On a temporal model for the Chikungunya disease: modeling, theory and numerics.

Reunion Island faced two episodes of Chikungunya, a vector-borne disease, in 2005 and in 2006. The latter was of unprecedented magnitude: one third of the population was infected. Until the severe episode of 2006, our knowledge of Chikungunya was very limited. The principal aim of our study is to propose a model, including human and mosquito compartments, that is associated to the time course of the first epidemic of Chikungunya. By computing the basic reproduction number R(0), we show there exists a disease-free equilibrium that is locally asymptotically stable if the basic reproduction number is less than 1. Moreover, we give a necessary condition for global asymptotic stability of the disease-free equilibrium. Then, we propose a numerical scheme that is qualitatively stable and present several simulations as well as numerical estimates of the basic reproduction number for some cities of Reunion Island. For the episode of 2005, R(0) was less than one, which partly explains why no outbreak appeared. Using recent entomological results, we investigate links between the episode of 2005 and the outbreak of 2006. Finally, our work shows that R(0) varied from place to place on the island, indicating that quick and focused interventions, like the destruction of breeding sites, may be effective for controlling the disease.