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An Allee effect makes per-capita growth less favorable at low abundance. A strong effect permits a stable zero state and a stable positive state separated by a finite threshold. Cooperative infections can exhibit the same positive-feedback structure: neither disease invades alone, but sufficiently many joint infections sustain both.

For two otherwise symmetric SIS models with recovery rate $\gamma$, primary transmission $\beta$ and secondary susceptibility multiplier $q$, split hosts into uninfected, singly infected and doubly infected classes. The displayed conditions give stable disease-free and joint endemic states, with an intermediate saddle: a strong Allee effect. The endemic susceptible fraction solves $(q-1)S^2-qS+{\mathcal{R}}_0^{-2}=0$. The smaller root is the stable high-prevalence state. Coinfected hosts recover each infection independently, so recovery from coinfection leads to a single-infection class rather than directly to complete susceptibility.

For sexually reproducing worms, an infected host need not contain both sexes. Independent random sexes split a Poisson total burden of mean $m$ into independent male and female Poisson distributions of mean $m/2$, giving the displayed probability of a fertile mixed-sex burden. For an arbitrary burden probability generating function $G$, this probability is $1-2G(1/2)+G(0)$. Its low-burden behaviour can introduce an Allee effect in transmission, so an asexual mean-burden invasion calculation is not automatically valid.

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.