2004 Special Year for Probability and its Applications

CMA National Research Symposium - A Celebration of Modelling and Applied Probability

All talks were held in Becker Hall at The Shine Dome, Gordon Street, ANU campus. Registration was in the foyer, and morning and afternoon teas were in the Jaeger Room.

The dinner on the evening of Tuesday 14 December was held in the Great Hall of University House, Balmain Crescent, ANU.

The program and abstracts are listed below. For a printable version of the schedule only, click here.

Program

(updated 8 December 2004)

Click on the title of a talk to view the abstract.

Tuesday 14 December 2004 - Applied Probability
9:00	Registration
Chair	Chris Heyde
9:15	Welcome to participants
9:30	Zari Rachev, Karlsruhe and UCSB A generalized heteroscedastic asset price process: properties, parameter estimation and pricing
10:30	Morning tea
Chair	Chris Heyde
11:00	Mark Westcott, CSIRO How many buses?
11:30	Tony Pettitt, QUT Statistical modelling for nosocomial infections: Estimating transmission rates for infection control and developing surveillance schemes
12:00	Yoshi Ito, Aichi-Gakuin Further geometric methods for the distribution of the sample correlation coefficient
12:30	Lunch break
Chair	Daryl Daley
14:00	Valerie Isham, UCL Macroparasite population models: persistence, population scale and cross-species interactions
15:00	Afternoon tea
Chair	Daryl Daley
15:30	Niels Becker, ANU Control of transmission with two types of infection
16:00	Kostya Borovkov, Melbourne On the asymptotic behaviour of a simple growing point process model
16:30	Jeff Hunter, Massey Mixing times and their application to perturbed Markov chains
17:00	Break
18:30	Symposium dinner in honour of Joe's birthday Great Hall, University House (Balmain Crescent, ANU)
Wednesday 15 December 2004 - Modelling
Chair	Bob Anderssen
09:30	John Blake, Birmingham Vigorous non-linear, non-spherical bubble dynamics: applications to biology, medicine, chemistry, physics and engineering
10:30	Morning tea
Chair	Steve Roberts
11:00	Linda Stals, ANU A plantation-nursery system
11:30	Glenn Fulford, QUT Spatial modelling of infectious diseases
12:00	Frank de Hoog, CSIRO Predicting winding stresses for wound coils with large deformations
12:30	Lunch break
Chair	Mike Osborne
14:00	Belinda Barnes, ANU An ecological framework linking scales based on self-thinning
14:30	Geoff Aldis, ADFA An integral equation model of the control of a smallpox outbreak
15:00	Afternoon tea
Chair	Markus Hegland
15:30	ANZIAM Canberra Branch Annual General Meeting

Abstracts

Click on the title of a talk to find its position in the program.

Geoff Aldis, ADFA
An integral equation model of the control of a smallpox outbreak

Smallpox is eradicated from the world-at-large but the risk of an accidental or criminal release remains. An integral equation model incorporates a realistic transmission kernel modelling such things as the finite incubation period, infectivity dependent on infection-age and the use of control measures. The population is structured with transmission occurring at home, at work, in the community and in hospital. Our results suggest that mass vaccination before or after an outbreak would be inefficient. Rapid identification and isolation of cases, quarantine of affected households and a public education campaign would be sufficient to bring an outbreak under control.

Belinda Barnes, ANU
An ecological framework linking scales based on self-thinning

This talk presents an overview for a generic, theoretical framework for vegetation modelling across scales, linking individual plants in patches with ecosystems. The model development was motivated by the need to understand carbon dynamics, and thus, since carbon scales with dry mass, the basis for the model is changing dry mass. Inclusion of a self-thinning mechanism connects the individual to the larger scale population, with the particular formulation scaling across space and time. Taking a dynamical systems approach provides the capacity for theoretical analysis, which in turn leads to an understanding of the impact of variation/disturbance at local scales on the ecosystem as a whole. The model predictions are compared with data from pine plantations, illustrating that, while the model is relatively simple and straightforward to apply, its predictions compare well with the data. Further, the modelling approach encapsulates ecological concepts, such as, competing species, the intermediate disturbance hypothesis and plant partitioning, which will be discussed and illustrated by example.

Niels Becker, ANU
Control of transmission with two types of infection

For several infectious diseases the probability of transmission and the severity of the infection are thought to depend on the 'dose' of pathogen ingested at the time of exposure. We compare the effectiveness of different vaccination strategies for the prevention of epidemics of such an infection in a community of households. Two types of vaccine response are considered. It is shown that the optimal vaccination strategy can depend on how individuals respond to vaccination and on the distribution of household size, and that the difference between the best and worst strategies is substantial.

John Blake, Birmingham
Vigorous non-linear, non-spherical bubble dynamics: applications to biology, medicine, chemistry, physics and engineering

The vigorous nonlinear motion of bubbles is characterised by high velocities, pressures and temperatures. It is often associated with the formation of a very high speed liquid jet which penetrates the far-side of the bubble, generating a toroidal bubble. The seminar will cover aspects of the physics of bubble formation, growth and collapse; discussion of global conservation of momentum and energy; the development of a boundary integral code for both simply and multiply connected bubbles together with example calculations which illustrate a wide range of phenomena. This generic theory has a wide range of practical applications to such areas as laser surgery, shock- wave lithotripters, diagnostic and therapeutic ultrasound, cavitation bubble luminescence, sonochemistry, underwater seismic surveying and underwater explosions. The last topic has assumed greater strategic security significance in recent times.

Kostya Borovkov, Melbourne
On the asymptotic behaviour of a simple growing point process model

We consider a finite simple point process in a finite-dimensional Euclidean space evolving in discrete time in the following way. Starting with an arbitrary initial configuration, at each time step a point is chosen at random from the process according to a certain distribution. Next, k new points are added to the process at locations, each obtained by adding an independent random vector to the location of the chosen "mother" point. The k "displacement vectors" are independent of each other and of the past evolution of the process, and follow a given common distribution, that can depend on the time step (while the value of k remains fixed over time). This process could be considered as a simple dynamic version of the von Neyman contagious point process that was used in biological applications (for example, in entomology, to model a situation when "larvae hatched from the eggs which are being laid in so-called masses")

Under mild moment conditions (uniform integrability and the existence of Cesaro limits for the sequences of the respective moments for the displacement vectors), we obtain the limiting behaviour of the distribution of the point last added to the process and also that of the normalized mean measure of the point process as time goes to infinity.

Frank de Hoog, CSIRO
Predicting winding stresses for wound coils with large deformations

Wound coils or rolls store essentially flat strip compactly without folding or cutting and typically, strip is wound and unwound a number of times before its end use. The variety of material that is wound into coils or rolls is very extensive and includes magnetic tape, paper, cellophane, plastics, fabric and metals such as aluminium and steel. Generally, coils are wound onto a core which typically consists of a mandrel and former though there are a number of other ways that coils or rolls can be wound, including surface winding, where the coil is nipped by an exterior roll which is driven, or a combination of centre and surface winding.

This talk examines centre winding of coils for which the displacement of the wraps can be large, under the assumption that the material is orthotropic. A radically different approach is taken in the analysis of residue stresses in the coils. Instead of the forward solution of determining the residue stresses from the given winding stress profile, as traditionally adopted in this field of research, an inverse solution is obtained in which the winding tension profile is determined from the prescribed residue stresses. This approach offers significant advantages in industrial applications, since the preferred residue stress distributions to prevent coiling problems, such as tight-bore collapses and coil slump, are often specified.

Glenn Fulford, QUT
Spatial modelling of infectious diseases

Spatial heterogeneity can be important in disease modelling, particularly animal diseases where animals migrate large distances. Using data on dispersal distances for juvenile possums a 1D spatial model of the spread of tuberculosis is developed, based on a metapopulation approach. A travelling wave in the density of infectious possums occurs. This model is compared to a nearest-neighbour approach for dispersal of juvenile possums.

Jeff Hunter, Massey
Mixing times and their application to perturbed Markov chains

A measure of the "mixing time" or "time to stationarity" in a finite irreducible discrete time Markov chain is considered. The statistic, η_i = ∑^m_j=1 m_ij π_j, where π_i is the stationary distribution and m_ij is the mean first passage time from state i to state j of the Markov chain, is shown to be independent of the state i that the chain starts in (so that η_i = η for all i), is minimal in the case of a periodic chain, yet can be arbitrarily large in a variety of situations. An application concerning the affect perturbations of transition probabilities have on the stationary distributions of Markov chains leads to a new bound, involving η, for the 1-norm of the difference between the stationary probability vectors of the original and the perturbed chain. When η is large the stationary distribution of the Markov chain is very sensitive to perturbations of the transition probabilities.

Valerie Isham, UCL
Macroparasite population models: persistence, population scale and cross-species interactions

Microparasite (e.g. bacterial, viral) infections are usually modelled by considering the numbers of hosts that are susceptible, latent, infected etc., the justification being that once a host is infected the parasites multiply rapidly within the host to reach an equilibrium level. In contrast, macroparasites do not reproduce directly within the host and spend part of their lifecycle externally, so that host's parasite load only increases through reinfection. Macroparasite infections thus present additional challenges to modellers because models must allow for the parasite lifecycle, the parasite load of each host and the reinfection process.

In this talk, I will illustrate the importance of stochasticity and heterogeneity in the context of some simple macroparasite models. In particular, it will be shown that stochastic effects play a vital role in determining the persistence or extinction of parasite strains, and that there are complicated interactions between stochasticity and the spatial scale of the host population. In most models, infection with a single parasite species is considered. In practice, infection with more than one species is not uncommon and the existence of cross-species immune interactions is debated. Some models for multi-species infections and their use in the identification of such interactions will be discussed.

Yoshi Ito, Aichi-Gakuin
Further geometric methods for the distribution of the sample correlation coefficient

By a geometric method Fisher (1915) has derived the joint p.d.f. of the sample standard deviations s₁, s₂ and the sample correlation coefficient r. By integrating it over {(s₁, s₂) | s₁ ≥ 0, s₂ ≥ 0} he obtained the exact distribution of the r. Ito and Izumi (2004) showed that the distribution of r can be obtained by an elementary geometric method from the joint p.d.f. without integrating it. In this talk, their method and two other similar methods will be compared.

Tony Pettitt, QUT
Statistical modelling for nosocomial infections: Estimating transmission rates for infection control and developing surveillance schemes

This talk describes work being carried out jointly by the speaker and postgraduate doctoral research students Marie Forrester, Ron Webster and Emma McBryde at QUT and is joint with the Princess Alexandra Hospital, Brisbane, and funded by an ARC Linkage grant.

The emergence of antibiotic resistance is considered by many to be one of the most important threats to human health in the 21st century. A limit of therapeutic options means that prevention will become increasingly important. Nosocomial, or hospital acquired, infections and especially multiple resistant organism (MRO) infections are currently a major health concern. A focus for infection control in hospitals is those wards where patients have a high morbidity such as the Intensive Care Unit (ICU), transplant and burns units.

Previous mathematical modelling of methicillin resistant Staphlococcus aureus (MRSA) and vancomycin resistant enterococci (VRE) infections in particular and MROs in hospitals and ICUs has been carried out using models of the transmission of the organism using an extension of the Ross-MacDonald model for vector borne diseases. The transmission of the organism from patient to patient is thought mainly to occur on the hands of health care workers (HCW). Infection control measures include hand washing of HCWs, the cohorting of patient and HCW, and isolation of patients. Knowledge about transmission parameters for such models is scarce.

We present here the statistical modelling of infectious disease data arising from the ICU in order to estimate transmission parameters associated with the transmission of the organism causing colonization, or carriage, rather than infection. The data are obtained as part of infection control measures and are typically complicated by partial observation. For example, swabs to detect colonization by MROs are taken from patients two times per week, so that the variable time of colonization is necessarily interval censored and subsequent modelling of quantities, such as hazard functions, needs to accommodate this. The patient numbers in the ICU remain approximately the same but patients remain in the unit for typically short periods of time. Standard statistical techniques become too complicated to implement effectively and a Bayesian approach using Markov chain Monte Carlo and augmented data sets provides a practical solution. The effects of isolating patients when known to be colonized, the sensitivity of the test for positive MRSA, and the incidence of MRSA on entry to the ICU are incorporated into statistical models using Bayesian methods. Using the models to analyse the data provides information on transmission rates from colonized patients, both isolated and not, to susceptible patients. The modelling provides a statistical basis for the assessment of various infection control interventions, such as isolation. Additionally, the transmission model can provide a basis for on-line surveillance of colonizations.

Zari Rachev, Karlsruhe and UCSB
A generalized heteroscedastic asset price process: properties, parameter estimation and pricing

The classical discrete Black and Scholes model for the stock price/firm value process (S_t)_t∈N

log(S_t) - log(S_t-1) = r + λσ + σε_t,  ε_t ∼^iid N(0, 1)

is generalized in several ways:

• Introduction of a GARCH volatility process
• The use of alternative non Gaussian distributions for the innovation process, especially the class of smoothly truncated stable distributions.
• Allowing stochastic interest rates and a generalized GARCH-in-mean term in the dynamic of the conditional mean.

These extensions enhance the explanatory power of the model significantly: Empirically observed properties of financial time series, like heavy tails, volatility clustering and leverage effect can be explained.

The first part of the talk is dedicated to the introduction of the price process and the smoothly truncated stable distributions. In the second part applications of the price process in the area of credit risk and option pricing are presented.

Joint work with Christian Menn, Karlsruhe.

Linda Stals, ANU
A plantation-nursery system

We consider a plantation system in which infected plants are replaced by plants from a nursery, also subject to a small probability of infection. Both a discrete and continuous Markov chain model of the plantation infection is proposed. Results are presented for different choices of parameters and initial conditions and some conclusions are drawn.

Mark Westcott, CSIRO
How many buses?

A colleague was investigating various designs for a bus terminal. Some of these led to interference between buses entering and leaving the terminal. One important question was whether this interference affected the theoretical maximum arrival rate the terminal could handle, for a given service rate, and whether this could be determined theoretically i.e. without simulation.

In this talk I shall discuss the maximum arrival rates for two types of interference, and give explicit results for Erlang and mixed exponential service time distributions. These come from fairly standard Markov chain methods. A potentially more general renewal theoretic method leads to an apparent paradox; under some circumstances, the seemingly more restrictive type of interference permits a higher maximum throughput.