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The deterministic SIR model with initial susceptible count $S_0$ grows initially exactly when $\beta S_0>\gamma$. Its basic reproduction number is $\beta S_0/\gamma$ under unnormalized mass-action infection. If that ratio is at most one, decreasing $S$ prevents later infectious growth.

The infectious count in the SIR model obeys $\dot{I}=I(\beta S-\gamma )$. At the initial state, $\dot{I}(0)=\beta N-\gamma$, so initial epidemic growth requires the epidemic threshold for the closed SIR modelHere ${\mathcal{R}}_0$ is the basic reproduction number in this mass-action infection convention. Necessity also follows globally: $S(t)$ is nonincreasing, so if $\beta N\le \gamma$, then $\beta S(t)-\gamma \le 0$ for all time and $I(t)$ never increases. When $\beta =0$, growth is impossible.

At an endemic equilibrium, $I_{\ast }>0$, so $\beta S_{\ast }=\mu +\nu$. Substitution into the susceptible equation givesThese values are physically positive exactly when $R_0>1$. The Jacobian matrix at this equilibrium isBoth eigenvalues have negative real parts, so the endemic equilibrium is locally asymptotically stable. Its decay exponents areThe regime $\nu \gg \mu$ means recovery is rapid compared with demographic replacement. For fixed $R_0>1$ away from the threshold, the exponents form a complex conjugate pair with frequency approximately $\sqrt{\mu \nu (R_0-1)}$. HenceStrictly, oscillations require $\mu (\mu +\nu )(R_0-1)>{\mu }^2{R}_0^2/4$; $\nu \gg \mu$ alone does not ensure this uniformly for every $R_0$ arbitrarily close to one or arbitrarily large.

Vaccinating a fraction $v$ of newborns replaces the susceptible input by $\mu N(1-v)$. The effective basic reproduction number is $R_v=(1-v)R_0$, and the same calculation replaces $R_0$ by $R_v$. Provided $R_v>1$ and the oscillatory approximation remains applicable, vaccination lengthens the oscillation period, because $R_v-1$ is smaller. Extremely near eradication the exponents become real and there is no oscillation period to extrapolate indefinitely.

Adding the three equations of the SIRS model givessoAt a fixed point, $I(bS-c)=0$. The disease-free equilibrium isFor an endemic equilibrium $I^{\ast }>0$, one has $S^{\ast }=c/b$ and $a{R}^{\ast }=c{I}^{\ast }$. Conservation then givesIt exists with positive infected population exactly whenequivalently when the basic reproduction number $bN/c$ exceeds one.

Eliminating $R=N-S-I$ yieldsAt the endemic equilibrium its Jacobian matrix isIts trace and determinant areTherefore the endemic equilibrium is locally asymptotically stable. It is a stable node whenand a stable focus when the inequality is reversed. Near a node, perturbations are sums of two nonoscillatory decaying exponentials. Near a focus, they execute damped oscillations with decay rate $-T/2$ and angular frequency $\sqrt{4D-T^2}/2$.

At an endemic equilibrium, $I^{\ast }>0$, so $0=\dot{I}=I^{\ast }(\beta S^{\ast }-\gamma )$ forcesThe equation $\dot{R}=0$ gives $R^{\ast }=\gamma I^{\ast }/\alpha$. The conservation law $S^{\ast }+I^{\ast }+R^{\ast }=N$ therefore yieldsA positive endemic state exists exactly when $\beta N>\gamma$, equivalently when the basic reproduction number ${\mathcal{R}}_0=\beta N/\gamma$ exceeds one.

Dividing the $I$ equation by the $S$ equation givesThereforeis a first integral. Define the basic reproduction number by $R_0=\beta S(0)/\gamma$. At the removal peak, $S=S(0)/R_0$, and henceand, using $S+I+R=N$,

This is the SIR model with waning immunity. The mass-action interaction term $\beta IS$ transfers people from the susceptible compartment to the infected compartment, $\nu I$ transfers infectives to the recovered compartment, and $fR$ returns recovered people to susceptibility as immunity wanes. Adding the equations givesso the total population $N$ is conserved.

When $f=0$ and infection is rare, $S\simeq N$ andThus $I$ initially decays if $\beta N<\nu$. Equivalently, the basic reproduction number is ${\mathcal{R}}_0=\beta N/\nu <1$, so the epidemic invasion threshold is not crossed and no epidemic occurs.

Under frequency-dependent mass-action kinetics, if $I(t)$ of the $N(t)$ individuals are infectious, the force of infection isOne newly infectious person in an otherwise susceptible population transmits at total rate approximately $\beta$ for mean infectious duration $1/\gamma$. Hence the basic reproduction number iswhere $d_I$ is the infectious, rather than latent-plus-infectious, duration.