Global Dynamics of a Periodic SEIRS Model with General Incidence Rate

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Abstract

We consider a family of periodic SEIRS epidemic models with a fairly general incidence rate of the form $S f (I)$ , and it is shown that the basic reproduction number determines the global dynamics of the models and it is a threshold parameter for persistence of disease. Numerical simulations are performed using a nonlinear incidence rate to estimate the basic reproduction number and illustrate our analytical findings.

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The epidemic threshold of vector-borne diseases with seasonality: the case of cutaneous leishmaniasis in Chichaoua, Morocco.

S Guernaoui, Nicolas Bacaër (2006)

Cutaneous leishmaniasis is a vector-borne disease transmitted to humans by sandflies. In this paper, we develop a mathematical model which takes into account the seasonality of the vector population and the distribution of the latent period from infection to symptoms in humans. Parameters are fitted to real data from the province of Chichaoua, Morocco. We also introduce a generalization of the definition of the basic reproduction number R (0) which is adapted to periodic environments. This R (0) is estimated numerically for the epidemic in Chichaoua; approximately 1.94. The model suggests that the epidemic could be stopped if the vector population were reduced by a factor approximately 3.76.

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Approximation of the basic reproduction number R0 for vector-borne diseases with a periodic vector population.

Nicolas Bacaër (2007)

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.