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In this study, a delay differential equation model of gene expression for both retroviruses and normal cell is proposed to study the dynamics of functional gene products. The model is categorised into two sub-models to understand the characteristics of a cell by incorporating time delays in the processes of gene expression. The first model which is for retroviruses, involves time delay in replication, transcription, reverse transcription and translation processes taking place in the cell, while in the second model which is for normal cell, the time delay in transcription and translation processes are incorporated. A numerical solution is obtained using semi-temporal data set. The impact of time delays on temporal concentration profile of DNA, mRNA and proteins have been analysed which gives better insight into the normal cell as well as retroviruses. Further, sensitivity analysis has been performed for both models to study the behaviour of gene expression in the cell. The results obtained from such models can be useful for biomedical applications.