Is one obligated to maintain ties with a biological brother who is a disbeliever, who hates hearing the Quran, mocks prayer and fasting and those who observe them?

There is no obligation to maintain ties of kinship with or show affection towards a disbelieving relative, as the bond between him and his Muslim relative is severed. Allah the Almighty said regarding Noah and his son: "Indeed, he is not of your family; indeed, he is [of] unrighteous conduct." And concerning Abraham and his father: "But when it became clear to him that he was an enemy to Allah, he disassociated himself from him." And Allah the Almighty said: "You will not find a people who believe in Allah and the Last Day loving those who oppose Allah and His Messenger, even if they were their fathers or their sons or their brothers or their kindred. Those - He has inscribed [true] faith in their hearts."

However, it is permissible to maintain ties with them in permissible worldly matters and to offer them gifts, especially if there is hope that this might incline them towards Islam. Allah the Almighty said: "Allah does not forbid you from those who do not fight you because of religion and do not expel you from your homes - from being righteous toward them and acting justly toward them. Indeed, Allah loves those who act justly." And concerning disbelieving parents: "And accompany them in [this] world with 'ma'ruf' (kindness and good conduct)." Al-Bukhari permitted maintaining ties with a polytheist parent or sibling as long as one does not show them affection in matters of religion.