, predictions of both intervention impact and selection for resistance were more modest than earlier work that did not consider such limitations. Definitive containment of transmission was unlikely but, where observed, achieved through early liberal post-exposure prophylaxis of known contacts of treated cases. Predictors of resistant strain dominance were high intrinsic fitness relative to the wild type virus, and early emergence in the course of the epidemic into a largely susceptible population, even when drug use was restricted to severe case treatment. Our work demonstrates the importance of consideration of 'real world' constraints in scenario analysis modeling, and highlights the utility of models to guide surveillance activities in preparedness and response.

A spatially continuous mean field theory of electrocortical activity.

A set of nonlinear continuum field equations is presented which describes the dynamics of neural activity in cortex. These take into account the most pertinent anatomical and physiological features found in cortex with all parameter values obtainable from independent experiment. Derivation of a white noise fluctuation spectrum from a linearized set of equations shows the presence of strong resonances that correspond to electroencephalographically observed 0.3-4 Hz (mammalian delta), 4-8 Hz (mammalian theta), 8-13 Hz (mammalian alpha) and >13 Hz (mammalian beta) activity. Numerical solutions of a full set of one-dimensional nonlinear equations include properties analogous to cortical evoked potentials, travelling waves at experimentally observed velocities, threshold type spike activity and limit cycle, chaotic and noise driven oscillations at the frequency of the mammalian alpha rhythm. All these types of behaviour are generated with parameters that are within ranges reported experimentally. The strong dependence of the phenomena observed on inhibitory-inhibitory interactions is demonstrated. These results suggest that the classically described alpha may be instantiated in a number of qualitatively distinct dynamical regimes, all of which depend on the integrity of inhibitory-inhibitory population interactions.