Birth–death processes
A birth–death process is a CTMC on integer-valued states in which the population changes by one individual at each event.
\[i\to i+1\quad\text{at rate }b(i),\qquad i\to i-1\quad\text{at rate }d(i).\]Short-time probabilities
\[P(i\to i+1)\approx b(i)\Delta t,\]\[P(i\to i-1)\approx d(i)\Delta t,\]\[P(i\to i)\approx1-[b(i)+d(i)]\Delta t.\]Biological uses
Birth–death processes can represent population growth and decline, infection and recovery counts, cell populations and other event-based biological systems.
Key idea. State-dependent birth and death rates translate biological event mechanisms into a continuous-time stochastic population model.