In a closed population of size $n$, a frequency-dependent SIRS epidemic has the displayed transitions at rates $\lambda SI/n$, $\gamma I$, and $\nu R$. The exponential distribution of infectious lifetimes and the constant immunity-loss hazard make the compartment counts a continuous-time Markov chain. Each jump preserves $S+I+R=n$.

In this discrete-time SIR model, each infected vertex is infectious for one step, can infect each susceptible neighbour with probability $\beta$, and is then permanently removed. The adjacency matrix of a graph describes the possible transmissions. If the initial removed set is empty, eventual removal and ever becoming infected are the same event.

Here $s(n)$ and $i(n)$ weight susceptibility and infectiousness of a farm with $n$ animals, with size distribution $P_n$ and recovery rate $\gamma$. The rank-one next-generation matrix has reproduction number ${\mathcal{R}}_0=(\beta /\gamma ){\sum }_{n}s(n)i(n)P_n$. Susceptibility proportional to size preferentially depletes large susceptible farms. Infectivity heterogeneity alone leaves an exactly homogeneous aggregate SIR trajectory for representative initial infections. When both weights equal size, the second rather than first size moment governs invasion.

For the closed SIR model with $\gamma >0$, the infectious compartment's fraction approaches $i_{\infty }=0$. One proof outline uses the bounded, increasing removed compartment's fraction: since $r(t)-r_0=\gamma {\int }_0^{t}i(u)du$, the total time-integrated $i$ is finite. The bounded derivative $di/dt$ from the SIR model equations prevents indefinitely recurring spikes, so $i(t)$ tends to zero.

For $s_0>0$, the final size relation for an epidemic follows by dividing the SIR model equations for $s$ and $r$ while $i>0$, giving $ds/dr=-(\beta /\gamma )s$. Integration then yieldsThis also follows from the SIR first integral and conservation of population. For the stipulated initial value problem, solveThe bisection method givesAbout $5.88\%$ of the original population remains in the susceptible compartment. The $94.12\%$ is the final removed compartment's fraction, including the initially $1\%$ in the infectious compartment. The newly acquired infection during the modelled period accounts for $s_0-s_{\infty }\approx 0.93120263520$ of the total population, not $1-s_{\infty }$. Dividing by $s_0$ instead gives the proportion of initially susceptible compartment members who acquire infection; its denominator must be stated.

At the SIR infection peak, the flow into the infectious compartment only balances its outflow; it does not stop. Further infection explains why the final susceptible compartment's fraction is well below $1/3$. Also $s_{\infty }=s_0{e}^{-R_0(r_{\infty }-r_0)}>0$ for $s_0>0$ and finite $R_0$: the closed SIR model does not require everybody to acquire infection before its infectious compartment empties in the long-time limit.

For $s>0$, the SIR model equations have the first integral of an ordinary differential equationHere $\ln$ is the natural logarithm. A short proof outline is to use the chain rule and substitute the SIR model equations:The SIR first integral gives $i$ directly from $s$, so it describes the curve in a phase portrait even before its time parametrization is known.

For frequency-dependent transmission, the closed SIR model is a system of ordinary differential equations for the fractions $s,i,r$:Here $\beta >0$ is the coefficient of frequency-dependent transmission, and $\gamma >0$ is the recovery rate, interpreted more generally as the removal rate. Both have units of inverse time. With a constant recovery rate, the associated individual time in the infectious compartment has an exponential distribution with mean $1/\gamma$; the SIR model does not require every person to spend exactly that long there. The fractions themselves have no units.^[ref]

For counts $S=Ns$, $I=Ni$, $R=Nr$, the same SIR model equations readThe factor $1/N$ keeps the same $\beta$ in both descriptions. Dropping it while keeping these units would fail dimensional analysis. Some existing SIR model variants use the unnormalized mass-action infection convention instead, including the SIR susceptible-recovered identity; their coefficient is $b=\beta /N$ in the notation used here.

Adding the SIR model equations gives $d(s+i+r)/dt=0$, a short proof of conservation of population. Nonnegative starting fractions stay nonnegative: the outflow from either the susceptible compartment or the infectious compartment vanishes when that group is empty, while $dr/dt\ge 0$. Thus the susceptible compartment can only shrink and the removed compartment can only grow.

For a concrete initial value problem, stipulateAt the start, the flow into the infectious compartment is $0.3\times 0.99\times 0.01=0.00297$ of the total per day, and the flow into the removed compartment is $0.1\times 0.01=0.00100$ of the total per day. Hence $di/dt=0.00197$ per day. If $N=1000$, these correspond to model rates of $2.97$ and $1.00$ people per day, and net growth $1.97$ people per day. A rate need not be an integer count of events on any particular day.

For this SIR model, the basic reproduction number isIt describes one introduced member of the infectious compartment in an otherwise wholly susceptible compartment. The instantaneous reproduction number at the actual starting state is $R_0{s}_0=2.97$, obtained by freezing the current conditions and the current susceptible compartment's fraction. More generally it is $R_{0}s(t)$ in this SIR model. The basic reproduction number is a ratio of two rates, so it has no units; $r_0$ is a different quantity, the initial removed compartment's fraction.

While $i>0$, its instantaneous relative growth isInitially this equals $0.3\times 0.99-0.1=0.197{\mathrm{day}}^{-1}$. Freezing $s\approx s_0$ gives the early exponential growth approximation $i(t)\approx 0.01e^{0.197t}$ with $t$ measured in days. It must fail as the susceptible compartment shrinks. An epidemic invasion threshold occurs at $R_0{s}_0=1$; if $i_0=0$, the deterministic SIR model stays at $i=0$ even above that threshold.

The removed compartment consists of people permanently excluded from both acquiring and passing on infection in the SIR model. Its count is $R(t)$ and its fraction is $r(t)=R(t)/N$. Removal can represent regained health with lasting immunity, or another permanent exclusion from acquiring or passing on infection; it need not mean being healthy again. Here there are no deaths. In a variant that counts deaths within $R$, conservation of population would concern the original cohort, rather than the number currently alive.^[ref]

The infectious compartment consists of people currently able to transmit infection. In the closed SIR model, $I(t)$ is their count and $i(t)=I(t)/N$ their fraction of the fixed total $N$. The SIR model has no separate delay between acquiring infection and entering the infectious compartment; the SEIR model adds such a stage.

The susceptible compartment consists of people who can acquire infection. In the closed SIR model, $S(t)$ is their count and $s(t)=S(t)/N$ their fraction of the fixed total $N$. Membership expresses the assumptions of the mathematical model, rather than measuring the strength of an individual's immunity.

Perturb the sinusoidal cellular flow by $ϵ\cos x\cos y\sin (\omega t)$ in its stream function. On $y=0$, parameterizing the unperturbed orbit with $q_0(0)=(X_0,0)$ gives $x(s)=2\arctan e^{s+a}$, $a=\log \tan (X_0/2)$. The heteroclinic Melnikov function for a periodic planar flow has integrand ${\mathrm{sech}}^2(t-t_0+a)\sin (\omega t)$. Its odd part vanishes and the Fourier transform of ${\mathrm{sech}}^2$ gives the displayed result. For each fixed nonzero frequency its simple zeros produce transverse intersections at sufficiently small perturbation, hence exchange of fluid lobes across the former cell boundary. The leading splitting is exponentially small at high frequency; that estimate is not uniform in frequency and perturbation size.

To maximize a log-likelihood, use energy $-\ell$ and Boltzmann distribution proportional to $e^{\ell /T}=L^{1/T}$ at temperature $T>0$. Reversing this sign favors likelihood minima and can make the probability density function nonnormalizable. For Poisson observations with total $s$ and sample size $m$, with respect to $d\lambda$ the normalized probability density function isIts mean is $(s+T)/m$ and variance $(sT+T^2)/m^2$, so it concentrates at the maximum-likelihood estimate $s/m$ as $T\downarrow 0$. Cooling of equilibrium laws and convergence of a particular changing-temperature chain are different statements; the latter requires mixing or a direct chain argument.

This simplicial notion is induced by an orientation of a vector space on the affine span of the simplex.

An orientation of a simplex is an ordering of its vertices modulo even permutations.

A simplicial complex is flag when every finite set of pairwise adjacent vertices spans a simplex.

Conversely, every flag complex is the clique complex of its one-skeleton, so these two descriptions specify the same class of complexes.

A triangulation of $S^k$ is regular in this antipodal-combinatorics convention when the antipodal map preserves its simplices and its nested coordinate equators $S^0\subset S^1\subset \cdots \subset S^{k-1}$ are subcomplexes. The two closed hemispheres at each stage are then triangulated balls. Barycentric subdivision preserves these conditions while making the mesh of a simplicial complex arbitrarily small. This convention is distinct from other meanings of regular triangulation in polyhedral geometry. Cambridge extremal-combinatorics notes use this convention.

A simplicial subdivision replaces the simplexes of a simplicial complex by finitely many smaller simplexes with the same underlying space, compatibly on common faces. A piecewise linear ball can be parametrized by a map linear on the simplexes after appropriate subdivisions of its domain and target.

A simplex tableau records the canonical constraint equations for a simplex basis and the canonical objective equation. For maximization use $z+{\sum }_j{\rho }_j{x}_j=z_0$, so a negative nonbasic objective-row coefficient is an improving direction. Constraint rows have an identity matrix in the basic columns. A simplex method pivot divides the leaving row by the pivot coefficient and eliminates the entering column from the other rows, including the objective row. The simplex ratio test chooses the largest feasible step. When all basic values and all ${\rho }_j$ are nonnegative, the objective equation directly proves $z\le z_0$ for every nonnegative feasible vector.

For $0<\epsilon <1/2$, these inequalities define a deformed cube with $2^n$ vertices of a polytope. Choose either the lower or upper bound recursively at each coordinate to obtain a vertex of a polytope; changing one choice gives an adjacent vertex of a polytope. When maximizing $x_n$, smallest-index improving-facet pivoting visits all $2^n$ vertices of a polytope: first traverse the lower last-coordinate face, then move to the upper face and traverse the preceding path in reverse. Thus this geometric version of the Bland pivoting rule can require $2^n-1$ pivots. Rescaling the coordinate $x_i$ to $y_i={\epsilon }^{2(n-i)}{x}_i$ makes the positive rate per unit freed coordinate equal to ${\epsilon }^{-(n-i)}$; Dantzig pivot rule then selects the same exponential path. Euclidean edge-length pricing is a different rule.

In maximization, select the entering nonbasic variable with greatest positive reduced cost, meaning objective increase per unit of that variable. Unlike geometric steepest-edge pricing, this rate depends on variable and constraint scaling. A rescaled Klee-Minty cube can give the same $2^n-1$-pivot path as smallest-index improving-facet pivoting.

For a feasible simplex dictionary, let $r_j>0$ be a nonbasic variable's improving reduced cost and let ${\theta }_j$ be its largest feasible increase from the simplex ratio test. The greatest improvement pivot rule chooses an entering variable maximizing $r_j{\theta }_j$. It measures the improvement over the entire edge, whereas the Dantzig pivot rule maximizes only $r_j$.

// Target: mathematical-optimization.bigb

In a feasible minimization simplex dictionary, the nonbasic variables are nonnegative. If every reduced cost ${\bar{c}}_j\ge 0$, the displayed objective identity proves $z\ge z_0$ for every feasible point. Thus the current basic feasible solution is globally optimal. If all reduced costs are strictly positive, equality forces every nonbasic variable to vanish and gives a unique basic solution, provided the dictionary determines all basic variables.