Conservation laws

A conservation law describes a quantity that remains unchanged even though it may move between different parts of a system.

The simplest way to understand this is to follow people moving between compartments in an epidemic model.

A simple population example

Suppose we have a closed population of exactly 1000 people. We divide the population into three groups:

\[ S(t)=\text{number of susceptible people}, \] \[ I(t)=\text{number of infectious people}, \] \[ R(t)=\text{number of recovered people}. \]

At the beginning, suppose

\[ S(0)=990,\qquad I(0)=10,\qquad R(0)=0. \]

The total population is therefore

\[ 990+10+0=1000. \]

Later, the numbers might be

\[ S(t)=900,\qquad I(t)=70,\qquad R(t)=30. \]

Now the total is still

\[ 900+70+30=1000. \]

What has happened? The individual compartment sizes have changed, but the total population has not. People have moved from one compartment to another; they have not been created or lost from the model.

Movement between compartments

Susceptible \(S\) → Infectious \(I\) → Recovered \(R\)

When one susceptible person becomes infected, \(S\) decreases by one and \(I\) increases by one. The total number of people is unchanged.

Similarly, when one infectious person recovers, \(I\) decreases by one and \(R\) increases by one. Again, the total is unchanged.

Therefore, at every time \(t\),

\[ \boxed{S(t)+I(t)+R(t)=N}, \]

where \(N=1000\) is the constant total population.

The assumptions behind conservation

The equation \(S+I+R=N\) is valid here because of the assumptions defining our model.

We assume that during the period considered:

there are no births;
there are no deaths;
nobody enters the population;
nobody leaves the population.

These assumptions make the population a closed system. Movement occurs inside the system, between \(S\), \(I\) and \(R\), but there is no flow across its boundary.

The same idea using rates

Because

\[ S(t)+I(t)+R(t)=N \]

and \(N\) is constant, differentiating with respect to time gives

\[ \frac{dS}{dt}+\frac{dI}{dt}+\frac{dR}{dt}=0. \]

The zero has a simple meaning: when all changes are added together, the gains and losses cancel.

An infection is a loss from \(S\) but an equal gain to \(I\). A recovery is a loss from \(I\) but an equal gain to \(R\). These are internal transfers, so they do not change the total population.

When the total is not conserved

Suppose births and deaths are now allowed. People can enter or leave the population, so the total population need not remain constant.

If \(B(t)\) is the birth rate and \(D(t)\) is the death rate, then a simple population balance is

\[ \frac{dN}{dt}=B(t)-D(t). \]

If migration is also included, immigration contributes an inflow and emigration contributes an outflow.

Thus conservation depends on what is inside the model and on the assumptions made about flows across the system boundary.

Key idea. Conservation means that movement within a system changes its parts but not its total. A total remains constant only when the model contains no net flow into or out of that total.