Plenary Lecture

Plenary Lecture

Limiting Behaviour of a SIS Epidemic Model with Environmental Stochasticity

Professor David Greenhalgh
Reader in Mathematics and Statistics
University of Strathclyde

Abstract: In this talk we extend the classical SIS (susceptible-infected-susceptible) epidemic model from a deterministic one to a stochastic one and formulate it as a stochastic differential equation (SDE) for I(t), the number of infectious individuals at time t. An SIS model is an epidemic model in which a typical individual starts off as susceptible, at some stage catches the disease and after an infectious period becomes susceptible again. Such models are often used for sexually transmitted diseases such as gonorrhoea, or bacterial diseases such as pneumococcus. We survey some relevant deterministic and stochastic models in the literature. We then formulate our basic model. The stochasticity is introduced as a Brownian motion in the disease transmission coefficient (equivalently in the contact rate of infected individuals). This models the effect of random environmental variation. After deriving the SDE for the spread of the disease we then prove that this SDE has a unique positive solution.
For the deterministic model classical results show that there is a unique threshold value R0D, the deterministic basic reproduction number, such that if R0D is less than or equal to one then the disease will die out and if R0D exceeds one then the disease tends to a unique endemic equilibrium. We show that for the stochastic model there is a smaller threshold value R0S and provided that a condition involving the variance of the stochastic noise is satisfied then the disease will die out almost surely (a.s.) for R0S<1. We conjecture that in fact the variance condition is not necessary. If R0S>1 then we show that the disease will fluctuate about a strictly positive level a.s. We discuss the connection between some limiting values of the stochastic threshold R0S and the deterministic threshold R0D. We then show that if R0S>1 the SDE SIS model has a unique non-zero stationary distribution and derive expressions for the mean and variance of this stationary distribution.
All the theoretical results are illustrated and confirmed by numerical simulations. We finish by discussing two real-life examples: first gonorrhoea amongst homosexuals and second pneumococcus amongst Scottish children under two years old.

Brief Biography of the Speaker:
David Greenhalgh graduated from Cambridge University, Cambridge, UK, in 1980 with a First Class Honours degree in Mathematics. In 1981 he took Part III Mathematics also at Cambridge University in which he gained a distinction. He remained at Cambridge for his PhD in Operational Research which he completed in 1984. His PhD thesis was entitled ‘Stochastic Models for Control of Epidemics’.
From Cambridge he moved to the Department of Pure and Applied Biology at Imperial College, London, UK, where he was awarded a Medical Research Council (MRC) Research Training Fellowship to work with Professor R. M. Anderson FRS, a leading international expert on epidemiology. He moved to the Department of Mathematics, Strathclyde University, Glasgow, UK in 1986 as a Lecturer. Since then he has been promoted to Senior Lecturer in 1997 and Reader in 2003. He currently holds the position of Reader in the recently formed Department of Mathematics and Statistics at Strathclyde University.
Dr. Greenhalgh has research interests in mathematical biology and epidemiology. He is an international expert in mathematical epidemiology and has around thirty years experience in this area. He has collaborated with world leading researchers in mathematics and epidemiology such as Professor Klaus Dietz (Germany), Professor Odo Diekmann (The Netherlands), Professor Istvan Gyori (Hungary) and Professor Xuerong Mao (Scotland). He has published around eighty papers in international refereed journals, seven book articles and over seventy conference papers. He is on the editorial board of fourteen international journals, two as Associate Editor. He has served, and currently still serves, on the UK Engineering and Physical Sciences (EPSRC) Mathematics Peer Review College and has served on many UK MRC Panels. These are two of the most prestigious grant giving bodies in the UK. He has also been awarded substantial research funding from a diverse range of sources. He has supervised seventeen research students, fifteen at PhD level. He is widely involved in the organisation of international conferences and has given over thirty invited talks, including plenary talks, at international meetings.

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