This system of differential equations corresponds to a compartmental model in epidemiology called SIRD:
\begin{aligned}&{\frac {dS}{dt}}=-{\frac {\beta IS}{N}}\\[6pt]&{\frac {dI}{dt}}={\frac {\beta IS}{N}}-\gamma I-\mu I\\[6pt]&{\frac {dR}{dt}}=\gamma I\\[6pt]&{\frac {dD}{dt}}=\mu I\end{aligned}
Note that:
\begin{aligned}&{\frac {dS}{dt}}+{\frac {dI}{dt}}+{\frac {dR}{dt}}+{\frac {dD}{dt}}=0\end{aligned}
Therefore:
\begin{aligned}&{\displaystyle S(t)+I(t)+R(t)+D(t)=N}\end{aligned}
If we set μ equal to zero, it simplifies to the SIR model:
\begin{aligned}&{\frac {dS}{dt}}=-\beta IS\\[6pt]&{\frac {dI}{dt}}=\beta IS-\gamma I\\[6pt]&{\frac {dR}{dt}}=\gamma I\end{aligned}
Note that:
\begin{aligned}&{\frac {dS}{dt}}+{\frac {dI}{dt}}+{\frac {dR}{dt}}=0\end{aligned}
Therefore:
\begin{aligned}&{\displaystyle S(t)+I(t)+R(t)=N}\end{aligned}
It can be further simplified to the SI model:
\begin{aligned}{\frac {dS}{dt}}&=-{\frac {\beta SI}{N}}+\gamma I\\[6pt]{\frac {dI}{dt}}&={\frac {\beta SI}{N}}-\gamma I\end{aligned}
Note that:
\begin{aligned}&{\frac {dS}{dt}}+{\frac {dI}{dt}}=0\end{aligned}
Therefore:
\begin{aligned}&{\displaystyle S(t)+I(t)=N}\end{aligned}
Its exact solution is a logistic function.
This is a logistic function:
\begin{aligned}&{\displaystyle f(x)={\frac {L}{1+e^{-k(x-x_{0})}}}}\end{aligned}
This is the exact solution:
\begin{aligned}&{\displaystyle I(t)={\frac {I_{\infty }}{1+Ve^{-\chi t}}}}\end{aligned}
\begin{aligned}&{\displaystyle I_{\infty }=(1-\gamma /\beta )N}\end{aligned}
\begin{aligned}&{\displaystyle \chi =\beta -\gamma }\end{aligned}
\begin{aligned}&{\displaystyle V=I_{\infty }/I_{0}-1}\end{aligned}
\begin{aligned}&{\displaystyle S(t)=N-I(t)}\end{aligned}
I conjectured that approximate solutions for the SIRD and SIR could involve a Gompertz function.
This is a formula for the Gompertz function:
\begin{aligned}&{\displaystyle f(t)=a\mathrm {e} ^{-b\mathrm {e} ^{-ct}}}\end{aligned}
And this is an alternative formula:
\begin{aligned}&{\displaystyle N(t)=N_{0}\exp(\ln(N_{\infty }/N_{0})(1-\exp(-bt)))}\end{aligned}
t is time
N_0 is the initial density of cells
N_inf is the plateau cell/population density
b is the initial rate of tumor growth
Specifically, I conjectured that:
For the SIRD, the approximations are:
- S as N minus a Gompertz function
- R as a Gompertz function
- D as a Gompertz function
For the SIR model:
- S as N minus a Gompertz function
- R as a Gompertz function
I would like to prove mathematically that the approximations are correct within some margin of error. Do I just add an O(t) to the function and plug in the system of equations and see what margin of error comes out?