Faith is a confirmed belief. The mental part is submission and the one who submit is Muslim. The spiritual part is belief, the one who believes is a believer. In Quran God talk to us as believers mainly and we say about ourselves Muslims since God know our heart. Muslim means that you acquired the reasons that make you submit. It is a qualitative adjective while believing is quantitative. Verses 28:52 and 28:53 describes the Christians and Jews who converted to Islam they believed but they described themselves as Muslims as a humility to God.
28:52 Those to whom We sent the Book before this,- they do believe in this (revelation):
الَّذِينَ آتَيْنَاهُمُ الْكِتَابَ مِن قَبْلِهِ هُم بِهِ يُؤْمِنُونَ
28:53 And when it is recited to them, they say: "We believe therein, for it is the Truth from our Lord: indeed we have been Muslims (bowing to Allah's Will) from before this.
وَإِذَا يُتْلَى عَلَيْهِمْ قَالُوا آمَنَّا بِهِ إِنَّهُ الْحَقُّ مِن رَّبِّنَا إِنَّا كُنَّا مِن قَبْلِهِ مُسْلِمِينَ
The disciples of Jesus used the verb believe in verse 5:111 but described themselves as Muslims.
5:111 "And behold! I inspired the disciples to have faith in Me and Mine Messenger. they said, 'We have faith, and do thou bear witness that we bow to Allah as Muslims'".
وَإِذْ أَوْحَيْتُ إِلَى الْحَوَارِيِّينَ أَنْ آمِنُواْ بِي وَبِرَسُولِي قَالُوَاْ آمَنَّا وَاشْهَدْ بِأَنَّنَا مُسْلِمُونَ
The Israelites believed but they asked God to die as Muslims. Thus faith can go up and down but the baseline is submitting have to be maintained. That is why people need to strengthen their faith but try always to speak to God of their submission despite the hardship and the fluctuation of their faith.
7:126 "But thou dost wreak thy vengeance on us simply because we believed in the Signs of our Lord when they reached us! Our Lord! pour out on us patience and constancy, and take our souls unto thee as Muslims (who bow to thy will)!
وَمَا تَنقِمُ مِنَّا إِلاَّ أَنْ آمَنَّا بِآيَاتِ رَبِّنَا لَمَّا جَاءتْنَا رَبَّنَا أَفْرِغْ عَلَيْنَا صَبْرًا وَتَوَفَّنَا مُسْلِمِينَ
Submission to God message since Adam is what God wants us to say. The first to submit are the prophets. Islam is the last submission. As believers we differ in the strength of our faith. To pray to God and praise him even at hard time prove your submission and strengthen your faith.