Drawing graphs of parametric equations

Question

the SIR model is an epidemiological model which sorts the whole population into three subclasses. x(t)(susceptibles), y(t)(infected), z(t)(recovered), with initial populations $x(0)= N_{1},y(0) = N_{2}, z(0)=N_{3}$ The solutions to the model are: \begin{equation} x=x_{0}u \end{equation} \begin{equation} y=-x_{0}+\frac{\gamma}{\beta}\ln u - \frac{C_{1}}{\beta} \end{equation} \begin{equation} z = -\frac{\gamma}{\beta}\ln u \end{equation} where $x_{0}, \gamma ,C_{1} \beta $ are arbitrary constants. And u is a function of time and it is equal to: \begin{equation} u = e^{-\frac{\beta}{\gamma}z} \end{equation} For example \begin{equation} u(0)=e^{-\frac{\beta}{\gamma}N_{3}} \end{equation} How would I graph the functions with time being the x axis and number of individuals represented on the y-axis?

Answer 1

I would suggest numerically solving the original ordinary system of differential equations (that gives rise to the parametrized solution of the SIR model). This is the easiest way to plot the variable over time.