Endemic equilibria

An endemic equilibrium is a steady state in which infection remains present at a positive level while the modelled compartment sizes no longer change.

Important distinction. “Equilibrium” means the compartment values are constant. “Endemic” means the infected compartment is still positive. It does not mean that no individual infections or recoveries are occurring.

What does equilibrium mean?

Suppose a model has state variables \(S(t)\) and \(I(t)\). At equilibrium,

\[\frac{dS}{dt}=0,\qquad \frac{dI}{dt}=0.\]

This means that the total flow into each compartment exactly balances the total flow out.

There are two important possibilities:

\[I^*=0\qquad\text{disease-free equilibrium},\]

\[I^*>0\qquad\text{endemic equilibrium}.\]

Why can infection stay constant?

Consider the SIS model:

\[S\longrightarrow I\longrightarrow S.\]

New infections enter the infectious class at rate

\[\beta\frac{SI}{N},\]

while recoveries leave it at rate

\[\gamma I.\]

If these two rates are equal, then the number of infectious individuals does not change:

\[\beta\frac{SI}{N}=\gamma I.\]

Intuition. Infection can remain at a constant population level even though people are still becoming infected and recovering. The level stays constant because the two flows balance.

Deriving the SIS endemic equilibrium

For a positive endemic equilibrium, \(I^*>0\). Starting from

\[0=\beta\frac{S^*I^*}{N}-\gamma I^*,\]

factor out \(I^*\):

\[0=I^*\left(\beta\frac{S^*}{N}-\gamma\right).\]

Because \(I^*>0\), the bracket must be zero:

\[\beta\frac{S^*}{N}=\gamma.\]

Hence

\[S^*=N\frac{\gamma}{\beta}=\frac{N}{R_0},\]

because \(R_0=\beta/\gamma\). Since \(S^*+I^*=N\),

\[\boxed{I^*=N\left(1-\frac{1}{R_0}\right)}.\]

Why must \(R_0>1\)?

For an endemic equilibrium, we need \(I^*>0\). Therefore,

\[1-\frac{1}{R_0}>0,\]

which requires

\[\boxed{R_0>1}.\]

If \(R_0=1\), then

\[I^*=N(1-1)=0.\]

So \(R_0=1\) is the threshold, not a positive endemic equilibrium in this simple SIS model.

How the endemic level changes with \(R_0\)

The graph represents

\[\frac{I^*}{N}=1-\frac{1}{R_0}.\]

At \(R_0=1\), the endemic infectious fraction is zero. As \(R_0\) increases above 1, the positive endemic equilibrium level increases in this simple SIS model.

Endemic equilibrium does not mean a fixed set of infected people

At endemic equilibrium, the number infectious can remain constant while the individuals in the infectious class change continuously. Some recover and return to \(S\), while others become newly infected and enter \(I\).

Example. If 10 people recover during a short period and 10 susceptible people become infected during the same period, then the infectious population can remain unchanged even though 20 individual transitions have occurred.

Does every epidemic model have an endemic equilibrium?

No. Its existence depends on the model structure. The basic closed SIR model without births or loss of immunity does not have a positive endemic equilibrium: recovered individuals accumulate and susceptible individuals are not replenished, so infection eventually disappears.

Models such as SIS, SIRS, or models with demographic renewal can support endemic equilibria because susceptible individuals are continually replenished.

Key idea. An endemic equilibrium is a positive steady infection level created by a balance of flows. In the simple SIS model, it exists only when \(R_0>1\); \(R_0=1\) is the threshold where the positive endemic equilibrium collapses to zero.