### BIFURCATION OF NONTRIVIAL PERIODIC SOLUTIONS FOR LEISHMANIASIS DISEASE MODEL

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#### Abstract

We develop an impulsive model for zoonotic visceral leishmaniasis disease on a population of dogs. The disease infects a population D of dogs. We determine the basic reproduction number R0, which depends on the vectorial capacity C. Our analysis focuses on the values of C which give either stability or instability of the disease-free equilibrium (DFE). If the vectorial capacity C is less than some threshold, we obtain the stability of DFE, which means that the disease is eradicated for any period of culling dogs. Otherwise, for C greater than the threshold, the period of culling must be in a limited interval. For the particular case, when the period of culling is equal to the threshold, we observe bifurcation phenomena, which means that the disease is installed. In our study of the exponential stability of the DFE we use the fixed point method, and for the bifurcation we use the Lyapunov-Schmidt method.

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DOI: https://doi.org/10.22190/FUMI1705583B

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