Andrew J. Black

Stochastic formulation of ecological models and their applications

The increasing use of computer simulation by theoretical ecologists started a move away from models formulated at the population level towards individual-based models. However, many of the models studied at the individual level are not analysed mathematically and remain defined in terms of a computer algorithm. This is not surprising, given that they are intrinsically stochastic and require tools and techniques for their study that may be unfamiliar to ecologists. Here, we argue that the construction of ecological models at the individual level and their subsequent analysis is, in many cases, straightforward and leads to important insights. We discuss recent work that highlights the importance of stochastic effects for parameter ranges and systems where it was previously thought that such effects would be negligible.

Stochastic amplification in an epidemic model with seasonal forcing

We study the stochastic susceptible-infected-recovered (SIR) model with time-dependent forcing using analytic techniques which allow us to disentangle the interaction of stochasticity and external forcing. The model is formulated as a continuous time Markov process, which is decomposed into a deterministic dynamics together with stochastic corrections, by using an expansion in inverse system size. The forcing induces a limit cycle in the deterministic dynamics, and a complete analysis of the fluctuations about this time-dependent solution is given. This analysis is applied when the limit cycle is annual, and after a period doubling when it is biennial. The comprehensive nature of our approach allows us to give a coherent picture of the dynamics which unifies past work, but which also provides a systematic method for predicting the periods of oscillations seen in whooping cough and measles epidemics.

Epidemiological consequences of household-based antiviral prophylaxis for pandemic influenza.

Antiviral treatment offers a fast acting alternative to vaccination; as such it is viewed as a first-line of defence against pandemic influenza in protecting families and households once infection has been detected. In clinical trials, antiviral treatments have been shown to be efficacious in preventing infection, limiting disease and reducing transmission, yet their impact at containing the 2009 influenza A(H1N1)pdm outbreak was limited. To understand this seeming discrepancy, we develop a general and computationally efficient model for studying household-based interventions. This allows us to account for uncertainty in quantities relevant to the 2009 pandemic in a principled way, accounting for the heterogeneity and variability in each epidemiological process modelled. We find that the population-level effects of delayed antiviral treatment and prophylaxis mean that their limited overall impact is quantitatively consistent (at current levels of precision) with their reported clinical efficacy under ideal conditions. Hence, effective control of pandemic influenza with antivirals is critically dependent on early detection and delivery ideally within 24 h.

Stochasticity in staged models of epidemics: quantifying the dynamics of whooping cough

Although many stochastic models can accurately capture the qualitative epidemic patterns of many childhood diseases, there is still considerable discussion concerning the basic mechanisms generating these patterns; much of this stems from the use of deterministic models to try to understand stochastic simulations. We argue that a systematic method of analysing models of the spread of childhood diseases is required in order to consistently separate out the effects of demographic stochasticity, external forcing and modelling choices. Such a technique is provided by formulating the models as master equations and using the van Kampen system-size expansion to provide analytical expressions for quantities of interest. We apply this method to the susceptible–exposed–infected–recovered (SEIR) model with distributed exposed and infectious periods and calculate the form that stochastic oscillations take on in terms of the model parameters. With the use of a suitable approximation, we apply the formalism to analyse a model of whooping cough which includes seasonal forcing. This allows us to more accurately interpret the results of simulations and to make a more quantitative assessment of the predictions of the model. We show that the observed dynamics are a result of a macroscopic limit cycle induced by the external forcing and resonant stochastic oscillations about this cycle.

Mixing times in evolutionary game dynamics

Without mutation and migration, evolutionary dynamics ultimately leads to the extinction of all but one species. Such fixation processes are well understood and can be characterized analytically with methods from statistical physics. However, many biological arguments focus on stationary distributions in a mutation-selection equilibrium. Here, we address the mixing time required to reach stationarity in the presence of mutation. We show that mixing times in evolutionary games have the opposite behavior from fixation times when the intensity of selection increases: in coordination games with bistabilities, the fixation time decreases, but the mixing time increases. In coexistence games with metastable states, the fixation time increases, but the mixing time decreases. Our results are based on simulations and the Wentzel-Kramers-Brillouin approximation of the master equation.

Computation of epidemic final size distributions

We develop a new methodology for the efficient computation of epidemic final size distributions for a broad class of Markovian models. We exploit a particular representation of the stochastic epidemic process to derive a method which is both computationally efficient and numerically stable. The algorithms we present are also physically transparent and so allow us to extend this method from the basic SIR model to a model with a phase-type infectious period and another with waning immunity. The underlying theory is applicable to many Markovian models where we wish to efficiently calculate hitting probabilities.

WKB calculation of an epidemic outbreak distribution

We calculate both the exponential and prefactor contributions in a WKB approximation of the master equation for a stochastic SIR model with highly oscillatory dynamics. Fixing the basic parameters of the model, we investigate how the outbreak distribution changes with the population size. We show that this distribution rapidly becomes highly non-Gaussian, acquiring large tails, indicating the presence of rare but large outbreaks, as the population is made smaller. The analytic results are found to be in excellent agreement with simulations until the systems become so small that the dynamics are dominated by fade-out of the disease.

Autonomic phenomena of temperature regulation in temporal lobe epilepsy

Cold shiver and piloerection are rare ictal signs in focal epilepsies. They are often associated with an epileptic seizure focus within the temporal lobe. In rare cases the phenomenon of piloerection has been reported to be confined to body parts ipsilateral to the seizure focus. In this multicentric study epilepsy patients with ictal cold shiver and/or piloerection were retrospectively asked to describe exactly location and spreading patterns of these signs as well as their temporal sequence in relation to other ictal signs. Clinical data, etiology of epilepsy, and seizure focus location were also assessed. In our patient group there was a high relationship to an epileptic focus within the left temporal lobe. Distinct spreading patterns or unilateral piloerection was not indicative of a focus in the ipsilateral temporal lobe as described previously. Our results suggest that phenomena of temperature dysregulation during epileptic seizures may be of value in the presurgical evaluation as they may be indicative of a left temporal lobe seizure focus.

Estimating a Markovian Epidemic Model Using Household Serial Interval Data from the Early Phase of an Epidemic

The clinical serial interval of an infectious disease is the time between date of symptom onset in an index case and the date of symptom onset in one of its secondary cases. It is a quantity which is commonly collected during a pandemic and is of fundamental importance to public health policy and mathematical modelling. In this paper we present a novel method for calculating the serial interval distribution for a Markovian model of household transmission dynamics. This allows the use of Bayesian MCMC methods, with explicit evaluation of the likelihood, to fit to serial interval data and infer parameters of the underlying model. We use simulated and real data to verify the accuracy of our methodology and illustrate the importance of accounting for household size. The output of our approach can be used to produce posterior distributions of population level epidemic characteristics.

Contact tracing and antiviral prophylaxis in the early stages of a pandemic: the probability of a major outbreak

Antiviral prophylaxis forms a significant component of health management plans for many countries around the world. A number of studies have shown that the delays typically encountered in distributing these antivirals to households, following the first infectious case, can result in their efficacy being severely reduced. Here, we investigate the use of contact tracing as a method to reduce the delays and hence mitigate the reduction in efficacy of antivirals. We assess the usefulness of contact tracing in terms of the probability of a major outbreak. It is found, with parameter distributions appropriate to the 2009 H1N1 pandemic and distributions reflecting commonly experienced delays, that standard contact tracing renders an outbreak impossible approximately one in five times compared with approximately one in ten times in its absence. A contact-tracing efficiency of 50% would see further improvements with an outbreak being impossible approximately one in four times, and a reduction of the median probability of a major outbreak from 0.41 to below 0.27.