No, the host cell is not destroyed in the lysogenic cycle. In this cycle, the virus integrates its genetic material into the host cell’s genome and lies dormant. This means that the viral DNA replicates along with the host cell’s DNA whenever the cell divides.
During the lysogenic cycle, the viral DNA can later be activated under certain conditions, leading to the lytic cycle where the host cell is ultimately destroyed. However, initially, the host cell remains healthy and functional, allowing it to continue to divide and propagate the viral DNA without any immediate harm. Only when the virus transitions from the lysogenic to the lytic cycle does the host cell become compromised and eventually destroyed.