This is a simple SIR model with an additional environmental compartment.
This model has the following compartments:
The processes being modeled are:
The flow diagram and the set of ordinary differential equations (ODE) which are used to implement this model are as follows:
Flow diagram for the model with environmental transmission.
\[\dot S =b - b_d S I - b_e S E - nS\] \[\dot I = b_d S I + b_e S E - g I -nI\] \[\dot R = g I - nR\] \[\dot E = p I - cE\]
Often the letter E is used for the exposed stage, which would then be called a SEIR model. I’m using the letter E here to denote environmental stage of the pathogen. While this could also be abbreviated to SIER (or maybe SIRE), if you see the terminology SEIR model somewhere, the authors usually refer to a model with the additional exposed stage, not an environmental stage. There are no naming rules for either model compartments or parameters, so you will see all kinds of letters used. They should all be explicitly specified by the authors such that there is no ambiguity.