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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.

An effective reproduction number describes transmission in the current population, accounting for available susceptibility and transmission conditions. The phrase is used with different time conventions, so a definition must specify whether it means an instantaneous reproduction number or a case reproduction number. The CDC uses a time-varying effective quantity $R_t$ to describe the direction of community transmission. ^[ref].

For the closed, homogeneous SIR model with constant $\beta ,\gamma$, the instantaneous convention givesThus the infectious compartment grows when $R_{\mathrm{eff}}>1$ and declines when $R_{\mathrm{eff}}<1$, provided $i>0$. At the SIR infection peak it equals one. This freezes the current susceptible fraction; it does not count a newly infected person's actual future secondary infections while $s$ continues to decrease. The distinction comes directly from the SIR model equations and the two time conventions above.

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.

Homogeneous mixing assumes that contacts sample the whole modelled population without differences between groups or locations. In the SIR model, the chance that a sampled contact belongs to the infectious compartment is $i=I/N$. This assumption makes a frequency-dependent transmission rule depend on the overall infectious compartment rather than a graph of individual contacts. Structured contacts require a different mathematical model, such as a discrete SIR epidemic on a graph.

In infection modelling, immunity is protection against infection or disease, with its strength and duration represented by the chosen host states. A SIR model has a permanently protected recovered state, whereas a SIS model has no lasting post-recovery protection.

An infection involves invasion and multiplication of an infectious agent in a host. Bacteria are one possible cause; an agent capable of causing disease is a pathogen. The National Cancer Institute defines infection through invasion and growth in the body. ^[ref].

In a SIR model, acquiring infection moves a member of the susceptible compartment into the infectious compartment. That immediate transition is a model assumption: the SEIR model instead includes an exposed stage before infectiousness. Specify these states when translating a biological process into compartmental models.

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.

A microparasite is typically modelled as multiplying rapidly within a host, so infection status rather than the exact organism count is the first between-host state variable. SIS model, SIR model and SEIR model descriptions are examples. Additional within-host structure can still be required.

A microparasite typically multiplies within an infected host on a much shorter time scale than between-host transmission. A useful first model classifies hosts by infection status. A macroparasite, such as an adult parasitic worm, instead requires a count of parasites per host: two infected hosts can carry very different within-host parasite burdens. These are modelling distinctions rather than a rule determined only by an organism's physical size.

For a microparasite with no lasting immunity, a homogeneous SIS model isThe basic reproduction number is $\beta /\gamma$, and the prevalence $I/N$ describes the infected-host fraction. A SIR model adds an immune class, and a SEIR model adds a latent class. Contact structure, age, susceptibility, recovery rate and infectiousness can require several such classes. Then a next-generation matrix or operator, rather than a single averaged contact rate, sets the invasion threshold. Stochastic epidemic models are relevant to early extinction and finite populations; network or spatial models represent nonuniform contact opportunities.

For a simple macroparasite burden model, let $p_j$ be the fraction of hosts carrying $j$ adult worms. Assume acquisitions at rate $\lambda =\beta L$, independent adult-worm losses at rate $\mu j$, and a number $L$ of infective stages in a well-mixed environmental reservoir. The immigration-death worm-burden model hasThis birth-death process needs the burden distribution or its probability generating function, not just an infected/not-infected partition. Its mean $m={\sum }_{j}j{p}_j$ satisfies $\dot{m}=\beta L-\mu m$. With host number $N$ fixed and each adult producing infective stages at rate $\sigma$, one possible environmental closure isFor this asexual, unsaturated transmission approximation, one worm's basic reproduction number is $(\sigma /\mu )\beta N/(d_L+\beta N)$. Growth above the invasion threshold requires additional density regulation if a finite endemic burden is sought. Sex-dependent mating would change this closure.

With constant external exposure $\lambda$, the equilibrium burden follows a Poisson distribution with mean $m=\lambda /\mu$, giving infected-host prevalence $1-e^{-m}$. If exposure rates vary across hosts as a gamma distribution, the Poisson-gamma mixture gives a negative binomial distribution with mean $m$ and aggregation parameter $k$:The same mean burden can therefore coexist with very different prevalence and highly infected subgroups. Aggregation of macroparasite burdens affects treatment coverage, burden-dependent host mortality, immunity and parasite fecundity. For sexually reproducing worms, hosts need both sexes for fertile output; a mean alone may not determine it. With independent equally likely sexes and Poisson burden, the probability of both sexes is $(1-e^{-m/2}{)}^2$, illustrating mating-limited macroparasite transmission and its low-burden bottleneck.

Both microparasite and macroparasite models need heterogeneous exposure and host responses. The special extra difficulty for macroparasites is translating a distributed worm burden, and possibly worm sex or developmental stage, into transmission and host harm. Conversely, complicated within-host microparasite processes can also require more than a binary host state when the simple fast-within-host approximation fails.

Use fractions of all farms, so $S_n+I_n+R_n=P_n$ and ${\sum }_n{P}_n=1$. Assume farm size is fixed, contacts mix randomly, infectious periods have size-independent exponential mean $1/\gamma$, and the farm-size distribution has finite moments $\mu ={\sum }_{n}n{P}_n$ and ${\mu }_2={\sum }_n{n}^2{P}_n$. Write $I={\sum }_n{I}_n$. The farm-size stratified SIR model with susceptibility proportional to size isHere $\beta$ includes the common contact-rate constant; using frequency-dependent mixing avoids introducing an extra total-farm factor. An infectious farm of any size generates $\beta n{P}_n/\gamma$ infections in initially susceptible size-$n$ farms. The next-generation matrix $K_{nm}=\beta n{P}_n/\gamma$ has rank one, giving the basic reproduction numberThe comparison SIR model replaces every size by $\mu$, so it has the same basic reproduction number and the same initial exponential growth rate $\beta \mu -\gamma$.

As the epidemic develops, put $\Lambda (t)=\beta {\int }_0^{t}I(u)du$. For negligible initial infection, $S_n=P_n{e}^{-n\Lambda }$. Large farms leave the susceptible class faster. If ${\mu }_S={\sum }_{n}n{S}_n/S$ with $S={\sum }_n{S}_n$, differentiation givesThe mean size of newly infected farms is ${\mu }_{\mathrm{inc}}={\sum }_n{n}^2{S}_n/{\sum }_{n}n{S}_n$, larger than the susceptible mean unless all sizes agree. If ${\mu }_I={\sum }_{n}n{I}_n/I$, thenRecovery cancels from this equation because its rate is size-independent. Starting with representative infectives, ${\mu }_I$ initially rises toward the size-biased mean ${\mu }_2/\mu$; later depletion of large susceptible farms lowers ${\mu }_{\mathrm{inc}}$ and usually lowers ${\mu }_I$. It is incorrect to assert that the mean size of infectives must decrease from every possible initial condition. In the growing linearized mode, $I_n\propto n{P}_n$, so its mean is already ${\mu }_2/\mu$.

The aggregate growth equation is $\dot{I}=I(\beta S{\mu }_S-\gamma )$. The falling susceptible mean adds a depletion effect absent from the homogeneous SIR model. The final size relation for an epidemic, with vanishing initial seed and attack fraction $A$, isBy Jensen's inequality, the right side is at most $1-e^{-{\mathcal{R}}_{0}A}$, so $A$ is no larger than the homogeneous attack fraction. For a common small representative seed the illustrated epidemic is flatter and infects fewer farms; exact pointwise ordering of infected fractions for every time and arbitrary initial seed is not a consequence of equality of ${\mathcal{R}}_0$.

The original sketch uses equally many farms of sizes 1 and 3, recovery rate $\gamma =1$, and a representative infected seed of $0.001$. Each homogeneous comparison has ${\mathcal{R}}_0=2$. The left, middle and right columns correspond to parts (a), (b) and (c). The top row shows infected farm fractions, and the bottom row distinguishes the mean sizes of susceptible and infected farms. These are illustrative trajectories under the stated model, not universal empirical outbreak curves.

With susceptibility independent of size and infectivity proportional to size, define $J={\sum }_{n}n{I}_n$. The farm-size stratified SIR model isAn infectious size-$m$ farm has next-generation column $K_{nm}=(\beta /\gamma )P_{n}m$. Its sole nonzero eigenvalue gives the basic reproduction numberagain equal to the comparison SIR model.

Since every susceptible farm experiences the same force of infection, all ratios $S_n/S_m$ remain fixed. For a representative initial seed, $S_n(0)=(1-i_0)P_n$ and $I_n(0)=i_0{P}_n$, uniqueness of the differential equations gives $S_n=P_{n}S$, $I_n=P_{n}I$, $R_n=P_{n}R$ for all time. Consequently $J=\mu I$ andFor this representative seed, the heterogeneous and homogeneous epidemics coincide exactly, and both susceptible and infected farm means stay at $\mu$. This is the overlapping middle-column sketch above, not merely an approximation based on equal reproduction numbers.

With a nonrepresentative seed the susceptible mean remains its initial value, but the infected mean need not. The influx of new infectives has the constant susceptible mean, soThus it is pulled toward ${\mu }_S$ while incidence continues. Infectivity variation alone does not selectively remove large susceptible farms. The exact homogeneous reduction depends on the representative initial condition; specifying only $P_n$ and ${\mathcal{R}}_0$ does not determine a trajectory for every possible seed.

Both weights now equal farm size, giving the farm-size stratified SIR modelThe next-generation matrix is $K_{nm}=(\beta /\gamma )n{P}_{n}m$, whose nonzero eigenvalue isThe variance matters because large farms are both more likely to acquire infection and more effective sources afterward. This positive correlation increases the basic reproduction number relative to the homogeneous SIR model. In particular the heterogeneous population can permit invasion when the homogeneous comparison is subcritical.

Put $\Lambda =\beta {\int }_0^{t}J(u)du$. Apart from the negligible initial seed, $S_n=P_n{e}^{-n\Lambda }$. The susceptible mean obeys ${\dot{\mu }}_S=-\beta J{\mathrm{Var}}_S(n)$. For infectives, the incidence mean is again ${\mu }_{\mathrm{inc}}={\sum }_n{n}^2{S}_n/{\sum }_{n}n{S}_n$, andThe growing mode $I_n\propto n{P}_n$ has mean ${\mu }_2/\mu$, not $\mu$. As the larger susceptible farms are depleted, the mean size of incoming infectives falls; the current infected mean follows it with a delay. These equations give the direction of change for any initial condition without imposing false monotonicity of the infected mean.

The weighted infected total satisfies $\dot{J}=(\beta {\sum }_n{n}^2{S}_n-\gamma )J$, so the effective reproduction number declines from $\beta {\mu }_2/\gamma$ as the susceptible size distribution changes. The final size relation for an epidemic becomesThe right-column sketch illustrates the larger initial growth rate and preferential depletion of large farms. The homogeneous comparison has size $\mu$ in both transmission factors; it must not be assigned the heterogeneous second moment. The general conclusions are enhanced invasion and selective removal of large susceptible farms. The final unweighted fraction of farms infected has no universal ordering relative to the homogeneous comparison: higher transmission and stronger heterogeneity-induced depletion compete, and the displayed final-size equations determine the answer for a specified distribution. For example, with $P_1=P_3=1/2$, the infinitesimal-seed attack fractions are approximately $0.367$ versus $0.176$ when the homogeneous reproduction number is $1.1$, but $0.740$ versus $0.797$ when it is $2$; the two comparisons have opposite ordering.

The basic reproduction number is the expected value of secondary infections caused by one typical infectious individual introduced into an otherwise susceptible population. For a heterogeneous model it is the dominant eigenvalue of the next-generation operator, so the appropriate infected-type distribution is part of “typical.” A small introduction grows when ${\mathcal{R}}_0>1$ and decays when ${\mathcal{R}}_0<1$, under the usual linear invasion assumptions.

The asymptotic susceptible level depends on the model. In a homogeneous SIS model with transmission rate $\beta$ and recovery rate $\gamma$, the positive endemic equilibrium has $S_{\ast }=1/{\mathcal{R}}_0$ when ${\mathcal{R}}_0=\beta /\gamma >1$. In a closed SIR model, however, the final size relation for an epidemic isFor an infinitesimal seed in a wholly susceptible population this becomes $-\log S_{\infty }={\mathcal{R}}_0(1-S_{\infty })$. The epidemic peak occurs at $S=1/{\mathcal{R}}_0$, and a nontrivial outbreak leaves $S_{\infty }<1/{\mathcal{R}}_0$. Thus $1/{\mathcal{R}}_0$ is an endemic or instantaneous invasion threshold, not a universal final susceptible fraction.

For cooperative infections, let $s,x,y,z$ be the fractions carrying neither infection, only $A$, only $B$, and both, with sum one. Let ${\lambda }_A={\beta }_A(x+z)$, ${\lambda }_B={\beta }_B(y+z)$, assume each infection independently recovers at rate $\gamma$, and multiply susceptibility to the second infection by $q>1$. One suitable pair of SIS models isRecovery of one infection from $z$ leads to the other single-infection class, not directly to $s$. The probability simplex is invariant. At the disease-free equilibrium the eigenvalues are ${\beta }_A-\gamma$, ${\beta }_B-\gamma$ and $-2\gamma$. Therefore failure of both isolated and low-density invasion requires ${\mathcal{R}}_A={\beta }_A/\gamma <1$, ${\mathcal{R}}_B={\beta }_B/\gamma <1$. Enhancement of secondary susceptibility does not alter these linear thresholds because coinfection is second order at introduction.

For completeness, a joint positive equilibrium must haveandTo obtain these, divide the equilibrium equations for total $A$ and total $B$ prevalence by their positive prevalences, yielding $s+qy={\mathcal{R}}_A^{-1}$ and $s+qx={\mathcal{R}}_B^{-1}$. Substitution into the $s$ balance gives the quadratic. Its discriminant must be nonnegative and the resulting $x,y,z$ positive. These conditions test equilibrium feasibility for unequal rates; stability must also be checked rather than inferred from the mere presence of a root.

A fully explicit Allee effect results in the symmetric permitted case ${\beta }_A={\beta }_B=\beta$, $\mathcal{R}=\beta /\gamma$. The joint equilibria haveTwo feasible positive equilibria coexist with the stable disease-free state exactly in the strict rangeIndeed at the fold the two roots coalesce, and at $\mathcal{R}=1$ the smaller-prevalence branch reaches the disease-free boundary. For $q\le 2$, roots at $\mathcal{R}<1$ do not give positive singly infected fractions. Strict inequalities exclude these marginal endpoints.

Here is the stability check underlying the cooperative SIS coinfection threshold. On the symmetric subspace put $v=x+z=y+z$ and $c=z$. ThenAt a positive equilibrium $c=q\mathcal{R}{v}^2/(1+q\mathcal{R}v)$. Its two-dimensional Jacobian matrix hasThe higher-prevalence branch has positive determinant and negative trace, while the lower-prevalence branch is a saddle equilibrium. The remaining antisymmetric mode, from $x-y$, has eigenvalue $-q\beta v<0$. Thus the smaller susceptible root $s_{-}$ is a stable joint endemic equilibrium, separated from the stable zero state by a finite seeding threshold. Neither infection can persist alone in this range. Strong cooperation can sustain both infections after a sufficiently large joint introduction even though neither invades from rarity. The numerical condition belongs to the specified secondary-susceptibility model; alternative cooperation mechanisms need their own threshold calculation.

In the SIR model, $S,I,R$ count susceptible, infectious, and recovered immune individuals. The closed population has $S+I+R=N+1$. Under homogeneous mixing, use the paper's mass-action infection convention: each susceptible has infection hazard function $\beta I$, so the population infection rate is $\beta SI$. Each infectious individual recovers with hazard function $\gamma$, giving population recovery rate $\gamma I$ and mean infectious duration $1/\gamma$.

The arrows are one-way: recovered individuals are immune, and there are no demographic entries or exits. The transition intensities attached to the arrows are population flow rates, rather than individual hazard functions.

The deterministic SIR model follows the ordinary differential equationsUse $\beta \ge 0$ and $\gamma >0$. The state space is $S,I,R\ge 0$, $S+I+R=N+1$; summing the three equations proves conservation of this population. Infection is supported only where $S>0$ and $I>0$, and recovery only where $I>0$. The rates vanish at the corresponding boundaries, preventing negative compartment sizes.

This convention has an unnormalized population infection rate $\beta SI$, as required by the printed threshold. A convention with infection rate $\beta SI/(N+1)$ would use a differently scaled parameter.

Whenever $I>0$, divide the first SIR model equation by the recovery equation:The SIR susceptible-recovered identity follows by integrating from $R=0,S=N$, giving susceptibles as a function of recovered individuals:Population conservation then gives the infectious count:These identities extend continuously to the limiting state where the infectious count vanishes.

The infectious count in the SIR model obeys $\dot{I}=I(\beta S-\gamma )$. At the initial state, $\dot{I}(0)=\beta N-\gamma$, so initial epidemic growth requires the epidemic threshold for the closed SIR modelHere ${\mathcal{R}}_0$ is the basic reproduction number in this mass-action infection convention. Necessity also follows globally: $S(t)$ is nonincreasing, so if $\beta N\le \gamma$, then $\beta S(t)-\gamma \le 0$ for all time and $I(t)$ never increases. When $\beta =0$, growth is impossible.

The SIR model isThe expected infectious duration is $d=1/\gamma$.