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An effective reproduction number describes transmission in the current population, accounting for available susceptibility and transmission conditions. The phrase is used with different time conventions, so a definition must specify whether it means an instantaneous reproduction number or a case reproduction number. The CDC uses a time-varying effective quantity $R_t$ to describe the direction of community transmission. ^[ref].

For the closed, homogeneous SIR model with constant $\beta ,\gamma$, the instantaneous convention givesThus the infectious compartment grows when $R_{\mathrm{eff}}>1$ and declines when $R_{\mathrm{eff}}<1$, provided $i>0$. At the SIR infection peak it equals one. This freezes the current susceptible fraction; it does not count a newly infected person's actual future secondary infections while $s$ continues to decrease. The distinction comes directly from the SIR model equations and the two time conventions above.

If the seed $v_0$ is positive, infection initially increases precisely whenIndeed, $v^{\prime }(0)=({\mathcal{R}}_0-1)v_0$. Susceptibility is nonincreasing, so if ${\mathcal{R}}_0\le 1$, the infectious count never starts an increasing phase. When ${\mathcal{R}}_0>1$, it grows until $u=1/{\mathcal{R}}_0$, the SIR infection peak, and then declines.

The basic reproduction number compares transmission and removal: one infectious individual generates infections at initial rate $\beta S_0$ and remains infectious for mean time $1/\gamma$. Their product $\beta S_0/\gamma$ is the expected number of secondary cases at that susceptible level. Above one, an infectious generation can replace itself with more cases. An exactly zero infectious seed remains zero regardless of this threshold.

For the closed SIR model with $\gamma >0$, the infectious compartment's fraction approaches $i_{\infty }=0$. One proof outline uses the bounded, increasing removed compartment's fraction: since $r(t)-r_0=\gamma {\int }_0^{t}i(u)du$, the total time-integrated $i$ is finite. The bounded derivative $di/dt$ from the SIR model equations prevents indefinitely recurring spikes, so $i(t)$ tends to zero.

For $s_0>0$, the final size relation for an epidemic follows by dividing the SIR model equations for $s$ and $r$ while $i>0$, giving $ds/dr=-(\beta /\gamma )s$. Integration then yieldsThis also follows from the SIR first integral and conservation of population. For the stipulated initial value problem, solveThe bisection method givesAbout $5.88\%$ of the original population remains in the susceptible compartment. The $94.12\%$ is the final removed compartment's fraction, including the initially $1\%$ in the infectious compartment. The newly acquired infection during the modelled period accounts for $s_0-s_{\infty }\approx 0.93120263520$ of the total population, not $1-s_{\infty }$. Dividing by $s_0$ instead gives the proportion of initially susceptible compartment members who acquire infection; its denominator must be stated.

At the SIR infection peak, the flow into the infectious compartment only balances its outflow; it does not stop. Further infection explains why the final susceptible compartment's fraction is well below $1/3$. Also $s_{\infty }=s_0{e}^{-R_0(r_{\infty }-r_0)}>0$ for $s_0>0$ and finite $R_0$: the closed SIR model does not require everybody to acquire infection before its infectious compartment empties in the long-time limit.

If $i_0>0$ and $s_0>\gamma /\beta$, the infectious compartment initially grows. The SIR model equations give its unique later maximum whenBefore this SIR infection peak, $\beta s-\gamma >0$; afterward it is negative because the susceptible compartment keeps shrinking. Substituting $s_{\ast }$ into the SIR first integral givesFor the stipulated initial value problem,Thus about $30.38\%$ of the modelled population belongs to the infectious compartment simultaneously at the SIR infection peak. This is a peak of the infectious compartment, not of incidence, whose flow is $\beta si$. If $s_0\le \gamma /\beta$, the maximum is already $i_0$ at the start and the interior SIR infection peak formula does not apply.

The numerical solution of a differential equation puts the SIR infection peak at about day $26.63258$ with $i\approx 0.303812682386$, agreeing with the SIR first integral to within $10^{-11}$. The accompanying generator checks nonnegative fractions and conservation of population without clipping or rescaling. For steps $0.4$, $0.2$ and $0.1$ day, the largest differences between successive numerical trajectories at shared times are approximately $1.086\times 10^{-7}$ and $6.985\times 10^{-9}$. Their ratio is $15.55$, close to the factor $16$ expected when a fourth-order Runge-Kutta method's step is halved.

The SIR model is a mathematical model of an epidemic that follows three groups: a susceptible compartment whose members can acquire infection, an infectious compartment whose members can pass it on, and a removed compartment whose members neither acquire nor pass on infection. The letters stand for susceptible, infectious and recovered or removed. Here we study the simplest closed, constant-parameter version, with homogeneous mixing and no births, deaths, migration or waning immunity. A person who leaves the infectious compartment never returns to the susceptible compartment in this version.^[ref]

The appeal of the SIR model is that one can explain a rise and fall in the infectious compartment without changing anyone's behavior: fewer opportunities for new infection eventually reverse the growth. The SIR first integral even determines the SIR infection peak without computing its time. These are useful deductions from explicit assumptions. The stipulated numbers below illustrate the mathematical model; they are neither estimates for an actual disease nor a forecast or medical advice.

Kermack and McKendrick's 1927 paper, "A contribution to the mathematical theory of epidemics", in Proceedings of the Royal Society A, is an early foundation for the SIR model. Its DOI is 10.1098/rspa.1927.0118, also recorded in the Royal Society's archive of contemporary referee reports.^[ref][ref]