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Let $P(t)=e^{Qt}$ be the transition probability matrix of the four-state continuous-time Markov chain, and put $N=\{S,R\}$ for a negative test and $P=\{E,I\}$ for a positive test. Starting susceptible at time zero, the likelihood contribution isEquivalently, with indicator diagonal matrices $D_N,D_P$ and the susceptible basis vector $e_S$ it isThis sums over every hidden state sequence compatible with the three test results.

Sample a finite-state continuous-time Markov chain with generator $Q$ at the independent arrival times of a rate-$\lambda$ Poisson process. The sampled states form a discrete-time Markov chain with transition matrixIts invariant distributions are exactly those of the continuous-time chain.

Choose two positions independently and uniformly from $\{1,\dots ,n\}$ and swap them. This Markov chain on the symmetric group holds with probability $1/n$ and assigns probability $2/n^2$ to each unordered transposition. Its stationary distribution is uniform, and its ordinary spectral gap is $2/n$ for $n\ge 2$. The latter follows from the Aldous spectral gap theorem: the single-label continuous-time Markov chain has rate $2/n^2$ between every pair of positions, whose mean-zero eigenvalues are all $-2/n$.

The stationary law of a two-resource linear flow network is normalizable exactly whenThese are the strict offered-load constraints on the two unit-capacity resources. Summing first over the through-route count reduces normalization to two geometric series, which diverge at equality. With positive arrival rates, the nonexplosive continuous-time Markov chain is then positive recurrent.

For the proportionally fair allocation on a linear flow network, independent Poisson flow arrivals at rates ${\nu }_r$ and exponential document sizes of rates ${\mu }_r$ give a continuous-time Markov chain with departures ${\mu }_r{n}_r{x}_r(n)$. Put ${\rho }_r={\nu }_r/{\mu }_r$. Under ${\rho }_0+{\rho }_i<1$ for each local route, the normalizing constant is $Z=(1-{\rho }_0{)}^{I-1}/{\prod }_i(1-{\rho }_0-{\rho }_i)$. The binomial coefficient weight has adjacent-state ratios equal to the aggregate service rates, proving detailed balance. For more general network topologies, proportional fairness need not give this reversible product form.

Stochastic chemical kinetics models molecular copy numbers as a continuous-time Markov chain. Each reaction has a state-dependent reaction propensity function and changes the state by its stoichiometric vector when it fires.

The stochastic SIR model is a continuous-time Markov chain with infection jump $(-1,1,0)$ at rate $\beta SI$ and recovery jump $(0,-1,1)$ at rate $\gamma I$. The state space consists of nonnegative integer compartment counts with conserved total population.

A sequential continuous-time Markov chain has transitions $1\to 2\to 3$ with rates $\lambda ,\nu$ and an absorbing state $3$. Its probability of ever leaving state $1$ by time $t$ is $1-e^{-\lambda t}$; its probability of occupying state $2$ is $\lambda (e^{-\lambda t}-e^{-\nu t})/(\nu -\lambda )$ for unequal rates.

For a finite-state continuous-time Markov chain, a transition intensity matrix has nonnegative off-diagonal rates $q_{rs}$ and $q_{rr}=-{\sum }_{s\ne r}{q}_{rs}$. Every row sums to zero. In the time-homogeneous case its transition probability matrix is the matrix exponential $P(t)=e^{tQ}$. An absorbing state has a zero row.

For a continuous-time Markov chain, $P_{ij}(t)={\mathbb{P}}_i(X_t=j)$ and the matrices satisfy $P(s+t)=P(s)P(t)$. On a finite state space with Q-matrix $Q$, $P(t)=e^{tQ}$.