Once one person in the family is sick, the others soon follow.
As soon as one person gets sick, everyone in the family gets sick too.
すぐに = quickly/immediately -> as soon as, once
風邪の人 = a person with a cold -> a person who is sick (with a cold)
家族の中 = in the family
感染する = to infect
と = if
もっと直訳すると、以下になります。
If there is a sick person in a family, the family will quickly be infected.
文法的ですけど、もっと自然の言い方はこの二つの例文です。
「Follow」は「follow suit」から来て、同じ状態になるまた同じことをするという意味いです。