# David GreenhalghUniversity of Strathclyde · Department of Mathematics and Statistics

David Greenhalgh

Bachelor of Arts (Mathematics), Cambridge, Certificate of Advanced Study in Mathematics (Cambridge), PhD Operational Research (Cambridge)

## About

119

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Introduction

Additional affiliations

January 1986 - February 2016

## Publications

Publications (119)

Mathematical modeling techniques have been used extensively during the human immunodeficiency virus (HIV) epidemic. Drug injection causes increased HIV spread in most countries globally. The media is crucial in spreading health awareness by changing mixing behavior. The published studies show some of the ways that differential equation models can b...

This article discusses two models, with two different needle assumptions for the transmission of hepatitis C virus (HCV) between people who inject drugs (PWIDs) who share needles and syringes. Our analysis demonstrates that the basic reproduction number R0 determines how the model behaves. R0=1 is a crucial threshold parameter which divides two qua...

Aedes mosquitoes were found to lay their eggs in the cryptic breeding sites. Eliminating cryptic and open breeding sites is essential in reducing dengue virus transmission. However, it is often challenging for health officers to assess these breeding sites which are usually missed during larval surveillance. The autodissemination approach may produ...

Background
Dengue is a significant public health issue that is caused by Aedes spp. mosquitoes. The current vector control methods are unable to effectively reduce Aedes populations and thus fail to decrease dengue transmission. Hence, there is an urgent need for new tools and strategies to reduce dengue transmission in a wide range of settings. In...

The new emergence and re-emergence of arbovirus infections transmitted by Aedes mosquitoes have been spreading across Southeast Asia, Central Africa, United States, tropical Oceania and has become a major of public health concern. These arbovirus diseases were found to have a similar vector, symptoms, and environments. The situation is complex due...

In this paper, we study a single serotype transmission model of dengue to determine the optimal vaccination age for Dengvaxia. The transmission dynamics are modelled with an age-dependent force of infection. The force of infection for each serotype is derived from the serological profile of dengue in Brazil without serotype distinction and from ser...

We investigate a model consisting of a predator population and both susceptible and infected prey populations. The predator can feed on either prey species but instead of choosing individuals at random the predator feeds preferentially on the most abundant prey species. More specifically we assume that the likelihood of a predator catching a suscep...

In this paper we introduce a single serotype transmission model, including an age-dependent mosquito biting rate, to find the optimal vaccination age against dengue in Brazil with Dengvaxia. The optimal vaccination age and minimal lifetime expected risk of hospitalisation are found by adapting a method due to Hethcote (Math Biosci 89:29–52). Any nu...

This is the first study to evaluate the efficacy of an autodissemination approach, as suggested by WHO. Therefore, the efficacy of an autodissemination approach in small-scale field trials against wild Aedes sp. population was evaluated in an urbanized setting, Malaysia. Lethal ovitraps enhanced with pyriproxyfen were used to control Aedes sp. popu...

In this paper we have adapted a delayed dengue model to Zika. By assuming that the epidemic starts by a single infected individual entering a disease-free population at some initial time t0 we have used the least squares parameter estimation technique in R to estimate the initial time t0 using observed Zika data from Brazil as well as the transmiss...

Mathematical modelling techniques are now being used by health organizations
worldwide to help understand the likely impact that intervention strategies treatment
options and combinations of these have on the prevalence and incidence of hepatitis
C virus (HCV) in the people who inject drugs (PWID) population. Studies of hepatitis C virus (HCV) infe...

In this paper, we will start off by introducing the classical Ross–Macdonald model for vector-borne diseases which we use to describe the transmission of dengue between humans and Aedes mosquitoes in Shah Alam, which is a city and the state capital of Selangor, Malaysia. We will focus on analysing the effect of using the Mosquito Home System (MHS),...

Mathematical modelling techniques are now being used by health organizations worldwide to help understand the likely impact that intervention strategies treatment options and combinations of these have on the prevalence and incidence of hepatitis C virus (HCV) in the people who inject drugs (PWID) population. In this talk, we develop a deterministi...

In this paper, we use the classical Ross-Macdonald model to analyze the effect of the Mosquito Home System (MHS), which is an example of an auto-dissemination trap, in controlling the spread of dengue in Malaysia in a high-rise condominium environment. By using the national dengue data from Malaysia, we are able to estimate λ which represents the i...

Background:
National or local laws, norms or regulations (sometimes and in some countries) require medical providers to report notifiable diseases to public health authorities. Reporting, however, is almost always incomplete. This is due to a variety of reasons, ranging from not recognizing the diseased to failures in the technical or administrati...

Background
Evidence of changing in biting and resting behaviour of the main malaria vectors has been mounting up in recent years as a result of selective pressure by the widespread and long-term use of insecticide-treated bed nets (ITNs), and indoor residual spraying. The impact of resistance behaviour on malaria intervention efficacy has important...

In this paper we study a mathematical model to analyse the optimal vaccination age against Dengue in Brazil. Data from Brazil are used to estimate the basic reproduction numbers for each of the four Dengue serotypes and then the optimal vaccination age is calculated using a method due to Hethcote [1]. The vaccine has different efficacies against ea...

SUMMARY The classical Ross–Macdonald model is often utilized to model vector-borne infections; however, this model fails on several fronts. First, using measured (or estimated) parameters, which values are accepted from the literature, the model predicts a much greater number of cases than what is usually observed. Second, the model predicts a sing...

A predator–prey model with disease amongst the prey and ratio-dependent functional response for both infected and susceptible prey is proposed and its features analysed. This work is based on previous mathematical models to analyse the important ecosystem of the Salton Sea in Southern California and New Mexico where birds (particularly pelicans) pr...

We discuss the effect of introducing telegraph noise, which is an example of an environmental noise, into the susceptible-infectious-recovered-susceptible (SIRS) model by examining the model using a finite-state Markov Chain (MC). First we start with a two-state MC and show that there exists a unique nonnegative solution and establish the condition...

Background:
Rio de Janeiro in Brazil will host the Summer Olympic Games in 2016. About 400,000 non-immune foreign tourists are expected to attend the games. As Brazil is the country with the highest number of dengue cases worldwide, concern about the risk of dengue for travelers is justified.
Methods:
A mathematical model to calculate the risk o...

In this paper we look at the two dimensional stochastic differential equation (SDE) susceptible-infected-susceptible (SIS) epidemic model with demographic stochasticity where births and deaths are regarded as stochastic processes with per capita disease contact rate depending on the population size. First we look at the SDE model for the total popu...

We introduce stochasticity into the deterministic differential equation model for the spread of HIV amongst people who inject drugs (PWIDs) studied by Greenhalgh and Hay (1997). This was based on the original model constructed by Kaplan (1989) which analyses the behaviour of HIV/AIDS amongst a population of PWIDs. We derive a stochastic differentia...

In this paper we discuss the stochastic differential equation (SDE) susceptible-infected-susceptible (SIS) epidemic model with demographic stoch-asticity. First we prove that the SDE has a unique nonnegative solution which is bounded above. Then we give conditions needed for the solution to become extinct. Next we use the Feller test to calculate t...

The spectral radius of the next generation matrix provides an expression for the basic reproduction number. Instead of calculating the dominant eigenvalue of the characteristic equation corresponding to the next generation matrix, a threshold parameter can be obtained by handling the coefficients of this equation. Here we prove two conjectures pres...

We propose and analyze a mathematical model to study the impact of awareness programs on an infectious disease outbreak. These programs induce behavioral changes in the population, which divide the susceptible class into two subclasses, aware susceptible and unaware susceptible. The system can have a disease-free equilibrium and an endemic equilibr...

New diagnoses of HIV infection were reported confidentially to the Public Health Laboratory Service AIDS Centre under a national voluntary surveillance scheme. Two sets of data drawn from the national data sets were made available to us for analysis, the first in 1991 and the second in 1994, by which time the replication of reports had been reduced...

In this paper we estimate the parameters in the stochastic SIS epidemic model by using pseudo-maximum likelihood estimation (pseudo-MLE) and least squares estimation. We obtain the point estimators and 100(1-α)% confidence intervals as well as 100(1-α)% joint confidence regions by applying least squares techniques. The pseudo-MLEs have almost the s...

The 7-valent pneumococcal conjugate vaccine (Prevenar(®), Wyeth; PCV7) was introduced to the UK paediatric immunisation schedule in 2006. This study investigates trends in serotypes and multi locus sequence types (STs) among cases of invasive pneumococcal disease (IPD) in Scotland prior to, and following, the introduction of PCV7.
Scottish Invasive...

In order to prevent the spread of the hepatitis C virus (HCV) amongst people who inject drugs (PWID), it is imperative that any injecting risk behaviour which may contribute to the transmission of disease has its role quantified. To inform public health organisations, mathematical modelling techniques were used to explore the risk of HCV infection...

Infection with HIV-1, degrading the human immune system and recent advances of drug therapy to arrest HIV-1 infection, has generated considerable research interest in the area. Bonhoeffer et al. (1997) [1], introduced a population model representing long term dynamics of HIV infection in response to available drug therapies. We consider a similar t...

Population systems are often subject to environmental noise. Motivated by Takeuchi et al. [7], we will discuss in this paper the effect of telegraph noise on the well-known SIS epidemic model. We establish the explicit solution of the stochastic SIS epidemic model, which is useful in performing computer simulations. We also establish the conditions...

Studies of hepatitis C virus (HCV) infection amongst injecting drug users (IDUs) have suggested that this population can be separated into two risk groups (naive and experienced) with different injecting risk behaviours. Understanding the differences between these two groups and how they interact could lead to a better allocation of prevention meas...

We examine a mathematical model for the transmission of Streptococcus Pneumoniae amongst young children when the carriage transmission coefficient depends on the serotype. Carriage means pneumococcal colonization. There are two sequence types (STs) spreading in a population each of which can be expressed as one of two serotypes. We derive the diffe...

In this paper we examine the effect of heterogenous mixing on the spread of HIV and AIDS amongst a population of injecting drug users. We consider heterogeneity in addicts' shooting gallery visiting rates, their syringe cleaning probabilities and their choice of shooting gallery. We discuss two models. In the first the size of the populations of th...

In this paper we develop and analyse a model for the spread of HIV/AIDS amongst a population of injecting drug users. We examine the impact on the spread of disease caused by allowing addicts to progress through three distinct stages of variable infectivity prior to the onset of full blown AIDS. We first state a three stage infectivity model of the...

For a mathematical model for the spread of HIV by sexual transmission in a heterosexual population we analyse the existence and stability of equilibrium solutions. The model is designed to investigate the effects of a fundamental constraint in any social/sexual mixing process for heterogeneous populations. The group contact constraint conserves the...

Mathematical modelling can provide valuable insights into the biological and epidemiological properties of infectious diseases
as well as the potential impact of intervention strategies employed by health organizations worldwide. In this paper, we develop
a deterministic, compartmental mathematical model to approximate the spread of the hepatitis C...

This paper discusses a simple mathematical model to describe the spread of Streptococcus pneumoniae. We suppose that the transmission of the bacterium is determined by multi-locus sequence type. The model includes vaccination and is designed to examine what happens in a vaccinated population if MLSTs can exist as both vaccine and non vaccine seroty...

Backward bifurcation is a relatively recent yet well-studied phenomenon associated with deterministic epidemic models. It allows for the presence of multiple subcritical endemic equilibria, and is generally found only in models possessing a reasonable degree of complexity. One particular aspect of backward bifurcation that appears to have been virt...

We describe associations between death from invasive pneumococcal disease (IPD) and particular serogroups and sequence types (STs) determined by multilocus sequence typing (MLST) using data from Scotland. All IPD episodes where blood or cerebrospinal fluid (CSF) culture isolates were referred to the Scottish Haemophilus, Legionella, Meningococcal a...

Streptococcus pneumoniae is a bacterium commonly found in the throat of young children. Pneumococcal serotypes can cause a variety of invasive and non-invasive diseases such as meningitis and pneumonia. In 2000 a vaccine was introduced in the USA that not only prevents vaccine type disease but has also been shown to eliminate carriage of the vaccin...

In this paper we extend the classical susceptible-infected-susceptible epidemic model from a deterministic framework to a stochastic one and formulate it as a stochastic differential equation (SDE) for the number of infectious individuals I(t). We then prove that this SDE has a unique global positive solution I(t) and establish conditions for extin...

This paper deals with the perturbation analysis of fuzzy linear systems. Three cases of perturbation are considered: (a) the right hand side is perturbed while the coefficient matrix remains unchanged; (b) the coefficient matrix is perturbed while the right hand side remains unchanged, and (c) both the coefficient matrix and the right hand side are...

In order to develop new ways to prevent Hepatitis C virus (HCV) transmission amongst injecting drug users (IDUs), it is necessary to fully understand the dynamics of this disease. We reviewed the evidence on three key areas of HCV transmission in this population: the rate of acute HCV infection amongst IDUs who have spontaneously resolved a previou...

We study an economy in which the rate of change of population depends on population policy decisions. This requires population as well as capital as state variables. By showing the algebraic relationship between the shadow price of the population and the shadow price of the per capita capital stock, we are still able to depict the optimal path and...

In this paper we develop previously studied mathematical models of the regulation of testosterone by luteinizing hormone and luteinizing hormone release hormone in the human body. We propose a delay differential equation mathematical model which improves on earlier simpler models by taking into account observed experimental facts. We show that our...

In this paper we study a predator-prey model with logistic growth in the prey population, where a disease spreads among the prey according to an susceptible-infected-susceptible (SIS) epidemic model. The predators do not consume infected prey. After a review of the literature we formulate the basic mathematical model. For simplicity, we work initia...

We study a mathematical model for the viral dynamics of HIV in an infected individual in the presence of HAART. The paper starts with a literature review and then formulates the basic mathematical model. An expression for R 0, the basic reproduction number of the virus under steady state application of HAART, is derived followed by an equilibrium a...

In order to obtain a reasonably accurate model for the spread of a particular infectious disease through a population, it may be necessary for this model to possess some degree of structural complexity. Many such models have, in recent years, been found to exhibit a phenomenon known as backward bifurcation, which generally implies the existence of...

In this paper we consider the phenomenon of backward bifurcation in epidemic modelling illustrated by an extended model for Bovine Respiratory Syncytial Virus (BRSV) amongst cattle. In its simplest form, backward bifurcation in epidemic models usually implies the existence of two subcritical endemic equilibria for R
0 < 1, where R
0 is the basic re...

In this paper we analyse a stochastic model representing HIV internal virus dynamics. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modelling. We show that the model established in this paper possesses non-negative solutions as this is essential in any population dynami...

A predator–prey model with transmissible disease in the prey species is proposed and analysed. The essential mathematical features are analysed with the help of equilibrium, local and global stability analyses and bifurcation theory. We find four possible equilibria. One is where the populations are extinct. Another is where the disease and predato...

Bootstrapping is used to estimate the effectiveness of different vaccination strategies for rubella in England and Wales. It is assumed that rubella infection follows the deterministic age-structured model discussed by Dietz and Schenzle (1985). The bootstrap is used to estimate percentile confidence intervals for the basic reproductive number and...

In this paper we introduce stochasticity into a model of AIDS and condom use via the technique of parameter perturbation which is standard in stochastic population modelling. We show that the model established in this paper possesses non-negative solutions as desired in any population dynamics. We also carry out a detailed analysis on asymptotic st...

In this paper we examine an age-structured partial differential equation compartmental model to predict minimal vaccination strategies to eliminate hepatitis A in Bulgaria. We describe the mathematical model and briefly summarise previous theoretical results. The basic reproduction number is a key parameter of the model. We consider proportional, a...

An SIRS epidemic model with general periodic vaccination strategy is analyzed. This periodic vaccination strategy is discussed first for an SIRS model with seasonal variation in the contact rate of period T = 1 year. We start with the case where the vaccination strategy and the contact rate have the same period and then discuss the case where the p...

In this paper we look at an SIRS epidemiological model with vaccination. Although immunity gained by experiencing the disease is permanent, vaccine-induced immunity is only temporary and a fixed time after vaccination individuals return to the susceptible class. The model is described and equilibrium and stability results are shown. There are three...

In this paper, a general periodic vaccination has been applied to control the spread and transmission of an infectious disease with latency. A SEIRS(1) epidemic model with general periodic vaccination strategy is analyzed. We suppose that the contact rate has period T, and the vaccination function has period LT, where L is an integer. Also we apply...

This paper uses two SIRS type epidemiological models to examine the impact on the spread of disease caused by vaccination when the immunity gained from such an intervention is not lifelong. This occurs, for example, in vaccination against influenza. We assume that susceptible individuals become immune immediately after vaccination and that immune i...

This paper presents an extension to the recently introduced class of nonlinear filters known as Aperture Filters. By taking a multiresolution approach, it can be shown that more accurate filtering results (in terms of mean absolute error) may be achieved compared to the standard aperture filter given the same size of training set. Most optimisation...

We consider a perturbation of the classical McKendrick-Von Foester equation originally discussed by E. N. Boulanger [J. Math. Biol. 32, No. 6, 521–533 (1994; Zbl 0803.92022)]. As we are dealing with population densities it is more natural to express the equations as integral equations. We establish existence and uniqueness of solutions under weaker...

This chapter discusses the uses of stochastic processes in epidemiology, mathematical modeling and the simulation of epidemics. The chapter presents several classical applications of stochastic processes in epidemic theory —namely, the Chain Binomial, followed by the simple and general stochastic epidemic models. It focuses on the application of st...

Aperture filters compose a recently introduced class of non-linear operators used in signal processing. Their operation involves filtering of signals that are observed within a window of finite width and height. They allow a tractable design of non-linear filters by reducing the search space. This paper presents an adaptation to the original design...

A SIRS epidemic model with general seasonal variation in the contact rate is analysed. This SIRS model has a unique disease free equilibrium (DFE) which is globally asymptotically stable when the basic reproductive number is less than or equal to one in value. Four childhood infectious diseases are studied (measles, chickenpox, mumps, and rubella)....

In this paper prediction methods are discussed in the context of developing an exception reporting system for laboratory reports. The detection of outbreaks and longer term trends is briefly addressed, before a consideration of data types and availability to be used in evaluating the prediction methods. Four general prediction methods are outlined...

In this paper we develop and analyse a model for the spread of HIV/AIDS amongst a population of injecting drug users. The model we discuss focuses on the transmission of HIV through the sharing of contaminated drug injection equipment and in particular we examine the mixing of addicts and needles when the AIDS incubation period is divided into thre...

We examine models of the spread of HIV amongst intravenous drug users in which infection with HIV results in a three stage incubation period prior to the onset of AIDS. Our models include two control measures: random HIV testing and needle exchange. We first give a brief literature review before discussing the complications which arise in modelling...

In this paper we examine the impact of condom use on the sexual transmission of human immunodeficiency virus (HIV) and acquired immune deficiency syndrome (AIDS) amongst a homogeneously mixing male homosexual population. We first derive a multi-group SIR-type model of HIV/AIDS transmission where the homosexual population is split into subgroups acc...

In this paper we examine the spread of HIV when this disease is transmitted through the random sharing of contaminated drug injection equipment. We first model the spread of disease using a standard set of behavioral assumptions discussed by Kaplan [1]. We demonstrate that deterministic and stochastic models based on these assumptions behave very s...

In this paper we develop and analyse a model for the spread of HIV/AIDS amongst a population of injecting drug users. We start off with a brief literature survey and review; this is followed by the derivation of a model which allows addicts to progress through three distinct stages of variable infectivity prior to the onset of full blown AIDS and w...

The basic reproduction number of an infection, R 0 , is the average number of secondary infections generated by a single typical infective individual in a totally susceptible population. It is directly related to the effort required to eliminate infection. We consider statistical methods for estimating R 0 from age-strati®ed serological survey data...

In this paper we discuss convergence properties for genetic algorithms. By looking at the effect of mutation on convergence, we show that by running the genetic algorithm for a sufficiently long time we can guarantee convergence to a global optimum with any specified level of confidence. We obtain an upper bound for the number of iterations necessa...

In this paper we extend the 'needles that kill' model discussed in Kaplan & O'Keefe (1993) to allow addicts to progress through three-stages of variable infectivity prior to the onset of full-blown AIDS, and where the class of infectious needles is split into three according to the different levels of infectivity in addicts. Given the structure of...

Many classical mathematical models for animal infections assume that all infected animals transmit the infection at the same rate, all are equally susceptible, and the course of the infection is the same in all animals. However for some infections there is evidence that seropositives may still transmit the infection, albeit at a lower rate. Animals...

We discuss a mathematical model for the spread of HIV/AIDS amongst a population of injecting intravenous drug-users who share injecting equipment in shooting galleries. The work is based on a model originally due to E.H. Kaplan [Rev. Inf. Diseases 11, 289-298 (1989)]. We shall start off with a brief description of the spread of HIV and AIDS in shoo...

Diagnoses of HIV infection are reported to the Public Health Laboratory Service (PHLS) by microbiologists through a voluntary confidential surveillance scheme. Names are not recorded on the database but the date of birth of the individual concerned is usually available. This paper discusses a statistical method to detect repeated counting of indivi...

Two SIR models for the spread of infectious diseases which were originally suggested by Greenhalgh & Das (1995, Theor. Popul. Biol. 47, 129-179; 1995, Mathematical Population Dynamics: Analysis of Heterogeneity, pp. 79-101, Winnipeg: Wuerz Publishing) are considered but with a time delay in the vaccination term. This reflects the fact that real vac...

Diagnoses of HIV infection are reported to the Public Health Laboratory Service (PHLS) AIDS Centre under a voluntary surveillance scheme. Names are not held in the data set, but the date of birth of the individual concerned is usually available. This paper describes a statistical method for identifying whether there are likely to be individuals rep...

The difference in transmissibility of HIV between heterosexual males and females in specific social contexts is known to play an important role in determining the form of HIV/AIDS epidemics across the globe. A fundamental constraint here is the conservation of the number of new partnerships formed between the sexes. We examine the impact of general...

In this article we examine the impact on the spread of HIV caused by regularly testing members of a needle sharing population for the presence of disease. We develop a model of injecting drug use which contains two classes of addicts, a class who are unaware of their infectious status and a class who are aware that they are infectious through havin...

In this paper we consider the amount of undetected replication in HIV infection diagnoses as reported to the Public Health Laboratory Service AIDS Centre, Colindale, London. These diagnoses are usually reported with the date of birth of the individual but no names held on the database. The PHLS cannot always tell whether two reports with the same d...

In this paper we develop and analyse a model for the spread of HIV/AIDS amongst a population of injecting drug users. Our work is based on a model originally due to Kaplan (1989, Rev. Inf. Diseases 11, 289-98). We start off with a brief literature survey and review; this is followed up by a detailed description of Kaplan's model. We then outline a...

In this paper, some SEIRS epidemiological models with vaccination and temporary immunity are considered. First of all, previously published work is reviewed. In the next section, a general model with a constant contact rate and a density-dependent death rate is examined. The model is reformulated in terms of the proportions of susceptible, incubati...