I am working on a research paper and developing a modified SIRS model with the following equations.
S˙ = −αS − β(1 − ξ)SA − βξSP + εP + δR + µ(P + R) + µ*A
P˙ = αS − (ε + γ + µ)P
A˙ = γP + σR + β(1 − ξ)SA + βξSP + νRA − (ζ + µ*)A
R˙ = ζA − νRA − (δ + σ + µ)R.
I am looking for a way to find the values of S˙, P˙, A˙, R˙, for S˙+P˙+A˙+R˙=1, so that I can use them as initial conditions for a graph. I have values for each parameter but will change with each new "data set". For example, one of them will have parameters:
α=0.813, ε=0.69105, β=0.15, ξ=0.74, γ=0.12195, ζ=0.112, δ=0.1, ν=0.2, σ=0.7, µ=.009571, and µ*=0.000458.
How should I go about doing this for accurate values of each and fairly quick work through?