Mathematical biology

The density of individuals at age $a$ evolves by transport with unit ageing speed and loss rate $\mu (a)$. New individuals enter at age zero through the integral birth boundary. Exponential separated profiles lead to the Euler-Lotka equation.

The mean lifetime number of offspring is the birth rate integrated against the probability of surviving to each age. For a continuous age model it is ${\int }_0^{\infty }b(a)\exp [-{\int }_0^a\mu (s)ds]da$. The Euler-Lotka equation compares this with one to locate positive population growth.

A fixed-size neutral population evolves by reproduction events in which one individual replaces another. Types are copied along directed reproduction arrows. Backward ancestral lineages merge when one such arrow joins them; the coalescence rate depends on the chosen time normalization.

Randomness in individual births, deaths and maturation makes a finite population fluctuate even in a fixed environment. Near a deterministic extinction threshold it can drive the population into an absorbing extinct state.

Mathematical epidemiology uses mathematical models and statistical inference to study the occurrence, transmission, and control of disease in populations.

Observed symptom-onset intensity is the convolution of past infection intensity with the incubation delay density: $\mu (t)={\int }_0^{\infty }h(t-s)f(s)ds$. Back-calculation infers the infection history from that relation. Include pre-observation infections unless their absence is known.

For equal-width time bins and infections assigned to bin-end time $t_i$, let $q_{\ell }=\mathrm{Pr}((\ell -1)\Delta \le T<\ell \Delta )$. The next onset-bin mean is ${\mu }_k={\sum }_{i<k}{h}_i{q}_{k-i}$. Assigning infections instead to bin starts shifts the index by one. Bin probabilities integrate a delay density; they are not the density itself.

Calendar time locates an event on a shared external timeline, in contrast with an individual's age or the time elapsed since an individual event.

Incidence measures the occurrence of new cases in a population during a specified period.

The infection age of an infected individual is the time elapsed since that individual's infection.

An imported infection is acquired outside the population or observation system being modelled and enters it as an external source of incidence.

An infectivity profile describes how an infected individual's transmission rate varies with infection age.

The generation interval is the time between infection of a source individual and infection of a secondary case caused by that source.

The generation-interval distribution gives the probability law of the generation interval. In discrete time, its probabilities $g_{\tau }$ form a nonnegative sequence summing to one.

The infectious disease renewal equation expresses current incidence as a convolution of past incidence with an infectivity profile, multiplied by a time-varying transmission level.

For incidence $I_t$ and discrete generation-interval probabilities $g_{\tau }$, the total infectiousness at time $t$ is ${\Lambda }_t={\sum }_{\tau \ge 1}{g}_{\tau }{I}_{t-\tau }$.

A time-varying reproduction number describes transmission at a specified calendar time while allowing transmission conditions to change during an epidemic.

The instantaneous reproduction number freezes the transmission conditions at calendar time $t$ and counts the expected secondary infections produced over a complete infectious lifetime under those conditions.

The case reproduction number is the expected number of secondary infections actually generated by a person infected at time $t$, allowing transmission conditions to change during that person's infectious lifetime.

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

The Lotka-Volterra predator-prey model couples exponential prey growth and predator mortality through a bilinear encounter term. Its positive coexistence equilibrium is surrounded by closed conserved-energy curves, so the populations oscillate periodically in the idealized model.

This Lotka-Volterra predator-prey model includes self-limitation of the prey through the term $-b{x}^2$, with positive constants $a,b,c,d,e$. A positive coexistence equilibrium point exists when $a>be/d$, at $x_{\ast }=e/d$, $y_{\ast }=(a-b{x}_{\ast })/c$. The Lyapunov functionsatisfies $\dot{V}=-bd(x-x_{\ast }{)}^2$: its predator-prey cross terms cancel. Its sublevel sets are compact in the positive quadrant. The only invariant subset of $\{\dot{V}=0\}$ is the coexistence equilibrium point, because staying on $x=x_{\ast }$ requires $y=y_{\ast }$. The LaSalle invariance principle therefore proves convergence to coexistence for every positive initial state.

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

For annual survival probabilities $1-{\mu }_i$, age-specific offspring numbers $b_a$, and a population multiplier $\gamma$, the discrete Euler-Lotka equation isIts value at $\gamma =1$ is the expected lifetime offspring count of a newborn under the fixed vital rates. A value above one implies a growing mode $\gamma >1$, while a value below one implies decline.

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.

A chemical reaction network specifies chemical species, reactions between them, and their stoichiometric vectors. It can be modelled deterministically by reaction-rate equations or stochastically by molecule-copy-number jumps.

A chemical species is a class of chemically identical entities treated as one component of a reaction system.

The mole fraction of species $i$ is $x_i=N_i/{\sum }_j{N}_j$, where $N_i$ is its amount in moles. It is dimensionless and equals the volume mixing ratio for an ideal gas mixture.

For an ideal gas mixture, the volume mixing ratio $x_i$ equals the mole fraction, number fraction, and ratio of partial to total pressure: $x_i=N_i/N=n_i/n=P_i/P$, with ${\sum }_i{x}_i=1$. This equivalence fails for general nonideal mixtures.

The stoichiometric vector of reaction $r$ records its net change in each chemical species. If the copy-number state is $x$, firing the reaction changes it to $x+{\nu }_r$.

A reaction-rate equation is an ordinary differential equation for continuously varying species concentrations. Under the law of mass action, its reaction rates are monomials in those concentrations.

Stochastic chemical kinetics models molecular copy numbers as a continuous-time Markov chain. Each reaction has a state-dependent reaction propensity function and changes the state by its stoichiometric vector when it fires.

Compartment-based stochastic diffusion represents molecular motion by nearest-neighbour jumps on a spatial lattice. Jump rates of order $D/h^2\pm v/(2h)$ converge to an advection--diffusion equation as compartment width $h$ tends to zero.

The asymmetric simple exclusion process permits at most one particle per lattice site and biases jumps in one direction. A mean-field hydrodynamic limit has advective flux proportional to $c(1-c)$ because both an occupied departure site and vacant arrival site are required.

An immigration--death process has constant birth rate $b$ and linear death rate $dy$ in state $y$. Its stationary distribution is Poisson with mean $b/d$.

A stochastic quasi-steady-state approximation replaces fast species by their stationary conditional distribution given the slow species. Averaging slow propensities over that distribution yields a reduced Markov jump process.

A slow-scale stochastic simulation algorithm applies the Gillespie algorithm to propensities averaged over fast conditional equilibrium, avoiding explicit simulation of every fast reaction.

The molecular copy number of a chemical species is the nonnegative integer number of its molecules in a specified reactor or compartment.

The reaction propensity $a_r(x)$ is the instantaneous rate at which reaction $r$ fires when the copy-number state is $x$. Conditional on the present state, its reaction clock has an exponential distribution of rate $a_r(x)$.

A power-law propensity uses the same reactant monomial as a deterministic law of mass action, scaled by reactor volume. For example, a bimolecular channel may be assigned $a(x)=\alpha x^2/V$. This convention differs at small copy number from the combinatorial propensity $\alpha x(x-1)/V$ and must therefore be stated explicitly.

For reactions with stoichiometric vectors ${\nu }_r$ and reaction propensity functions $a_r$, the probability mass function $p(x,t)$ obeysIt is the forward operator of a Markov jump process equation for the copy-number continuous-time Markov chain.

The Gillespie algorithm samples an exact path of a well-mixed stochastic reaction network. At state $x$, let $a_0(x)={\sum }_r{a}_r(x)$; draw a waiting time with exponential distribution of rate $a_0$, choose reaction $r$ with probability $a_r/a_0$, and update $x\leftarrow x+{\nu }_r$.

A moment hierarchy is the coupled family of equations obtained by applying a Markov jump-process generator to powers of the state. Nonlinear reaction propensity functions make a moment of one order depend on higher-order moments, so the hierarchy generally does not close after finitely many equations.

For a Markov jump-process generator $L$, differentiating an integrable observable gives $d\mathbb{E}[f(X_t)]/dt=\mathbb{E}[Lf(X_t)]$, when the generator identity and interchange are justified. Choosing powers gives equations for moments. These need not form a closed finite system.

A moment closure replaces higher moments in a moment hierarchy by functions of retained lower moments. It converts an infinite hierarchy into a finite approximate system.

A central-moment closure sets selected high-order central moments to zero. For example, setting the third central moment of $X$ to zero gives

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 upper multiplier-$-1$ threshold of $x\mapsto sx+kx{e}^{-\gamma x}$, set $u=\gamma (x-x_{\ast })$. The second iterate is $u-2(s^2-s/2+1/6)u^3+O(u^4)$. Since its cubic damping coefficient is positive for $0\le s<1$, this endpoint is locally asymptotically stable despite the inconclusive linear multiplier.

At the positive equilibrium,The fixed point is strictly linearly stable whenThe upper endpoint is also locally asymptotically stable, as the nonlinear stability at the Ricker flip threshold shows; hence the full local asymptotic-stability interval includes that endpoint. The lower endpoint has no positive equilibrium.

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 $\cos (\pi t/(2\tau ))$.

Forlinearization about $x=1$ gives $u^{\prime }=\alpha [u(t)-2u(t-T)]$. The equilibrium is stable forand loses stability through imaginary roots $\lambda =\pm i\sqrt{3}\alpha$ at equality.

For $x^{\prime }(t)=\alpha x(t-T)[1-x(t)]$, linearization about $x=1$ is $u^{\prime }=-\alpha u$ because the delayed perturbation appears only in a quadratic term. The equilibrium is therefore locally asymptotically stable for every delay $T\ge 0$.

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

Infection passed from a parent to its offspring. In a population model, the corresponding infected births remain in the infected compartment rather than entering the susceptible compartment.

Infection transferred between individuals in the same generation, often modeled by a law of mass action term proportional to susceptible and infected populations. This contrasts with vertical transmission from parent to offspring.

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.

If every transition transfers individuals between compartments without births, deaths, immigration, or emigration, the sum of all compartment sizes is constant.

Disease prevalence is the proportion of a population that is infected at a specified time. In a susceptible-infective model with total population $N=S+I$, it is $\theta =I/N$.

The SI model divides a closed population into susceptible and permanently infectious compartments. Under frequency-dependent homogeneous mixing,and conservation of population gives $S+I=N$.

If $S(0)=N\theta /(1+\theta )$ and $I(0)=N/(1+\theta )$, thenAfter centering time at the incidence peak $t_{\ast }={\beta }^{-1}\log \theta$, the incidence curve is $N\beta /[4{\cosh }^2\{\beta (t-t_{\ast })/2\}]$.

The plant-disease modelhas total population $N=S+I$ and disease prevalence $\theta =I/N$ satisfyingThus infection does not alter total-population growth. The invasion threshold at the carrying capacity $N=1$ is $\beta =1$.

Removing susceptible and infected plants at the same per-capita rate $k$ changes the reduced equations toFor $0\le k<1$, the surviving population has carrying capacity $1-k$, and infection invades precisely when $\beta (1-k)>1$.

The SEIR model divides a population into susceptible, exposed, infectious, and recovered compartments.

The force of infection is the instantaneous infection hazard experienced by a susceptible individual. Under homogeneous frequency-dependent mixing it is $\beta I(t)/N$.

For a closed homogeneous epidemic, integrating $dS/dR$ relates the ultimately susceptible population to cumulative infections through an exponential equation.

The SIR model divides a closed population into susceptible, infectious, and recovered compartments. Infection transfers people from $S$ to $I$, and recovery transfers them from $I$ to $R$.

The deterministic SIR model with initial susceptible count $S_0$ grows initially exactly when $\beta S_0>\gamma$. Its basic reproduction number is $\beta S_0/\gamma$ under unnormalized mass-action infection. If that ratio is at most one, decreasing $S$ prevents later infectious growth.

For the closed deterministic SIR model with $S(0)=S_0$, $R(0)=0$, infection rate $\beta SI$ and recovery rate $\gamma I$, division of the two flow equations gives $S(t)=S_0{e}^{-\beta R(t)/\gamma }$. Population conservation then expresses $I$ in terms of $R$.

The stochastic SIR model is a continuous-time Markov chain with infection jump $(-1,1,0)$ at rate $\beta SI$ and recovery jump $(0,-1,1)$ at rate $\gamma I$. The state space consists of nonnegative integer compartment counts with conserved total population.

With one initial infectious individual, two events require an infection as the first event. If it occurs at time $s$, its density is $\beta N{e}^{-(\beta N+\gamma )s}$ and the next total event rate is $2\beta (N-1)+2\gamma$. Integrating this density times the chance of another event before a fixed horizon gives the two-event probability.

A chain-binomial epidemic model uses discrete time, with conditionally independent binomial distributions for infection and recovery counts given the current compartments. It requires valid probability parameters; replacing $1-e^{-\beta I\delta t}$ by $\beta I\delta t$ is a small-step approximation.

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 $N_c=(\mu +\nu )/\beta$.