This is a simple 2 species (hosts and vectors) model, using the basic SIR framework as follows.
The model has the following compartments:
We assume that once a vector is infected, it stays infected until it dies. Therefore, recovered vectors are not included in the model.
The processes being modeled are:
We envision hosts to be humans and vectors to be e.g. mosquitos. We further assume the time scale for the model is short enough that we can ignore births and deaths for human hosts but not vectors. This is likely a decent approximation as long as our simulation time is not more than few years.
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 vector-borne transmission model.
\[\dot S_h = - b_1 S_h I_v + w R_h\] \[\dot I_h = b_1 S_h I_v - g I_h\] \[\dot R_h = g I_h - w R_h\] \[\dot S_v =b - b_2 S_v I_h - n S_v\] \[\dot I_v = b_2 S_v I_h - n I_v\]