The evolution
and transmission of a virus
A simplified model for the evolution and transmission of a mutating virus is
described. The essential features of the model are: within host evolution; host
heterogeneity; between host transmission and dynamic viral diversity. The virus
replicates within the host, generating variants through random mutation. A virus
with a mutation in an epitope that would be otherwise recognised by the host
has an advantage in being able to escape at least that component of the immune
response. On the other hand, a virus with a mutation in a non-recognised epitope
bears a cost. Replication of the virus is modulated by the cytotoxic T lymphocyte
(CTL) response of the host. The strength of the response depends on the human
leukocyte alleles (HLA) of class I expressed by the host. Hence, for the purpose
of the model we divide the human host population into types, where those of
the same type are HLA-similar with regard to their response to the virus types
represented in the model. We model the transmission of the virus using an SIR
model, with no recovery resulting in immunity and infectious individuals having
an increased death rate. Hence we are motivated by the transmission of the HIV
virus. The final complication in the model gives rise to dynamic viral diversity
via selection by CTLs at the time of transmission between HLA-disparate individuals.
An inoculum of virus transmitted to a host expressing different HLA class I
alleles to the donor will give rise to an outgrowth of the quasispecies that
is most fit in the recipient host. This is on a much shorter time-scale (measured
in days) than the evolution of the virus within the host resulting from random
mutation. Development and analysis of the model is part of an ongoing collaboration
with Prof. Angela McLean, University of Oxford.
Strategies
for minimising the risk of importing exotic diseases
The danger of a global influenza pandemic is now greater than ever before. The
global epidemic of severe acute respiratory syndrome (SARS) in 2003, and concerns
that terrorists may use an infectious agent such as smallpox as a weapon, had
previously demonstrated the need to determine control strategies for exotic
infections. The prior determination of such strategies, and the use of mathematical
models to assist this, are hampered by the obvious lack of data. We propose
an integral equation model of Kermack-McKendrick type that may be used to compare
strategies based on the isolation of infectious individuals and targeted vaccination.
The model structures the incidence of infection according to the location of
exposure, and requires some knowledge of the infectivity kernel and the initial
rate of exponential increase of cases. The model’s use in the design of
strategies to minimize the risks of pandemic influenza, SARS and smallpox in
a previously unexposed community is demonstrated.
Roberts, M. G., Baker, M., Jennings, L. C., Sertsou,
G. & Wilson, N. A model for the spread and control of pandemic influenza
in an isolated geographical region. Journal of the Royal Society Interface In
press.
Aldis, G. K. & Roberts, M.
G. 2005 An integral equation model for the control of a smallpox outbreak. Mathematical
Biosciences 195, 1-22.
Roberts, M. G. 2004 Modelling strategies for minimizing
the impact of an imported exotic infection. Proceedings of the Royal Society,
Series B 271, 2411-2415.
Vaccination
strategies for the control of infectious diseases
A model, developed
for the Ministry of Health in 1996, successfully predicted the 1997 measles
epidemic. A control policy, initiated in response to the model, swiftly contained
it. It was determined that further epidemics would occur unless the timing of
the second measles vaccination was changed from 11 years to 5 years of age.
The vaccination schedule was changed for 2001. A similar model was used to investigate
the vaccination strategies for pertussis.
Roberts, M. G. & Tobias, M. I. 2000 Predicting and
preventing measles epidemics in New Zealand: Application of a mathematical model.
Epidemiology and Infection 124, 279-287.
The type
reproduction number and the characterisation of reservoirs of infection
We propose a new threshold quantity for the analysis of the epidemiology of
infectious diseases. The quantity is similar in concept to the familiar basic
reproduction number, R0, but it singles out particular host types instead of
providing a criterion that is uniform for all host types. Using this methodology
we are able to identify the long-term effects of disease-control strategies
for particular subgroups of the population, to estimate the level of control
necessary when targeting control effort at a subset of host types, and to identify
host types that constitute a reservoir of infection. These insights cannot be
obtained by using R0 alone.
Roberts, M. G. & Heesterbeek, J. A. P. 2003 A new
method for estimating the effort required to control infectious diseases. Proceedings
of the Royal Society, Series B 270, 1359-1364.
Strategies
for the control of bovine tuberculosis in possums
A model was developed that described the epidemiology and control of tuberculosis
in possum populations. The model was extended to reflect the spatial interaction
of possums and Tb on interconnected habitat patches, and new threshold quantities
for diseases in metapopulations were derived. A similar model was employed to
describe the dynamics and control of brucellosis in American bison herds. A
model, motivated by the search for a possum biocontrol agent, was used to describe
the interaction of multiple parasite species in wild animal populations, including
threshold quantities determining when a species can invade or when an existing
species is ejected from a parasite community.
Fulford, G. R., Roberts, M. G. & Heesterbeek, J. A.
P. 2002 The metapopulation dynamics of an infectious disease: tuberculosis in
possums. Theoretical Population Biology 61, 15-29.
Kao, R. R. & Roberts, M. G. 1999 A comparison of wildlife control and cattle
vaccination as methods for the control of bovine tuberculosis. Epidemiology
and Infection 122, 505-519.
Kao, R. R., Roberts, M. G. & Ryan, T. J. 1997 A model of bovine tuberculosis
control in domesticated cattle herds. Proceedings of the Royal Society, Series
B 264, 1069-1076.
Roberts, M. G. 1999 A Kermack-McKendrick model applied to an infectious disease
in a natural population. IMA Journal of Mathematics Applied in Medicine and
Biology 16, 319-332.
Roberts, M. G. & Kao, R. R. 1998 The dynamics of an infectious disease in
a population with birth pulses. Mathematical Biosciences 149, 23-36.
Allocation
of resources for the management of diabetes, physical inactivity and obesity
Multistate life-tables may be used to describe the age-dependent epidemiology
of chronic conditions in a population. The flows between the states are described
by systems of differential equations known as IPRM models, because
they may be solved to express the incidence of a condition in terms of prevalences,
remission rates and relative risks of mortality, all of which may be functions
of age and obtained from population surveys. These models may be used to predict
the effects of proposed interventions on measures of population health. They
have been applied to a range of conditions of importance in New Zealand: diabetes,
physical inactivity and obesity; and used to explore the interaction between
diabetes and obesity.
Roberts, M. G. & Tobias, M. I. 2001 The use of multistate life-table models
for improving population health. IMA Journal of Mathematics Applied in Medicine
and Biology 18, 119-130.
Tobias, M. I. & Roberts, M. G. 2001 Modelling physical activity: a multi-state
life-table approach. Australian and New Zealand Journal of Public Health 25,
141-148.
The sustainable
control of intestinal nematode parasites of sheep and the management of anthelmintic
resistance
A model, devised in collaboration with the Zoology Department, University of
Cambridge, explained the relative influences of climate and acquired immunity
on the observed temporal patterns of the parasite population. A new threshold
quantity (Q0) for helminth population dynamics was devised. An analysis of the
discrete/continuous dynamical system that describes parasite epidemiology found
it to exhibit chaotic behaviour in some regions of parameter space. The methodology
was used to describe the epidemiology of sheep nematodes in New Zealand, Australia
and the United Kingdom, and extended to provide criteria for the development
of anthelmintic resistance under different control strategies.
Heesterbeek, J. A. P. & Roberts, M. G. 1995 Threshold
quantities for helminth infections. Journal of Mathematical Biology 33, 415-434.
Kao, R. R., Leathwick, D. M., Roberts, M. G. & Sutherland, I. A. 2000 Nematode
parasites of sheep: a survey of epidemiological parameters and their application
in a simple model. Parasitology 121, 85-103.
Roberts, M. G. 1999 The immunoepidemiology of nematode
parasites of farmed animals: A mathematical approach. Parasitology Today 15,
246-251.
Roberts, M. G. & Heesterbeek, J. A. P. 1995 The dynamics
of nematode infections of farmed ruminants. Parasitology 110, 493-502.
Roberts, M. G. & Heesterbeek, J. A. P. 1998 A simple parasite model with
complicated dynamics. Journal of Mathematical Biology 37, 272-290.