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Herbert W. Hethcote developed mathematical models of infectious-disease transmission. His University of Iowa research history emphasizes the interaction between infectious and susceptible populations.^[ref]

His The Mathematics of Infectious Diseases by Herbert W. Hethcote brings together models, thresholds and their assumptions. In a constant-population SIS model, recovery returns individuals to susceptibility; the infectious fraction satisfies a scalar nonlinear equation whose equilibria reveal an invasion threshold. The example shows how a biological transition diagram becomes a differential equation and then a stability question.^[ref]

An SIS model divides a host population into a susceptible compartment and an infectious compartment. Recovery returns hosts directly to susceptibility, without lasting immunity or a separate recovered stage. Let $N$ be a constant population size and $I(t)$ the infectious fraction, so the infectious count is $NI$ and the susceptible count is $N(1-I)$.

For homogeneous frequency-dependent transmission at rate $\beta$ and recovery at rate $\gamma >0$, the count-level infection rate is $\beta (NI)N(1-I)/N$. Dividing the net count rate by $N$ givesThe disease-free state is $I=0$. For ${\mathcal{R}}_0=\beta /\gamma >1$, the positive endemic equilibrium is $I_{\ast }=1-1/{\mathcal{R}}_0$. The derivative of the right-hand side at $I_{\ast }$ is $-(\beta -\gamma )<0$, proving local stability. If ${\mathcal{R}}_0<1$, every positive fraction decreases toward zero; at ${\mathcal{R}}_0=1$, $\dot{I}=-\beta I^2$ gives algebraic decay. The scalar model illustrates a transmission/recovery threshold; heterogeneous contacts or different infectious-duration assumptions require other equations.^[ref]

The epidemiological abbreviation is distinct from the singular-isothermal-sphere abbreviation in galaxy dynamics.