Approximation of the Basic Reproduction Number R 0 for Vector-Borne Diseases with a Periodic Vector Population

Bacaër, Nicolas

doi:10.1007/s11538-006-9166-9

Cited by 234 publications

(189 citation statements)

References 24 publications

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“…The vectorial reproduction ratio of the non-autonomous model (2.1), denoted by R t 0 , can be computed using the technique in Bacaer (2009Bacaer ( , 2007, Bacaer and Guernaoui (2006), Bacaer and Abdurahman (2008), Bacaer et al (2011) and Wang and Zhao (2008). Using the notation in Bacaer and Guernaoui (2006), Bacaer and Ouifki (2007) and Wang and Zhao (2008) …”

Section: Trivial Solutionmentioning

confidence: 99%

Mathematical assessment of the role of temperature and rainfall on mosquito population dynamics

Abdelrazec

Gumel

2016

J. Math. Biol.

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A new stage-structured model for the population dynamics of the mosquito (a major vector for numerous vector-borne diseases), which takes the form of a deterministic system of non-autonomous nonlinear differential equations, is designed and used to study the effect of variability in temperature and rainfall on mosquito abundance in a community. Two functional forms of eggs oviposition rate, namely the Verhulst-Pearl logistic and Maynard-Smith-Slatkin functions, are used. Rigorous analysis of the autonomous version of the model shows that, for any of the oviposition functions considered, the trivial equilibrium of the model is locally-and globally-asymptotically stable if a certain vectorial threshold quantity is less than unity. Conditions for the existence and global asymptotic stability of the non-trivial equilibrium solutions of the model are also derived. The model is shown to undergo a Hopf bifurcation under certain conditions (and that increased density-dependent competition in larval mortality reduces the likelihood of such bifurcation). The analyses reveal that the Maynard-Smith-Slatkin oviposition function sustains more oscillations than the Verhulst-Pearl logistic function (hence, it is more suited, from ecological viewpoint, for modeling the egg oviposition process). The non-autonomous model is shown to have a globally-asymptotically stable trivial periodic solution, for each of the oviposition functions, when the associated reproduction threshold is less than unity. Furthermore, this model, in the absence of density-dependent mortality rate for larvae, has a unique and globally-asymptotically stable periodic solution under certain conditions. Numerical simulations of the non-autonomous model, using mosquito surveillance and weather data from the Peel region of Ontario, Canada, show a peak mosquito abundance for temperature and rainfall values in the range [20−25] • C and [15-35] mm, respectively. These ranges are recorded in the Peel region between July and August (hence, this study suggests that anti-mosquito control effects should be intensified during this period).

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Section: Trivial Solutionmentioning

confidence: 99%

Mathematical assessment of the role of temperature and rainfall on mosquito population dynamics

Abdelrazec

Gumel

2016

J. Math. Biol.

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“…Bacaër, 2007). Williams and Dye (1997) describe how this is related to changes in incidence and how it will affect the net reproductive ratio and its thresholds.…”

Section: Seasonality Of Sandfliesmentioning

confidence: 99%

Progress in the Mathematical Modelling of Visceral Leishmaniasis

Rock¹,

Quinnell²,

Medley³

et al. 2016

Advances in Parasitology

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The leishmaniases comprise a complex of diseases characterised by clinical outcomes that range from self-limiting to chronic, disfiguring and stigmatising, to life-threatening. Diagnostic methods, treatments, and vector and reservoir control options exist, but deciding the most effective interventions requires a quantitative understanding of the population level infection and disease dynamics. The effectiveness of any set of interventions has to be determined within the context of operational conditions, including economic and political commitment. Mathematical models are the best available tools for studying quantitative systems crossing disciplinary spheres (biology, medicine, economics) within environmental and societal constraints.In 2005, the World Health Assembly and government health ministers of India, Nepal, and Bangladesh signed a Memorandum of Understanding to eliminate the life threatening form of leishmaniasis, visceral leishmaniasis (VL), on the Indian subcontinent by 2015 through a combination of early case detection, improved treatments, and vector control. The elimination target is <1/10,000 population at the district or subdistrict level compared to the current 20/10,000 in the regions of highest transmission.Towards this goal, this chapter focuses on mathematical models of VL, and the biology driving those models, to enable realistic predictions of the * Corresponding author Email address: orin.courtenay@warwick.ac.uk (Orin Courtenay)Preprint submitted to Advances in Parasitology July 18, 2016 best combination of interventions. Several key issues will be discussed which have affected previous modelling of VL and the direction future modelling may take. Current understanding of the natural history of disease, immunity (and loss of immunity), as well as stages of infection and their durations are considered particularly for humans, but also for dogs. Asymptomatic and clinical infection are discussed in the context of their relative roles in Leishmania transmission, as well as key components of the parasite-sandfly-vector interaction and intervention strategies including diagnosis, treatment and vector control. Gaps in current biological knowledge and potential avenues to improve model structures and mathematical predictions are identified. Underpinning the marriage between biology and mathematical modelling, the content of this chapter represents the first step towards developing the next generation of models for VL.

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“…Formally defining and interpreting R 0 in the presence of periodic forcing of parameters requires considerable mathematical care [26,27]; however, what is important for our purposes here is that the threshold for disease spread is determined by the more easily defined basic reproduction number for the timeaveraged system [28]-i.e. the autonomous system in which b(t) is replaced by kbl-and this is what we shall always mean when referring to 'R 0 '.…”

Section: The Erlang-distributed Epidemic Modelsmentioning

confidence: 99%

Effects of the infectious period distribution on predicted transitions in childhood disease dynamics

Krylova

Earn

2013

J. R. Soc. Interface.

107

140

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The population dynamics of infectious diseases occasionally undergo rapid qualitative changes, such as transitions from annual to biennial cycles or to irregular dynamics. Previous work, based on the standard seasonally forced 'susceptible-exposed-infectious -removed' (SEIR) model has found that transitions in the dynamics of many childhood diseases result from bifurcations induced by slow changes in birth and vaccination rates. However, the standard SEIR formulation assumes that the stage durations (latent and infectious periods) are exponentially distributed, whereas real distributions are narrower and centred around the mean. Much recent work has indicated that realistically distributed stage durations strongly affect the dynamical structure of seasonally forced epidemic models. We investigate whether inferences drawn from previous analyses of transitions in patterns of measles dynamics are robust to the shapes of the stage duration distributions. As an illustrative example, we analyse measles dynamics in New York City from 1928 to 1972. We find that with a fixed mean infectious period in the susceptible -infectious -removed (SIR) model, the dynamical structure and predicted transitions vary substantially as a function of the shape of the infectious period distribution. By contrast, with fixed mean latent and infectious periods in the SEIR model, the shapes of the stage duration distributions have a less dramatic effect on model dynamical structure and predicted transitions. All these results can be understood more easily by considering the distribution of the disease generation time as opposed to the distributions of individual disease stages. Numerical bifurcation analysis reveals that for a given mean generation time the dynamics of the SIR and SEIR models for measles are nearly equivalent and are insensitive to the shapes of the disease stage distributions.

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Approximation of the Basic Reproduction Number R 0 for Vector-Borne Diseases with a Periodic Vector Population

Cited by 234 publications

References 24 publications

Mathematical assessment of the role of temperature and rainfall on mosquito population dynamics

Mathematical assessment of the role of temperature and rainfall on mosquito population dynamics

Progress in the Mathematical Modelling of Visceral Leishmaniasis

Effects of the infectious period distribution on predicted transitions in childhood disease dynamics

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