$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Example: A nonlinear ODE model from epidemiology
Spreading of a disease (e.g., a flu) can be modeled by
a \( 2\times 2 \) ODE system
$$
\begin{align*}
S^{\prime} &= -\beta SI\\
I^{\prime} &= \beta SI - \nu I
\end{align*}
$$
Here:
- \( S(t) \) is the number of people who can get ill (susceptibles)
- \( I(t) \) is the number of people who are ill (infected)
- Must know \( \beta >0 \) (danger of getting ill) and
\( \nu >0 \) (\( 1/\nu \): expected recovery time)
