2.1 Ordered set

Let $(E, \leq )$ be a (partially) ordered set. Whenever it exists, the supremum of $A\subset E$ , denoted by $\sup (A)$ , is the least upper bound of A: for all $x\in A$ , $x \leq \sup (A)$ and if for some $z\in E$ one has $x\leq z$ for all $x\in A$ , then $\sup (A) \leq z$ . A collection $(x_i)_{i \in \mathcal {I}}$ of elements of E is an antichain if for all distinct $i, j \in \mathcal {I}$ , the elements $x_i$ and $x_j$ are not comparable for the order relation.

2.2 Banach space and Banach lattice

Let $(X, \left \lVert \,\cdot \,\right \rVert )$ be a complex Banach space not reduced to $\{0\}$ . An operator T on X is a bounded linear (and thus continuous) map from X to itself. If $Y \subset X$ is a closed subspace of X such that $T(Y) \subset Y$ , we denote $T|_Y$ the restriction of T to the subspace Y, that is, an operator on the Banach space $(Y, \left \lVert \,\cdot \,\right \rVert )$ . The operator norm of T is given by

(2.1) $$ \begin{align} \left\lVert\,T\,\right\rVert_X= \sup\left\{ \left\lVert\,Tx\,\right\rVert\, \colon\, x \in X \text{ s.t. } \left\lVert\,x\,\right\rVert = 1\right\}, \end{align} $$

its spectrum by $\operatorname {\mathrm {Sp}}(T) = \{\lambda \in {\mathbb {C}}\, \colon \, T - \lambda \operatorname {\mathrm {Id}} \text { is not bijective}\}$ , where $\operatorname {\mathrm {Id}}$ is the identity operator on X. If $\lambda \in {\mathbb {C}}$ and $x\in X\backslash \{0\}$ satisfy $Tx=\lambda x$ , then the element x is an eigenvector of T and $\lambda $ , which belongs to $\operatorname {\mathrm {Sp}}(T) $ , is an eigenvalue of T. The spectral radius of T is defined by (see [Reference Rudin47, Theorem 18.9])

(2.2) $$ \begin{align} \rho(T) = \sup\left\{ |\lambda| \,\colon\, \lambda \in \operatorname{\mathrm{Sp}} (T)\right\} =\lim_{n\rightarrow \infty } \left\lVert\,T^n\,\right\rVert_X^{1/n}. \end{align} $$

By convention, we set $T^0=\operatorname {\mathrm {Id}}$ . The spectral radius is commutative in the sense that if T and S are two operators on X, we have

(2.3) $$ \begin{align} \rho(TS) = \rho(ST). \end{align} $$

We define the spectral bound of the operator T by

(2.4) $$ \begin{align} s(T) = \sup \{{\mathcal{R}e}(\lambda) \,\colon\, \lambda \in \operatorname{\mathrm{Sp}}(T)\}. \end{align} $$

Let $X^\star $ denote the (continuous or topological) dual Banach space of X, that is, the set of all the continuous linear forms on X. For $x\in X$ , $x^\star \in X^\star $ , let $\langle x^\star , x \rangle $ denote the duality product and the norm of $x^\star $ in $X^\star $ is defined by $\left \lVert \,x^\star \,\right \rVert =\sup \{\langle x^\star , x \rangle \, \colon \, \left \lVert \,x\,\right \rVert =1\}$ . For an operator T, the dual operator $T^\star $ on $X^\star $ is defined by $\langle T^\star x^\star , x \rangle =\langle x^\star , Tx \rangle $ for all $x\in X$ , $x^\star \in X^\star $ . It is well known that $\left \lVert \,T^\star \,\right \rVert _{X^\star }=\left \lVert \,T\,\right \rVert _X$ and $\operatorname {\mathrm {Sp}}(T^\star ) = \operatorname {\mathrm {Sp}}(T)$ .

An ordered real Banach space $(X, \left \lVert \,\cdot \,\right \rVert , \leq )$ is a real Banach space $(X, \left \lVert \,\cdot \,\right \rVert )$ with an order relation $\leq $ satisfying:

1. (1) for any $x, y, z \in X, \lambda \geq 0$ such that $x \leq y$ , we have $x + z \leq y + z$ and $\lambda x \leq \lambda y$ .

For any $x \in X$ , we define $|x| = \sup (\{x, -x\})$ the supremum of x and $-x$ whenever it exists. Following [Reference Schaefer48, Section 2], the ordered Banach space $(X, \left \lVert \,\cdot \,\right \rVert , \leq )$ is a Banach lattice if:

1. (1) for any $x, y \in X$ , there exists a supremum of x and y in X;

2. (2) for any $x, y \in X$ so that $|x| \leq |y|$ , we have $\left \lVert \,x\,\right \rVert \leq \left \lVert \,y\,\right \rVert $ .

Let $(X, \left \lVert \,\cdot \,\right \rVert , \leq )$ be a real Banach lattice. We denote $X_+ = \{x \in X \,\colon \, x \geq 0 \}$ the positive cone of X. Recall it is a closed set. We shall also consider the dual cone $X^\star _+=\{x^\star \in X^\star \, \colon \, \langle x^\star , x \rangle \geq 0 \text { for all } x\in X_+\}$ . A linear map T on X is positive if $T(X_+) \subset X_+$ . According to [Reference Aliprantis and Burkinshaw2, Theorem 4.3] positive linear maps on Banach lattices are bounded (and thus are operators).

If S and T are two operators on X, we write $T \leq S$ if the operator $S - T$ is positive. If the operators $T, S$ , and $S-T$ are positive, then we have, see [Reference Marek37, Theorem 4.2]:

(2.5) $$ \begin{align} \rho(T) \leq \rho(S). \end{align} $$

Any real Banach lattice X and any operator T on X admits a natural complex extension. The spectrum of T will be identified as the spectrum of its complex extension and denoted by $\operatorname {\mathrm {Sp}}(T)$ , furthermore, by [Reference Abramovich and Aliprantis1, Lemma 6.22], the spectral radius of the complex extension is also given by $\lim _{n\rightarrow \infty } \left \lVert \,T^n\,\right \rVert _X^{1/n}$ , with $\left \lVert \,\cdot \,\right \rVert _X$ still defined by (2.1). Moreover, by [Reference Abramovich and Aliprantis1, Corollary 3.23], if T is positive (seen as an operator on the real Banach lattice X), then T and its complex extension have the same norm.

2.3 Lebesgue spaces and essential limits

Let $(\Omega , \mathcal {G}, \mu )$ be a measured space with $\mu $ a finite measure. For any $\mathcal {A} \subset \mathcal {G}$ , we denote by $\sigma (\mathcal {A})$ the $\sigma $ -field generated by $\mathcal {A}$ . If f and g are two real-valued measurable functions defined on $\Omega $ , we write $f \leq g$ a.e. (resp. $f = g$ a.e.) when $\mu (\{f> g\}) = 0$ (resp. $\mu (\{f \neq g \}) = 0$ ), and denote $\operatorname {\mathrm {supp}}(f )= \{f \neq 0\}$ the support of f. We say that a real-valued measurable function f is nonnegative when $f \geq 0$ a.e., we say that f is positive, denoted $f> 0$ a.e., when $\mu (\{f \leq 0\}) = 0$ , and we say that f is bounded if there exists $M \geq 0$ such that $|f| \leq M$ a.e. In what follows, the inclusion of measurable sets will be understood up to sets of $\mu $ -zero measure. More precisely, if $A, B \subset \Omega $ are measurable sets, we write $A \subset B$ a.e. (resp. $A = B$ a.e.) when a.e. (resp. a.e.). Let $L^0(\Omega , \mathcal {G}, \mu )$ , simply denoted $L^0$ , be the set of $[-\infty , + \infty ]$ -valued measurable functions defined on $\Omega $ , where functions which are a.e. equal are identified. The elements and of $L^0$ denote the functions which are a.e. equal, respectively, to 1 and to 0. For the sake of clarity, we will omit to write a.e. in the proofs.

Let $(f_t)_{t\in T}$ be a family of measurable functions defined on $\Omega $ taking values in $[-\infty , +\infty ]$ . We recall that $f^*=\operatorname *{\mbox {ess sup}}_{t\in T} f_t$ is a measurable function such that $f_t\leq f^*$ a.e. for all $t\in T$ and if f is measurable function such that $f_t\leq f$ a.e. for all $t\in T$ then $f ^*\leq f$ a.e. (if T is at most countable, then one can take $\operatorname *{\mbox {ess sup}}_{t\in T} f_t=\sup _{t\in T} f_t$ ). We now consider $T={\mathbb {R}}_+$ . Let $(f_t)_{ t\in {\mathbb {R}}_+}$ be a non-decreasing family, in the sense that for all $t\leq s$ we have $f_t\leq f_s$ a.e., then if $(t_n)_{ n\in {\mathbb {N}}}$ is a sequence converging to $+\infty $ , we have that the sequence $(f_{t_n})_{ n\in {\mathbb {N}}}$ converges a.e. toward $f^*=\operatorname *{\mbox {ess sup}}_{t\in {\mathbb {R}}_+} f_t$ , and thus we shall simply write $f^*=\operatorname *{\mbox {lim ess}}_{t\rightarrow +\infty } f_t$ . We leave to the reader the definition of $\operatorname *{\mbox {ess inf}}$ and the limit of a non-increasing sequence of measurable functions. For the family $(f_t)_{ t\in {\mathbb {R}}_+}$ , we consider $f^*_t=\operatorname *{\mbox {ess sup}}_{s\geq t} f_s$ for all $t\in {\mathbb {R}}_+$ , and get that the sequence $(f^*_t)_{t\in {\mathbb {R}}^+}$ is non-increasing and write $\operatorname *{\mbox {lim ess sup}}_{t\rightarrow \infty } f_t= \operatorname *{\mbox {lim ess}}_{t\rightarrow \infty } f^*_t$ . We define in a similar way $\operatorname *{\mbox {lim ess inf}}_{t\rightarrow \infty } f_t$ . Notice that if $g_t=f_t$ a.e. for all $t\in {\mathbb {R}}_+$ , then the essential supremum/infimum limits of $(f_t)_{t\in {\mathbb {R}}^+}$ and $(g_t)_{t\in {\mathbb {R}}^+}$ are a.e. equal. Therefore, the essential supremum/infimum limits of sequences is well defined on the space $L^0$ . We say the sequence of functions $(f_t)_{ t\in {\mathbb {R}}_+}$ in $L^0$ essentially converges if $\operatorname *{\mbox {lim ess sup}}_{t\rightarrow \infty } f_t=\operatorname *{\mbox {lim ess inf}}_{t\rightarrow \infty } f_t$ in $L^0$ , and write $\operatorname *{\mbox {lim ess}}_{t\rightarrow +\infty } f_t$ for this common limit (which is an element of $L^0$ ). When considering $T={\mathbb {N}}$ instead of $T={\mathbb {R}}_+$ , the analog of the essential convergence is the a.e. convergence, that is, the usual convergence in $L^0$ .

For a measurable function f, we write $\mu (f)=\int f\, \mathrm {d} \mu =\int _\Omega f(x)\, \mu (\mathrm {d} x)$ the integral of f with respect to $\mu $ when it is well defined. When f is measurable and a.e. finite and nonnegative, we denote $f \mu $ the measure on $(\Omega , \mathcal {G})$ defined by for any measurable set A. For , the Lebesgue space $L^p(\Omega , \mathcal {G}, \mu )$ is the set of all real-valued measurable functions $f\in L^0$ defined on $\Omega $ whose $L^p$ -norm, $\left \lVert \,f\,\right \rVert _p = \mu (|f|^p)^{1/p}$ if $p<+\infty $ and $\left \lVert \,f\,\right \rVert _\infty = \operatorname *{\mbox {ess sup}}(|f|)$ if $p=+\infty $ , is finite. When there is no ambiguity we shall simply write $L^p(\Omega )$ , $L^p(\mu )$ or $L^p$ for $L^p(\Omega , \mathcal {G}, \mu )$ . The Banach space $L^p$ endowed with the usual order $f\leq g$ , that is, $\mu (\{f>g\})=0$ , is a Banach lattice. The positive cone $L^p_+$ is the subset of $L^p$ of nonnegative functions; it is normal (as the norm $\left \lVert \,\cdot \,\right \rVert $ is monotonic, that is, $0\leq f\leq g$ implies $\left \lVert \,f\,\right \rVert _p\leq \left \lVert \,g\,\right \rVert _p$ , see [Reference Deimling15, Proposition 19.1]) and reproducing (that is, $L^p_+ -L^p_+=L^p$ ). Since the supports of two functions, which are a.e. equal are also a.e. equal, we get that the support of $f\in L^p$ is well defined up to the a.e. equality; it will still be denoted by $\operatorname {\mathrm {supp}}(f)$ . For $p\in [1, +\infty )$ , the dual of $L^p$ is $L^q$ where $1/p+1/q=1$ , with the duality product $\langle g, f \rangle = \int fg\, \mathrm {d} \mu $ for $f \in L^p$ and $ g \in L^q$ (for $p=1$ , we use that the measure $\mu $ is finite).

For any $f \in L^\infty $ , we denote by $M_f$ the multiplication by f, which can be seen as an operator on $L^p$ for $p \in [1,+\infty ]$ . For $A \in \mathcal {G}$ a measurable set, we denote:

(2.6)

Let T be an operator on $L^p$ . The projection of T on A, denoted $T_A$ , is the operator defined by

(2.7) $$ \begin{align} T_A=M_A \, T\, M_A, \end{align} $$

and, if $\mu (A)>0$ , we denote by $T|_A$ the restriction of the operator $T_A$ to $L^p(A)$ , where the set A is endowed with the trace of $\mathcal {G}$ on A and the measure $\mu |_A (\cdot )=\mu (A\cap \cdot )$ .

We now assume that $\mu (\Omega )>0$ , so that $L^p(\Omega )$ is not reduced to a singleton. When there is no ambiguity on the operator T, we simply write $\rho (A)$ for the spectral radius of $T_A$ (and of $T|_A$ when $\mu (A)>0$ ). In particular, we have $\rho (\Omega )=\rho (T)$ and $\rho (A) = 0$ if $\mu (A) = 0$ . If the operator T is positive, we also have that

$$\begin{align*}A \subset B \quad \Longrightarrow \quad \rho(A) \leq \rho(B). \end{align*}$$

2.4 Decomposition of positive operators on $L^p$ , with $p\in (1,+\infty )$

Recall the measure $\mu $ is finite and nonzero. We recall the atomic decomposition from Schwartz [Reference Schwartz50] of a positive operator T on $L^p$ (see also [Reference Delmas, Lefki and Zitt19]). A measurable set $A \in \mathcal {G}$ is T-invariant, or simply invariant when there is no ambiguity, if $M_{A^c} T M_A = 0$ , which, see [Reference Delmas, Lefki and Zitt19, Equation (7)], is equivalent to

(2.8) $$ \begin{align} \langle g , Tf \rangle=0, \end{align} $$

for all $f \in L^p_+$ and $ g \in L^q_+$ such that $\operatorname {\mathrm {supp}}(f) \subset A$ a.e. and $\operatorname {\mathrm {supp}}(g) \subset A^c$ a.e. The operator T is irreducible if its only invariant sets are a.e. equal to $\Omega $ or $\emptyset $ . A measurable set A with positive measure is irreducible if the operator $T|_A$ on $L^p(A)$ is irreducible. The future of a set $A \subset \Omega $ , denoted $\mathcal {F}(A)$ , is the smallest invariant set that contains A. If $\mathcal {C}$ is an at most countable collection of subsets of $\Omega $ , then we denote by $\mathcal {F}(\mathcal {C})$ the future of the union of the elements of $\mathcal {C}$ :

(2.9) $$ \begin{align} \mathcal{F}(\mathcal{C})= \mathcal{F}\left( \bigcup_{A\in \mathcal{C}} A\right) = \bigcup_{A\in \mathcal{C}} \mathcal{F}(A), \end{align} $$

where the last equality is [Reference Delmas, Lefki and Zitt19, Lemma 3.13].

A set is admissible if it belongs to the $\sigma $ -field generated by the invariant sets. An atom is a minimal admissible set with a positive measure (that is, A is an atom if A is admissible, $\mu (A)>0$ and if B is an admissible set such that $B\subset A$ a.e. then a.e. $B=\emptyset $ or $B=A$ ), and we identify two atoms that are a.e. equal. In particular, if the set A is an atom and B is admissible then we have

$$\begin{align*}\mu(A \cap B)>0 \implies A\subset B \quad\text{a.e.} \end{align*}$$

According to [Reference Delmas, Lefki and Zitt19, Theorem 1], a measurable set is admissible and irreducible with positive measure if and only if it is an atom. Since the atoms have positive measure, we get that the set of atoms (up to the a.e. equality), $\mathfrak {A}$ , is at most countable. We shall also consider the (at most countable) set of nonzero atoms:

$$\begin{align*}\mathfrak{A}^*=\{ A \in \mathfrak{A}\, \colon\, \rho(A)>0\}. \end{align*}$$

The relation $\preccurlyeq $ on $\mathfrak {A}$ , defined by $A \preccurlyeq B$ if $\mathcal {F}(A) \subset \mathcal {F}(B)$ a.e. (or equivalently $A \subset \mathcal {F}(B)$ a.e.) is an order relation. We end this section by noticing that antichains of atoms are characterized by their future.

Lemma 2.1 (Antichains with same future)

Let $\mathcal {C}$ and $\mathcal {C}'$ be two antichains of atoms. Then, we have

$$\begin{align*}\mathcal{F}(\mathcal{C})=\mathcal{F}(\mathcal{C}')\quad \Longleftrightarrow \quad \mathcal{C}=\mathcal{C}'. \end{align*}$$

2.5 Power compact operators on $L^p$

A linear map T on a Banach space is compact if the image of the unit ball is relatively compact; it is then bounded. An operator T on a Banach space is power compact if there exists $k \in {\mathbb {N}}^*$ such that $T^k$ is compact. We recall some well-known properties of power compact operators (see [Reference Dunford and Schwartz21, Reference König31] for instance).

It is well known that the spectral radius (and more generally the spectra) is a continuous function on the set of compact operators with respect to the operator norm (see [Reference Newburgh41, Theorem 11]). We shall, however, use a weaker result from Anselone [Reference Anselone3]. We say a family $\mathcal {V}$ of operators on X is collectively compact if $\bigcup _{V\in \mathcal {V}} V(B)$ is relatively compact, where B is the unit ball of X. The following result is a direct consequence of Proposition 4.1 and Theorem 4.16 in [Reference Anselone3].

We give a result on compact operators in Lebesgue space. Recall that $\mu $ is a nonzero finite measure on $\Omega $ , and that $L^p$ denote $L^p(\Omega , \mathcal {G}, \mu )$ .

We recall in our framework some classical results, see [Reference Delmas, Lefki and Zitt19, Theorem 6.2 and Lemma 6.5].

Following [Reference Delmas, Dronnier and Zitt18, Lemma 4.2], we now state a technical result based on Collatz–Wielandt inequality and the Krein–Rutman theorem, giving a bound on the spectral radius given a strict supersolution of the eigenvalue equation.

Lemma 2.6 (Supersolutions and spectral radius)

Let S be a positive operator on $L^p$ with $p \in (1, + \infty )$ . If there exists $\lambda>0$ and a non-null nonnegative function $v \in L^p_+$ such that $Sv \geq \lambda v$ , then:

$$\begin{align*}\rho(S)\geq \rho(S_A) \geq \lambda, \end{align*}$$

where $A=\operatorname {\mathrm {supp}}(v)$ . Furthermore, if S is power compact, then we have

1. (1) If $Sv= \lambda v$ on A, then $\rho (S_A) = \lambda $ .

2. (2) If $Sv - \lambda v$ is positive on A, then $\rho (S_A)> \lambda $ .

Proof We first note that $\rho (S)\geq \rho (S_A)$ by (2.5). Multiplying the inequality $Sv\geq \lambda v$ by yields that $S_A v \geq \lambda v$ . The fact that $\rho (S_A)\geq \lambda $ , and thus $\rho (S) \geq \lambda $ , is then a direct consequence of the Collatz–Wielandt inequality [Reference Förster and Nagy22, Propositions 2.1 and 2.2].

We now assume that S is power compact. Since $(S_A)^\star = (S^\star )_A$ , we shall denote them simply by $S^\star _ A$ . Since $S_A\leq S$ , we deduce from Lemma 2.4 that the operator $S_A$ , and thus $S_A^\star $ , is power compact. We apply the Krein–Rutman theorem (Theorem 2.5 (1)) to the power compact operator $S_A^\star $ , which has a positive spectral radius $\rho (S_A^\star ) = \rho (S_A)>0$ : there exists $w \in L^q_+$ a non-null eigenfunction of $S_A^\star $ related to the eigenvalue $\rho (S_A)$ . Since , we deduce that $\operatorname {\mathrm {supp}}(w) \subset A$ and thus $\langle w,v \rangle>0$ .

We now consider the following duality product:

(2.11) $$ \begin{align} \left\langle w,S_Av - \lambda v\right\rangle = \left\langle S_A^\star w,v\right\rangle - \lambda\left\langle w,v\right\rangle = (\rho(S_A) - \lambda)\left\langle w,v\right\rangle. \end{align} $$

Under the hypothesis that $Sv = \lambda v$ on $A=\operatorname {\mathrm {supp}}(v)$ , the left-hand side of (2.11) is zero, and thus $\rho (S_A)=\lambda $ . This gives Point (1).

To prove Point (2), since $S_Av-\lambda v$ is by hypothesis positive on A, and is nonnegative and not identically equal to zero, we get $\left \langle w,S_Av - \lambda v\right \rangle>0$ . Then use (2.11) again to get that $\rho (S_A)>\lambda $ .

2.6 Operators related to the SIS model

Let $(\Omega , \mathcal {G}, \mu )$ be a measured space with finite nonzero measure $\mu $ , that is, $\mu (\Omega )\in (0, +\infty )$ . Recall we simply write $L^p$ for $L^p(\Omega , \mathcal {G}, \mu )$ . Observe that $L^\infty \subset L^p$ for $p \in (1,+\infty )$ and that the identity map $\iota $ from $L^\infty $ to $L^p$ is a bounded injection.

Let $p\in (1, +\infty )$ and $\gamma \in L^\infty $ such that $\gamma>0$ a.e. Let T be a positive operator on $L^\infty $ such that (1.5) holds for all $f\in L^\infty $ . In particular, as $L^\infty $ is dense in $L^p$ (for the $L^p$ -norm), we can extend T by density into a bounded linear map $\hat {T}$ on $L^p$ . Recall that $M_{1/\gamma }$ denotes the multiplication by $1/\gamma $ . Notice that under Assumption 2, see (1.6), the linear map $TM_{1/\gamma }$ , denoted by $T_{1/\gamma }$ , can be seen as a bounded linear map on $L^\infty $ , and it can also be extended by density into a bounded linear map $\hat {T}_{1/\gamma }$ on $L^p$ (see also Figure 3).

Figure 3:

Operators related to T in the SIS model $(T, \gamma , \varphi )$ .

4.1 Equilibria, supports, and spectral radius

For any $g \in \Delta $ , let $L_g$ denote the compact operator on $L^p$ defined by

(4.1) $$ \begin{align} L_g = M_{\varphi(g)} \, \hat{T}_{1/\gamma}. \end{align} $$

This operator is associated with the linearization $M_{\varphi (g)} T - \gamma $ of the dynamics near g in the same way as $\hat {T}_{1/\gamma }$ is associated with the linearization $T - \gamma $ of the dynamics near $0$ (as $\varphi (0)=1$ ). Notice that when $(T, \gamma , \varphi )$ satisfy Assumption 2, then $(M_{\varphi (g)} T, \gamma , \varphi )$ also does. It is immediate to check that for $g\in \Delta $ :

(4.2) $$ \begin{align} g\text{ is an equilibrium} \quad \Longleftrightarrow\quad L_g \, (\gamma g)=\gamma g. \end{align} $$

As a consequence of Point (3), we directly get the following corollary.

Proof of Lemma 4.1

Since $\varphi (g) T(g) = \gamma g$ , $\varphi (g)$ does not vanish on $\operatorname {\mathrm {supp}}(g)$ . On the complement set, $g=0$ so $\varphi (g)=1$ does not vanish either. Since $\varphi $ is nonnegative, we get Point (1).

Point (2) is a direct consequence of the characterization of invariant sets given by (2.8) as $1/\gamma $ and $\varphi (g)$ are positive by Point (1) and Theorem 2.5 (2) on nonzero atoms. Point (3) is a direct consequence of (4.2), [Reference Delmas, Lefki and Zitt19, Lemma 3.6] (which state that the support of the eigenfunction $\gamma g$ is $L_g$ -invariant) and Point (2).

Point (1) follows directly from Lemma 2.6 with $S=L_g$ , $v=\gamma g$ , and $\lambda = 1$ .

The proof of Point (2) follows similar lines. Let $h\leq g$ be two equilibria with . The two eigenvalue equations written for h and g yield $ (L_g-L_h) (\gamma g) + L_h(\gamma (g-h)) = \gamma (g-h)$ , which we rewrite as

$$\begin{align*}L_h (\gamma(g-h)) = \gamma (g-h) + (\varphi(h)-\varphi(g)) Tg. \end{align*}$$

On the right-hand side, $\varphi (h) - \varphi (g)$ and $Tg$ are nonnegative, and by strict monotonicity of $\varphi $ they are both positive on $B = \operatorname {\mathrm {supp}}(g-h) \subset \operatorname {\mathrm {supp}}(g)$ . If B is empty, then $g=h$ and we are back to Point (1). If not, we apply Lemma 2.6 (2) to $S = L_h$ , $\lambda = 1$ , and which is nonnegative: $\rho ((L_h)_B)> 1$ . If B was a subset of $\operatorname {\mathrm {supp}}(h)$ this would imply $\rho ((L_h)_{\operatorname {\mathrm {supp}}(h)})>1$ , a contradiction with Point (1). So B is not a subset of $\operatorname {\mathrm {supp}}(h)$ , or in other words $\operatorname {\mathrm {supp}}(h) \neq \operatorname {\mathrm {supp}}(g)$ .

Finally, let us prove Point (3). Clearly, $h\leq g$ implies that $\operatorname {\mathrm {supp}}(h) \subset \operatorname {\mathrm {supp}}(g)$ . To prove the reverse implication, let us assume that $\operatorname {\mathrm {supp}}(h)\subset \operatorname {\mathrm {supp}}(g)$ . Since $F( \max (g,h)) \geq 0$ , by Lemmas 8.5 (2) and 8.6, the semi-flow starting from $\max (g,h)$ is non-decreasing and converges to an equilibrium $\tilde {g}$ , which therefore satisfies $\max (g,h) \leq \tilde {g}$ . Since $\operatorname {\mathrm {supp}}(\max (g,h)) = \operatorname {\mathrm {supp}}(g)$ is invariant by Point (3), we deduce from Lemma 3.5 (3) that $\operatorname {\mathrm {supp}}(\tilde g)\subset \operatorname {\mathrm {supp}}(g)$ and thus $\operatorname {\mathrm {supp}}(\tilde {g}) = \operatorname {\mathrm {supp}}(g)$ . Since $g\leq \tilde {g}$ , by the previous point, the functions g and $\tilde {g}$ must be equal, so $g = \max (g,h)$ , or in other words $h\leq g$ .

4.2 Maximum equilibria and critical vaccination

Recall the notations of Section 2.6. Let A be a measurable set. Notice the linear map $(\hat {T}_{1/\gamma })_A = \widehat {(T_A)}_{1/\gamma }$ is an operator on $L^p$ . Following [Reference Delmas, Dronnier and Zitt17], we then define the basic reproduction number of A as the spectral radius of this operator $R_0(A) = \rho ((\hat {T}_{1/\gamma })_A)$ , and simply write $R_0$ for $R_0(\Omega )$ . Notice that, by (2.5), the map $A \mapsto R_0(A)$ is non-decreasing, that is, for any $A, B$ measurable sets with $A \subset B$ a.e., we have $R_0(A) \leq R_0(B)$ .

The following result generalizes [Reference Delmas, Dronnier and Zitt17, Theorems 4.7 and 4.13], and is proved similarly, see Section 9 for details.

In the next remarks and examples, we explore the uniformity of the convergence in Point (1) and Proposition 3.1 (3). See also the summarized results from Proposition 5.5.

Remark 4.4 (Exponential rate of convergence to when $R_0<1$ and $\operatorname *{\mbox {ess inf}} \gamma> 0$ )

Assume $(T, \gamma , \varphi )$ satisfies Assumption 2, $R_0<1$ and $\operatorname *{\mbox {ess inf}} \gamma> 0$ . By Lemma 9.2, we also have $s(T - \gamma ) < 0$ . Then, mimicking the proof of [Reference Delmas, Dronnier and Zitt17, Theorem 4.6] and using that $1-\varphi \geq 0$ , we get that for all $c \in (0, -s(T - \gamma ))$ , there exists $\theta \in {\mathbb {R}}_+$ such that, for all $h \in \Delta $ , $t \geq 0$ , we have

$$\begin{align*}\left\lVert\,\phi(t,h)\,\right\rVert_\infty \leq \theta \left\lVert\,h\,\right\rVert_\infty \, \mathrm{e}^{-ct}. \end{align*}$$

Example 4.6 (A uniform convergence when $\operatorname *{\mbox {ess inf}} \gamma =0$ )

If $\operatorname *{\mbox {ess sup}} g^*=1$ and $\operatorname *{\mbox {ess inf}} \gamma = 0$ it is possible for the semi-flow

to converge to $g^*$ in $L^\infty $ . Consider the particular case: $\Omega =(0, 1]$ with $\mu =\nu + \delta _1$ , where $\nu $ is the Lebesgue measure and $\delta _1$ the Dirac mass at $1$ ;

for all $f\in L^p$ ; $\gamma (x)=x/2$ and $\varphi (r)=1-r$ . In this case, $\{1\}$ is the only atom, and $R_0(\{1\})=2$ . We get $g^*(x)=1/(1+x)$ and thus $\operatorname *{\mbox {ess sup}} g^*=1$ . (Notice that $g^*(0+)=\operatorname *{\mbox {ess sup}} g^*$ and $\gamma (0+)=\operatorname *{\mbox {ess inf}} \gamma $ .) Elementary calculus give that, for $t\geq 0$ ,

and for $x\in (0, 1)$ :

so that

. So the semi-flow

converges to $g^*$ in $L^\infty $ .

We now focus on critical vaccination. Let $\varphi _0$ defined by $\varphi _0(r)=1 -r$ and $\mathrm {T}_k$ the kernel operator from Remark 1.4 with k and $\gamma $ satisfying (1.7) so that $(\mathrm {T}_k, \gamma , \varphi _0)$ satisfies Assumption 2. Let $\eta \in \Delta $ seen as a perfect vaccination strategy: the SIS model $(\mathrm {T}_k M_\eta , \gamma , \varphi _0)$ (which indeed satisfies Assumption 2) corresponds to the initial SIS model, where for $x \in \Omega $ , a proportion $1-\eta (x)$ of the population is vaccinated and thus does not spread the disease (see [Reference Delmas, Dronnier and Zitt16] and references therein). In this setting, vaccinating the population amounts to replace the measure $\mu $ by $\eta \mu $ .

Motivated by this example, we shall consider the effective reproduction number defined by

$$\begin{align*}R_e(\eta) = \rho( \hat{T}_{1/\gamma} M_\eta) , \end{align*}$$

for $\eta \in \Delta $ (notice that $(TM_\eta , \gamma , \varphi )$ satisfies Assumption 2 and $\hat {T}_{1/\gamma } M_\eta =\widehat {(TM_n)}_{1/\gamma }$ ). Following [Reference Delmas, Dronnier and Zitt16], we shall be interested in critical vaccination $\eta $ for which $R_e(\eta ) = 1$ . It is observed in [Reference Delmas, Dronnier and Zitt18] that for the SIS model $(\mathrm {T}_k, \gamma , \varphi _0)$ , the vaccination strategy $\eta =\varphi _0(g^*)$ is critical. We generalize this result (with a shorter proof based on the fact that $R_e(\varphi (g))=\rho (L_g)$ , see (4.1)) for more general operators T and functions $\varphi $ .

Theorem 4.7 (Equilibria and critical vaccination)

Let $(T, \gamma , \varphi )$ that satisfy Assumption 2. Let $h \in \Delta $ be an equilibrium. Then, we have $h = g^* \, \Longleftrightarrow \, R_e(\varphi (h)) \leq 1$ . If furthermore $R_0> 1$ , then we have

$$\begin{align*}h = g^* \, \Longleftrightarrow \, R_e(\varphi(h)) = 1. \end{align*}$$

Proof First, remark that $\rho (L_g) = R_e(\varphi (g))$ , where $L_g$ is defined by (4.1).

If $R_0 \leq 1$ , then by Proposition 4.3, we have . As $R_e(\varphi (0)) = R_0 \leq 1$ by Assumption 2, we directly get the equivalence $h = g^* \, \Longleftrightarrow \, R_e(\varphi (h)) \leq 1$ .

If $R_0> 1$ , then we have by Proposition 4.3. According to Lemma 4.1 (1), if , we have $R_e(\varphi (h))=\rho (L_h)\geq 1$ , and by (2.5) and (2) that if $h\neq g^*$ then $\rho (L_h) \geq \rho \big ((L_h)_A\big )>1$ with $A={\operatorname {\mathrm {supp}}(g^*)}$ .

To complete the proof, that is, $R_e(\varphi (g^*))=1$ , we shall assume that $\rho (L_h)>1$ and show that $h\neq g^*$ . Informally, the idea is to follow the unstable direction at the equilibrium h to construct a trajectory leading to another equilibrium. Since $L_h$ is a positive compact operator, thanks to the Krein–Rutman theorem (Theorem 2.5 (1)), we can consider an eigenvector of $L_h$ related to $\rho (L_h)$ . Since the set $A=\operatorname {\mathrm {supp}}(h)$ is invariant by Lemma 4.1 (3), we have

(4.3)

If was equal to , $\rho (L_h)> 1$ would be an eigenvalue of $(L_h)_A$ , contradicting Lemma 4.1 (1). As A is invariant and , multiplying (4.3) by gives , showing that is an eigenvector of $(L_h)_{A^c}$ , so $\rho ((L_h)_{A^c})= \rho (L_h)>1$ .

Since $(L_h)_{A^c} = M_{\varphi (h)} (\hat {T}_{1/\gamma })_{A^c} = \widehat {(T_{A^c})}_{1/\gamma }$ (as $h=0$ on $A^c$ and $\varphi (0)=1$ ), we may apply Proposition 9.3 with $T=T_{A^c}$ : there exists a $\lambda>0$ and such that $T_{A^c} w - \gamma w = \lambda w$ . Without loss of generality, we can assume that $\left \lVert \,w\,\right \rVert _\infty $ is small enough to ensure that $ w \in \Delta $ and, as $\varphi $ is continuous with $\varphi (0)=1$ , that $\varphi ( \left \lVert \,w\,\right \rVert _\infty ) \geq 1- \delta $ with $\delta>0$ small enough so that $\delta (\lambda + \left \lVert \,\gamma \,\right \rVert _\infty ) \leq \lambda $ . Note that $\operatorname {\mathrm {supp}}(w) \subset A^c$ . Since , and $A=\operatorname {\mathrm {supp}}(h)$ is invariant, we get $ \varphi (h+ w) Th = \varphi (h) Th$ . Since h is an equilibrium, we obtain that

$$ \begin{align*} F(h + w) = \varphi(h+ w) T(h+ w) - \gamma (h+ w) &= \varphi(h+ w) Tw - \gamma w \\ &\geq (1-\delta) T_{A^c}w - \gamma w \\ &\geq \left( \lambda - \delta(\lambda + \left\lVert\,\gamma\,\right\rVert_\infty ) \right) \, w\\ & \geq 0. \end{align*} $$

By Lemmas 8.5 (2) and 8.6, this implies that the trajectory starting from $h+ w$ converges monotonously to an equilibrium g. Since $h\leq h+ w \leq g\leq g^*$ and , we get that $h\neq g^*$ as claimed.

4.3 Equilibria and antichains of atoms

We now focus on the characterization of equilibria. We recall from (2.9) that the future of an antichain $\mathcal {C}$ of atoms (which is at most countable) is given by $ \mathcal {F}(\mathcal {C}) = \mathcal {F} \left ( \bigcup _{A \in \mathcal {C}} A \right )= \bigcup _{A \in \mathcal {C}} \mathcal {F} \left ( A \right )$ . The set of supercritical atoms:

$$\begin{align*}\mathfrak{A}^{\mathrm{sup}}=\{A\in \mathfrak{A}\, \colon\, R_0(A)>1\} \end{align*}$$

is finite by [Reference Delmas, Lefki and Zitt19, Lemma 6.5]. We say an antichain $\mathcal {C}$ of atoms is supercritical if all its elements are supercritical atoms, that is, $\mathcal {C}\subset \mathfrak {A}^{\mathrm {sup}}$ . We denote by $\mathfrak {S}$ the (finite) set of supercritical antichains. For a set A, let $\mathcal {C}_A$ denote the (possibly empty) supercritical antichain given by the maximal elements of $\{B \in \mathfrak {A}^{\mathrm {sup}}\, \colon \, B \subset A \text { a.e.}\}$ . Notice that when A is admissible, we get by (2.10) that $\mathcal {C}_A$ is non-empty if and only if $R_0(A)>1$ . For $h\in \Delta $ , we simply write $\mathcal {C}_h$ for $\mathcal {C}_{\operatorname {\mathrm {supp}}(h)}$ .

The following theorem generalizes the uniqueness result of Proposition 4.3 when the operator T is not necessarily quasi-irreducible. Recall that, by Lemma 4.1 (3), equilibria are characterized by their support.

Theorem 4.8 (Equilibria and supercritical antichains are in bijection)

Let $(T, \gamma , \varphi )$ satisfy Assumption 2. The set of the equilibria and the set of supercritical antichains are in bijection through the equivalent relations:

(4.4) $$ \begin{align} \operatorname{\mathrm{supp}}(g)= \mathcal{F}(\mathcal{C}) \quad \Longleftrightarrow \quad \mathcal{C}_g= \mathcal{C} , \end{align} $$

where $g\in \Delta $ is an equilibrium and $\mathcal {C}\in \mathfrak {S}$ a supercritical antichain. Furthermore, if , then the equilibrium g is the maximal equilibrium of $\mathcal {F}(\mathcal {C}_g)$ .

We divide the proof in two lemmas.

Lemma 4.9 (Support of an equilibrium and related supercritical antichain)

If $g \in \Delta $ is an equilibrium, then we have $\operatorname {\mathrm {supp}}(g) = \mathcal {F}\left (\mathcal {C}_g\right )$ a.e. In particular, if g and h are two equilibria, we get

$$\begin{align*}\mathcal{C}_g=\mathcal{C}_h \Longleftrightarrow g=h. \end{align*}$$

Proof As the set $\operatorname {\mathrm {supp}}(g)$ is invariant by Lemma 4.1 (3) and as every element of $\mathcal {C}_g$ is included in $\operatorname {\mathrm {supp}}(g)$ , we have $\mathcal {F}(\mathcal {C}_g) \subset \operatorname {\mathrm {supp}}(g)$ .

By construction of $\mathcal {C}_g$ , every atom $B\subset \operatorname {\mathrm {supp}}(g)$ with $R_0(B)>1$ is included in $\mathcal {F}(\mathcal {C}_g)$ . This implies by (2.10) that, with $A=\operatorname {\mathrm {supp}}(g) \cap \mathcal {F}(\mathcal {C}_g)^c$ an invariant (by Lemma 4.1 (3)) and thus admissible set:

$$\begin{align*}R_0(A) = \max_{ B\subset A, \, B\in \mathfrak{A}} R_0(B) \leq 1. \end{align*}$$

Then Lemma 3.4 (3) gives that is an equilibrium of $\mathcal {F}(\mathcal {C}_g)^c$ and of A. Then, Proposition 4.3 (1) (with the SIS model $(T_A, \gamma , \varphi )$ ) implies that , that is, $\operatorname {\mathrm {supp}}(g) \subset \mathcal {F}(\mathcal {C}_g)$ . Thus, we get $\operatorname {\mathrm {supp}}(g)=\mathcal {F}(\mathcal {C}_g) $ . The second part of the lemma is then a direct consequence of Corollary 4.2 and Lemma 2.1.

Let $\mathcal {E}\subset \Delta $ denote the set of equilibria of the SIS model $(T, \gamma , \varphi )$ .

4.4 Monatomicity and order relation via equilibria

Let $(T, \gamma , \varphi )$ that satisfy Assumption 2. Consider the SIS model $(T, \lambda \gamma , \varphi )$ with recovery rate $ \gamma $ multiplied by a real parameter $\lambda>0$ . The reproduction number of a measurable set A for this model is $\rho ((\hat {T}_{1/\lambda \gamma })_A)= \rho ((\hat {T}_{1/ \gamma })_A\, M_{1/\lambda } )=R_0(A)/\lambda $ . We deduce from Theorem 4.8 that the number of equilibria of the SIS model $(T, \lambda \gamma , \varphi )$ is decreasing with $\lambda $ .

We say that the operator T on $L^p$ for $p \in (1, +\infty )$ is monatomic if it has exactly one nonzero atom, that is, $\operatorname {\mathrm {card}}(\mathfrak {A}^*) = 1$ . Monatomicity is a natural extension of (quasi-)irreducibility (see [Reference Delmas, Lefki and Zitt19, Remark 1.2] and references therein). We complete the characterization of monatomic operator T given in [Reference Delmas, Lefki and Zitt19, Theorem 2] using the number of equilibria of the SIS models $(T, \lambda \gamma , \varphi )$ .