From Population Mechanisms to Birth And Death Rates

From Events to Population Rates

A population model starts with mechanisms: an individual may give birth at a certain hazard, die at a certain hazard, or take part in an interaction that can change the population. A continuous-time population process does not record which person acted; it records the resulting population size. The modeling step is to add the rates of all mechanisms that produce the same change in that size. In particular, an individual-level hazard is not automatically the rate for the whole population.

Work on a probability space \((\Omega,\mathcal{F},\mathbb{P})\). Let \((N_t)_{t\ge0}\) be a continuous-time Markov chain with state space \(S\subseteq\{0,1,2,\ldots\}\), where \(N_t=n\) means there are \(n\) individuals at time \(t\). Use the natural filtration \(\mathcal{F}_t=\sigma(N_s:0\le s\le t)\). A rate \(r\) means an event occurs at instantaneous hazard \(r\), conditional on the current state and the specified event mechanism. We assume rates are finite at each state considered. For a birth-and-death model, each event changes the count by one: a birth takes \(n\) to \(n+1\), and a death takes \(n\) to \(n-1\).

These are population-level rates. If each of \(n\) individuals has a birth hazard \(b_n\), then the total birth rate is \(n b_n\), provided each birth adds one individual and the individual hazards are the stated conditional hazards. If each individual has a death hazard \(d_n\), the total death rate is \(n d_n\). The per-capita rates \(b_n,d_n\) may depend on the current population size. Distinct mechanisms leading to the same change must also be added: for example, ordinary deaths and interaction-caused deaths both contribute to \(q_{n,n-1}\).

Why Rates Add

The elementary calculation behind rate aggregation is the competing-clock calculation. Suppose that, at a fixed state, there are finitely many possible event mechanisms. Mechanism \(\ell\) has an exponential waiting time with rate \(r_\ell\), and the clocks are conditionally independent given the current state. Let \(r=\sum_\ell r_\ell\). Then the probability that no event occurs during an interval of length \(h\) is the product of the probabilities that none of the clocks rings. The probability that a particular mechanism rings first is proportional to its rate.

Proof. Write \(T_\ell\) for the waiting time of mechanism \(\ell\), and \(T=\min_\ell T_\ell\). For \(t\ge0\), independence and the exponential survival formula give \(\mathbb{P}(T>t)=\prod_\ell e^{-r_\ell t}=e^{-rt}\). This is the survival function of an exponential waiting time of rate \(r\); if \(r=0\), every \(r_\ell=0\) and no event occurs. When \(r>0\), the probability density that mechanism \(\ell\) rings at time \(t\) and every other clock is still silent is \(r_\ell e^{-r_\ell t}\prod_{k\ne\ell}e^{-r_k t}=r_\ell e^{-rt}\). Integrating over \(t\ge0\) gives \(\int_0^\infty r_\ell e^{-rt}\,dt=r_\ell/r\). If several mechanisms lead to the same state \(j\), the events that each is first are disjoint, so their probabilities add. Equivalently, their first-event densities add to \((\sum_{\ell:\,\ell\text{ leads to }j}r_\ell)e^{-rt}\). The rate for that transition is therefore the sum of those mechanism rates. \(\square\)

For the nearest-neighbor population models used below, the small-time interpretation of these rates can also be checked directly. From a fixed state \(n\), the only possible first destinations are \(n-1\) and \(n+1\), and the rates at those states are finite. Let \(r_n=\lambda_n+\mu_n\), and let \(M\) be the maximum total rate at the possible first destinations. The chance of a first jump by time \(h\) is at most \(r_n h\), since \(1-e^{-r_n h}\le r_n h\). Conditional on a first jump, the chance of another jump in the remaining time, which is at most \(h\), is at most \(Mh\). Thus the chance of two or more jumps by time \(h\) is at most \(r_n M h^2\). This local bound justifies ignoring multiple jumps in the first-order rate calculation for these nearest-neighbor models. It uses their finite set of possible first destinations; it is not an assertion about arbitrary countable-state chains.

Worked Examples

Rate Translation as a Modeling Rule

The examples use two different sources of population-wide rates. In the first, many individual clocks have the same constant per-capita rate, so the total is the number of individuals times that rate. In the second, pairwise mechanisms contribute a number of clocks proportional to the number of possible pairs. As the population changes, the number of active clocks changes too; this is one way a constant mechanism at the individual or pair level creates a state-dependent population rate.

More generally, to construct rates from a proposed mechanism, first specify the current state and the event that occurs. Then count the eligible individuals, pairs, or other event opportunities; multiply by the hazard per opportunity; and assign the resulting rate to the state transition caused by that event. If multiple mechanisms cause the same transition, add their rates. Finally, set the diagonal generator entry to minus the sum of all off-diagonal rates in that row. This procedure is local: it determines the instantaneous rates at each state, not by itself long-term behavior or the existence of a non-explosive process on an unbounded state space.

Common mistake. If each of four individuals has birth hazard \(\tfrac15\), it is incorrect to set the population birth rate to \(\tfrac15\). That is the hazard for one individual. The population has four birth clocks, so the total rate is \(4(\tfrac15)=\tfrac45\). The same distinction applies to deaths. Also, adding per-capita rates before accounting for the number of individuals can give the wrong answer: in the pairwise example, the interaction rate is based on six pairs, not four individuals each with a full \(\tfrac1{10}\) hazard.