Likelihood From Infection and Recovery Event Histories

From an Observed Epidemic Path to a Likelihood

When infection and recovery times are observed, the data contain more than the final number of cases. Each event identifies which transition occurred, and each interval without an event tells us that the process avoided all possible transitions during that time. A continuous-time likelihood must account for both pieces. The resulting formula has a useful structure: event counts multiply the rates, while the time spent at risk appears through integrated rates.

Fix a population size \(N\). On a probability space \((\Omega,\mathcal{F},\mathbb{P})\), let \((X_t)_{t\ge0}\), with \(X_t=(S_t,I_t,R_t)\), be a continuous-time Markov chain on \( \mathcal{S}=\{(s,i,r):s,i,r\text{ are nonnegative integers and }s+i+r=N\}\). Use its natural filtration \((\mathcal{F}_t)_{t\ge0}\), the information generated by the process through time \(t\). The infection and recovery transitions are \((s,i,r)\to(s-1,i+1,r)\) and \((s,i,r)\to(s,i-1,r+1)\), respectively. Their rates are \(\beta si/N\) and \(\gamma i\), where the parameters \(\beta,\gamma\ge0\). At a state where a transition is impossible, its rate is zero.

Write \(Y_j\) for the state just after event \(j\), and set \(Y_0=X_0\). Between events the state is constant. If the state on an interval is \((s,i,r)\), the total event rate is \(\beta si/N+\gamma i\). The probability of no event for a duration \(u\) is the exponential of minus \(u\) times this total rate. If an event then occurs, its type-specific rate supplies the corresponding event-density factor.

Proof. On any interval during which the state is fixed, the two possible event mechanisms have constant rates \(a=\beta si/N\) and \(b=\gamma i\). Their waiting times can be represented by independent exponential clocks with those rates. Thus the probability that neither clock rings during a duration \(u\) is \(e^{-au}e^{-bu}=e^{-(a+b)u}\). The density that the first event occurs after duration \(u\) and is an infection is \(a e^{-(a+b)u}\): the infection clock has density \(ae^{-au}\), while the recovery clock has survival probability \(e^{-bu}\). The recovery-event density is \(b e^{-(a+b)u}\), by the same calculation with the clocks interchanged. If one or both rates are zero, these formulas still apply; an event with rate zero has density zero.

Apply these factors successively to each observed waiting interval and event, and then include the probability of no event from \(t_m\) to \(T\). The rate factors at infection events contribute \(\beta si/N\) each, and those at recovery events contribute \(\gamma i\) each. Their parameter-dependent parts therefore multiply to \(\beta^{n_{\mathrm{inf}}}\gamma^{n_{\mathrm{rec}}}\); the state-dependent factors do not depend on the parameters. The no-event factors over all intervals multiply to the exponential of minus the integral of the total rate. Since the state is constant between events, that integrated rate is \(\int_0^T(\beta S_tI_t/N+\gamma I_t)\,dt=\beta A+\gamma B\). Multiplying the event and no-event factors gives the stated likelihood. If an observed event has zero rate, its event factor is zero, as asserted. \(\square\)

This derivation also explains the role of observation: an event contributes its instantaneous rate, while every interval contributes evidence against events that could have occurred. The integrals \(A\) and \(B\) are the total infection and recovery exposures. They summarize the time at risk in the fully observed history.

Maximum-Likelihood Estimates

Proof. With \(A,B>0\), the log likelihood for positive parameters is \(\ell(\beta,\gamma)=n_{\mathrm{inf}}\log\beta+n_{\mathrm{rec}}\log\gamma-\beta A-\gamma B\), up to a constant. It is a sum of a function of \(\beta\) and a function of \(\gamma\), so each can be maximized separately. For a parameter \(\theta\) with event count \(n\) and exposure \(C>0\), the relevant function is \(g(\theta)=n\log\theta-C\theta\). If \(n>0\), its derivative is \(n/\theta-C\), which is positive for \(\theta<n/C\), zero at \(n/C\), and negative for \(\theta>n/C\). Thus its unique maximum is \(n/C\). If \(n=0\), the likelihood factor is \(e^{-C\theta}\), which decreases as \(\theta\) increases and is maximized over \(\theta\ge0\) at zero. Over strictly positive \(\theta\), it approaches its supremum as \(\theta\) decreases to zero but does not attain it. Applying these conclusions to \((n,C)=(n_{\mathrm{inf}},A)\) and \((n_{\mathrm{rec}},B)\) proves the result. \(\square\)

The positive-exposure conditions matter. If an exposure is zero and its event count is also zero, that parameter does not appear in the likelihood and cannot be estimated from that history. If exposure is zero but the corresponding event count is positive, the displayed likelihood formula has no finite maximizer. For an ordinary compatible SIR history with finite rates, an event of a given type requires positive exposure of that type before the event; this degenerate case signals an inconsistent history or an unsuitable likelihood specification.

Combining Independent Histories and Avoiding a Pitfall

For independent epidemics observed under the same parameter values, their likelihoods multiply. Equivalently, add their infection counts, recovery counts, infection exposures, and recovery exposures. The same rate estimates then use the pooled counts divided by the pooled exposures. This pooling is valid for fully observed histories under the stated model; it does not replace a model for partially observed events.

Common pitfall. Multiplying only the rates at observed events omits the evidence from waiting. In the first example, that incorrect calculation gives only \(\tfrac{24}{25}\beta^2\gamma\). It grows without bound as \(\beta\) increases, so it cannot yield the finite estimate supported by the data. The correct likelihood also includes \(\exp(-37\beta/20-13\gamma/4)\), which accounts for the probability of no extra event during the observed intervals and produces the finite estimates \(\tfrac{40}{37}\) and \(\tfrac4{13}\).

The likelihood also clarifies what the data can and cannot identify. Infection events inform \(\beta\) through their count, while time with susceptible and infectious individuals present informs it through \(A\). Recoveries inform \(\gamma\) through their count, while infectious person-time informs it through \(B\). A final epidemic size alone generally does not provide these event times and exposures, so it does not supply this complete-history likelihood.