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A network describing how gene products affect the production of other gene products. Positive and negative regulatory interactions can generate multistability, oscillations, and noise-sensitive transitions.

// Target: mathematical-biology.bigb

A food web records resource-consumer interactions among species. A linear food chain is a special case. Positive interaction coefficients alone do not imply persistence: the per-capita growth balances must admit feasible positive long-term mean abundances.

An Allee effect makes per-capita growth less favorable at low abundance. A strong effect permits a stable zero state and a stable positive state separated by a finite threshold. Cooperative infections can exhibit the same positive-feedback structure: neither disease invades alone, but sufficiently many joint infections sustain both.

A metapopulation describes occupied habitat patches with local colonization and extinction. The patch occupancy variable differs from population abundance within a patch.

An epidemic is disease occurrence above the expected level for a specified population, place and period. The comparison needs a baseline: an increase in incidence alone does not establish an epidemic, and cases need not be increasing throughout an outbreak. For example, if a hypothetical community normally records two cases per week, a fall from twenty cases to eight still leaves occurrence above that baseline.^[ref]

An epidemic curve plots cases against time, commonly using symptom onset. A common-source outbreak reflects shared exposure, whereas a propagated outbreak reflects transmission between hosts; these mechanisms can also combine.^[ref]

In a closed homogeneous SIR model, let $s$ and $i$ be susceptible and infectious fractions, with transmission coefficient $\beta$ and recovery rate $\gamma >0$. The equations giveFor $i>0$, the infectious compartment grows when the effective reproduction number ${\mathcal{R}}_{0}s$ exceeds one and shrinks below one. This is a dynamical growth threshold, distinct from comparison with an observed baseline. An epidemic invasion threshold concerns growth after a small introduction; a final size relation for an epidemic concerns total susceptible depletion. An endemic equilibrium instead has continuing transmission balanced by the population's other processes.

The FitzHugh-Nagumo model represents an excitable cell by a fast activation variable and a slower recovery variable. A common nondimensional form is $\dot{V}=V-V^3/3-W+I$, $\dot{W}=\epsilon (V+a-bW)$. Its cubic activation response and recovery feedback can produce an excitable resting state or sustained oscillations through bifurcations. Particular reductions and travelling-wave equations can have additional state variables; their signs and parameter conventions must be stated separately.

By Darij Grinberg.

Online text: www.cip.ifi.lmu.de/~grinberg/t/23s/lec6.pdf

Cited passages: Lecture 6, p. 1, Theorem 1.1.2(a) and isolated-vertex note; p. 2, trivial trail convention; Lecture 6, pp. 1–3, Theorem 1.1.2 and the note about isolated vertices.

Online text: web.stanford.edu/class/math220a/handouts/waveequation1.pdf

Cited passages: Wave Equation in R, section 5.4.1, printed pages 7–8.

Cited passages: Stanford Math 220A, Section 5.1; Stanford Math 220A, Sections 5.2–5.4.