Publications by authors named "Driessche P"

94 Publications

Dynamics of an age-of-infection cholera model.

Math Biosci Eng 2013 Oct-Dec;10(5-6):1335-49

Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada.

A new model for the dynamics of cholera is formulated that incorporates both the infection age of infectious individuals and biological age of pathogen in the environment. The basic reproduction number is defined and proved to be a sharp threshold determining whether or not cholera dies out. Final size relations for cholera outbreaks are derived for simplified models when input and death are neglected.
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http://dx.doi.org/10.3934/mbe.2013.10.1335DOI Listing
July 2014

A cholera model in a patchy environment with water and human movement.

Math Biosci 2013 Nov 16;246(1):105-12. Epub 2013 Aug 16.

Department of Epidemiology, University of Michigan, Ann Arbor, MI 48109, United States; Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States. Electronic address:

A mathematical model for cholera is formulated that incorporates direct and indirect transmission, patch structure, and both water and human movement. The basic reproduction number R0 is defined and shown to give a sharp threshold that determines whether or not the disease dies out. Kirchhoff's Matrix Tree Theorem from graph theory is used to investigate the dependence of R0 on the connectivity and movement of water, and to prove the global stability of the endemic equilibrium when R0>1. The type/target reproduction numbers are derived to measure the control strategies that are required to eradicate cholera from all patches.
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http://dx.doi.org/10.1016/j.mbs.2013.08.003DOI Listing
November 2013

Sensitive dependence on initial conditions in gene networks.

Chaos 2013 Jun;23(2):025101

Department of Mathematics and Statistics, University of Victoria, PO Box 3060, STN CSC, Victoria, British Columbia V8W 3R4, Canada.

Active regulation in gene networks poses mathematical challenges that have led to conflicting approaches to analysis. Competing regulation that keeps concentrations of some transcription factors at or near threshold values leads to so-called singular dynamics when steeply sigmoidal interactions are approximated by step functions. An extension, due to Artstein and coauthors, of the classical singular perturbation approach was suggested as an appropriate way to handle the complex situation where non-trivial dynamics, such as a limit cycle, of fast variables occur in switching domains. This non-trivial behaviour can occur when a gene regulates multiple other genes at the same threshold. Here, it is shown that it is possible for nonuniqueness to arise in such a system in the case of limiting step-function interactions. This nonuniqueness is reminiscent of but not identical to the nonuniqueness of Filippov solutions. More realistic gene network models have sigmoidal interactions, however, and in the example considered here, it is shown numerically that the corresponding phenomenon in smooth systems is a sensitivity to initial conditions that leads in the limit to densely interwoven basins of attraction of different fixed point attractors.
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http://dx.doi.org/10.1063/1.4807480DOI Listing
June 2013

Relations between deterministic and stochastic thresholds for disease extinction in continuous- and discrete-time infectious disease models.

Math Biosci 2013 May 1;243(1):99-108. Epub 2013 Mar 1.

Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, USA.

Thresholds for disease extinction provide essential information for control, eradication or management of diseases. Through relations between branching process theory and the corresponding deterministic model, it is shown that the deterministic and stochastic thresholds are in agreement for discrete-time and continuous-time infectious disease models with multiple infectious groups. Branching process theory can be applied in conjunction with the deterministic model to give additional information about disease extinction. These relations are illustrated, analytically and numerically, in two settings, a general stage-structured model and a vector-host model applied to West Nile virus in mosquitoes and birds.
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http://dx.doi.org/10.1016/j.mbs.2013.02.006DOI Listing
May 2013

The importance of contact network topology for the success of vaccination strategies.

J Theor Biol 2013 May 29;325:12-21. Epub 2013 Jan 29.

Department of Mathematics and Statistics, University of Victoria, Victoria BC, Canada V8W 3R4.

The effects of a number of vaccination strategies on the spread of an SIR type disease are numerically investigated for several common network topologies including random, scale-free, small world, and meta-random networks. These strategies, namely, prioritized, random, follow links and contact tracing, are compared across networks using extensive simulations with disease parameters relevant for viruses such as pandemic influenza H1N1/09. Two scenarios for a network SIR model are considered. First, a model with a given transmission rate is studied. Second, a model with a given initial growth rate is considered, because the initial growth rate is commonly used to impute the transmission rate from incidence curves and to predict the course of an epidemic. Since a vaccine may not be readily available for a new virus, the case of a delay in the start of vaccination is also considered in addition to the case of no delay. It is found that network topology can have a larger impact on the spread of the disease than the choice of vaccination strategy. Simulations also show that the network structure has a large effect on both the course of an epidemic and the determination of the transmission rate from the initial growth rate. The effect of delay in the vaccination start time varies tremendously with network topology. Results show that, without the knowledge of network topology, predictions on the peak and the final size of an epidemic cannot be made solely based on the initial exponential growth rate or transmission rate. This demonstrates the importance of understanding the topology of realistic contact networks when evaluating vaccination strategies.
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http://dx.doi.org/10.1016/j.jtbi.2013.01.006DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7094094PMC
May 2013

Extending the type reproduction number to infectious disease control targeting contacts between types.

J Math Biol 2013 Nov 2;67(5):1067-82. Epub 2012 Sep 2.

Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 3R4, Canada,

A new quantity called the target reproduction number is defined to measure control strategies for infectious diseases with multiple host types such as waterborne, vector-borne and zoonotic diseases. The target reproduction number includes as a special case and extends the type reproduction number to allow disease control targeting contacts between types. Relationships among the basic, type and target reproduction numbers are established. Examples of infectious disease models from the literature are given to illustrate the use of the target reproduction number.
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http://dx.doi.org/10.1007/s00285-012-0579-9DOI Listing
November 2013

Impact of heterogeneity on the dynamics of an SEIR epidemic model.

Math Biosci Eng 2012 Apr;9(2):393-411

Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., Canada.

An SEIR epidemic model with an arbitrarily distributed exposed stage is revisited to study the impact of heterogeneity on the spread of infectious diseases. The heterogeneity may come from age or behavior and disease stages, resulting in multi-group and multi-stage models, respectively. For each model, Lyapunov functionals are used to show that the basic reproduction number R0 gives a sharp threshold. If R0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable and the disease dies out from all groups or stages. If R0 > 1, then the disease persists in all groups or stages, and the endemic equilibrium is globally asymptotically stable.
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http://dx.doi.org/10.3934/mbe.2012.9.393DOI Listing
April 2012

Reproduction numbers for infections with free-living pathogens growing in the environment.

J Biol Dyn 2012 28;6:923-40. Epub 2012 Jun 28.

Department of Veterinary Integrative Biosciences, College of Veterinary Medicine and Biomedical Sciences, Texas A&M University, College Station, TX 77843, USA.

The basic reproduction number ℛ(0) for a compartmental disease model is often calculated by the next generation matrix (NGM) approach. When the interactions within and between disease compartments are interpreted differently, the NGM approach may lead to different ℛ(0) expressions. This is demonstrated by considering a susceptible-infectious-recovered-susceptible model with free-living pathogen (FLP) growing in the environment. Although the environment could play different roles in the disease transmission process, leading to different ℛ(0) expressions, there is a unique type reproduction number when control strategies are applied to the host population. All ℛ(0) expressions agree on the threshold value 1 and preserve their order of magnitude. However, using data for salmonellosis and cholera, it is shown that the estimated ℛ(0) values are substantially different. This study highlights the utility and limitations of reproduction numbers to accurately quantify the effects of control strategies for infections with FLPs growing in the environment.
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http://dx.doi.org/10.1080/17513758.2012.693206DOI Listing
December 2012

Cholera models with hyperinfectivity and temporary immunity.

Bull Math Biol 2012 Oct 3;74(10):2423-45. Epub 2012 Aug 3.

Department of Mathematics and Statistics, University of Victoria, BC, Canada.

A mathematical model for cholera is formulated that incorporates hyperinfectivity and temporary immunity using distributed delays. The basic reproduction number R(0) is defined and proved to give a sharp threshold that determines whether or not the disease dies out. The case of constant temporary immunity is further considered with two different infectivity kernels. Numerical simulations are carried out to show that when R(0) > 1, the unique endemic equilibrium can lose its stability and oscillations occur. Using cholera data from the literature, the quantitative effects of hyperinfectivity and temporary immunity on oscillations are investigated numerically.
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http://dx.doi.org/10.1007/s11538-012-9759-4DOI Listing
October 2012

Mathematical Modeling of Viral Zoonoses in Wildlife.

Nat Resour Model 2012 Feb 30;25(1):5-51. Epub 2011 Dec 30.

Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409 U.S.A.

Zoonoses are a worldwide public health concern, accounting for approximately 75% of human infectious diseases. In addition, zoonoses adversely affect agricultural production and wildlife. We review some mathematical models developed for the study of viral zoonoses in wildlife and identify areas where further modeling efforts are needed.
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http://dx.doi.org/10.1111/j.1939-7445.2011.00104.xDOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3358807PMC
February 2012

Effective degree household network disease model.

J Math Biol 2013 Jan 18;66(1-2):75-94. Epub 2012 Jan 18.

Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 3R4, Canada.

An ordinary differential equation (ODE) epidemiological model for the spread of a disease that confers immunity, such as influenza, is introduced incorporating both network topology and households. Since most individuals of a susceptible population are members of a household, including the household structure as an aspect of the contact network in the population is of significant interest. Epidemic curves derived from the model are compared with those from stochastic simulations, and shown to be in excellent agreement. Expressions for disease threshold parameters of the ODE model are derived analytically and interpreted in terms of the household structure. It is shown that the inclusion of households can slow down or speed up the disease dynamics, depending on the variance of the inter-household degree distribution. This model illustrates how households (clusters) can affect disease dynamics in a complicated way.
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http://dx.doi.org/10.1007/s00285-011-0502-9DOI Listing
January 2013

Vaccination against 2009 pandemic H1N1 in a population dynamical model of Vancouver, Canada: timing is everything.

BMC Public Health 2011 Dec 14;11:932. Epub 2011 Dec 14.

Division of Mathematical Modeling, University of British Columbia Centre for Disease Control, 655 West 12th Avenue, V5Z 4R4 Vancouver, British Columbia, Canada.

Background: Much remains unknown about the effect of timing and prioritization of vaccination against pandemic (pH1N1) 2009 virus on health outcomes. We adapted a city-level contact network model to study different campaigns on influenza morbidity and mortality.

Methods: We modeled different distribution strategies initiated between July and November 2009 using a compartmental epidemic model that includes age structure and transmission network dynamics. The model represents the Greater Vancouver Regional District, a major North American city and surrounding suburbs with a population of 2 million, and is parameterized using data from the British Columbia Ministry of Health, published studies, and expert opinion. Outcomes are expressed as the number of infections and deaths averted due to vaccination.

Results: The model output was consistent with provincial surveillance data. Assuming a basic reproduction number = 1.4, an 8-week vaccination campaign initiated 2 weeks before the epidemic onset reduced morbidity and mortality by 79-91% and 80-87%, respectively, compared to no vaccination. Prioritizing children and parents for vaccination may have reduced transmission compared to actual practice, but the mortality benefit of this strategy appears highly sensitive to campaign timing. Modeling the actual late October start date resulted in modest reductions in morbidity and mortality (13-25% and 16-20%, respectively) with little variation by prioritization scheme.

Conclusion: Delays in vaccine production due to technological or logistical barriers may reduce potential benefits of vaccination for pandemic influenza, and these temporal effects can outweigh any additional theoretical benefits from population targeting. Careful modeling may provide decision makers with estimates of these effects before the epidemic peak to guide production goals and inform policy. Integration of real-time surveillance data with mathematical models holds the promise of enabling public health planners to optimize the community benefits from proposed interventions before the pandemic peak.
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http://dx.doi.org/10.1186/1471-2458-11-932DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3280345PMC
December 2011

Global dynamics of cholera models with differential infectivity.

Math Biosci 2011 Dec 2;234(2):118-26. Epub 2011 Oct 2.

Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3R4.

A general compartmental model for cholera is formulated that incorporates two pathways of transmission, namely direct and indirect via contaminated water. Non-linear incidence, multiple stages of infection and multiple states of the pathogen are included, thus the model includes and extends cholera models in the literature. The model is analyzed by determining a basic reproduction number R0 and proving, by using Lyapunov functions and a graph-theoretic result based on Kirchhoff's Matrix Tree Theorem, that it determines a sharp threshold. If R0≤1, then cholera dies out; whereas if R0>1, then the disease tends to a unique endemic equilibrium. When input and death are neglected, the model is used to determine a final size equation or inequality, and simulations illustrate how assumptions on cholera transmission affect the final size of an epidemic.
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http://dx.doi.org/10.1016/j.mbs.2011.09.003DOI Listing
December 2011

Pandemic influenza: Modelling and public health perspectives.

Math Biosci Eng 2011 Jan;8(1):1-20

Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada.

We describe the application of mathematical models in the study of disease epidemics with particular focus on pandemic influenza. We outline the general mathematical approach and the complications arising from attempts to apply it for disease outbreak management in a real public health context.
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http://dx.doi.org/10.3934/mbe.2011.8.1DOI Listing
January 2011

The impact of maturation delay of mosquitoes on the transmission of West Nile virus.

Math Biosci 2010 Dec 7;228(2):119-26. Epub 2010 Sep 7.

Centre for Disease Modeling, Laboratory of Mathematical Parallel Systems, Department of Mathematics and Statistics, York University, Toronto, ON, Canada M3J 1P3.

We formulate and analyze a delay differential equation model for the transmission of West Nile virus between vector mosquitoes and avian hosts that incorporates maturation delay for mosquitoes. The maturation time from eggs to adult mosquitoes is sensitive to weather conditions, in particular the temperature, and the model allows us to investigate the impact of this maturation time on transmission dynamics of the virus among mosquitoes and birds. Numerical results of the model show that a combination of the maturation time and the vertical transmission of the virus in mosquitoes has substantial influence on the abundance and number of infection peaks of the infectious mosquitoes.
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http://dx.doi.org/10.1016/j.mbs.2010.08.010DOI Listing
December 2010

Impact of group mixing on disease dynamics.

Math Biosci 2010 Nov 27;228(1):71-7. Epub 2010 Aug 27.

Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W3R4, Canada.

A general mathematical model is proposed to study the impact of group mixing in a heterogeneous host population on the spread of a disease that confers temporary immunity upon recovery. The model contains general distribution functions that account for the probabilities that individuals remain in the recovered class after recovery. For this model, the basic reproduction number R₀ is identified. It is shown that if R₀ < 1, then the disease dies out in the sense that the disease free equilibrium is globally asymptotically stable; whereas if R₀ > 1, this equilibrium becomes unstable. In this latter case, depending on the distribution functions and the group mixing strengths, the disease either persists at a constant endemic level or exhibits sustained oscillatory behavior.
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http://dx.doi.org/10.1016/j.mbs.2010.08.008DOI Listing
November 2010

Control design for sustained oscillation in a two-gene regulatory network.

J Math Biol 2011 Apr 27;62(4):453-78. Epub 2010 Apr 27.

Department of Mathematics and Statistics, University of Victoria, PO BOX 3060, STN CSC, Victoria, BC, Canada.

Control strategies for gene regulatory networks have begun to be explored, both experimentally and theoretically, with implications for control of disease as well as for synthetic biology. Recent work has focussed on controls designed to achieve desired stationary states. Another useful objective, however, is the initiation of sustained oscillations in systems where oscillations are normally damped, or even not present. Alternatively, it may be desired to suppress (by damping) oscillations that naturally occur in an uncontrolled network. Here we address these questions in the context of piecewise-affine models of gene regulatory networks with affine controls that match the qualitative nature of the model. In the case of two genes with a single relevant protein concentration threshold per gene, we find that control of production terms (constant control) is effective in generating or suppressing sustained oscillations, while control of decay terms (linear control) is not effective. We derive an easily calculated condition to determine an effective constant control. As an example, we apply our analysis to a model of the carbon response network in Escherichia coli, reduced to the two genes that are essential in understanding its behavior.
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http://dx.doi.org/10.1007/s00285-010-0343-yDOI Listing
April 2011

Effective degree network disease models.

J Math Biol 2011 Feb 24;62(2):143-64. Epub 2010 Feb 24.

Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada.

An effective degree approach to modeling the spread of infectious diseases on a network is introduced and applied to a disease that confers no immunity (a Susceptible-Infectious-Susceptible model, abbreviated as SIS) and to a disease that confers permanent immunity (a Susceptible-Infectious-Recovered model, abbreviated as SIR). Each model is formulated as a large system of ordinary differential equations that keeps track of the number of susceptible and infectious neighbors of an individual. From numerical simulations, these effective degree models are found to be in excellent agreement with the corresponding stochastic processes of the network on a random graph, in that they capture the initial exponential growth rates, the endemic equilibrium of an invading disease for the SIS model, and the epidemic peak for the SIR model. For each of these effective degree models, a formula for the disease threshold condition is derived. The threshold parameter for the SIS model is shown to be larger than that derived from percolation theory for a model with the same disease and network parameters, and consequently a disease may be able to invade with lower transmission than predicted by percolation theory. For the SIR model, the threshold condition is equal to that predicted by percolation theory. Thus unlike the classical homogeneous mixing disease models, the SIS and SIR effective degree models have different disease threshold conditions.
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http://dx.doi.org/10.1007/s00285-010-0331-2DOI Listing
February 2011

A multigroup model for a heterosexually transmitted disease.

Math Biosci 2010 Apr 4;224(2):87-94. Epub 2010 Jan 4.

Department of Mathematics and Statistics, University of Victoria P.O. BOX 3060 STN CSC, Victoria, BC, Canada.

A multigroup model is considered for a disease transmitted heterosexually in which there is a core group that has higher sexual activity than other groups. In order to develop control strategies to eradicate the disease, type reproduction numbers are determined. A graph-theoretic interpretation of type reproduction numbers for a star network with a core group at the center is established. Using these general formulas for the star network, a two-group model consisting of a core and a non-core group is considered. If disease can persist in the core group in isolation, then the amount of increase of treatment or decrease of the contact rate of infectious males (or females) in the core group that is required to eradicate the disease is expressed in terms of model parameters. If disease can persist but not in either group in isolation, then the amount of reduction of the connection between the two groups needed to eradicate the disease is determined. These two cases are illustrated with parameters applicable to gonorrhea in the US.
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http://dx.doi.org/10.1016/j.mbs.2009.12.008DOI Listing
April 2010

A graph-theoretic method for the basic reproduction number in continuous time epidemiological models.

J Math Biol 2009 Oct 2;59(4):503-16. Epub 2008 Dec 2.

Department of Mathematical and Statistical Sciences, Centre for Mathematical Biology, University of Alberta, Edmonton, AB T6G 2G1, Canada.

In epidemiological models of infectious diseases the basic reproduction number 'R(0) is used as a threshold parameter to determine the threshold between disease extinction and outbreak. A graph-theoretic form of Gaussian elimination using digraph reduction is derived and an algorithm given for calculating the basic reproduction number in continuous time epidemiological models. Examples illustrate how this method can be applied to compartmental models of infectious diseases modelled by a system of ordinary differential equations. We also show with these examples how lower bounds for 'R(0) can be obtained from the digraphs in the reduction process.
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http://dx.doi.org/10.1007/s00285-008-0240-9DOI Listing
October 2009

The control of vector-borne disease epidemics.

J Theor Biol 2008 Nov 30;255(1):16-25. Epub 2008 Jul 30.

Department of Fisheries and Wildlife, Oregon State University, 104 Nash Hall, Corvallis, OR 97331-3803, USA.

The theoretical underpinning of our struggle with vector-borne disease, and still our strongest tool, remains the basic reproduction number, R(0), the measure of long term endemicity. Despite its widespread application, R(0) does not address the dynamics of epidemics in a model that has an endemic equilibrium. We use the concept of reactivity to derive a threshold index for epidemicity, E(0), which gives the maximum number of new infections produced by an infective individual at a disease free equilibrium. This index describes the transitory behavior of disease following a temporary perturbation in prevalence. We demonstrate that if the threshold for epidemicity is surpassed, then an epidemic peak can occur, that is, prevalence can increase further, even when the disease is not endemic and so dies out. The relative influence of parameters on E(0) and R(0) may differ and lead to different strategies for control. We apply this new threshold index for epidemicity to models of vector-borne disease because these models have a long history of mathematical analysis and application. We find that both the transmission efficiency from hosts to vectors and the vector-host ratio may have a stronger effect on epidemicity than endemicity. The duration of the extrinsic incubation period required by the pathogen to transform an infected vector to an infectious vector, however, may have a stronger effect on endemicity than epidemicity. We use the index E(0) to examine how vector behavior affects epidemicity. We find that parasite modified behavior, feeding bias by vectors for infected hosts, and heterogeneous host attractiveness contribute significantly to transitory epidemics. We anticipate that the epidemicity index will lead to a reevaluation of control strategies for vector-borne disease and be applicable to other disease transmission models.
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http://dx.doi.org/10.1016/j.jtbi.2008.07.033DOI Listing
November 2008

Oscillations in a patchy environment disease model.

Math Biosci 2008 Sep 16;215(1):1-10. Epub 2008 May 16.

Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V8N 3R4.

For a single patch SIRS model with a period of immunity of fixed length, recruitment-death demographics, disease related deaths and mass action incidence, the basic reproduction number R(0) is identified. It is shown that the disease-free equilibrium is globally asymptotically stable if R(0)<1. For R(0)>1, local stability of the endemic equilibrium and Hopf bifurcation analysis about this equilibrium are carried out. Moreover, a practical numerical approach to locate the bifurcation values for a characteristic equation with delay-dependent coefficients is provided. For a two patch SIRS model with travel, it is shown that there are several threshold quantities determining its dynamic behavior and that travel can reduce oscillations in both patches; travel may enhance oscillations in both patches; or travel can switch oscillations from one patch to another.
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http://dx.doi.org/10.1016/j.mbs.2008.05.001DOI Listing
September 2008

A model for influenza with vaccination and antiviral treatment.

J Theor Biol 2008 Jul 26;253(1):118-30. Epub 2008 Feb 26.

Department of Mathematics, University of Manitoba, Winnipeg, Man., Canada.

Compartmental models for influenza that include control by vaccination and antiviral treatment are formulated. Analytic expressions for the basic reproduction number, control reproduction number and the final size of the epidemic are derived for this general class of disease transmission models. Sensitivity and uncertainty analyses of the dependence of the control reproduction number on the parameters of the model give a comparison of the various intervention strategies. Numerical computations of the deterministic models are compared with those of recent stochastic simulation influenza models. Predictions of the deterministic compartmental models are in general agreement with those of the stochastic simulation models.
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http://dx.doi.org/10.1016/j.jtbi.2008.02.026DOI Listing
July 2008

Case fatality proportion.

Bull Math Biol 2008 Jan 18;70(1):118-33. Epub 2007 Aug 18.

Department of Mathematics and Statistics, University of Victoria, Victoria, Canada.

A precise definition of case fatality proportion for compartmental disease transmission models with disease induced mortality rate is given. This is applied in classical epidemic modeling frameworks to models with multiple infectious stages, with multi-groups, with spatial patches, and with age of infection. It is shown that the case fatality proportion is the sum over all stages of the product of the probability of dying from the disease at a given stage and the probability of surviving to that stage. The derived expressions for case fatality can be used to estimate the disease induced death rates from more readily available data.
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http://dx.doi.org/10.1007/s11538-007-9243-8DOI Listing
January 2008

Modeling diseases with latency and relapse.

Math Biosci Eng 2007 Apr;4(2):205-19

Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada, V8W 3P4.

A general mathematical model for a disease with an exposed (la tent) period and relapse is proposed. Such a model is appropriate for tuberculosis, including bovine tuberculosis in cattle and wildlife, and for herpes. For this model with a general probability of remaining in the exposed class, the basic reproduction number R(0) is identified and its threshold property is discussed. In particular, the disease-free equilibrium is proved to be globally asymptotically stable if R(0) < 1. If the probability of remaining in the exposed class is assumed to be negatively exponentially distributed, then R(0) = 1 is a sharp threshold between disease extinction and endemic disease. A delay differential equation system is obtained if the probability function is assumed to be a step-function. For this system, the endemic equilibrium is locally asymptotically stable if R(0) > 1, and the disease is shown to be uniformly persistent with the infective population size either approaching or oscillating about the endemic level. Numerical simulations (for parameters appropriate for bovine tuberculosis in cattle) with R(0) > 1 indicate that solutions tend to this endemic state.
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http://dx.doi.org/10.3934/mbe.2007.4.205DOI Listing
April 2007

A final size relation for epidemic models.

Math Biosci Eng 2007 Apr;4(2):159-75

Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada.

A final size relation is derived for a general class of epidemic mod els, including models with multiple susceptible classes. The derivation depends on an explicit formula for the basic reproduction number of a general class of disease transmission models, which is extended to calculate the basic reproduction number in models with vertical transmission. Applications are given to specific models for influenza and SARS.
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http://dx.doi.org/10.3934/mbe.2007.4.159DOI Listing
April 2007

Dispersal delays, predator-prey stability, and the paradox of enrichment.

Theor Popul Biol 2007 Jun 6;71(4):436-44. Epub 2007 Mar 6.

Biology Department, MS #34, Woods Hole Oceanographic Institution, Woods Hole, MA 02543-1049, USA.

It takes time for individuals to move from place to place. This travel time can be incorporated into metapopulation models via a delay in the interpatch migration term. Such a term has been shown to stabilize the positive equilibrium of the classical Lotka-Volterra predator-prey system with one species (either the predator or the prey) dispersing. We study a more realistic, Rosenzweig-MacArthur, model that includes a carrying capacity for the prey, and saturating functional response for the predator. We show that dispersal delays can stabilize the predator-prey equilibrium point despite the presence of a Type II functional response that is known to be destabilizing. We also show that dispersal delays reduce the amplitude of oscillations when the equilibrium is unstable, and therefore may help resolve the paradox of enrichment.
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http://dx.doi.org/10.1016/j.tpb.2007.02.002DOI Listing
June 2007

Periodicity in piecewise-linear switching networks with delay.

J Math Biol 2007 Aug 23;55(2):271-98. Epub 2007 Mar 23.

Department of Mathematics and Statistics, University of Victoria, BC, Canada.

Gene regulatory networks and neural networks can be modeled by piecewise-linear switching systems of differential equations, known as Glass networks. These biological networks exhibit delays in regulatory activity, for example, transcription, translation and spatial transport in gene networks, and transmission delays in neural networks. Such delays may be significant in determining their dynamical behavior. Here Glass networks with a discrete delay are introduced and analyzed. Fixed points away from thresholds are straightforward to identify, even in the presence of delays, so the focus of this work is on cyclic patterns of switching. Under a condition that ensures an unambiguous pattern of switching, it is shown by means of a fractional linear mapping that delayed Glass networks have a periodic orbit for all positive finite delays. Furthermore, an algorithm is presented to locate the periodic orbit for a given cycle, to determine whether the periodic orbit is locally asymptotically stable, and to check if it is unique. In addition, the complete dynamics of the two-dimensional delayed Glass network is provided: if there is a cycle of four orthants, then there exists a unique globally stable limit cycle; whereas if there is a black wall, then across the wall there exists a unique limit cycle that is globally stable with respect to the associated orthants. This behavior is in contrast to the non-delayed case, in which spiralling approach to fixed points on threshold boundaries can occur.
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http://dx.doi.org/10.1007/s00285-007-0084-8DOI Listing
August 2007

Impact of travel between patches for spatial spread of disease.

Bull Math Biol 2007 May 21;69(4):1355-75. Epub 2007 Feb 21.

Department of Applied Mathematics, National Chung Hsing University, Taichung, Taiwan, ROC.

A multipatch model is proposed to study the impact of travel on the spatial spread of disease between patches with different level of disease prevalence. The basic reproduction number for the ith patch in isolation is obtained along with the basic reproduction number of the system of patches, Re(0). Inequalities describing the relationship between these numbers are also given. For a two-patch model with one high prevalence patch and one low prevalence patch, results pertaining to the dependence of Re(0) on the travel rates between the two patches are obtained. For parameter values relevant for influenza, these results show that, while banning travel of infectives from the low to the high prevalence patch always contributes to disease control, banning travel of symptomatic travelers only from the high to the low prevalence patch could adversely affect the containment of the outbreak under certain ranges of parameter values. Moreover, banning all travel of infected individuals from the high to the low prevalence patch could result in the low prevalence patch becoming diseasefree, while the high prevalence patch becomes even more disease-prevalent, with the resulting number of infectives in this patch alone exceeding the combined number of infectives in both patches without border control. Under the set of parameter values used, our results demonstrate that if border control is properly implemented, then it could contribute to stopping the spatial spread of disease between patches.
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http://dx.doi.org/10.1007/s11538-006-9169-6DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7088731PMC
May 2007

Modeling relapse in infectious diseases.

Math Biosci 2007 May 7;207(1):89-103. Epub 2006 Oct 7.

Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3P4.

An integro-differential equation is proposed to model a general relapse phenomenon in infectious diseases including herpes. The basic reproduction number R(0) for the model is identified and the threshold property of R(0) established. For the case of a constant relapse period (giving a delay differential equation), this is achieved by conducting a linear stability analysis of the model, and employing the Lyapunov-Razumikhin technique and monotone dynamical systems theory for global results. Numerical simulations, with parameters relevant for herpes, are presented to complement the theoretical results, and no evidence of sustained oscillatory solutions is found.
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http://dx.doi.org/10.1016/j.mbs.2006.09.017DOI Listing
May 2007
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