Joseph Gani

The Maki-Thompson rumour model: a detailed analysis

Abstract The general solution for stochastic rumour models has recently been derived by Pearce (Math. Comput. Modelling. (2000)) for both the Daley–Kendall and Maki–Thompson cases. This paper concentrates on the simpler Maki–Thompson model, which presents some particular problems of its own. A general solution is set out, and illustrated by an example starting with 1 spreader and 2 ignorants; in this case, the probabilities of 2, 1 and 0 surviving ignorants are 0, 1/4 and 3/4 respectively.

Death and Birth-Death and Immigration Processes with Catastrophes

This paper explores an alternative approach starting from first principles, to the derivation of probability generating functions (pgfs) of death, birth-death and immigration processes in continuous time, subject to random catastrophes. A more elementary version of the general method proposed by Economou and Fakinos (2003) is presented. We examine the simple death process, the survival of susceptibles in a carrier-borne epidemic, the birth-death and immigration process, the unbiased random walk and the barber shop queue, all of them subject to random catastrophes occurring as a Poisson process. The stationary pgfs and the expected values of the processes are derived.

The spread of AIDS among interactive transmission groups

We consider the spread of an AIDS epidemic among N interacting communities (cities, say), each having at least one of the major four HIV transmission groups: 1.(i) homosexual/bisexual men, 2.(ii) blood transfusion recipients, 3.(iii) intravenous drug users, or 4.(iv) heterosexuals. Our model consists of a system of 4N differential equations (d.e.s). We show that as N -> ~, the number of infectives in each community converges to the unique solution of a Liouville type stochastic differential equation (s.d.e.).

A note on three stochastic processes with immigration

Three stochastic processes, the birth, death and birth-death processes, subject to immigration can be decomposed into the sum of each process in the absence of immigration and an independent process. We examine these independent processes through their probability generating functions (pgfs) and derive their expectations.

A Simple Approach to the Integrals under Three Stochastic Processes

A simplified approach to the integral under the stochastic path for each of a birth, death, and immigration process is presented. The geometric method described is intended to provide intuitive understanding for the known expressions of the Laplace-Stieltjes transforms of the integral of the process.

An Immigration-Death Process with a Time-Dependent Delay

In this article, a non homogeneous immigration-death process with delay in the death rate parameter is considered. A process with a time-dependent delay is developed and shown to result in a non homogeneous Poisson process.

Fifty Years of Statistics at the Australian National University, 1952–2002

The Australian National University (ANU) was founded in 1946; it was designed to develop postgraduate training and research within Australia, and to attract to Canberra eminent academics, Australian or otherwise, then pursuing careers overseas. Details of the University’s origins and development may be found in Foster and Varghese’s book The Making of the Australian National University. Sir Keith Hancock, one of the initial Academic Advisers to the ANU, suggested that the University’s Research School of Social Sciences (RSSS), of which he became Director in 1955, should include a statistician of broad interests who could assist the social scientists with their quantitative research. Such an appointee could act as a statistical consultant within the School, but would also be free to pursue independent research in statistics. In the event, the decision was taken to create a separate Department of Statistics within the Research School of Social Sciences. In January 1952, six years after the foundation of the ANU, the first Professor of Statistics was appointed in the RSSS. He was P. A. P. Moran (Pat, later FAA 1962, FRS 1975), then a Senior Research Fellow at the Institute of Statistics, Oxford University. His initial interests had been of a mathematical nature, but his scope had broadened to encompass statistical problems in time series, stochastic processes and their applications, animal population dynamics, sampling methods and ecology. He might have expected to provide lectures in statistical methods to the economists and social scientists in the RSSS, as well as to the biologists and medical scientists in the John Curtin School of Medical Research (JCSMR); his main efforts, however, were to be directed towards his personal research and the training of postgraduate students. At the time, there were several wellknown statisticians working in Australia, among them E. J. G. Pitman at the University of Tasmania, M. H. Belz at the University of Melbourne, E. A. Cornish at the CSIRO Division of Mathematical Statistics in Adelaide, J. B. Douglas at the New South Wales Institute of Technology (later the University of New South Wales) in Sydney, and H. O. Lancaster at the School of Public Health and Tropical Medicine of the University of Sydney (appointed the first Professor of Mathematical Statistics at Sydney in 1959). Belz had founded the first Department of Statistics in Australia at the University of Melbourne in 1948; this department provided undergraduate courses (one of which was to inspire E. J. Hannan), but statisticians with postgraduate training remained in short supply until the ANU Department of Statistics began to award doctorates. In 1952, all the PhDs in Statistics teaching in Australian universities (e.g. H. A. David at the University of Melbourne) had earned their doctorates overseas. ANU’s example in postgraduate training was soon followed by several of the other Australian universities.

A PCR Birth Process Subject to an Enzyme Death Process

The PCR process is modelled as a birth process for DNA strands, subject to a death process for enzymes. The case of 1 DNA strand and 1 enzyme is considered first, and generalized to the n strand and b enzyme case. Finally an approximate process is considered which gives a good approximation for the expected number of DNA strands.

Random Breakage of a Rod into Unit Lengths.

Summary In this article we consider the random breakage of a rod into L unit elements and present a Markov chain based method that tracks intermediate breakage configurations. The probability of the time to final breakage for L = 3, 4, 5 is obtained and the method is shown to extend in principle, beyond L = 5.

DETERMINISTIC AND STOCHASTIC MODELS FOR THE SPREAD OF CHOLERA

In this note, we study deterministic and stochastic models for the spread of cholera. The deterministic model for the total number of cholera cases fits the observed total number of cholera cases in some recent outbreaks. The stochastic model for the total number of cholera cases leads to a binomial type distribution with a mean that agrees with the deterministic model. doi:10.1017/S1446181110000027