On the biological interpretation of a definition for the parameter R 0 in periodic population models

Bacaër, Nicolas; Dads, El Hadi Ait

doi:10.1007/s00285-011-0479-4

Cited by 95 publications

(63 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Given an N -periodic system, a first approach of concept of basic reproduction number leads us to define this measure as the basic reproduction number of its ICAS, that is, R e 0 = ρ(F e (I − T e ) −1 ), see [8,9]. Another way is to define it using its N IS formulation, that is, [13].…”

Section: Preliminaries and Statement Of Workmentioning

confidence: 99%

“…Then, the threshold value for v such that the model of vaccination becomes asymptotically stable is 1 − 1 R e 0 , see [9].…”

Section: Preliminaries and Statement Of Workmentioning

confidence: 99%

“…In this case, in the mathematical model appears periodic functions or matrices which hamper the analysis of the problem. The basic reproduction number associated with a model with periodic coefficients of period N has been defined by different authors, see [8,9,10,11,12,13]. In particular, in [8,9] this concept is introduced as the spectral radius of a matrix which contains all the information for a period N and in [13] the authors define N reproduction numbers associated with the system.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A study on vaccination models for a seasonal epidemic process

Cantó¹,

Coll²,

Sánchez³

2014

Applied Mathematics and Computation

View full text Add to dashboard Cite

ElsevierCantó Colomina, B.; Coll, C.; Sánchez, E. (2014). A study on vaccination models for a seasonal epidemic process. Applied AbstractIn this paper seasonal epidemiological processes are considered and a strategy of periodic vaccination is proposed. The invariant formulations associated with an N -periodic system and the reproduction numbers associated with them are considered. New measures to study the stability of the system are introduced. Moreover, these new reproduction numbers help us to establish conditions on the periodic vaccination rates in the vaccination program. Finally, an SIR model is showed and a comparison between the results obtained using constant or periodic vaccination program is analyzed.

show abstract

Section: Preliminaries and Statement Of Workmentioning

confidence: 99%

“…Then, the threshold value for v such that the model of vaccination becomes asymptotically stable is 1 − 1 R e 0 , see [9].…”

Section: Preliminaries and Statement Of Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A study on vaccination models for a seasonal epidemic process

Cantó¹,

Coll²,

Sánchez³

2014

Applied Mathematics and Computation

View full text Add to dashboard Cite

show abstract

“…Another example in this area is the seeming clash between the R 0 -concepts for periodic environments introduced by Bacaër and Guernaoui (2006) and by Cushing and Ackleh (2011) (see also Bacaër and Ait Dads, 2012). Basically Bacaër and Guernaoui consider ordinary births, with and extended birth state description also including the phase of the cycle at which individuals are born, whereas Cushing and Ackleh consider generalized individuals of their own making, the construction of which, mathematically interesting though it may be, were it only for its unexpectedness, provides little in the way of inspiration for biological thought.…”

Section: How Does All This Help Mathematical Biology?mentioning

confidence: 99%

On the concept of individual in ecology and evolution

Metz

2012

J. Math. Biol.

View full text Add to dashboard Cite

Part of the art of theory building is to construct effective basic concepts, with a large reach and yet powerful as tools for getting at conclusions. The most basic concept of population biology is that of individual. An appropriately reengineered form of this concept has become the basis for the theories of structured populations and adaptive dynamics. By appropriately delimiting individuals, followed by defining their states as well as their environment, it become possible to construct the general population equations that were introduced and studied by Odo Diekmann and his collaborators. In this essay I argue for taking the properties that led to these successes as the defining characteristics of the concept of individual, delegating the properties classically invoked by philosophers to the secondary role of possible empirical indicators for the presence of those characteristics. The essay starts with putting in place as rule for effective concept engineering that one should go for relations that can be used as basis for deductive structure building rather than for perceived ontological essence. By analysing how we want to use it in the mathematical arguments I then build up a concept of individual, first for use in population dynamical considerations and then for use in evolutionary ones. These two concepts do not coincide, and neither do they on all occasions agree with common intuition-based usage.

show abstract

“…The works of R 0 by Diekmann, Heesterbeek, and Metz [9] and by van den Driessche and Watmough [32] have found numerous applications in the study of various infectious disease models. There are also quite a few investigations on R 0 for population models in a periodic environment; see, e.g., Bacaër and Guernaoui [4], Wang and Zhao [35], Thieme [31], Bacaër and Ait Dads [2,3], Inaba [13], and references therein. Recently, the theory of the basic reproduction ratio has been developed by Zhao [44] for periodic and time-delayed population models with compartmental structure.…”

Section: Basic Reproduction Ratiomentioning

confidence: 99%

Dynamics of a Time-Delayed Lyme Disease Model with Seasonality

Wang¹,

Zhao²

2017

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

Abstract. In this paper, we propose a time-delayed Lyme disease model incorporating the climate factors. We obtain the existence of a disease-free periodic solution under some additional conditions. Then we introduce the basic reproduction ratio R0 and show that under the same set of conditions, R0 serves as a threshold parameter in determining the global dynamics of the model; that is, the disease-free periodic solution is globally attractive if R0 < 1; the system is uniformly persistent and admits a positive periodic solution if R0 > 1. Numerically, we study the Lyme disease transmission in Long Point, Ontario, Canada. Our simulation results indicate that Lyme disease is endemic in this region if no further intervention is taken. We find that Lyme disease will die out in this area if we decrease the recruitment rate of larvae, which implies that we can control the disease by preventing tick eggs from hatching into larvae.

show abstract

On the biological interpretation of a definition for the parameter R 0 in periodic population models

Cited by 95 publications

References 35 publications

A study on vaccination models for a seasonal epidemic process

A study on vaccination models for a seasonal epidemic process

On the concept of individual in ecology and evolution

Dynamics of a Time-Delayed Lyme Disease Model with Seasonality

Contact Info

Product

Resources

About