Transition Rates in a Stochastic SIR Epidemic

From Population Counts to Event Rates

A stochastic SIR model describes random changes in the numbers of susceptible, infectious, and removed people. Its state records those counts, while its transition rates specify how quickly infection and recovery events occur. Once the rates are known at the current state, competing exponential waiting times give the probability that each kind of event happens next. This tutorial focuses on that state-by-state calculation; it does not require an approximation for the early epidemic.

Consider a closed population of fixed size \(N\). Each person is in exactly one of three classes: susceptible, infectious, or removed. Removed people have recovered or otherwise left the infectious population and do not become susceptible again. On a probability space \((\Omega,\mathcal{F},\mathbb{P})\), let \(X_t=(S_t,I_t,R_t)\), \(t\ge0\), be a continuous-time Markov chain with state space \(D_N=\{(s,i,r):s,i,r\text{ are nonnegative integers and }s+i+r=N\}\). We use the natural filtration \(\mathcal{F}_t=\sigma(X_u:0\le u\le t)\). The model assumes that, conditional on the current state, the rates of the next infection and recovery depend only on that state.

For the infection mechanism, suppose each infectious person makes contacts at rate \(c\), each contact transmits infection with probability \(p\), and a contacted person is chosen uniformly from the \(N\) people. Define \(\beta=cp\). With \(i\) infectious people, the total rate of contacts that successfully reach a susceptible person is \(\beta i(s/N)\). For recovery, assume that each infectious person recovers at rate \(\gamma\), independently of the others. The total recovery rate is therefore \(\gamma i\).

The factor \(s/N\) is the probability that a uniformly selected contact is susceptible. Thus infection requires both \(s>0\) and \(i>0\). Recovery requires \(i>0\). The count \(r\) does not directly enter either rate, but it is part of the state and enforces the fixed-population constraint. Another common model uses an infection rate proportional to \(si\) without division by \(N\); that is a different parameterization, and its rate parameter should not be silently substituted for \(\beta\) here.

The Generator and Its Row Sums

For a continuous-time Markov chain, the generator records the instantaneous transition rates. Write \(q_{xy}\) for the generator entry from state \(x\) to state \(y\), and \(Q=(q_{xy})\). For distinct states \(x,y\), the entry is the rate of a direct one-event transition from \(x\) to \(y\). The diagonal entry is minus the total rate of leaving \(x\). For the SIR model, set \(a(s,i,r)=\lambda(s,i,r)+\mu(s,i,r)\). Then \(q_{x,x}=-a(x)\), and the only possible nonzero off-diagonal entries are \(q_{x,(s-1,i+1,r)}=\lambda(x)\) and \(q_{x,(s,i-1,r+1)}=\mu(x)\), when those transitions are possible.

Proof. Fix \(x=(s,i,r)\). The only possible off-diagonal transitions are infection and recovery, so their rates sum to \(\lambda(x)+\mu(x)=a(x)\); a transition whose rate is zero contributes nothing. The diagonal is defined to be \(-a(x)\). Therefore the row sum is \(-a(x)+\lambda(x)+\mu(x)=0\). If \(a(x)=0\), nonnegativity of both rates implies \(\lambda(x)=\mu(x)=0\), so the diagonal and every off-diagonal entry are zero. \(\square\)

This rule is useful both for constructing the generator and for checking arithmetic. A negative diagonal is not an additional event: it balances the rates of leaving the state. A state with no possible infection or recovery is absorbing, and its generator row is all zero.

Which Event Happens Next?

At a fixed state \(x\), write \(\lambda=\lambda(x)\), \(\mu=\mu(x)\), and \(a=\lambda+\mu\). Under the continuous-time Markov model, the infection and recovery clocks are exponential with their respective rates. When a rate is zero, its clock never rings. The next-event time is the first ring of the clocks that have positive rates. The following calculation gives both the waiting-time distribution and the event type probabilities.

Proof. Suppose first that \(a>0\). Let \(W_I\) and \(W_R\) be the infection and recovery clock times. For \(u\ge0\), independence and the exponential survival probabilities give \(\mathbb{P}(W>u)=\mathbb{P}(W_I>u,W_R>u)=e^{-\lambda u}e^{-\mu u}=e^{-au}\). A zero-rate clock has survival probability one, so the same calculation covers \(\lambda=0\) or \(\mu=0\). For \(t\ge0\), the density that infection is the first event at time \(t\) is the infection-clock density \(\lambda e^{-\lambda t}\) multiplied by the probability that the recovery clock has not rung, \(e^{-\mu t}\). It is therefore \(\lambda e^{-at}\). Integrating over all \(t\ge0\) gives \(\int_0^\infty\lambda e^{-at}\,dt=\lambda/a\); integrating from \(0\) to \(h\) gives \(\lambda(1-e^{-ah})/a\). Replacing \(\lambda\) by \(\mu\) gives the recovery formulas. These two event probabilities sum to one because \((\lambda+\mu)/a=1\). If \(a=0\), nonnegativity implies both rates are zero, so neither clock rings. \(\square\)

In particular, the chance of any event in the next \(h\) time units is \(1-e^{-ah}\) when \(a>0\), and the remaining probability \(e^{-ah}\) is the chance that the state has not changed. Conditional on an event occurring within that interval, the event-type probabilities are still \(\lambda/a\) and \(\mu/a\): dividing each first-event-within-\(h\) probability by \(1-e^{-ah}\) gives those ratios when \(h>0\). If \(h=0\), the conditioning event has probability zero, so that conditional statement is not defined.

Common Mistakes and Model Checks

A frequent error is to use \(\beta i\) as the infection rate, counting contacts made by infectious people without accounting for the chance that a contact reaches a susceptible person. In the first example, that incorrect calculation would give \(\tfrac12\cdot2=1\), rather than the correct rate \(\tfrac12\cdot2\cdot(7/10)=\tfrac7{10}\). It would then produce the incorrect total rate \(1+\tfrac12=\tfrac32\) and the incorrect infection-next probability \(1/(\tfrac32)=\tfrac23\). The correction is to include the susceptible fraction \(s/N\) under the stated uniform-contact mechanism.

Other checks help catch errors before using the probabilities:

• Confirm \(s+i+r=N\) in every state and ensure each event destination preserves that sum.
• Calculate both event rates separately before adding them to obtain the total rate.
• Check that event-type probabilities add to one when the total rate is positive.
• Check the generator row by adding its negative diagonal to all off-diagonal rates.
• Handle zero rates directly: when \(a=0\), ratios with denominator \(a\) are not used; the process has no next event.

The state-dependent rates are the basic local description of this stochastic epidemic model. They determine the waiting time and next transition without saying that the trajectory is stationary: the rates generally change as the counts change. They also make the boundary behavior explicit. With no infectious people, there can be neither new infections nor recoveries; with no susceptible people but some infectious people, only recovery events remain possible.