Compartmental model

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

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.