Publications in Refereed Journals

There exists an interplay between vaccine coverage, disease prevalence, and the vaccinating behaviour of individuals. Moreover, because of herd immunity, there is also a strategic interaction between individuals when they are deciding whether or not to vaccinate, since the probability that an individual becomes infected depends upon how many other individuals are vaccinated. To understand this potentially complex interplay, a game dynamic model is developed in which individuals adopt strategies according to an imitation dynamic (a learning process), and base vaccination decisions on disease prevalence and perceived risks of vaccines and disease. The model predicts that oscillations in vaccine uptake are more likely in populations where individuals imitate others more readily or where vaccinating behaviour is more sensitive to changes in disease prevalence. Oscillations are also more likely when the perceived risk of vaccines is high. The model reproduces salient features of the time evolution of vaccine uptake and disease prevalence during the whole-cell pertussis vaccine scare in England & Wales during the 1970s. This suggests that using game theoretical models to predict, and even manage, the population dynamics of vaccinating behaviour may be feasible. C.T. Bauch, J. Lloyd-Smith, M. Coffee and A. Galvani (2005). 'Dynamically modeling SARS and respiratory EIDS: past, present, future'. Epidemiology , in press. Abstract. The emergence and rapid global spread of the SARS coronavirus in 2002-2003 The emergence and rapid global spread of the SARS coronavirus in 2002-2003 prompted efforts by modelers to characterize SARS epidemiology and inform control policies. Here we overview and discuss emerging infectious disease (EID) modeling, provide a critical survey of SARS modeling literature, and discuss promising future directions for research. We reconcile discrepancies between published model-based estimates of the basic reproductive number R0 (a crucial epidemiological parameter), discuss insights regarding SARS control measures that have emerged uniquely from a modeling approach, and argue that high priorities for future modeling of SARS and similar respiratory EIDs should include informing quarantine policy and better understanding the impact of population heterogeneity on transmission patterns. C.T. Bauch (2005). 'The spread of infectious diseases in spatially structured populations: an invasory pair approximation'. Mathematical Biosciences , in press. Abstract. The invasion of new species and the spread of emergent infectious diseases in spatially structured populations has stimulated the study of explicit spatial models such as cellular automata, network models and lattice models. However, the analytic The invasion of new species and the spread of emergent infectious diseases in spatially structured populations has stimulated the study of explicit spatial models such as cellular automata, network models and lattice models. However, the analytic intractability of these models calls for the development of tractable mathematical approximations to spatial models such as moment closure approximations. Here we explore moment closure approximations for the invasion of an SIS epidemic on a regular lattice. We use moment closure methods to derive an expression for the basic reproductive number, R0, in a lattice population. On lattices, R0 should be bounded above by the number of neighbors per individual. However, we show that conventional pair approximations actually predict unbounded growth in R0 with increasing transmission rates. To correct this problem, we propose an `invasory' pair approximation which yields a relatively simple expression for R0 that remains bounded above and also predicts R0 values from lattice model simulations more accurately than conventional pair and triple approximations. The invasory pair approximation is applicable to any spatial model, since it exploits characteristics of invasions that are common to all spatially structured populations. C.T. Bauch and D.J.D. Earn (2004). 'Vaccination and the theory of games'. Proceedings of the National Academy of Sciences 101: 13391-13394. Abstract. Voluntary vaccination policies for childhood diseases present parents with a subtle challenge: if a sufficient proportion of the population is already immune, either naturally or by vaccination, then even the slightest risk associated with vaccination will outweigh the risk from infection. As a result, individual self-interest might preclude complete eradication of a vaccine-preventable disease. We show that a formal game theoretical analysis of this problem leads to new insights that help to explain human decision-making with respect to vaccination. Increases in perceived vaccine risk will Voluntary vaccination policies for childhood diseases present parents with a subtle challenge: if a sufficient proportion of the population is already immune, either naturally or by vaccination, then even the slightest risk associated with vaccination will outweigh the risk from infection. As a result, individual self-interest might preclude complete eradication of a vaccine-preventable disease. We show that a formal game theoretical analysis of this problem leads to new insights that help to explain human decision-making with respect to vaccination. Increases in perceived vaccine risk will tend to induce larger declines in vaccine uptake for pathogens that cause more secondary infections (such as measles and pertussis). After a vaccine scare, even if perceived vaccine risk is greatly reduced, it will be relatively difficult to restore prescare vaccine coverage levels. C.T. Bauch and M. Anand (2004). 'The role of mathematical models in ecological restoration and management'. International Journal of Ecology and Environmental Sciences 30: 117-122. Abstract. Mathematical modelling has played an increasingly important role in biology and related fields for the better part of a century. Here, we show the role that modelling can play in understanding and managing forest ecosystems. In particular, we illustrate how mathematical analysis of a simplified forest growth model predicts properties such Mathematical modelling has played an increasingly important role in biology and related fields for the better part of a century. Here, we show the role that modelling can play in understanding and managing forest ecosystems. In particular, we illustrate how mathematical analysis of a simplified forest growth model predicts properties such as the time required recover from perturbation, and the response to various levels of harvesting intensity. The development and analysis of realistic forest models can thus serve as a valuable aid for ecorestoration and ecosystem management efforts. C.T. Bauch, A.P. Galvani and D.J.D. Earn. (2003). 'Group-interest versus self-interest in smallpox vaccination policy'. Proceedings of the National Academy of Sciences 100: 10564-10567. Abstract. The recent threat of bioterrorism has fueled debate on smallpox vaccination policy for the United States. Certain policy proposals call for voluntary mass vaccination; however, if individuals decide whether to vaccinate according to self-interest, the level of herd immunity achieved may differ from what is best for the population as a whole. We present a synthesis of game theory and epidemic modeling that formalizes this conflict between self-interest and group interest and shows that voluntary vaccination is unlikely to reach the group-optimal level. This shortfall results in a substantial increase in expected mortality after an attack. The recent threat of bioterrorism has fueled debate on smallpox vaccination policy for the United States. Certain policy proposals call for voluntary mass vaccination; however, if individuals decide whether to vaccinate according to self-interest, the level of herd immunity achieved may differ from what is best for the population as a whole. We present a synthesis of game theory and epidemic modeling that formalizes this conflict between self-interest and group interest and shows that voluntary vaccination is unlikely to reach the group-optimal level. This shortfall results in a substantial increase in expected mortality after an attack. C.T. Bauch and D.J.D. Earn (2003). 'Transients and Attractors in Epidemics'. Proceedings of the Royal Society of London B 270: 1573-1578. Abstract. Historical records of childhood disease incidence reveal complex dynamics. For measles, a simple model has indicated that epidemic patterns represent attractors of a Historical records of childhood disease incidence reveal complex dynamics. For measles, a simple model has indicated that epidemic patterns represent attractors of a nonlinear dynamic system and that transitions between different attractors are driven by slow changes in birth rates and vaccination levels. The same analysis can explain the main features of chickenpox dynamics, but fails for rubella and whooping cough. We show that an additional (perturbative) analysis of the model, together with knowledge of the population size in question, can account for all the observed incidence patterns by predicting how stochastically sustained transient dynamics should be manifested in these systems. C.T. Bauch and A.P. Galvani (2003). 'Using Network Models to Approximate Spatial Point Process Models'. Mathemat ica l Biosciences 181: 101-114. Abstract. Spatial effects are fundamental to ecological and epidemiological systems, yet the incorporation of space into models is potentially complex. Fixed-edge network models (i.e. networks where each edge has the same fixed strength of interaction) are widely used to study spatial processes but they make simplistic assumptions about spatial Spatial effects are fundamental to ecological and epidemiological systems, yet the incorporation of space into models is potentially complex. Fixed-edge network models (i.e. networks where each edge has the same fixed strength of interaction) are widely used to study spatial processes but they make simplistic assumptions about spatial scale and structure. Furthermore, it can be difficult to parameterize such models with empirical data. By comparison, spatial point-process models are often more realistic than fixed-edge network models, but are also more difficult to analyze. Here we develop a moment closure technique that allows us to define a fixed-edge network model which predicts the prevalence and rate of epidemic spread of a continuous spatial point-process epidemic model. This approach provides a systematic method for accurate parameterization of network models using data from continuously distributed populations (such as data on dispersal kernels). Insofar as point-process models are accurate representations of real spatial biological systems, our example also supports the view that network models are realistic representations of space. C.T. Bauch and D.J.D. Earn (2003). 'Interepidemic Intervals in Forced and Unforced SEIR Models', in 'Dynamical Systems and Their Applications in Biology' (eds. S. Ruan, G. Wolkowicz, J. Wu), Fields Institute Communications 36: 33-44. Abstract. Many infectious diseases give rise to recurrent epidemics. The time interval between epidemics is consequently an important property that epidemiologists and public health offcials would like to be able to predict. Accurate estimates have been made for certain diseases by associating the observed interepidemic interval with the natural period of damped oscillations near the stable equilibrium solution of the standard (unforced) SEIR model. For childhood infections, this successful prediction is surprising because seasonal variation in contact rates (due to school terms) is known to have significant effects on patterns of disease incidence. Here, we show that the natural damping period of transients near the annual attractor of the seasonally forced SEIR model is usually wellapproximated by the damping period obtained without forcing. This explains why naive calculations of interepidemic intervals have yielded accurate results in certain cases. However, the unforced approximation cannot be justified if the forced model has a nonannual attractor with a non-negligible basin of attraction, as is typically the case for measles; consequently, agreement between the interepidemic interval predicted by the unforced model for measles and real measles time series, appears to be coincidental. Many infectious diseases give rise to recurrent epidemics. The time interval between epidemics is consequently an important property that epidemiologists and public health offcials would like to be able to predict. Accurate estimates have been made for certain diseases by associating the observed interepidemic interval with the natural period of damped oscillations near the stable equilibrium solution of the standard (unforced) SEIR model. For childhood infections, this successful prediction is surprising because seasonal variation in contact rates (due to school terms) is known to have significant effects on patterns of disease incidence. Here, we show that the natural damping period of transients near the annual attractor of the seasonally forced SEIR model is usually wellapproximated by the damping period obtained without forcing. This explains why naive calculations of interepidemic intervals have yielded accurate results in certain cases. However, the unforced approximation cannot be justified if the forced model has a nonannual attractor with a non-negligible basin of attraction, as is typically the case for measles; consequently, agreement between the interepidemic interval predicted by the unforced model for measles and real measles time series, appears to be coincidental. C.T. Bauch (2002). 'A Versatile ODE Approximation to a Network Model for Sexually Transmitted Disease Spread'. Journal of Mathematical Biology 45(5): 375-395. Abstract. We develop a moment closure approximation (MCA) to a network model of We develop a moment closure approximation (MCA) to a network model of sexually transmitted disease (STD) spread through a steady/casual partnership network. MCA has been used previously to approximate static, regular lattices, whereas application to dynamic, irregular networks is a new endeavour, and application to sociologically-motivated network models has not been attempted. Our goals are 1) to investigate issues relating to the application of moment closure approximations to dynamic and irregular networks, and 2) to understand the impact of concurrent casual partnerships on STD transmission through a population of predominantly steady monogamous partnerships. We are able to derive a moment closure approximation for a dynamic irregular network representing sexual partnership dynamics, however, we are forced to use a triple approximation due to the large error of the standard pair approximation. This example underscores the importance of doing error analysis for moment closure approximations. We also find that a small number of casual partnerships drastically increases the prevalence and rate of spread of the epidemic. Finally, although the approximation is derived for a specific network model, we can recover approximations to a broad range of network models simply by varying model parameters which control the structure of the dynamic network. Thus our moment closure approximation is very flexible in the kinds of network models it can approximate. C.T. Bauch and D. Rand (2000). 'A Moment Closure Model for Sexually Transmitted Disease Spread Through a Concurrent Partnership Network'. Proceedings of the Royal Society of London B 267: 2019-2027. Abstract. A moment closure model of sexually transmitted disease spread through a concurrent partnership network is developed. The model employs pair approximations of higher-order correlations to derive equations of motion in terms of numbers of pairs and singletons. The model is derived from an underlying stochastic process of partnership network formation and disease transmission. The model is analysed numerically, and the final size and time evolution are considered for various levels of concurrency, as measured by the concurrency index κ3 of Kretzschmar and Morris. Additionally, a new A moment closure model of sexually transmitted disease spread through a concurrent partnership network is developed. The model employs pair approximations of higher-order correlations to derive equations of motion in terms of numbers of pairs and singletons. The model is derived from an underlying stochastic process of partnership network formation and disease transmission. The model is analysed numerically, and the final size and time evolution are considered for various levels of concurrency, as measured by the concurrency index κ3 of Kretzschmar and Morris. Additionally, a new way of calculating R0 for spatial network models is developed. It is found that concurrency signicantly increases R0 and the final size of a sexually transmitted disease, with some interesting exceptions.