2003, Can't connect to MySQL server on 'mysql' ([Errno 111] Connection refused

Question

I'm building a FastAPI app with MySQL, SQLAlchemy, Gunicorn and Docker. I keep getting the Error 111 Connection refused when I FIRST TIME run docker-compose build and docker-compose up in the service app container.

Following the previous instructions on SO I have

1. Corrected the host in the connection string to be the same as the service name (mysql)
2. Set my variables under environment in docker-compose.yml

It worked BUT the catch is only after I run my app container for the SECOND TIME ONWARD.

For some unknown reason I always got Error 111 on the first time running docker-compose up

Below is my docker-compose.yml

version: '3.8'
services:
  mysql:
      image: mysql:8.0
      restart: always
      environment:
        MYSQL_ROOT_PASSWORD: password
        MYSQL_PASSWORD: password
        MYSQL_DATABASE: database
  app:
    image: first-production-image
    build: 
      context: ${APP_HOME}
      dockerfile: Dockerfile
      target: dev
      args:
        - APP_HOME=${APP_HOME}
    environment:
      - PORT=8000
      - DB_CONN=${DB_CONN}
    volumes:
      - mysql_volume:/var/lib/mysql
    ports:
      - "80:8000"
    depends_on:
      - mysql
volumes:
  mysql_volume:

My .env file containing the connection string:

APP_HOME=/backend/app
DB_CONN="mysql+pymysql://root:password@mysql:3306/database"

and where I make the connection to the database with SQLAlchemy:

from sqlalchemy import create_engine

SQLALCHEMY_DATABASE_URL = os.getenv("DB_CONN")
engine = create_engine(SQLALCHEMY_DATABASE_URL)
models.Base.metadata.create_all(bind=engine)

Below is my Dockefile:

FROM python:3.10-slim-bullseye as base

ARG APP_HOME
WORKDIR ${APP_HOME}
ENV PYTHONDONTWRITEBYTECODE 1
ENV PYTHONUNBUFFERED 1
COPY ./requirements.txt ./
RUN pip install --no-cache-dir --upgrade pip -r ./requirements.txt
COPY . .

FROM base as dev
EXPOSE 8000
CMD exec gunicorn -w 4 -k uvicorn.workers.UvicornWorker main:app --bind 0.0.0.0:8000 --preload

FROM base as prod
CMD exec gunicorn --bind :$PORT --workers 4 --worker-class uvicorn.workers.UvicornWorker --threads 8 main:app

I don't understand why I have to restart my app container to get my app running properly. Could you enlighten me on this? Thank you!