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

Institut de Recherche pour le Développement (IRD), 32 avenue Henri Varagnat, 93143 Bondy Cedex, France.
(Impact Factor: 1.39). 04/2007; 69(3):1067-91. DOI: 10.1007/s11538-006-9166-9
Source: PubMed

ABSTRACT

The main purpose of this paper is to give an approximate formula involving two terms for the basic reproduction number R(0) of a vector-borne disease when the vector population has small seasonal fluctuations of the form p(t) = p(0) (1+epsilon cos(omegat - phi)) with epsilon < 1. The first term is similar to the case of a constant vector population p but with p replaced by the average vector population p(0). The maximum correction due to the second term is (epsilon(2)/8)% and always tends to decrease R(0). The basic reproduction number R(0) is defined through the spectral radius of a linear integral operator. Four numerical methods for the computation of R(0) are compared using as example a model for the 2005/2006 chikungunya epidemic in La Réunion. The approximate formula and the numerical methods can be used for many other epidemic models with seasonality.

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• "In order to reflect the effect of demographic behavior of individuals, scholars have recognized that age-structured epidemic models are more realistic, since any disease prevention policy depends on the age structure of host population, and instantaneous death and infection rates depend on the age. Since the pioneer work of McKendrick [3], many authors have studied various age-structured epidemic models [2, 7–12]. "
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ABSTRACT: We formulate an age-structured SIS epidemic model with periodic parameters, which includes host population and vector population. The host population is described by two partial differential equations, and the vector population is described by a single ordinary differential equation. The existence problem for endemic periodic solutions is reduced to a fixed point problem of a nonlinear integral operator acting on locally integrable periodic functions. We obtain that if the spectral radius of the Fréchet derivative of the fixed point operator at zero is greater than one, there exists a unique endemic periodic solution, and we investigate the global attractiveness of disease-free steady state of the normalized system.
Journal of Applied Mathematics 02/2015; 2015:1-12. DOI:10.1155/2015/838312 · 0.72 Impact Factor
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• "In what follows, we use the definition of Bacaër and Guernaoui [4] (see also Bacaër [5]) and the general calculation method in Wang and Zhao [36] to evaluate the basic reproduction number R 0 for system (3.1). Then we analyze the threshold dynamics of system (3.1). "
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ABSTRACT: Based on the classical Ross-Macdonald model, in this paper we propose a periodic malaria model to incorporate the effects of temporal and spatial heterogeneity on disease transmission. The temporal heterogeneity is described by assuming that some model coefficients are time-periodic, while the spatial heterogeneity is modeled by using a multi-patch structure and assuming that individuals travel among patches. We calculate the basic reproduction number [Formula: see text] and show that either the disease-free periodic solution is globally asymptotically stable if [Formula: see text] or the positive periodic solution is globally asymptotically stable if [Formula: see text]. Numerical simulations are conducted to confirm the analytical results and explore the effect of travel control on the disease prevalence.
Discrete and Continuous Dynamical Systems - Series B 12/2014; 19(10). DOI:10.3934/dcdsb.2014.19.3133 · 0.77 Impact Factor
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• "Unlike linear stability analysis, Floquet theory has been somewhat neglected in ecology and epidemiology, especially with reference to realistic, spatially structured problems. However , seasonal fluctuations of environmental factors are crucial determinants of many ecological and epidemiological problems, notably including invasion conditions for competing species (Klausmeier 2008) or pathogenic organisms (Heesterbeek and Roberts 1995a, b; Bacaër 2007; Wang and Zhao 2008; Bacaër and Ait Dads 2012). In this framework , the quantity that actually controls pathogen invasion is the largest Floquet exponent ξ max of the linearized system describing epidemiological dynamics. "
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ABSTRACT: The transmission of waterborne pathogens is a complex process that is heavily linked to the spatial characteristics of the underlying environmental matrix as well as to the temporal variability of the relevant hydroclimatological drivers. In this work, we propose a time-varying, spatially explicit network model for the dynamics of waterborne diseases. Applying Floquet theory, which allows to extend results of local stability analysis to periodic dynamical systems, we find conditions for pathogen invasion and establishment in systems characterized by fluctuating environmental forcing, thus extending to time-varying contexts the generalized reproduction numbers recently obtained for spatially explicit epidemiology of waterborne disease. We show that temporal variability may have multifaceted effects on the invasion threshold, as it can either favor pathogen invasion or make it less likely. Moreover, environmental fluctuations characterized by distinctive geographical signatures can produce diversified, highly nontrivial effects on pathogen invasion. Our study is complemented by numerical simulations, which show that pathogen establishment is neither necessary nor sufficient for large epidemic outbreaks to occur in time-varying environments. Finally, we show that our framework can be used to reliably characterize the early geography of epidemic outbreaks triggered by fluctuating environmental conditions.
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