Bulletin of Mathematical Biology

, Volume 69, Issue 3, pp 1067–1091

# Approximation of the Basic Reproduction Number R0 for Vector-Borne Diseases with a Periodic Vector Population

• Nicolas Bacaër
Original Article

## 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+ε cos (ωt − φ)) with ε ≪ 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 (ε2/8)% 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.

## Keywords

Epidemics Basic reproduction number Seasonality

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