Conservation laws

A conservation law states that a specified total quantity remains constant while its components may change. Such relationships can simplify a biological model and provide an important check on its consistency.

A compartment example

Suppose a closed population is divided into susceptible, infectious and recovered individuals. If there are no births, deaths or migration, then

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

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

Individuals can move between the compartments, so \(S(t)\), \(I(t)\) and \(R(t)\) may all change. Their sum remains \(N\).

Differential form

Differentiating the conservation relationship gives

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

This expresses the same idea in terms of rates: gains in some compartments are exactly balanced by losses from others.

When a quantity is not conserved

Conservation depends on the model boundary and assumptions. If births, deaths or migration are included, population size may no longer be constant. Instead, its change must equal the net flow across the system boundary.

Key idea. A conservation law is not imposed merely for mathematical convenience. It must follow from the biological structure and assumptions of the model.