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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.

For frequency-dependent transmission with homogeneous mixing, each member of the susceptible compartment experiences force of infection $\lambda =\beta I/N=\beta i$. Consequently the total flow into the infectious compartment is $\beta SI/N$ people per unit time. The constant $\beta$ combines the contact frequency and the chance that a relevant contact causes infection, and has units of inverse time. Holding the infectious compartment's fraction fixed holds each person's force of infection fixed when $N$ changes. An unnormalized mass-action infection convention instead writes the total flow as $bSI$, where $b=\beta /N$ has units of inverse person-time; these two coefficients must not be confused.

In the SIR model, $S,I,R$ count susceptible, infectious, and recovered immune individuals. The closed population has $S+I+R=N+1$. Under homogeneous mixing, use the paper's mass-action infection convention: each susceptible has infection hazard function $\beta I$, so the population infection rate is $\beta SI$. Each infectious individual recovers with hazard function $\gamma$, giving population recovery rate $\gamma I$ and mean infectious duration $1/\gamma$.

The arrows are one-way: recovered individuals are immune, and there are no demographic entries or exits. The transition intensities attached to the arrows are population flow rates, rather than individual hazard functions.

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.

Write $N=S+I$. Inthe term $N$ represents production of healthy new plants by both healthy and diseased plants at unit per-capita rate. The term $-NS$ is density-dependent loss of healthy plants through competition, and $-\beta IS$ is loss through mass-action infection. Inthe term $-NI$ is the same density-dependent loss for diseased plants, while $+\beta IS$ is the matching gain when healthy plants become diseased. Thus $\beta$ is the transmission-rate coefficient per healthy-diseased encounter.

Adding the equations cancels the infection terms and gives the logistic differential equationThe disease therefore changes how the population is divided between healthy and diseased plants but has no effect on the total-population dynamics: infected plants reproduce at the same rate, all offspring enter the healthy class, and infection causes no additional death in this model.

For frequency-dependent transmission, the closed SIR model is a system of ordinary differential equations for the fractions $s,i,r$:Here $\beta >0$ is the coefficient of frequency-dependent transmission, and $\gamma >0$ is the recovery rate, interpreted more generally as the removal rate. Both have units of inverse time. With a constant recovery rate, the associated individual time in the infectious compartment has an exponential distribution with mean $1/\gamma$; the SIR model does not require every person to spend exactly that long there. The fractions themselves have no units.^[ref]

For counts $S=Ns$, $I=Ni$, $R=Nr$, the same SIR model equations readThe factor $1/N$ keeps the same $\beta$ in both descriptions. Dropping it while keeping these units would fail dimensional analysis. Some existing SIR model variants use the unnormalized mass-action infection convention instead, including the SIR susceptible-recovered identity; their coefficient is $b=\beta /N$ in the notation used here.

Adding the SIR model equations gives $d(s+i+r)/dt=0$, a short proof of conservation of population. Nonnegative starting fractions stay nonnegative: the outflow from either the susceptible compartment or the infectious compartment vanishes when that group is empty, while $dr/dt\ge 0$. Thus the susceptible compartment can only shrink and the removed compartment can only grow.

For a concrete initial value problem, stipulateAt the start, the flow into the infectious compartment is $0.3\times 0.99\times 0.01=0.00297$ of the total per day, and the flow into the removed compartment is $0.1\times 0.01=0.00100$ of the total per day. Hence $di/dt=0.00197$ per day. If $N=1000$, these correspond to model rates of $2.97$ and $1.00$ people per day, and net growth $1.97$ people per day. A rate need not be an integer count of events on any particular day.

For this SIR model, the basic reproduction number isIt describes one introduced member of the infectious compartment in an otherwise wholly susceptible compartment. The instantaneous reproduction number at the actual starting state is $R_0{s}_0=2.97$, obtained by freezing the current conditions and the current susceptible compartment's fraction. More generally it is $R_{0}s(t)$ in this SIR model. The basic reproduction number is a ratio of two rates, so it has no units; $r_0$ is a different quantity, the initial removed compartment's fraction.

While $i>0$, its instantaneous relative growth isInitially this equals $0.3\times 0.99-0.1=0.197{\mathrm{day}}^{-1}$. Freezing $s\approx s_0$ gives the early exponential growth approximation $i(t)\approx 0.01e^{0.197t}$ with $t$ measured in days. It must fail as the susceptible compartment shrinks. An epidemic invasion threshold occurs at $R_0{s}_0=1$; if $i_0=0$, the deterministic SIR model stays at $i=0$ even above that threshold.

The equationscombine mass-action infection with an exponential density-dependent birth model. Susceptibles reproduce at per-capita rate $b{e}^{-aS}$, while infectives have no reproductive term and die at total per-capita rate $d+\delta$.