Course: Self-Study

Textbook: Mathematical Models in Population Biology and Epidemiology

In the model considered here the population is divided into susceptibles (S), infectives (I), isolated or quarantined individuals (Q), and recovered individuals (R), for whom permanent immunity is assumed. Let N denote the total population i.e. $N=S+I+Q+R$, and let $A = S + I + R$ denote the active (nonisolated) individuals. The model takes the form:

$$\begin{align}

\frac{dS}{dt}&=\mu N-\mu S− \sigma S \frac{I}{A}\\

\frac{dI}{dt}&=-(\mu + \gamma)I+\sigma S \frac{I}{A}\\

\frac{dQ}{dt}&=-(\mu+\xi)Q+\gamma I\\

\frac{dR}{dt}&=−\mu R+\xi Q \\

A&=S+I+R

\end{align}$$

￼￼￼￼￼￼All newborns are assumed to be susceptible. $\mu$ is the per capita mortality rate, $\sigma$ is the per capita infection rate of an average susceptible individual provided everybody else is infected, $\gamma$ is the rate at which individuals leave the infective class, and $\xi$ is the rate at which individuals leave the isolated class; all are positive constants.

Question:Rescale the model by: $\tau=\sigma t$, $u=\frac{S}{A}$, $y=\frac{I}{A}$, $q=\frac{Q}{A}$, $z=\frac{R}{A}$. Rearrange your new model as follows:

$$\begin{align}

\dot{y}&=y(1-\nu -\theta -y-z+\theta y-(\nu +\zeta)q)\\

\dot{q}&=(1+q)(\theta y-(\nu+\zeta)q) \\

\dot{z}&=\zeta q-\nu z +z(\theta y-(\nu+\zeta)q).

\end{align}$$

Express the new parameters in terms of the old parameters. Check that all the new parameters and variables are dimensionless.

I would like some help on rescaling the original model because I have no idea on what it means in this case. I originally thought that I could just multiply by $\frac{1}{A}$ because that would 'scale' the system I have, but when doing so I cannot rearrange it to look like the second model due to the added parameters $\nu$, $\theta$, and $\zeta$. Any guidance would be appreciated.