Praise be to Allah.
It is not permissible to stay with a husband who is like this, because by neglecting the salaah (prayer) he is a kaafir, and it is not permissible for a Muslim woman to marry a kaafir or stay with him. Allaah says (interpretation of the meaning):
“… if you ascertain that they [women] are true believers, send them not back to the disbelievers, they are not lawful (wives) for the disbelievers nor are the disbelievers lawful (husbands) for them…” [al-Mumtahinah 60:10]
The marriage between you and him is null and void, and there can be no nikaah (marriage) between you unless Allaah guides him and he repents to Allaah and comes back to Islam. In that case the marriage will remain valid. The husband’s conduct is not correct, and it seems to me that he is suffering from some kind of (mental) illness, namely paranoia and waswaas (insinuating whispers of the Shaytaan) from which some people suffer with regard to their worship and their dealings with others. Nothing will get rid of this sickness but remembering Allaah (dhikr), seeking refuge in Him and putting one’s trust in Him to get rid of it. What matters is that for your part, you have to leave him and not stay with him, because he is a kaafir and you are a believer. With regard to your husband, we advise him to come back to his religion and to seek refuge with Allaah from the accursed Shaytaan, and to try to recite some beneficial du’aa’s with which to rid his heart of this waswaas. We ask Allaah to help him. And Allaah knows best.