This paper presents a mathematical model to describe the spread of flavescence dorée, a disease caused by the bacterium , which is transmitted by the insect vector in grapevine crops. The key contribution of this work is the derivation of conditions under which positive periodic solutions exist. These conditions are based on the assumption that key factors such as recruitment rates, disease transmission, and vector infectivity vary periodically, thus reflecting seasonal changes. The existence of these periodic solutions is proven using the degree theory, and numerical examples are provided to support the theoretical findings. This model aims to enhance the understanding of the epidemiological dynamics of flavescence dorée and contribute to developing better control strategies to manage the disease in grapevines.
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