A patchy model for Chikungunya-like diseases
Keywords:Patch, Chikungunya, Dengue, Movements, Disease free equilibrium, Basic reproduction number, Endemic equilibrium, Local and Global Stability.
We consider a n-patches model, to study the impact of human population movements between cities (patches) in the spread of Chikungunya or even Dengue diseases. In previous works, it was showed that the basic reproduction number can vary from place to place, but this result was obtained without taking into account human movements. We provide a theoretical study of the patchy model, and derive the basic reproduction number, which may depend on Human movement rates between the patches and on local population sizes. We show that the basic reproduction number is bounded by the maximum of local basic reproduction number. We also show that there exists a disease-free equilibrium (DFE) that is locally asymptotically stable whenever the basic reproduction number is less than 1. Under suitable assumptions, DFE is even globally asymptotically stable. We emphasize that Human movements are of particular importance to evaluate the spreading or not of Chikungunya or Dengue diseases, and thus movement rates have to be estimated very accurately. We confirm also the importance of the local basic reproduction numbers and show that even local field interventions can be of benefit to control/reduce the spread of a disease. A complete analytical study for a 2-patches model and several examples are provided to illustrate our conclusion.
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