Mathematical biology

A per-capita quantity is a total quantity divided by the number of individuals in the population.

An age-structured population model tracks a density $n(a,t)$ transported toward greater age, reduced by age-dependent mortality, and replenished at age zero by births.

For birth rate $b(a)$, death rate $d(a)$ and exponential population growth rate $r$, the Euler-Lotka equation is

The law of mass action assigns an elementary reaction a rate constant times the product of its reactant concentrations, with each concentration raised to its stoichiometric multiplicity.

Enzyme kinetics models how free enzyme, substrate, enzyme-substrate complexes, and product concentrations evolve through a reaction mechanism.

The Michaelis-Menten rate law isIt is linear at low substrate concentration and saturates at $V_{\max }$ at high concentration.

When enzyme complexes relax much faster than substrate concentrations, the quasi-steady-state approximation sets the time derivatives of the fast complexes to zero and solves their algebraic balance equations.

For sequential binding with dimensionless parameters $\alpha ,\beta ,\gamma$, the reduced product rate has the formIt is quadratic at low substrate concentration and saturates at high concentration.

Autocatalysis is a reaction mechanism in which a product promotes its own production.

An autocatalytic biochemical switch uses bistability and a transient input to move a system from one stable concentration to another. A saddle-node bifurcation supplies a threshold above which the low-concentration equilibrium disappears.

Forthe zero-input stable states are $0$ and $(k+\sqrt{k^2-4})/2$. The switching threshold is the saddle-node bifurcation determined by $f(g_c,s_c)={\partial }_{g}f(g_c,s_c)=0$ on the low-concentration branch.

For $k\gg 1$, the low-concentration saddle node of the saturating autocatalytic switch has

The equationhas an unstable zero equilibrium and a stable positive equilibrium $K$. Its per-capita density correction is quadratic rather than the logistic model's linear correction.

If a population $N$ has per-capita birth rate $b{e}^{-aN}$ and per-capita death rate $d$, thenWhen $b>d$, its positive equilibrium isand the derivative of the scalar vector field there is $-ad{N}^{\ast }<0$, so the equilibrium is locally asymptotically stable.

The population iterationcombines annual survival with density-dependent recruitment. Its positive equilibrium iswhich exists when $k>1-r$.

At the positive equilibrium,The fixed point is locally asymptotically stable exactly when

The scalar equation $n^{\prime }(t)=rn(t-\tau )$ makes the present growth rate depend on the population one delay time earlier. A history function on $[-\tau ,0]$ is required in place of a single initial value.

Given the solution on one interval of length $\tau$, the delayed term is known on the next interval, where the delay equation becomes an ordinary differential equation. Repeating this process constructs the solution interval by interval.

Substitution of $n(t)=e^{\lambda t}$ into $n^{\prime }(t)=rn(t-\tau )$ gives the transcendental characteristic equation

Purely imaginary characteristic roots produce oscillatory solutions. In particular, $r\tau =-\pi /2$ gives $\lambda =\pm i\pi /(2\tau )$ and the real solution $\cos (\pi t/(2\tau ))$.

A cross-infection model's disease-free equilibrium becomes a saddle when the product of infection gains exceeds the product of recovery rates.

The Kramers-Moyal expansion Taylor-expands the gain terms of a jump-process master equation in the jump size. Keeping its first two terms yields a Fokker-Planck equation whose drift and diffusion coefficients are the first two infinitesimal jump moments.

For jumps $n\to n+1$ at rate $\lambda$ and $n\to n-2$ at rate $\beta n^2$, the diffusion approximation hasThe stable deterministic population is $n_{\ast }=\sqrt{\lambda /(2\beta )}$. Its linear-noise stationary approximation is normal with mean $n_{\ast }$ and variance $3n_{\ast }/4$.

For a jump process with rate $W(n,r)$ from state $n$ to $n+r$, probability balance givesThe first term is inflow from $n-r$ and the second is outflow from $n$.

A compartmental model divides a conserved population into states and uses differential equations for transition rates.

Under mass action, random pairwise encounters between populations of sizes $X$ and $Y$ occur at a rate proportional to $XY$.

A spatial SIR-type model combines compartment transitions with diffusion in the compartments whose members move appreciably.

Mass-action transmission at unit nondimensional rate transfers population from susceptible to infected compartments at rate $SI$.

A Laplacian term in the infected-population equation models unbiased random migration of infective individuals.

A disease-free equilibrium is an equilibrium of an epidemic model at which every infected compartment is zero.

An endemic equilibrium is an equilibrium at which infection persists, so at least one infected compartment is positive.

The equationscombine mass-action infection with an exponential density-dependent birth model. Susceptibles reproduce at per-capita rate $b{e}^{-aS}$, while infectives have no reproductive term and die at total per-capita rate $d+\delta$.

For $b>d$, the disease-free equilibrium isIt is linearly unstable precisely when

When the disease-free equilibrium is unstable, the positive equilibrium satisfiesIts Jacobian has negative trace and positive determinant, so the equilibrium is locally asymptotically stable.

The basic reproduction number ${\mathcal{R}}_0$ is the expected number of secondary infections caused by one infective in an otherwise susceptible population.

A rare infection grows initially when ${\mathcal{R}}_0>1$ and decays when ${\mathcal{R}}_0<1$.

With susceptible, infective, and immune populations $X,Y,Z$, equal per-capita birth and death rate $\mu$, mass-action transmission coefficient $\beta$, and recovery rate $\nu$, the equations areThe conserved population is $N=X+Y+Z$, and the threshold population is $N_c=(\mu +\nu )/\beta$.

When $N>N_c=(\mu +\nu )/\beta$, the positive equilibrium isOn the invariant plane $X+Y+Z=N$, the $(X,Y)$ Jacobian at this equilibrium has negative trace $-(\mu +\beta Y^{\ast })$ and positive determinant $\beta Y^{\ast }(\mu +\nu )$, so both eigenvalues have negative real part and the equilibrium is locally asymptotically stable.

For susceptible, infective, and recovered populations $S,I,R$, mass-action infection coefficient $\beta$, recovery rate $\nu$, and immunity-loss rate $f$, the SIRS equations areThey conserve $S+I+R=N$ and have basic reproduction number ${\mathcal{R}}_0=\beta N/\nu$.

When $\beta N>\nu$ and $f>0$, the positive equilibrium isWriting $a=\beta I^{\ast }>0$, the reduced $(S,I)$ Jacobian has characteristic polynomialBoth roots have negative real part. Its discriminant is negative for sufficiently small $f$ and positive for sufficiently large $f$, giving respectively a stable focus and a stable node.