SIR model using SymPy

I attempted to replicate some of the simple analytical results presented in the book, Mathematical Epidemiology by Brauer et al.

\[ \begin{align} \mathrm{d}S/\mathrm{d}t &= -\beta I S \\ \mathrm{d}I/\mathrm{d}t &= \beta I S - \gamma I\\ \end{align} \] The first part is simply to compute \(dI/dS\).

from sympy import *

R_0, b, g, dIdt, dSdt, S, I = symbols('R_0 b g dIdt dSdt S I')

dSdt = - b*S*I
dIdt = + b*S*I - g*I
dSdI = dIdt / dSdt #-(I*S*b - I*g)/(I*S*b)

# b <- R0*g
simplify(-(I*S*R_0*g - I*g)/(I*S*R_0*g)) 

-1 + 1/(R_0*S)

The second part is integrate the equation, \(\text{d}I/\text{d}S\)

S, I = symbols("S I", cls=Function)
b, g, t, R_0, S0, I0 = symbols("b g t R_0 S0 I0")

eq = Eq(I(t).diff(t), - S(t).diff(t) + (1/R_0)*(1/S(t))*S(t).diff(t))

integrate(eq, t)

Eq(I(t), -S(t) + log(S(t))/R_0)