Early Epidemic Growth and the Basic Reproduction Number

From SIR Transitions to Early Growth

At the start of an epidemic, most of the population may still be susceptible. An infectious person then has nearly the same opportunity to transmit as any other infectious person, and infections can be approximated as births in a branching population. This approximation links two useful quantities: the rate at which the number of infectious people initially grows, and the expected number of secondary infections caused by one infectious person.

Consider a fixed population of size \(N\), where \(N\) is a positive integer. On a probability space \((\Omega,\mathcal{F},\mathbb{P})\), let \((S_t,I_t,R_t)_{t\ge0}\) be a continuous-time Markov chain with state space \(S_N=\{(s,i,r)\in\mathbb{Z}_{\ge0}^3:s+i+r=N\}\). Use the natural filtration \(\mathcal{F}_t=\sigma((S_u,I_u,R_u):0\le u\le t)\). Assume infection and recovery are the only transitions, with rates \(\beta si/N\) and \(\gamma i\), respectively, where \(\beta\ge0\) and \(\gamma>0\). Thus an infection changes \((s,i,r)\) to \((s-1,i+1,r)\), while a recovery changes it to \((s,i-1,r+1)\).

When \(S_t\) is close to \(N\), an infectious person generates new infections at rate \(\beta S_t/N\), approximately \(\beta\), and recovers at rate \(\gamma\). The early epidemic can therefore be approximated by a linear birth-death process: each infectious individual gives rise to new infectious individuals at rate \(\beta\) and is removed at rate \(\gamma\). A birth adds one infectious person while the parent remains infectious; a death removes one infectious person. This is an approximation to the SIR process, not an exact description of it, because in the SIR model the susceptible count decreases.

Expected Secondary Infections

In the wholly susceptible approximation, suppose one infectious person makes infectious contacts according to a Poisson process of rate \(\beta\), independently of an infectious lifetime \(L\) with exponential distribution of rate \(\gamma\). The conditional expected number of contacts, given \(L\), is \(\beta L\). Consequently, the expected number of secondary infections is determined by the mean infectious duration.

Proof. Conditional on \(L\), the number of infectious contacts has mean \(\beta L\). Taking expectations gives \(R_0=\mathbb{E}[\beta L]=\beta\mathbb{E}[L]\). For an exponential lifetime of rate \(\gamma>0\), its density is \(\gamma e^{-\gamma t}\) for \(t\ge0\), so \(\mathbb{E}[L]=\int_0^\infty t\gamma e^{-\gamma t}\,dt=1/\gamma\), by integration by parts. Therefore \(R_0=\beta/\gamma\). When \(\beta=0\), the same calculation gives \(R_0=0\). \(\square\)

The expected number of infections is a lifetime total. By contrast, early growth compares the rate of new infections with the rate of removals at a given time. The birth-death approximation gives a precise relation between that net rate and \(R_0\).

Mean Growth in the Branching Approximation

Proof. First start the branching process with one individual, and write \(m(t)\) for its expected population at time \(t\). The branching property says that, conditional on the population at time \(t\), each individual starts an independent family whose expected size after a further time \(h\) is \(m(h)\). Taking expectations gives \(m(t+h)=m(t)m(h)\). The first moments are finite: the process is dominated in population size by the pure-birth process with birth rate \(\beta\) and no deaths, whose mean at each finite time is finite.

Over a short interval of length \(h\), starting from one individual, the probability of one birth is \(\beta h+o(h)\), the probability of one death is \(\gamma h+o(h)\), and the probability of two or more events is \(o(h)\). A birth changes the population from \(1\) to \(2\), while a death changes it from \(1\) to \(0\). Therefore \(m(h)=1+(\beta-\gamma)h+o(h)\). Using \(m(t+h)=m(t)m(h)\), we obtain the right derivative \(m'(t)=(\beta-\gamma)m(t)\), with \(m(0)=1\). Solving this differential equation gives \(m(t)=e^{(\beta-\gamma)t}\). With \(i_0\) independent initial lineages, expectations add, so \(\mathbb{E}[I_t]=i_0e^{(\beta-\gamma)t}\). Finally, \(\beta-\gamma=\gamma(\beta/\gamma-1)=\gamma(R_0-1)\). \(\square\)

The sign of \(r\) describes the direction of change in the mean under the approximation. If \(R_0>1\), then \(r>0\); if \(R_0=1\), then \(r=0\); and if \(R_0<1\), then \(r<0\). These statements concern the expected population in the branching approximation. They do not say that every individual epidemic follows a smooth exponential curve.

What the Finite-Population Model Says Exactly

The branching approximation is useful because it has a simple exponential mean, but the finite-population SIR model has an exact drift formula that shows where the approximation enters. For a state \((s,i,r)\), the infection event increases \(I_t\) by one and the recovery event decreases it by one. The generator applied to the function \(f(s,i,r)=i\) is therefore \(\beta si/N-\gamma i\). Since the SIR state space is finite, this yields the exact expectation identity below.

Proof. In state \((s,i,r)\), the infection transition has rate \(\beta si/N\) and changes \(f\) by \(+1\); the recovery transition has rate \(\gamma i\) and changes \(f\) by \(-1\). Thus the generator is \(\mathcal{L}f(s,i,r)=(\beta si/N)(1)+(\gamma i)(-1)=\beta si/N-\gamma i\). For a finite-state continuous-time Markov chain, the derivative of the expectation of a function equals the expectation of its generator applied to that function. Applying this identity to \(f(S_t,I_t,R_t)=I_t\) gives the stated equation. \(\square\)

If \(S_t\) is replaced by \(N\), the drift becomes approximately \((\beta-\gamma)I_t\), which motivates the early-growth equation. The replacement is not an identity: even if \(\mathbb{E}[S_t]\) is known, in general \(\mathbb{E}[S_tI_t]\) is not the product \(\mathbb{E}[S_t]\mathbb{E}[I_t]\). As susceptible people are infected, their depletion reduces the per-infective infection rate.

Common Pitfall: Subtracting Rates to Get \(R_0\)

A common mistake is to calculate the basic reproduction number as \(\beta-\gamma\). For example, with \(\beta=\tfrac{3}{10}\) and \(\gamma=\tfrac15\), this subtraction gives \(\tfrac{1}{10}\). That is the early growth rate per day, not \(R_0\): it has units of inverse time, whereas \(R_0\) is a dimensionless expected count. The correct calculation is \(R_0=\beta/\gamma=(\tfrac{3}{10})/(\tfrac15)=\tfrac32\). The two quantities are related by \(r=\gamma(R_0-1)\), not by \(r=R_0\).

Keep the model's scope in view. The formula \(R_0=\beta/\gamma\) assumes a wholly susceptible population, constant per-person infection rate \(\beta\), and exponential recovery rate \(\gamma>0\). When the susceptible fraction is approximately \(s/N\) and treated as fixed, the corresponding early reproduction ratio is \(\beta(s/N)/\gamma\), and the approximate growth rate is \(\beta(s/N)-\gamma\). The finite-population SIR drift remains the exact expression involving \(S_tI_t\); the branching approximation is appropriate only while susceptible depletion is small.