In this paper, we first introduce a system consists of two dissipative cavities in which there exists a two-level atom in each cavity. These two cavities are linked via a field-field interaction term. Due to the presence of dissipation in the cavities we are lead to use the Gardiner-Collett Hamiltonian for the considered model. In the continuation, the exact analytical solution of the wave function for the system has been obtained. Finally, we use the concurrence as a measure for the investigation of the degree of entanglement between atoms for different initial states of the atomic system. The results show that for two initial atomic states and for non-zero values of cavities decay rate, the downfall of entanglement is clearly seen. Whereas, in the absence of dissipation, an oscillatory behaviour of entanglement around a stationary value of entanglement is seen. However, the entanglement sudden death is seen for a separable atomic initial state and for non-zero values of cavities decay rate.

Keywords: : Dissipative systems, Jaynes-Cummings\'s model, Gardiner-Collett Hamiltonian, Quantum entanglement.

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