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P van den Driessche

Publications and source records attributed to P van den Driessche.

32 records · Page 2Linked to original sources
PubMed1995

An SIS epidemic model with variable population size and a delay.

The SIS epidemiological model has births, natural deaths, disease-related deaths and a delay corresponding to the infectious period. The thresholds for persistence, equilibria and stability are determined. The persistence of the disease combined with the disease-related deaths can cause the population size to decrease to zero, to remain finite, or to grow exponentially with a smaller growth rate constant. For some parameter values, the endemic infective-fraction equilibrium is asymptotically stable, but for other parameter values, it is unstable and a surrounding periodic solution appears by Hopf bifurcation.

Disease Outbreaks↗
PubMed1993

A disease transmission model in a nonconstant population.

A general SIRS disease transmission model is formulated under assumptions that the size of the population varies, the incidence rate is nonlinear, and the recovered (removed) class may also be directly reinfected. For a class of incidence functions it is shown that the model has no periodic solutions. By contrast, for a particular incidence function, a combination of analytical and numerical techniques are used to show that (for some parameters) periodic solutions can arise through homoclinic loops or saddle connections and disappear through Hopf bifurcations.

Communicable Diseases↗
PubMed1993

An epidemiological model for HIV/AIDS with proportional recruitment.

A model for HIV transmission is formulated for a homosexual population of varying size, with recruitment into the susceptible class proportional to the active population size and with stages of progression to AIDS. Analysis of this model includes identifying the threshold that determines whether the disease dies out or proportions remain endemic and establishing criteria that determine whether the population size decays asymptotically exponentially to zero or grows asymptotically exponentially to infinity. In an analogous heterosexual model, the conservation of heterosexual contacts is shown to imply that this two-sex model reduces to the one-sex model.

Female↗
PubMed1993

Waves of extinction from sterile insect release.

Employing theoretical models, we show that the sterile insect release method (SIRM) can give rise to traveling waves of fertile insects. The velocity of these waves depends critically upon the density of sterile insects (and thus upon the rate at which sterile individuals are being released into the environment). When the sterile density is sufficiently low, the traveling wave advances, giving rise to an invasion. However, when the sterile density exceeds a critical threshold, the wave reverses direction, thereby retreating and giving rise to local extinction. Both analytical (regular perturbation) methods and numerical methods are used to determine wave velocities and wave profiles. This is the first time traveling wave solutions have been shown for the SIRM.

Animals↗
PubMed1992

A threshold result for an epidemiological model.

A threshold parameter R0 is identified for an SIRS epidemiological model which has nonlinear incidence and a distributed delay for transfer out of the removed class. For R0 less than 1, the disease free equilibrium is proved to be the global attractor for all solutions.

Communicable Diseases↗
PubMed1991

Some epidemiological models with nonlinear incidence.

Epidemiological models with nonlinear incidence rates can have very different dynamic behaviors than those with the usual bilinear incidence rate. The first model considered here includes vital dynamics and a disease process where susceptibles become exposed, then infectious, then removed with temporary immunity and then susceptible again. When the equilibria and stability are investigated, it is found that multiple equilibria exist for some parameter values and periodic solutions can arise by Hopf bifurcation from the larger endemic equilibrium. Many results analogous to those in the first model are obtained for the second model which has a delay in the removed class but no exposed class.

Epidemiology↗
PubMed1990

Analysis of a disease transmission model in a population with varying size.

An S----I----R----S epidemiological model with vital dynamics in a population of varying size is discussed. A complete global analysis is given which uses a new result to establish the nonexistence of periodic solutions. Results are discussed in terms of three explicit threshold parameters which respectively govern the increase of the total population, the existence and stability of an endemic proportion equilibrium and the growth of the infective population. These lead to two distinct concepts of disease eradication which involve the total number of infectives and their proportion in the population.

Epidemiology↗
PubMed1989

An epidemiological model with a delay and a nonlinear incidence rate.

An epidemiological model with both a time delay in the removed class and a nonlinear incidence rate is analysed to determine the equilibria and their stability. This model is for diseases where individuals are first susceptible, then infected, then removed with temporary immunity and then susceptible again when they lose their immunity. There are multiple equilibria for some parameter values, and, for certain of these, periodic solutions arise by Hopf bifurcation from the large nontrivial equilibrium state.

Epidemiologic Methods↗
PubMed1981

Stability analysis for models of diseases without immunity.

A cyclic, constant parameter epidemiological model is described for a closed population divided into susceptible, exposed and infectious classes. Distributed delays are introduced and the model is formulated as two coupled Volterra integral equations. The delays do not change the general nature of thresholds or asymptotic stability; in all cases considered the disease either dies out, or approaches an endemic steady state.

Bacterial Infections↗
PubMedDate not supplied

Implication of Ariaal sexual mixing on gonorrhea.

Recent research on sexual mixing in populations of sub-Saharan Africa raises the question as to whether STDs can persist in these populations without the presence of a core group. A mathematical model is constructed for the spread of gonorrhea among the Ariaal population of Northern Kenya. A formula for the basic reproduction number R(0) (the expected number of secondary infections caused by a single new infective introduced into a susceptible population) is determined for this population in the absence of a core group. Survey data taken in 2003 on sexual behavior from the Ariaal population are used in the model which is formulated for their age-set system including four subpopulations: single and married, female and male. Parameters derived from the data, and other information from sub-Saharan Africa are used to estimate R(0). Results indicate that, even with the elevating effect of the age-set system, the disease should die out since R(0) < 1. Thus, the persistence of gonorrhea in the population must be due to factors not included in the model, for example, a core group of commercial sex workers or concurrent partnerships.

Female↗
PubMedDate not supplied

Epidemiological models with varying population size and dose-dependent latent period.

Two epidemiological models in a variable-size population are considered. Both have a dose-dependent latent period that depends on the proportion of infectious individuals in the population. In the SEIS model, the proportion of infectious individuals tends to either zero or a stable endemic value. In the SEIRS model, for diseases that confer temporary immunity, periodicity can arise by a Hopf bifurcation.

Communicable Diseases↗
PubMedDate not supplied

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission.

A precise definition of the basic reproduction number, R0, is presented for a general compartmental disease transmission model based on a system of ordinary differential equations. It is shown that, if R0<1, then the disease free equilibrium is locally asymptotically stable; whereas if R0>1, then it is unstable. Thus, R0 is a threshold parameter for the model. An analysis of the local centre manifold yields a simple criterion for the existence and stability of super- and sub-threshold endemic equilibria for R0 near one. This criterion, together with the definition of R0, is illustrated by treatment, multigroup, staged progression, multistrain and vector-host models and can be applied to more complex models. The results are significant for disease control.

Animals↗