= Mathematical biology
{wiki}

= Per capita
{parent=Mathematical biology}
{wiki}

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

= Age-structured population model
{parent=Mathematical biology}
{wiki=McKendrick–von_Foerster_equation}

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.

= Euler-Lotka equation
{c}
{parent=Age-structured population model}
{wiki}

For birth rate $b(a)$, death rate $d(a)$ and exponential population growth rate $r$, the Euler-Lotka equation is
$$
1=\int_0^\infty b(a)\exp\left(-ra-\int_0^ad(s)ds\right)da.
$$

= Law of mass action
{parent=Mathematical biology}
{wiki}

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
{parent=Mathematical biology}
{wiki}

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

= Michaelis-Menten equation
{c}
{parent=Enzyme kinetics}
{wiki=Michaelis%E2%80%93Menten_kinetics}

The Michaelis-Menten rate law is
$$
v=\frac{V_{\max}s}{K_M+s}.
$$
It is linear at low substrate concentration and saturates at $V_{\max}$ at high concentration.

= Quasi-steady-state approximation
{parent=Enzyme kinetics}
{wiki=Quasi-steady-state_assumption}

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.

= Quasi-steady rate law for a two-substrate allosteric enzyme
{parent=Quasi-steady-state approximation}

For sequential binding with dimensionless parameters $\alpha,\beta,\gamma$, the reduced product rate has the form
$$
\frac{dp}{dt}
=A\frac{u^2}{\alpha+u+(\beta/\gamma)u^2}.
$$
It is quadratic at low substrate concentration and saturates at high concentration.

= Autocatalysis
{parent=Mathematical biology}
{wiki}

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

= Autocatalytic biochemical switch
{parent=Autocatalysis}

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.

= Saturating autocatalytic switch
{parent=Autocatalytic biochemical switch}

For
$$
\dot g=s+k\frac{g^2}{1+g^2}-g,
\qquad k>2,
$$
the zero-input stable states are $0$ and $(k+\sqrt{k^2-4})/2$. The switching threshold is the <saddle-node bifurcation> determined by $f(g_c,s_c)=\partial_gf(g_c,s_c)=0$ on the low-concentration branch.

= Strong-autocatalysis switching threshold
{parent=Saturating autocatalytic switch}

For $k\gg1$, the low-concentration saddle node of the <saturating autocatalytic switch> has
$$
g_c\sim\frac1{2k},
\qquad
s_c\sim\frac1{4k}.
$$

= Cubic saturation population model
{parent=Mathematical biology}

The equation
$$
\dot n=\alpha n-\beta n^3
=\alpha n\left(1-\frac{n^2}{K^2}\right),
\qquad K=\sqrt{\alpha/\beta},
$$
has 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.

= Exponential density-dependent birth model
{parent=Mathematical biology}

If a population $N$ has per-capita birth rate $be^{-aN}$ and per-capita death rate $d$, then
$$
\dot N=N\left(be^{-aN}-d\right).
$$
When $b>d$, its positive <equilibrium of an autonomous differential equation>[equilibrium] is
$$
N^*=\frac1a\log\frac bd,
$$
and the derivative of the scalar vector field there is $-adN^*<0$, so the equilibrium is locally <asymptotic stability>[asymptotically stable].

= Survival-augmented Ricker map
{parent=Mathematical biology}

The population iteration
$$
x_{n+1}=x_n\bigl(r+ke^{-\lambda x_n}\bigr),
\qquad 0\leq r<1,
$$
combines annual survival with density-dependent recruitment. Its positive equilibrium is
$$
x_*=\frac1\lambda\log\frac{k}{1-r},
$$
which exists when $k>1-r$.

= Stability interval of the survival-augmented Ricker equilibrium
{parent=Survival-augmented Ricker map}

At the positive equilibrium,
$$
g'(x_*)=1-(1-r)\log\frac{k}{1-r}.
$$
The fixed point is locally asymptotically stable exactly when
$$
1-r<k<(1-r)e^{2/(1-r)}.
$$

= Malthusian delay differential equation
{parent=Mathematical biology}
{c}
{wiki=Delay_differential_equation}

The scalar equation $n'(t)=rn(t-\tau)$ 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.

= Method of steps for a delay differential equation
{parent=Malthusian delay differential equation}

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.

= Characteristic equation of a delay differential equation
{parent=Malthusian delay differential equation}

Substitution of $n(t)=e^{\lambda t}$ into $n'(t)=rn(t-\tau)$ gives the transcendental characteristic equation
$$\lambda=re^{-\lambda\tau}.$$

= Periodic solution of a scalar delay equation
{parent=Characteristic equation of a delay differential equation}

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

= Epidemic threshold
{parent=Mathematical biology}

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

= Kramers-Moyal expansion
{c}
{parent=Mathematical biology}
{wiki=Kramers–Moyal_expansion}

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.

= Constant-birth pair-annihilation process
{parent=Kramers-Moyal expansion}

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

= Kramers-Moyal master equation
{parent=Kramers-Moyal expansion}

For a jump process with rate $W(n,r)$ from state $n$ to $n+r$, probability balance gives
$$
\partial_tP(n,t)=\sum_r
\big[P(n-r,t)W(n-r,r)-P(n,t)W(n,r)\big].
$$
The first term is inflow from $n-r$ and the second is outflow from $n$.

= Compartmental model
{parent=Mathematical biology}
{wiki=Compartmental_models_in_epidemiology}

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

= Mass-action interaction
{parent=Compartmental model}

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

= Spatial SIR model
{parent=Compartmental model}

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

= Mass-action infection
{parent=Spatial SIR model}

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

= Diffusion of infectives
{parent=Spatial SIR model}

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

= Disease-free equilibrium
{parent=Compartmental model}

A disease-free equilibrium is an <equilibrium of an autonomous differential equation>[equilibrium] of an epidemic model at which every infected compartment is zero.

= Endemic equilibrium
{parent=Compartmental model}

An endemic equilibrium is an <equilibrium of an autonomous differential equation>[equilibrium] at which infection persists, so at least one infected compartment is positive.

= Susceptible-infective model with exponential density-dependent birth
{parent=Compartmental model}

The equations
$$
\dot S=S\left(be^{-aS}-\beta I-d\right),
\qquad
\dot I=I\left(\beta S-d-\delta\right)
$$
combine <mass-action infection> with an <exponential density-dependent birth model>. Susceptibles reproduce at per-capita rate $be^{-aS}$, while infectives have no reproductive term and die at total per-capita rate $d+\delta$.

= Disease-free equilibrium of the susceptible-infective model with exponential birth
{parent=Susceptible-infective model with exponential density-dependent birth}

For $b>d$, the disease-free equilibrium is
$$
(S,I)=\left(\frac1a\log\frac bd,0\right).
$$
It is linearly unstable precisely when
$$
\frac1a\log\frac bd>\frac{d+\delta}{\beta}.
$$

= Endemic equilibrium of the susceptible-infective model with exponential birth
{parent=Susceptible-infective model with exponential density-dependent birth}

When the disease-free equilibrium is unstable, the positive equilibrium satisfies
$$
S^*=\frac{d+\delta}{\beta},
\qquad
\beta I^*+d=be^{-aS^*}.
$$
Its Jacobian has negative trace and positive determinant, so the equilibrium is locally asymptotically stable.

= Basic reproduction number
{title2=$\mathcal R_0$}
{parent=Compartmental model}
{wiki}

The basic reproduction number $\mathcal R_0$ is the expected number of secondary infections caused by one infective in an otherwise susceptible population.

= Epidemic invasion threshold
{parent=Basic reproduction number}

A rare infection grows initially when $\mathcal R_0>1$ and decays when $\mathcal R_0<1$.

= SIR model with demography and permanent immunity
{c}
{parent=Compartmental model}

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 are
$$
X'=\mu N-\beta XY-\mu X,
\qquad Y'=\beta XY-(\mu+\nu)Y,
\qquad Z'=\nu Y-\mu Z.
$$
The conserved population is $N=X+Y+Z$, and the threshold population is $N_c=(\mu+\nu)/\beta$.

= Endemic equilibrium of the SIR model with demography
{c}
{parent=SIR model with demography and permanent immunity}

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

= SIR model with waning immunity
{c}
{parent=Compartmental model}
{wiki=SIRS_model}

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 are
$$
S'=fR-\beta IS,
\qquad I'=\beta IS-\nu I,
\qquad R'=\nu I-fR.
$$
They conserve $S+I+R=N$ and have basic reproduction number $\mathcal R_0=\beta N/\nu$.

= Endemic equilibrium of the SIR model with waning immunity
{c}
{parent=SIR model with waning immunity}

When $\beta N>\nu$ and $f>0$, the positive equilibrium is
$$
S^*=\frac\nu\beta,
\qquad
I^*=\frac{f(\beta N-\nu)}{\beta(f+\nu)},
\qquad
R^*=\frac{\nu(\beta N-\nu)}{\beta(f+\nu)}.
$$
Writing $a=\beta I^*>0$, the reduced $(S,I)$ Jacobian has characteristic polynomial
$$
\lambda^2+(f+a)\lambda+a(f+\nu).
$$
Both 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.