For the zombie plague disease, I thought perhaps it should be treated in a similar manner as a death save (i.e. - a race between 3 successes or 3 fails to recover or die respectively). Unlike Walking Dead, this disease does not automatically kill the infected, but with the exhaustion mechanic together with CON damage, it is a very deadly disease. Use with caution against your players.
Between the Horde Zombie damage mechanic and the Plague Rot disease, your players will crap their pants whenever they come across zombies in the future. Even normal zombies will be that much more frightening because they won't be able to tell them apart.
Zombie Plague Rot
When a player is infected by plague zombie rot, they must make a CON saving throw 8 hours after the initial infection and each day thereafter. On a failure, the PC takes a permanent 1d6 CON damage (with HP maximum permanently lowered accordingly) and 2 levels of exhaustion. (The new save is made using the adjusted CON score). Due to the nature of the disease, exhaustion may not be recovered while the PC is infected (except magically). After 3 failed saves, the PC dies and will rise as a Plague Zombie in 1d12 hours.
After 3 successful saves, the PC is healed of the disease, but still suffers the effects of the CON and HP loss. A Lesser Restoration spell will not automatically cure the disease, but may be used to grant the PC one automatic save toward recovery and heal 1 level of exhaustion. Three castings of the Lesser Restoration spell may be used to cure the disease, but does not restore lost CON or HP maximum. A Greater Restoration spell will cure the disease completely and restore the permanently lost CON and HP maximum.
After infection, the wound location will have a red, nasty rash around the wound (even if HP healed). It will be apparent to the PC that some kind of infection is active.
Failed save 1: Red rash spreads from site of wound to other parts of the body. PC will have fever, sweats and chills. Exhaustion level 2 effects set in. PC is tired, listless, and moving slower as energy is sapped fighting the infection.
Failed save 2: PC will look horrible. Pale, rashy, sweating profusely, shivering constantly, and barely able to fight or perform simple tasks (Disadvantage on everything from Exhaustion level 4).
Failed save 3: The PC will look dead. Because he is. In a short time, he will look undead.
If the PC makes their 3rd successful save, they will begin to recover exhaustion levels normally. The fever will break and the rash will recede over the course of several days.