3.1. Proof of Theorem3.1

3.1.1. Proof of Theorem3.1: Items 1, 2 and 3

We can easily check that the operator $\mathcal{A}$ is a Hille–Yosida operator and the nonlinear map $F$ defined in (3.1) is positive, continuous and locally Lipschitz on $X_0$ . Then, standard results can be applied to provide the existence and uniqueness of a mild solution to (3.2) (see [Reference Magal and Ruan29, Reference Thieme41, Reference Thieme43]). The Volterra formulation is also well known (see [Reference Iannelli23, Reference Webb45] for more details).

For the boundedness, let us set

\begin{align*} E(t)= \int _\Omega (E_{0}(t,x) + E_{1}(t,x)) {\textrm {d}} x, \quad A(t)= \int _\Omega \int _0^\infty \frac {1}{\tau (x)} (A_{0}(t,a,x) + A_{1}(t,a,x)) {\textrm {d}} a {\textrm {d}} x. \end{align*}

Then, by (2.3)–(2.4), it comes

\begin{align*} \dot E(t)= & H(E(t)) \int _{\Omega } \int _{\Omega }\int _0^\infty (m_0(x,y) r_0(a,y) A_0(t,a,y) + m_1(x,y) r_1(a,y) A_1(t,a,y)) \mathrm{d}a{\textrm {d}} y {\textrm {d}} x \\ & \qquad - \int _{\Omega } (\mu _0(x)+ \gamma _0(x)) E_{0}(t,x) {\textrm {d}} x - \int _{\Omega } (\mu _1(x)+ \gamma _1(x)) E_{1}(t,x) {\textrm {d}} x, \\ \dot A(t)= & \int _{\Omega } (\gamma _0(x) E_0(t,x) + \gamma _1(x) E_1(t,x)) {\textrm {d}} x - \int _{\Omega } \int _0^\infty \frac {1}{\tau (x)} (d_0(a,x)A_0(t,a,x) + d_1(a,x)A_1(t,a,x)) {\textrm {d}} a {\textrm {d}} x . \end{align*}

By Assumption2.1, we find that

(3.3) \begin{align} \dot E(t) \leq & \nu H(E(t))A(t) - \zeta E(t) , \nonumber \\ \dot A(t) \leq & \alpha E(t) - \beta A(t), \end{align}

where $ \nu = \|\tau \|_{\infty } \max \big \{ \|r_0\|_{\infty }\int _{\Omega } \sup _y m_0(x,y) {\textrm {d}} x, \|r_1\|_{\infty }\int _{\Omega } \sup _y m_1(x,y) {\textrm {d}} x \big \}, \zeta = \min \big(\underline {\mu }_0 + \underline {\gamma }_0, \underline {\mu }_1 {+} \underline {\gamma }_1 \big)$ , $\alpha =\max \left \{\|\gamma _0\|_{\infty }, \|\gamma _1\|_{\infty } \right \}$ and $\beta = \min \left \{ \inf _{(a,x) \in (0,\infty )+\times \Omega } d_0 , \inf _{(a,x) \in (0,\infty )+\times \Omega } d_1 \right \}$ . Here, $\underline {f}\,:\!=\, \inf \limits _{x \in \Omega }f$ .

Let $\bar c \gt 0$ , and set $W= E+ \bar c A$ . Estimates (3.3) give

(3.4) \begin{equation} \dot W(t) \le \left (\bar c \alpha - \zeta \right ) E(t)+ \left (\nu H\left ( E(t)\right ) -\bar c\beta \right ) A(t). \end{equation}

Since the function $H$ , introduced in (2.2), is decreasing and takes values in $(0,1)$ , we have

\begin{equation*} \nu H\left ( E(t)\right ) - \bar c \beta \lt 0 \quad \text{ iff } \quad E(t) \gt H^{-1} \left ( \frac {\bar c\beta }{\nu } \right )\!, \end{equation*}

where, of course, the above estimate holds on the necessary condition $\frac {\bar c \beta }{\nu } \in (0,1)$ . Thus, by choosing $\bar c$ , such that

(3.5) \begin{equation} 0 \lt \bar c \lt \min \left ( \frac {\zeta }{ \alpha }, \frac {\nu }{\beta } \right )\!, \end{equation}

estimate (3.4) leads to $\dot W(t)\lt 0$ as soon as $E(t) \gt H^{-1} \left ( \frac {\bar c\beta }{\nu } \right )$ . From where we find that $E$ is ultimately bounded and for all $t$ ,

\begin{equation*} E(t) \le H^{-1} \left ( \frac {\bar c\beta }{\nu } \right )\!, \end{equation*}

with $\bar c$ satisfying (3.5).

Finally, by (3.3), we find for all $t\ge 0$ :

\begin{equation*} A(t) \le A(0) e^{-t\beta }+ \frac {\alpha }{\beta } H^{-1} \left ( \frac {\bar c\beta }{\nu } \right ) \left (1- e^{-t\beta }\right )\!. \end{equation*}

This ends the proof of items 1, 2 and 3 of Theorem3.1.

3.1.2. Proof of Theorem3.1: Item 4

We first introduce the following proposition:

Proposition 3.1. Let Assumptions 2.1 and 2.2 hold. Then, the semiflow $\{\Phi (t)\}_{t\geq 0}$ induced by the Cauchy problem (3.2) satisfies, $\Phi =\Phi _1+\Phi _2$ , such that

1. 1. For each $t\gt 0$ , $\Phi _1(t) \,:\, X_{0+} \to X$ , maps bounded sets of $X_{0+}$ into relatively compact sets of $X$ ;

2. 2. There exists $\zeta \,:\, [0,+\infty )\times [0,+\infty ) \to [0,+\infty )$ such that for each $\epsilon \gt 0$ , $\displaystyle \lim _{t\to +\infty }\zeta (t,\epsilon )\to 0$ and, if $\boldsymbol{w}_{0}\in X_{0+}$ with $\Vert \boldsymbol{w}_{0} \Vert _{X} \leq \epsilon$ , then $\Vert \Phi _2(t) \boldsymbol{w}_{0}\Vert _{X} \leq \zeta (t,\epsilon )$ for all $t\geq 0$ .

Before providing the proof of Proposition3.1, note that, by [Reference Hale18, Lemma 3.2.3], this proposition concludes the proof of Theorem3.1, item 4.

Proof of Proposition 3.1. The nonlinear map $F$ , given by (3.1), is such that $F=F_1+F_2$ , with

(3.6) \begin{equation} F_1 \begin{pmatrix} \boldsymbol{u} \\ \boldsymbol{0}_{L^{1}} \\ \boldsymbol{v} \end{pmatrix} = \begin{pmatrix} h(\boldsymbol{u}) \displaystyle \int _0^{\infty }\mathcal{B}[\boldsymbol{v}](a,\cdot ){\textrm {d}} a \\ \boldsymbol{0}_{L^{1}} \\ 0_{L^{1}((0,\infty ) \times \Omega ,{\mathbb{R}}^2)} \end{pmatrix}, \text{ and } \quad F_2 \begin{pmatrix} \boldsymbol{u} \\ \boldsymbol{0}_{L^{1}} \\ \boldsymbol{v} \end{pmatrix} = \begin{pmatrix} \boldsymbol{0}_{L^{1}} \\ \mathcal{C}\boldsymbol{u} \\ 0_{L^{1}((0,\infty ) \times \Omega ,{\mathbb{R}}^2)} \end{pmatrix}\!. \end{equation}

By Assumptions2.1 and 2.2, $F_1$ maps bounded sets of $X_{0+}$ into a relatively compact set of $X_{0}$ (e.g. see [Reference Djidjou-Demasse, Ducrot and Fabre13]). Let $B \subseteq X_{0+}$ be a bounded subset of $X_{0+}$ . Note that for any $\boldsymbol{w}_0\in B$ , the integrated solution $t\in [0,+\infty )\mapsto \Phi (t)\boldsymbol{w}_0$ of the Cauchy problem (3.2) writes, for all $t\ge 0$ :

(3.7) \begin{equation} \Phi (t)\boldsymbol{w}_0=T_{\mathcal A_0}(t)\boldsymbol{w}_0+ \lim _{\lambda \to +\infty } \int _0^t T_{\mathcal A_0}(t-s)\lambda (\lambda -\mathcal A)^{-1}F(\Phi (s)\boldsymbol{w}_0) {\textrm {d}} s, \end{equation}

where $\mathcal A_0 \,:\, \mathcal{D}(\mathcal A_0)\subset X_0 \to X$ is the part of $\mathcal A$ on $X_0$ , defined by

\begin{equation*} \mathcal A_0 \boldsymbol{w}\,:\!=\,\mathcal A \boldsymbol{w},\ \forall \boldsymbol{w} \in \mathcal{D}(\mathcal A_0)=\{ \boldsymbol{w} \in \mathcal{D}(A) \,:\, \mathcal A \boldsymbol{w} \in X_0\}, \end{equation*}

and $\{T_{\mathcal A_0}(t)\}_t$ is the $C_0$ -semigroup generated by $\mathcal A_0$ .

By [Reference Thieme42, Theorem 3.14] and [Reference Magal and Ruan28, Lemma 2.1], note that $\mathsf{s}(\mathcal A_0)=\mathsf{s}(\mathcal A)\lt 0$ , where $\mathsf{s}(G)$ is the spectral bound of $G$ defined by:

\begin{equation*} \mathsf{s}(G)\,:\!=\,\sup \{ \Re (\lambda ) \,:\, \lambda \text{ is in the spectral set of } G\}. \end{equation*}

Consequently, if $\omega _{\mathcal A}\in ({-}\mathsf{s}(\mathcal A_0),0)$ , there exists a constant $M_{\mathcal A}\geq 1$ such that:

(3.8) \begin{equation} \Vert T_{\mathcal A_0}(t) \Vert _{\mathcal{L}(X_0)}\leq M_{\mathcal A} e^{-\omega _{\mathcal A} t},\ \forall t\geq 0. \end{equation}

Define, for each $\boldsymbol{w}_0 \in B$ , the maps $t\mapsto \hat {\Phi }(t)\boldsymbol{w}_0$ , $t \mapsto \tilde {\Phi }(t)\boldsymbol{w}_0$ and $t \mapsto \check {\Phi }(t)\boldsymbol{w}_0$ by:

(3.9) \begin{align} & \hat {\Phi }(t)\boldsymbol{w}_0 = \lim _{\lambda \to +\infty } \int _0^t T_{\mathcal A_0}(t-s)\lambda (\lambda -\mathcal A)^{-1}F_1(\Phi (s)\boldsymbol{w}_0) {\textrm {d}} s, \end{align}

(3.10) \begin{align} & \tilde {\Phi }(t)\boldsymbol{w}_0 =\lim _{\lambda \to +\infty } \int _0^t T_{\mathcal A_0}(t-s)\lambda (\lambda -\mathcal A)^{-1}F_2(\Phi (s)\boldsymbol{w}_0) {\textrm {d}} s, \end{align}

(3.11) \begin{align} & \check {\Phi }(t)\boldsymbol{w}_0 =T_{\mathcal A_0}\Phi (t)\boldsymbol{w}_0 . \end{align}

The uniqueness of the integrated solution (3.7) and (3.9)–(3.11) gives: for each $\boldsymbol{w}_0 \in B$ ,

\begin{equation*} \Phi (t)\boldsymbol{w}_0=\hat {\Phi }(t)\boldsymbol{w}_0+\check {\Phi }(t)\boldsymbol{w}_0+\tilde {\Phi }(t)\boldsymbol{w}_0,\ \forall t\geq 0. \end{equation*}

By items 1, 2 and 3 of Theorem3.1, we can find a positive constant $\ell _0\,:\!=\,\ell _0(B)\gt 0$ such that

(3.12) \begin{equation} \sup _{t\geq 0, \boldsymbol{w}_0 \in B} \Vert \Phi (t)\boldsymbol{w}_0 \Vert _{ X} \leq \ell _0. \end{equation}

Furthermore, by estimate (3.8), the equality (3.11) and the boundedness of $B$ , we can find a positive constant $\ell _1\,:\!=\,\ell _1(B)\gt 0$ such that

(3.13) \begin{equation} \sup _{\boldsymbol{w}_0 \in B} \Vert \check {\Phi }(t)\boldsymbol{w}_0 \Vert _{X} \leq \ell _1 e^{-\omega _{\mathcal A} t},\ \forall t\geq 0. \end{equation}

Proposition3.1 is direct consequence of the following lemma:

Lemma 3.1. Let Assumptions 2.1 and 2.2 be satisfied. Then,

1. 1. For each $T\gt 0$ , the set $S_B =\{\hat {\Phi }({\cdot})\boldsymbol{w}_0 \in C([0,T],X_0) \,:\, \boldsymbol{w}_0 \in B\}$ is relatively compact in $C([0,T],X_0)$ .

2. 2. The nonlinear maps $\{\tilde {\Phi }(t)\}_{t\geq 0}$ defined in (3.10) has the form $\tilde {\Phi } =\tilde {\Phi }_1+\tilde {\Phi }_2$ , such that

   1. (i) For each $t\gt 0$ , $\tilde {\Phi }_1(t) \,:\, X_{0+} \to X$ , maps $B$ into relatively compact sets of $X$ ;

   2. (ii) There exists a constant $ \ell = \ell (B)\gt 0$ such that $\Vert \tilde {\Phi }_2(t)\boldsymbol{w}_0\Vert _{X} \leq \ell e^{- \omega _{\mathcal A} t}$ , for all $t\geq 0$ and $\boldsymbol{w}_0\in B$ ; with $\omega _{\mathcal A} \in ({-}\mathsf{s}(\mathcal A_0),0)$ .

Therefore, to conclude the proof of Proposition3.1, and consequently, the proof of Theorem3.1 (item 4), we will provide additional details on the proof of Lemma3.1.

Proof of Lemma 3.1: item 1. Our aim is to prove that for all $t\geq 0$ , the set $S_B(t)$ defined by $S_B(t) =\{\hat {\Phi }(t)\boldsymbol{w}_0\,:\, \boldsymbol{w}_0 \in B\}$ is compact in $X_0$ and $S_B$ is an equicontinuous family. Because $F_1(\Phi (s)\boldsymbol{w}_0) \in X_0$ for all $s\geq 0$ , $\boldsymbol{w}_0 \in B$ , (3.9) rewrites

\begin{equation*} \hat {\Phi }(t)\boldsymbol{w}_0=\int _0^t T_{\mathcal A_0}(t-s) F_1(\Phi (s)\boldsymbol{w}_0) {\textrm {d}} s, \end{equation*}

and by (3.12), we introduce the bounded set $B_0 =\{ \Phi (t)\boldsymbol{w}_0 \,:\, t\geq 0, \boldsymbol{w}_0 \in B \}$ . Thus, the compactness of $F_1$ gives that $F_1(B_0)$ is relatively compact. Since $(s,y)\mapsto T_{\mathcal A_0}(s)y$ is continuous, it comes that

\begin{equation*} \{T_{\mathcal A_0}(t-s) F_1(\Phi (s)\boldsymbol{w}_0) \,:\, s \in [0,t], \boldsymbol{w}_0 \in B\} \subset \{T_{\mathcal A_0}(t-s)F_1(y) \,:\, s \in [0,t], y \in B_0\} \end{equation*}

is relatively compact. Consequently, Mazur’s theorem (see, e.g. [Reference Brezis9, Corollary 3.8] or the original paper by Mazur [Reference Mazur31]) implies that for all $ t \gt 0$ , the set $ S_B(t)$ is relatively compact.

It remains to prove that $S_B$ is equicontinuous. Let $0\leq t_0\leq t\leq T$ , then

\begin{equation*} \begin{array}{llll} \hat {\Phi }(t)\boldsymbol{w}_0-\hat {\Phi }(t_0)\boldsymbol{w}_0&=& \int _{t_0}^t T_{\mathcal A_0}(t-s) F_1(\Phi (s)\boldsymbol{w}_0) {\textrm {d}} s+ \int _{0}^{t_0} [T_{\mathcal A_0}(t-s)-T_{\mathcal A_0}(t_0-s)] F_1(\Phi (s)\boldsymbol{w}_0) {\textrm {d}} s\\[5pt] &=& \int _{t_0}^t T_{\mathcal A_0}(t-s) F_1(\Phi (s)\boldsymbol{w}_0) {\textrm {d}} s + \int _{0}^{t_0} [T_{\mathcal A_0}(t-t_0+s)-T_{\mathcal A_0}(s)] F_1(\Phi (t_0-s)\boldsymbol{w}_0) {\textrm {d}} s. \end{array} \end{equation*}

The above equality leads to

\begin{equation*} \Vert \hat {\Phi }(t)\boldsymbol{w}_0-\hat {\Phi }(t_0)\boldsymbol{w}_0\Vert _{X} \leq \sup _{y \in B_0} \Vert F_1(y) \Vert _{X} \int _{t_0}^t M_{\mathcal A} e^{-\omega _{\mathcal A} (t-s)} {\textrm {d}} s+ t_0 \sup _{s\in [0,t_0],y \in F_1(B_0)}\Vert T_{\mathcal A_0}(t-t_0+s)y-T_{\mathcal A_0}(s)y\Vert _{X}. \end{equation*}

The equicontinuity of $S_B$ holds from the above estimate and the relative compactness of $F_1(B_0)$ .

Proof of Lemma 3.1: item 2. For $t\ge 0$ and $\boldsymbol{w}_0 \in B$ , set $\Phi (t)\boldsymbol{w}_0= (\boldsymbol{u}(t,\cdot ), 0_{L^{1}}, \boldsymbol{v}(t,\cdot ,\cdot ))$ , as well as $\hat \Phi (t)\boldsymbol{w}_0= (\boldsymbol{\hat u}(t,\cdot ), 0_{L^{1}}, \boldsymbol{\hat v}(t,\cdot ,\cdot ))$ , $\tilde \Phi (t)\boldsymbol{w}_0= (\boldsymbol{\tilde u}(t,\cdot ), 0_{L^{1}}, \boldsymbol{\tilde v}(t,\cdot ,\cdot ))$ and $\check \Phi (t)\boldsymbol{w}_0= (\boldsymbol{\check u}(t,\cdot ), 0_{L^{1}}, \boldsymbol{\check v}(t,\cdot ,\cdot ))$ . Therefore,

(3.14) \begin{equation} \boldsymbol{f}= \boldsymbol{\hat f} + \boldsymbol{\tilde f}+ \boldsymbol{\check f}, \text{ for } \quad \boldsymbol{f} \in \{\boldsymbol{u}, \boldsymbol{v} \}. \end{equation}

By estimates (3.12) and (3.13), we, respectively, have, $\forall t\geq 0$ :

(3.15) \begin{equation} \begin{cases} \sup _{\boldsymbol{w}_0 \in B} \Vert \boldsymbol{u}(t,\cdot ) \Vert _{L^1} \leq \ell _0 ,\\ \sup _{\boldsymbol{w}_0 \in B} \Vert \boldsymbol{\check u}(t,\cdot ) \Vert _{L^1} \leq \ell _1 e^{-\omega _{\mathcal A} t}. \end{cases} \end{equation}

By (3.6) and (3.10), we have

\begin{equation*} \begin{cases} \partial _t \boldsymbol{\tilde u}(t,\cdot ) = - \mathcal{N} \boldsymbol{\tilde u}(t,\cdot ),\\ \boldsymbol{\tilde v}(t,0,\cdot ) =\mathcal{C} \boldsymbol{u}(t,\cdot ),\\ \displaystyle (\partial _t+\partial _a) \boldsymbol{\tilde v}(t,\cdot ,\cdot ) = - \mathcal{D} \boldsymbol{\tilde v}, \end{cases} \end{equation*}

with the initial condition $\boldsymbol{\tilde u}(0,\cdot )= \boldsymbol{0}_{L^1}$ , and $\boldsymbol{\tilde v}(0,\cdot ,\cdot )= \boldsymbol{0}_{L^1}$ . It then comes, for all $t\ge 0$ :

(3.16) \begin{equation} \boldsymbol{\tilde u}(t,\cdot ) \equiv \boldsymbol{0}_{L^1}, \text{ and } \quad \boldsymbol{\tilde v}(t,a,\cdot ) = \boldsymbol{1}_{[0,t]}(a) \boldsymbol{\Pi }(0,a,\cdot )\,\mathcal{C}({\cdot})\,\boldsymbol{u}(t-a,\cdot ). \end{equation}

Thus, by (3.14) and (3.16), we find

\begin{equation*} \boldsymbol{u}= \boldsymbol{\hat u} + \boldsymbol{\check u}. \end{equation*}

Therefore, (3.16) gives

\begin{equation*} \boldsymbol{\tilde v}(t,a,\cdot ) = \boldsymbol{1}_{[0,t]}(a) \boldsymbol{\Pi }(0,a,\cdot )\,\mathcal{C}({\cdot}) \left ( \boldsymbol{\hat u}(t-a,\cdot )+ \boldsymbol{\check u}(t-a,\cdot ) \right ) \,:\!=\, \boldsymbol{\tilde v}_1[\boldsymbol{w}_0](t,a,\cdot )+ \boldsymbol{\tilde v}_2[\boldsymbol{w}_0](t,a,\cdot ), \end{equation*}

with $\boldsymbol{\tilde v}_1[\boldsymbol{w}_0](t,a,\cdot )= \boldsymbol{1}_{[0,t]}(a) \boldsymbol{\Pi }(0,a,\cdot )\,\mathcal{C}({\cdot}) \boldsymbol{\hat u}(t-a,\cdot )$ , and $\boldsymbol{\tilde v}_2[\boldsymbol{w}_0](t,a,\cdot )= \boldsymbol{1}_{[0,t]}(a) \boldsymbol{\Pi }(0,a,\cdot )\,\mathcal{C}({\cdot}) \boldsymbol{\check u}(t-a,\cdot )$ .

We now show that the set $\{\boldsymbol{\tilde v}_1[\boldsymbol{w}_0](t,\cdot ,\cdot ) \}_{\boldsymbol{w}_0\in B}$ is relatively compact in $L^1((0,\infty )\times \Omega )$ . Let $\{\boldsymbol{\tilde v}_{1,n}(t,a,\cdot )= \boldsymbol{1}_{[0,t]}(a) \boldsymbol{\Pi }(0,a,\cdot )\,\mathcal{C}({\cdot}) \boldsymbol{\hat u}_n(t-a,\cdot ) \}_{ n\in \mathbb{N}}$ a bounded sequence in $\left \{\boldsymbol{\tilde v}_1[\boldsymbol{w}_0](t,\cdot ,\cdot )\right \}_{\boldsymbol{w}_0\in B}$ . By Lemma3.1 (item 1.), the set $\{\boldsymbol{\hat u}_n(t,\cdot ) \}_{ n\in \mathbb{N}}$ is compact in $C\left ([0,t], L^1({\mathbb{R}})\right )$ . As a result, we can find $\boldsymbol{\bar {\hat u}} \in C\left ([0,t], L^1({\mathbb{R}})\right )$ , such that $\boldsymbol{\hat u}_n \to \boldsymbol{\bar {\hat u}}$ in $C\left ([0,t], L^1({\mathbb{R}})\right )$ , as $n\to \infty$ . Thus, $\boldsymbol{\tilde v}_{1,n}(t,a,\cdot ) \to \boldsymbol{1}_{[0,t]}(a) \boldsymbol{\Pi }(0,a,\cdot )\,\mathcal{C}({\cdot}) \boldsymbol{\bar {\hat u}}(t-a,\cdot )$ , in $L^1((0,\infty )\times \Omega )$ , as $n\to \infty$ .

Finally, by (3.15), we can find a positive constant $\ell =\ell (B)$ , such that $\Vert \boldsymbol{\tilde v}_2[\boldsymbol{w}_0](t,\cdot ,\cdot ) \Vert _{L^1} \le \ell e^{-\omega _{\mathcal{A}}t},$ for all $t\ge 0$ . This ends the proof of Lemma3.1.

3.2. The steady states

The next results are concerned with the existence of the steady states of the system (3.2), which always exhibits a mosquito-free steady state given by $\mathcal{E}^0 = (0,0,0,0)^{T}$ .

The existence of fully-exposed and co-existence steady states of System (3.2) is strongly related to the linear operator $\boldsymbol{\mathcal{L}}$ defined by:

(3.17) \begin{equation} \boldsymbol{\mathcal{L}} \,:\,(L^1(\Omega ))^2\ni \boldsymbol{\varphi }({\cdot}) \longmapsto \int _{\Omega }\boldsymbol{m}(\cdot ,y)\boldsymbol{\theta }(y)\boldsymbol{\varphi }(y) {\textrm {d}} y\in (L^1(\Omega ))^2, \end{equation}

where $\boldsymbol{\theta }\,:\, \Omega \to {\mathbb{R}}_+^2$ :

(3.18) \begin{equation} \boldsymbol{\theta }(x) = \left (\int _0^\infty \boldsymbol{r}(a,x) \boldsymbol{\Pi }(0,a,x) {\textrm {d}} a\right )\mathcal{C}(x)\mathcal{N}^{-1}(x). \end{equation}

Note that

\begin{equation*} \boldsymbol{ \mathcal{L}} = \begin{pmatrix} (1-c) \mathcal{L}_0 & \quad 0 \\ c \mathcal{L}_0 & \quad \mathcal{L}_1 \end{pmatrix}\!, \end{equation*}

with $\mathcal{L}_j$ ,s the linear operators defined by:

(3.19) \begin{equation} \mathcal{L}_{j}\,:\,L^1(\Omega )\ni \varphi ({\cdot}) \longmapsto \int _{\Omega }m_{j}(\cdot ,y)\theta _{j}(y)\varphi (y) {\textrm {d}} y\in L^1(\Omega ),\quad \mbox{for} \quad j\in \{0,1\}, \end{equation}

with

(3.20) \begin{equation} \theta _{j}(x) = \frac {\tau (x)\gamma _{j}(x)}{\mu _{j}(x) + \gamma _{j}(x)} \int _0^\infty r_{j}(a,x) \pi _{j}(0,a,x) {\textrm {d}} a, \quad j\in \{0,1\}. \end{equation}

The quantity $\theta _{j}(x)$ can be interpreted as the reproductive number (or fitness) of an AFM with the insecticide resistance level $x$ in insecticide state $j$ . Note that $\theta _{j}(x)$ – hereafter referred to as the fitness function – represents the fitness of an AFM with insecticide resistance level $x$ within the $j$ – insecticide state. In the above expression, the survival probability $\pi _{j}(0, a, x) = e^{-\int _{0^{a}} d_{j}(\sigma , x) , d\sigma }$ of an AFM with IR level $x$ up to age $a$ , multiplied by $r_j(a, x)$ , represents the contribution of the AFM to egg production at age $a$ . Integrating this product over all ages $a$ gives the total number of eggs effectively produced by the AFM during its lifetime. Note that under Assumption2.1, the positive functions $\theta _{j} \,:\, \Omega \mapsto {\mathbb{R}}_+$ are continuous. The vector-valued fitness function $\boldsymbol{\theta }(x)$ then represents the overall fitness of AFMs with insecticide resistance level $x$ . Finally, note that here, $\theta _{j}$ is not derived directly from the next-generation operator approach, but it can be shown that this corresponds to the basic reproduction number when applying the next-generation operator approach [Reference Djidjou-Demasse, Ducrot and Fabre13, Reference Pane, Richard, Seydi and Djidjou-Demasse38].

We then have the following result.

Before going through the proof of Theorem3.2, observe that the positive steady state $\mathcal{E}^*$ is closely related to the principal eigenfunction of the linear operators $\mathcal{L}$ for any given probability kernel $m$ satisfying Assumptions2.1 and 2.2. The profile of the steady state $\mathcal{E}^*$ with respect to $x \in \Omega$ can be explicitly characterized when mutation kernels $m_j$ depends on a small positive parameter $\varepsilon$ (with $\varepsilon \ll 1$ ) using the scaling form:

\begin{equation*} m_j^\varepsilon (x)=\varepsilon ^{-1}m_j\big(\varepsilon ^{-1} x\big), \end{equation*}

where $\varepsilon \gt 0$ represents the mutation variance in the phenotypic space. Specifically, for small $\varepsilon$ , the eigenfunction $\psi _j$ concentrates on the set $\mathcal{M}_j$ defined by:

\begin{equation*} \mathcal M_j= \left \{x\in \Omega \,:\, \theta _j(x)= \|\mathcal \theta _j\|_\infty \right \}\!. \end{equation*}

This set, $\mathcal{M}_j$ , is known as the set of evolutionary attractors (or dominant strains) in the classical adaptive dynamics framework (e.g. [Reference Geritz, Metz, Kisdi and Meszéna17, Reference Metz, Geritz, Meszena, Jacobs and van Heerwaarden32]). Moreover, when $\theta _j$ is at least $\mathcal{C}^1$ with a finite number of maxima, it has been shown in [Reference Djidjou-Demasse, Ducrot and Fabre13] that these dominant strains coincide with the set $\mathcal{M}_j$ .

Denoting by $\mathcal{L}_j^\varepsilon$ the operator $\mathcal{L}_j$ modified by replacing the kernel $m_j$ with $m_j^\varepsilon$ , it follows from Theorem 2.2 in [Reference Djidjou-Demasse, Ducrot and Fabre13] that the spectral radius $r\left (\mathcal{L}_j^\varepsilon \right )$ satisfies, for sufficiently small $\varepsilon$ :

\begin{equation*} r\left (\mathcal{L}_j^\varepsilon \right ) = \left (\theta _j(x^*)\right )^2 + \mathcal{O}(\varepsilon ), \quad \text{for all } x^* \in \mathcal M_j.\end{equation*}

From this estimate, we deduce that $\textrm {sign}\left [r(\boldsymbol{\mathcal{L}}) - 1\right ] = \textrm {sign}\left [ \max \left ( (1-c)\theta _0^2(x^*), \theta _1^2(x^*) \right ) - 1\right ]$ , for all $x^* \in \mathcal{M}_j$ and sufficiently small $\varepsilon$ . Furthermore, if $\varepsilon \ll 1$ , $\mathcal{M}_j = \{x_j^*\}$ and $\max \left ( (1-c)\theta _0^2(x_j^*), \theta _1^2(x_j^*) \right ) \gt 1$ , then the eigenfunction $\psi _j^*$ is concentrated around the evolutionary attractor $x_j^*$ in the IR space $\Omega$ and $\lim _{\varepsilon \to 0}\int _{\Omega } w(x) \psi _j^*(x){\textrm {d}} x= w\left (x_j^*\right )$ for any continuous function $w \in \mathcal C\left (\Omega \right )$ . For a detailed analysis of this concentration phenomenon, see Theorem 2.3 in [Reference Djidjou-Demasse, Ducrot and Fabre13].

Consequently, for sufficiently small $\varepsilon$ , by (3.21)–(3.22), the total number of FM exposed and unexposed at the positive steady state $\mathcal{E}^*$ rewrite

\begin{align*} A_0^*&= p\left [\mathcal{E}^{*}\right ](1-c) \frac {\tau (x_0^*) \gamma _0(x_0^*) \bar \pi _0(x_0^*)}{\mu _0(x_0^*)+\gamma _0(x_0^*)} , \\[5pt] A_1^*&= p\left [\mathcal{E}^{*}\right ]c \frac {\tau (x_0^*)\gamma _0(x_0^*) \bar \pi _1(x_0^*)}{\mu _1(x_0^*)+\gamma _1(x_0^*)} + p\left [\mathcal{E}^{*}\right ] \frac {\tau (x_1^*) \gamma _1(x_1^*) \bar \pi _1(x_1^*)}{\mu _1(x_1^*)+\gamma _1(x_1^*)} . \end{align*}

3.2.1. Proof of Theorem3.2

Recall that in this section, we assume that $c$ is a constant function. The steady states of system (2.5) are obtained by solving the following system:

(3.23) \begin{equation} \begin{cases} \displaystyle H\left ( E^* \right ) \int _\Omega \int _0^\infty m_{0}(x,y) r_{0}(a,y) A_{0}^{*}(a,y){\textrm {d}} a {\textrm {d}} y -\left ( \mu _{0}(x) + \gamma _{0}(x) \right ) E_{0}^{*}(x) = 0, \\ A_{0}^{*}(0,x) = (1-c)\tau (x)\gamma _{0}(x)E_{0}^{*}(x), \\ \partial _a A_{0}^{*}(a,x) = - d_{0}(a,x)A_{0}^{*}(a,x), \\ \displaystyle H\left ( E^* \right ) \int _\Omega \int _0^\infty m_{1}(x,y) r_{1}(a,y) A_{1}^{*}(a,y){\textrm {d}} a {\textrm {d}} y -\left ( \mu _{1}(x) + \gamma _{1}(x) \right ) E_{1}^{*}(x) = 0, \\ A_{1}^{*}(0,x) = c\tau (x)\gamma _{0}(x)E_{0}^{*}(x) + \tau (x)\gamma _{1}(x)E_{1}^{*}(x), \\ \partial _a A_{1}^{*}(a,x) = - d_{1}(a,x)A_{1}^{*}(a,x), \end{cases} \end{equation}

where

\begin{equation*} E^{*} = \int _{\Omega } \left (E_{0}^{*}(x) + E_{1}^{*}(x) \right ){\textrm {d}} x, \end{equation*}

and

(3.24) \begin{equation} H(E^{*}) = \left ( 1 + E^{*} \right )^{-\kappa }, \kappa \gt 0. \end{equation}

1. 1. It is obvious that $\mathcal{E}^0 = (0,0,0,0)^T$ is a trivial solution of (3.23) without any condition.

2. 2. In an environment where all mosquitoes are exposed to the insecticide (i.e. $c=1$ ), system (3.23) becomes

   (3.25) \begin{equation} \begin{cases} \displaystyle H\left ( E^* \right ) \int _\Omega \int _0^\infty m_{0}(x,y) r_{0}(a,y) A_{0}^{*}(a,y){\textrm {d}} a {\textrm {d}} y -\left ( \mu _{0}(x) + \gamma _{0}(x) \right ) E_{0}^{*}(x) = 0, \\[3pt] A_{0}^{*}(0,x) = 0, \\ \partial _a A_{0}^{*}(a,x) = - d_{0}(a,x)A_{0}^{*}(a,x), \\[3pt] \displaystyle H\left ( E^* \right ) \int _\Omega \int _0^\infty m_{1}(x,y) r_{1}(a,y) A_{1}^{*}(a,y){\textrm {d}} a {\textrm {d}} y -\left ( \mu _{1}(x) + \gamma _{1}(x) \right ) E_{1}^{*}(x) = 0, \\[3pt] A_{1}^{*}(0,x) = \tau (x)\gamma _{0}(x)E_{0}^{*}(x) + \tau (x)\gamma _{1}(x)E_{1}^{*}(x), \\ \partial _a A_{1}^{*}(a,x) = - d_{1}(a,x)A_{1}^{*}(a,x). \end{cases} \end{equation}
   Solving (3.25) for $A_{0}^{*}$ yields $A_{0}^{*}\equiv 0$ , which in turn implies that $E_{0}^{*}\equiv 0$ . Thus, system (3.25) rewrites as:
   (3.26) \begin{equation} \begin{cases} \displaystyle H\left ( E^* \right ) \int _\Omega \int _0^\infty m_{1}(x,y) r_{1}(a,y) A_{1}^{*}(a,y){\textrm {d}} a {\textrm {d}} y -\left ( \mu _{1}(x) + \gamma _{1}(x) \right ) E_{1}^{*}(x) = 0, \\[3pt] A_{1}^{*}(0,x) = \tau (x)\gamma _{1}(x)E_{1}^{*}(x), \\ \partial _a A_{1}^{*}(a,x) = - d_{1}(a,x)A_{1}^{*}(a,x), \end{cases} \end{equation}
   where
   (3.27) \begin{equation} E^{*} = \int _{\Omega } E_{1}^{*}(x) {\textrm {d}} x. \end{equation}
   Solving (3.26) for $A_{1}^{*}$ gives
   (3.28) \begin{equation} A_{1}^{*}(a,x)= \tau (x)\gamma _{1}(x)\pi _{1}(0,a,x)E_{1}^{*}(x). \end{equation}
   Replacing (3.28) in the first equation of (3.26) yields
   (3.29) \begin{equation} H\left ( E^* \right ) \int _\Omega \int _0^\infty m_{1}(x,y) r_{1}(a,y) \tau (y)\gamma _{1}(y)\pi _{1}(0,a,y)E_{1}^{*}(y){\textrm {d}} a {\textrm {d}} y =\left ( \mu _{1}(x) + \gamma _{1}(x) \right ) E_{1}^{*}(x). \end{equation}
   By setting $\bar {E_{1}^{*}}(x)= \left ( \mu _{1}(x) + \gamma _{1}(x) \right )E_{1}^{*}(x)$ , (3.29) becomes
   \begin{equation*}\int _\Omega \int _0^\infty m_{1}(x,y) \frac {\tau (y)\gamma _{1}(y)}{\mu _{1}(y) + \gamma _{1}(y)} r_{1}(a,y)\pi _{1}(0,a,y)\bar {E}_{1}^{*}(y){\textrm {d}} a {\textrm {d}} y = \frac {1}{H(E^*)}\bar {E}_{1}^{*}(x),\end{equation*}
   which leads to
   (3.30) \begin{equation} \int _\Omega m_{1}(x,y) \theta _{1}(y) \bar {E_{1}^{*}}(y){\textrm {d}} y = \frac {1}{H(E^*)}\bar {E}_{1}^{*}(x), \end{equation}
   where $\theta _{1}$ is given by
   \begin{equation*} \theta _1(y)= \frac {\tau (y)\gamma _{1}(y)}{\mu _{1}(y) + \gamma _{1}(y)} \int _0^\infty r_{1}(a,y)\pi _{1}(0,a,y) {\textrm {d}} a. \end{equation*}

   Equality (3.30) rewrites as

   \begin{equation*} \mathcal{L}_{1}[\bar {E}_{1}^{*}](x)= \frac {1}{H(E^*)}\bar {E}_{1}^{*}(x), \end{equation*}
   where $\mathcal{L}_{1}$ is the linear operator given by (3.19). Thus, the existence of $E_{1}^{*} \gt 0$ is strongly related to the spectral property of operator $\mathcal{L}_{1}$ . Since operator $\mathcal{L}_{j}$ , for $j \in \{0,1\}$ , is positive, bounded, irreducible and compact, then Frobenius theorem applies (e.g. see [Reference Djidjou-Demasse, Ducrot and Fabre13]), and there exists an eigenfunction $\phi _{j} \in L_{+}^{1}(\Omega )$ normalized by $\|\phi \|_{L^1}= 1$ such that
   \begin{equation*} \mathcal{L}_{j}[\phi _{j}]= r\left ( \mathcal{L}_{j} \right )\phi _{j}. \end{equation*}
   From (3.30), we find that
   (3.31) \begin{equation} r(\mathcal{L}_{j})= \frac {1}{H(E^*)} \ , E_{1}^{*}(x)= p [\mathcal{E}^{R}]\frac {\phi _{1}(x)}{\mu _{1}(x) + \gamma _{1}(x)}, \ \text{and} \ A_{1}^{*}(a,x)= p [\mathcal{E}^{R}]\frac {\tau (x)\gamma _{1}(x)\phi _{1}(x)}{\mu _{1}(x) + \gamma _{1}(x)}\pi _{1}(0,a,x), \end{equation}
   for some constant $p [\mathcal{E}^{R}]$ that we compute as follows:

   from (3.31) and (3.24), one has $\displaystyle \frac {1}{r(\mathcal{L}_{1})}= H(E^*)= (1+E^*)^{-\kappa }$ .

   On the other hand, (3.27) leads to $\displaystyle E^*= p [\mathcal{E}^{R}]\int _{\Omega } \frac {\phi _{1}(x)}{\mu _{1}(x) + \gamma _{1}(x)} {\textrm {d}} x$ ,

   from which we obtain:

   \begin{equation*} p [\mathcal{E}^{R}]= \big( r(\mathcal{L}_{1})^{\frac {1}{K}} - 1 \big) \left ( \int _{\Omega } \frac {\phi _{1}(x)}{\mu _{1}(x) + \gamma _{1}(x)} {\textrm {d}} x \right )^{-1}. \end{equation*}
   It follows that $p [\mathcal{E}^{R}]\gt 0$ if and only if $r(\mathcal{L}_{1})\gt 1$ .

   Finally, system (2.5) has a fully-exposed steady state when $r(\mathcal{L}_{1})\gt 1$ such that $\mathcal{E}^R= (0,0,E_{1}^R(x),A_{1}^R(a,x))^T$ with

   \begin{equation*} E_{1}^{R}(x)= p [\mathcal{E}^{R}] \frac {\phi _{1}(x)}{\mu _{1}(x)+\gamma _{1}(x)} \quad \mbox{and}\quad A_{1}^{R}(a,x)= p [\mathcal{E}^{R}] \frac {\tau (x)\gamma _1(x)\phi _{1}(x)}{\mu _{1}(x)+\gamma _{1}(x)}\pi _{1}(0,a,x). \end{equation*}

3. 3. Here, we assume that $E_{i}^{*}$ and $A_{i}^{*}$ are nonzero functions, for $i \in \{0,1\}$ .

   By setting $u^{*}(x)= \left ( E_{0}^{*}(x), E_{1}^{*}(x)\right )^T$ and $v^{*}(a,x)= \left ( A_{0}^{*}(a,x), A_{1}^{*}(a,x)\right )^T$ , system (3.23) rewrites as:

   (3.32) \begin{equation} \left \{ \begin{array}{rl} h(u^*) \int _0^\infty \mathcal{B}[v^*](a,x){\textrm {d}} a - \mathcal{N}(x)u^*(x) & = \ 0, \\ v^*(0,x) & = \ \mathcal{C}(x)u^*(x), \\ \partial _a v^*(a,x) & = \ - \mathcal{D}(a,x)v^*(a,x). \end{array} \right . \end{equation}

   Solving (3.32) for $v^*$ yields

   (3.33) \begin{equation} v^*(a,x)= \boldsymbol{\Pi }(0,a,x)\mathcal{C}(x)u^*(x). \end{equation}

   Replacing (3.33) in (3.32) gives

   (3.34) \begin{equation} h(u^*) \int _{\Omega }\int _0^\infty \boldsymbol{m}(x,y)\boldsymbol{\Pi }(0,a,y)\mathcal{C}(y)u^*(y){\textrm {d}} a{\textrm {d}} y = \mathcal{N}(x)u^*(x). \end{equation}
   Define $\bar {u}^*(x)= \mathcal{N}(x)u^*(x)$ . Then, (3.34) becomes
   \begin{equation*}\int _{\Omega }\int _0^\infty \boldsymbol{m}(x,y)\boldsymbol{\Pi }(0,a,y)\mathcal{C}(y)\mathcal{N}^{-1}(y)\bar {u}^*(y){\textrm {d}} a{\textrm {d}} y =\frac {1}{h(u^*)} \bar {u}^*(x).\end{equation*}
   Consequently, we obtain
   (3.35) \begin{equation} \int _{\Omega }\boldsymbol{m}(x,y)\boldsymbol{\theta }(y)\bar {u}^*(y) {\textrm {d}} y = \frac {1}{h(u^*)} \bar {u}^*(x), \end{equation}
   where $\boldsymbol{\theta }(x)$ is given by
   \begin{equation*}\boldsymbol{\theta }(x) = \left (\int _0^\infty \boldsymbol{r}(a,x) \boldsymbol{\Pi }(0,a,x) {\textrm {d}} a\right )\mathcal{C}(x)\mathcal{N}^{-1}(x).\end{equation*}
   Equality (3.35) rewrites as:
   (3.36) \begin{equation} \boldsymbol{\mathcal{L}} [\bar {u}^*](x)= \frac {1}{h(u^*)} \bar {u}^*(x), \end{equation}
   with $\boldsymbol{\mathcal{L}}$ the linear operator given by (3.17). Therefore, the existence of $\bar {u}^* \gt 0$ is strongly related to the spectral property of operator $\boldsymbol{\mathcal{L}}$ . Since operator $\boldsymbol{\mathcal{L}}$ is positive, compact and irreducible, then Krein-Rutman theorem applies (see [Reference Du15]) and ensures the existence of an eigenfunction $\boldsymbol{\psi }^* \in L^1(\Omega , {\mathbb{R}}^2)$ , $\boldsymbol{\psi }^*\gt 0$ and normalized (ie. $\|\boldsymbol{\psi }^*\|_{L^1\times L^1}= 1$ ) such that:
   \begin{equation*} \boldsymbol{\mathcal{L}}\left [ \boldsymbol{\psi }^* \right ](x)= r(\boldsymbol{\mathcal{L}}) \boldsymbol{\psi }^*(x). \end{equation*}
   We deduce from (3.36) that
   (3.37) \begin{equation} r(\boldsymbol{\mathcal{L}})= \frac {1}{h(u^*)} = \left (1+ \bar h(u^*) \right )^\kappa\! ; \end{equation}
   (3.38) \begin{equation} u^*(x)= p\left [\mathcal{E}^{*}\right ]\mathcal{N}^{-1}(x)\boldsymbol{\psi }^*(x) ; \end{equation}
   and
   (3.39) \begin{equation} v^*(a,x)= p\left [\mathcal{E}^{*}\right ]\boldsymbol{\Pi }(0,a,x)\mathcal{C}(x) \mathcal{N}^{-1}(x)\boldsymbol{\psi }^*(x), \end{equation}
   for some constant $p\left [\mathcal{E}^{*}\right ]$ . Equality (3.37) leads to $\bar h(u^*) = r(\boldsymbol{\mathcal{L}})^{\frac {1}{\kappa }}-1$ . Furthermore, from (3.38), one has:
   \begin{equation*} \bar h(u^*) = p\left [\mathcal{E}^{*}\right ] \bar h \left (\mathcal{N}^{-1}({\cdot})\boldsymbol{\psi }^*({\cdot}) \right )\!. \end{equation*}
   Finally, we obtain:
   \begin{equation*} p\left [\mathcal{E}^{*}\right ]= \big( r(\boldsymbol{\mathcal{L}})^{\frac {1}{\kappa }}-1 \big) \bar h \left (\mathcal{N}^{-1}({\cdot})\boldsymbol{\psi }^*({\cdot}) \right )^{-1}, \end{equation*}
   with $p\left [\mathcal{E}^{*}\right ]\gt 0$ if and only if $r(\boldsymbol{\mathcal{L}})\gt 1$ .

   Notice that $ \boldsymbol{ \mathcal{L}} = \begin{pmatrix} (1-c) \mathcal{L}_0 & 0 \\ c \mathcal{L}_0 & \mathcal{L}_1 \end{pmatrix}.$ Thus, it follows that $r(\boldsymbol{\mathcal{L}})= \max \left ( (1-c)r(\mathcal{L}_{0}), r(\mathcal{L}_1)\right )$ .

4.1. Technical materials

The first technical result reads as:

Proof of Lemma 4.1. To prove that $\mathsf{s}(\mathcal{V}[\overline {\mathcal{E}}])\lt 0$ , it is sufficient to prove that $\mathcal{V}[\overline {\mathcal{E}}]$ is resolvent positive.

Let $\lambda \in {\mathbb{C}}$ , $(u, 0_{L^1}, v)^T \in X_0$ and $(\varphi , \phi , \psi )^T \in X$ such that $\left ( \lambda \mathrm{I}_{\mathrm{d}} - \mathcal{V}[\bar {\mathcal{E}}] \right ) \left(\begin{array}{c} u \\ 0_{L^1} \\ v \end{array}\right)= \left(\begin{array}{c} \varphi \\ \phi \\ \psi \end{array}\right)$ . One has:

(4.3) \begin{equation} \begin{cases} \left ( \lambda \mathrm{I}_{\mathrm{d}} + \mathcal{N}(x) \right )u(x) + \kappa Dh(\bar {u}) (u) \int _0^\infty \mathcal{B}[\bar {v}](a,x) {\textrm {d}} a = \varphi (x), \\ v(0,x) = \phi (x), \\ \partial _a v(a,x) + \left ( \lambda \mathrm{I}_{\mathrm{d}} + \mathcal{D}(a,x) \right )v(a,x) = \psi (a,x). \end{cases} \end{equation}

Solving (4.3) for $v$ yields

(4.4) \begin{equation} v(a,x) = e^{-\lambda a} \boldsymbol{\Pi }(0,a,x)\phi (x) + \int _0^a e^{-\lambda (a-s)} \boldsymbol{\Pi }(s,a,x) \psi (s,x) {\textrm {d}} s, \end{equation}

for $\displaystyle \Re (\lambda ) \gt - \min \big\{\! \inf _{(0,\infty ) \times \Omega } d_{0}(\cdot ,\cdot ), \inf _{(0,\infty ) \times \Omega } d_{1}(\cdot ,\cdot ) \big\}$ .

Replacing (4.4) in (4.3) leads to

(4.5) \begin{align} u(x) & + \kappa h^{1+\frac {1}{\kappa }}(\bar {u}) \left ( \int _{\Omega } \mathbf{1}_2 u(y) {\textrm {d}} y \right ) b(\lambda , \bar v)(x) + \kappa h^{1+\frac {1}{\kappa }}(\bar {u}) f(\lambda ,\bar u,\bar v,\phi ,\psi ) b(\lambda , \bar v)(x) \nonumber \\ &= \left ( \lambda \mathrm{I}_{\mathrm{d}} + \mathcal{N}(x) \right )^{-1}\varphi (x), \ \text{for } \Re (\lambda ) \gt - \min \Big\{ \sup _{x \in \Omega } \{\mu _{0}(x) + \gamma _{0}(x)\} , \sup _{x \in \Omega } \{\mu _{1}(x) + \gamma _{1}(x)\} \Big\}, \end{align}

wherein we have set

\begin{align*} b(\lambda , \bar v)(x)= & \left ( \lambda \mathrm{I}_{\mathrm{d}} + \mathcal{N}(x) \right )^{-1} \left ( \int _0^\infty \mathcal{B}[\bar {v}](a,x) {\textrm {d}} a \right )\!,\\ f(\lambda ,\bar u,\bar v,\phi ,\psi )= & \int _0^\infty \int _{\Omega } \boldsymbol{\mathcal{T}}(y) \left ( e^{-\lambda a} \boldsymbol{\Pi }(0,a,y)\phi (y) + \int _0^a e^{-\lambda (a-s)} \boldsymbol{\Pi }(s,a,y) \psi (s,y) {\textrm {d}} s \right ) {\textrm {d}} y {\textrm {d}} a . \end{align*}

Therefore, by (4.5), we find that

\begin{equation*} \int _{\Omega } \mathbf{1}_2 u(y) {\textrm {d}} y = \left ( 1+ \kappa h^{1+\frac {1}{\kappa }}(\bar {u}) \int _{\Omega } b(\lambda , \bar v)(x) {\textrm {d}} x \right )^{-1} \int _{\Omega } \mathbf{1}_2 f(\lambda ,\bar u,\bar v)(x) {\textrm {d}} x, \end{equation*}

with

\begin{align*} & f(\lambda ,\bar u,\bar v)(x)= \left ( \lambda \mathrm{I}_{\mathrm{d}} + \mathcal{N}(x) \right )^{-1}\varphi (x)- \kappa h^{1+\frac {1}{\kappa }}(\bar {u}) f(\lambda ,\bar u,\bar v,\phi ,\psi ) b(\lambda , \bar v)(x). \end{align*}

We then find that

\begin{align*} u(x) & = \left ( \lambda \mathrm{I}_{\mathrm{d}} + \mathcal{N}(x) \right )^{-1}\varphi (x) - \kappa h^{1+\frac {1}{\kappa }}(\bar {u}) f(\lambda ,\bar u,\bar v,\phi ,\psi ) b(\lambda , \bar v)(x) \\ & \quad -\kappa h^{1+\frac {1}{\kappa }}(\bar {u}) \left [ \left ( 1+ \kappa h^{1+\frac {1}{\kappa }}(\bar {u}) \int _{\Omega } b(\lambda , \bar v)(y) {\textrm {d}} y \right )^{-1} \int _{\Omega } \mathbf{1}_2 f(\lambda ,\bar u,\bar v)(y) {\textrm {d}} y \right ] b(\lambda , \bar v)(x) . \end{align*}

Finally, we have

(4.6) \begin{equation} \left ( \lambda \mathrm{I}_{\mathrm{d}} - \mathcal{V}[\overline {\mathcal{E}}]\right )^{-1} \begin{pmatrix} \varphi \\ \phi \\ \psi \end{pmatrix} = \begin{pmatrix} \left ( \lambda \mathrm{I}_{\mathrm{d}} + \mathcal{N}(x) \right )^{-1}\varphi (x) - h(\lambda , a, \bar {u}, \bar {v}, \phi , \psi )\\ 0_{L^1(\Omega , {\mathbb{R}}^2)} \\[4pt] e^{-\lambda a} \boldsymbol{\Pi }(0,a,x) \phi (x) + \int _0^a e^{-\lambda (a-s)} \boldsymbol{\Pi }(s,a,x) \psi _{i}(s,x) {\textrm {d}} s \end{pmatrix}\!, \text{ for } \lambda \gt - \lambda _0, \end{equation}

where

\begin{align*} h(\lambda , a, \bar {u}, \bar {v}, \phi , \psi ) & = \kappa h^{1+\frac {1}{\kappa }}(\bar {u}) f(\lambda ,\bar u,\bar v,\phi ,\psi ) b(\lambda , \bar v)(x) \\ & \quad + \kappa h^{1+\frac {1}{\kappa }}(\bar {u}) \left [ \left ( 1+ \kappa h^{1+\frac {1}{\kappa }}(\bar {u}) \int _{\Omega } b(\lambda , \bar v)(y) {\textrm {d}} y \right )^{-1} \int _{\Omega } \mathbf{1}_2 f(\lambda ,\bar u,\bar v)(y) {\textrm {d}} y \right ] b(\lambda , \bar v)(x), \end{align*}

and $\displaystyle \lambda _0 = \min \big\{ \inf _{(0,\infty ) \times \Omega } d_{0}(\cdot ,\cdot ) , \inf _{(0,\infty ) \times \Omega } d_{1}(\cdot ,\cdot ), \sup _{x \in \Omega } \{\mu _{0}(x) + \gamma _{0}(x)\} , \sup _{x \in \Omega } \{\mu _{1}(x) + \gamma _{1}(x)\} \big\}$ .

For $\bar {v}= 0$ , it is straight forward to show that $(\lambda \mathrm{I}_{\mathrm{d}} - \mathcal{V}[\overline {\mathcal{E}}])^{-1}(\varphi , \phi , \psi )^T \in X_{0,+}$ for any $(\varphi , \phi , \psi )^T \in X_+.$ It follows that $\mathcal{V}[\overline {\mathcal{E}}]$ is resolvent positive for $\bar {v}= 0$ and $\lambda \gt - \lambda _0$ .

We now set

(4.7) \begin{equation} \mathcal{S}_{\lambda } = \mathsf{r}(\mathcal{G}[\overline {\mathcal{E}}](\lambda \textrm {I}_{\textrm {d}}-\mathcal{V}[\overline {\mathcal{E}}])^{-1}),\quad \forall \lambda \gt \mathsf{s}(\mathcal{V}[\overline {\mathcal{E}}]). \end{equation}

Thanks to [Reference Thieme42, Theorem 3.4], we have

\begin{equation*} \mathsf{sgn}(\mathcal{S}_{\lambda } -1) = \mathsf{sgn}(\lambda +\mathsf{s}(\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}])),\quad \forall \lambda \gt \mathsf{s}(\mathcal{V}[\overline {\mathcal{E}}]). \end{equation*}

Since $(\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}])$ and $(\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}])_0$ , the part of $(\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}])$ in $X_0$ has the same spectral set [Reference Magal and Ruan28, Lemma 2.1], we have

\begin{equation*} \mathsf{sgn}(\mathcal{S}_{\lambda } -1) = \mathsf{sgn}(\lambda +\mathsf{s}(\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}]))=\mathsf{sgn}(\lambda +\mathsf{s}((\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}])_0)),\quad \forall \lambda \gt \mathsf{s}(\mathcal{V}[\overline {\mathcal{E}}]). \end{equation*}

Let us go through more explicit characterization of $\mathcal{S}_\lambda$ . For that ends, let us introduce the Banach space

\begin{equation*} X_1= L^{1}(\Omega ,{\mathbb{R}}^2) \times L^{1}(\Omega ,{\mathbb{R}}^2) \times 0_{L^{1}((0,\infty ) \times \Omega ,{\mathbb{R}}^2)}. \end{equation*}

Notice that $\mathcal{G}[\overline {\mathcal{E}}](\lambda \textrm {I}_{\textrm {d}}-\mathcal{V}[\overline {\mathcal{E}}])^{-1}$ has the same spectral radius as its restriction on $X_1$ [Reference Djidjou-Demasse, Goudiaby and Seydi14, Lemma 2.2]. We then introduce the linear operator $\mathcal{M}_\lambda \,:\, X_1 \to X_1$ by

\begin{equation*} \mathcal{M}_\lambda (\varphi ,\phi ,0)= \mathcal{G}[\bar {\mathcal{E}}] \left ( \lambda \mathrm{I}_{\mathrm{d}} - \mathcal{V}[\bar {\mathcal{E}}] \right )^{-1}(\varphi ,\phi ,0). \end{equation*}

For $\lambda \gt \mathsf{s}(\mathcal{V}[\overline {\mathcal{E}}])$ and $(\varphi ,\phi ,0) \in X_1$ , we have, from (4.2) and (4.6):

(4.8) \begin{equation} \mathcal{M}_\lambda (\varphi ,\phi ,0)= \mathcal{G}[\bar {\mathcal{E}}] \left ( \lambda \mathrm{I}_{\mathrm{d}} - \mathcal{V}[\bar {\mathcal{E}}] \right )^{-1} \begin{pmatrix} \varphi \\ \phi \\ 0 \end{pmatrix}= \begin{pmatrix} \mathcal{F}_\lambda [\phi ] \\ \mathcal{H}_\lambda [\varphi ]\\ 0 \end{pmatrix}\!, \end{equation}

with

\begin{align*} & \mathcal{F}_\lambda [\phi ](x)= h(\bar u) \int _0^\infty \mathcal{B} \left [ e^{-\lambda a} \boldsymbol{\Pi }(0,a,x) \phi (x) \right ] {\textrm {d}} a,\\ & \mathcal{H}_\lambda [\varphi ](x)= \mathcal{C}(x) \left (\lambda \mathrm{I}_{\mathrm{d}}+ \mathcal{N}(x) \right )^{-1} \varphi (x). \end{align*}

By (4.8), it comes

(4.9) \begin{equation} \left ( \mathcal{G}[\bar {\mathcal{E}}] \left ( \lambda \mathrm{I}_{\mathrm{d}} - \mathcal{V}[\bar {\mathcal{E}}] \right )^{-1} \right )^2 \begin{pmatrix} \varphi \\ \phi \\ 0 \end{pmatrix}= \begin{pmatrix} \mathcal{F}_\lambda \circ \mathcal{H}_\lambda [\varphi ] \\ \mathcal{H}_\lambda \circ \mathcal{F}_\lambda [\phi ] \\ 0 \end{pmatrix}\!. \end{equation}

Because $\mathsf{r}(\mathcal{H}_\lambda \circ \mathcal{F}_\lambda )=\mathsf{r}(\mathcal{F}_\lambda \circ \mathcal{H}_\lambda )$ and $\mathsf{r}\big(\big( \mathcal{G}[\bar {\mathcal{E}}] \left ( \lambda \mathrm{I}_{\mathrm{d}} - \mathcal{V}[\bar {\mathcal{E}}] \right )^{-1} \big)^2 \big)=\mathsf{r}\big( \mathcal{G}[\bar {\mathcal{E}}] \left ( \lambda \mathrm{I}_{\mathrm{d}} - \mathcal{V}[\bar {\mathcal{E}}] \right )^{-1} \big)^2$ , it follows from (4.7) and (4.9) that

\begin{equation*} \mathcal{S}_{\lambda }=\sqrt {\mathsf{r}\big( \mathcal{G}[\bar {\mathcal{E}}] \left ( \lambda \mathrm{I}_{\mathrm{d}} - \mathcal{V}[\bar {\mathcal{E}}] \right )^{-1} \big)^2}=\sqrt {\mathsf{r}(\mathcal{H}_\lambda \circ \mathcal{F}_\lambda )}. \end{equation*}

From the above discussion, we have the following results.

Note that, since

\begin{align*} \mathcal{F}_0 \circ \mathcal{H}_0 [\varphi ](x) & = \mathcal{F}_0 \left [\mathcal{C}({\cdot}) \mathcal{N}^{-1}({\cdot}) \varphi \right ](x) = h(\bar {u}) \boldsymbol{\mathcal{L}}[\varphi ](x), \end{align*}

it comes

\begin{equation*} \mathcal{S}_0= \sqrt {\mathsf{r}(\mathcal{H}_0 \circ \mathcal{F}_0)}= \sqrt {h(\bar {u})\mathsf{r}(\boldsymbol{\mathcal{L}})}= \sqrt {h(\bar {u}) \max \{(1-c)\mathsf{r}(\mathcal{L}_0), \mathsf{r}(\mathcal{L}_1)}\}.\end{equation*}

Proof of Lemma 4.2. The first item of Lemma4.2 is a direct consequence of the fact that $X$ is an abstract L space (see [Reference Thieme42, Theorem 3.14]). The second item is a consequence of Proposition4.1, item 1.

We still need to prove item 3 of the lemma. Assume $\mathcal{S}_0\gt 1$ . Since $\mathcal{V}[\overline {\mathcal{E}}]$ is a Hille–Yosida operator, the map $\lambda \in (\mathsf{s}(\mathcal{V}[\overline {\mathcal{E}}]),\infty ) \to \mathcal{M}_\lambda \in \mathcal{L}(X_1)$ is continuous. Recalling that $\mathcal{M}_\lambda ^2$ is a compact operator for all $\lambda \gt \mathsf{s}(\mathcal{V}[\overline {\mathcal{E}}])$ , we find that $\lambda \in (\mathsf{s}(\mathcal{V}[\overline {\mathcal{E}}]),\infty ) \to \mathsf{r}\left (\mathcal{M}_\lambda ^2\right )$ is continuous (e.g. see [Reference Degla12, Theorem 2.1]). Furthermore, since $\mathcal{V}[\overline {\mathcal{E}}]$ is a Hille–Yosida linear operator, one has $\mathcal{M}_\lambda ^2 \to 0_{\mathcal{L}(X_1)}$ when $\lambda \to \infty$ . This implies that $\mathcal{S}_{\lambda }=\sqrt {\mathsf{r}\left (\mathcal{M}_ \lambda ^2\right )}\to 0$ when $\lambda \to \infty$ . Since $\mathcal{S}_{0}\gt 1$ , we infer from the intermediate values theorem that there exists $\lambda \gt 0$ such that

\begin{equation*} \mathcal{S}_{\lambda }=1 \Longleftrightarrow \mathsf{r}\left (\mathcal{M}_\lambda ^2\right )=1. \end{equation*}

By Proposition4.1, item 1, it comes

\begin{equation*} \lambda =\mathsf{s}((\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}])_0)=\mathsf{s}(\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}])\gt 0. \end{equation*}

With similar arguments as in the proof of Theorem3.1 (item 4), we can find that the part $(\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}])_0$ of $(\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}])$ in $X_0$ generated a $C_0$ -semigroup $\{T_{(\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}])_0}(t)\}_{t\ge 0}$ such that

\begin{equation*} T_{(\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}])_0}(t)=W_1(t)+W_2(t), \end{equation*}

where $W_1(t)$ is a compact linear operator for all $t\gt 0$ , and $W_{2}(t)$ verified

\begin{equation*} \Vert W_2(t) \Vert _{\mathcal{L}( X_0)} \leq \ell e^{-\eta t},\quad \forall t\geq 0, \end{equation*}

with $\ell$ and $\eta$ , two positive constants. Consequently, by the same arguments as in [Reference Webb46, Proposition 2.4], it comes

\begin{equation*} \omega _{0,ess}((\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}])_0)\leq \limsup _{t\to +\infty } \dfrac {\ln (\Vert W_2(t) \Vert _{\mathcal{L}( X_0)})}{t}\leq -\eta , \end{equation*}

with $\omega _{0,ess}((\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}])_0)$ the essential growth bound of $(\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}])_0$ .

Finally, since $\lambda =\omega ((\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}])_0)\gt 0$ , Proposition4.1 (item 1) gives that $\omega _{0,ess}((\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}])_0)\lt \omega ((\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}])_0)=\lambda =\mathsf{s}((\mathcal{G}[\overline {\mathcal{E}}]+\mathcal{V}[\overline {\mathcal{E}}])_0)$ . The result follows directly from [Reference Webb46, Proposition 2.5] or [Reference Magal and Ruan29, Proposition 4.6.5].

4.2. Asymptotic behaviour

In this section, we present results concerning the asymptotic behaviour of System (2.3)–(2.4). These results include the non-persistence of both exposed and unexposed mosquito populations when $\max \{(1-c)\mathsf{r}(\mathcal{L}_0), \mathsf{r}(\mathcal{L}_1)\}\lt 1$ , as well as the non-persistence of the unexposed mosquito population when $(1 - c)r(\mathcal{L}_0) \lt 1$ .

Proof of Theorem 4.1. By Lemma4.2, we have

\begin{equation*} \mathsf{sgn}(\mathcal{S}_{0} -1) = \mathsf{sgn}(\mathsf{s}(G[\mathcal{E}^0])_0) = \mathsf{sgn}(\omega (G[\mathcal{E}^0])_0), \end{equation*}

with $\mathcal{S}_0 = \sqrt { \max \{(1-c)\mathsf{r}(\mathcal{L}_0), \mathsf{r}(\mathcal{L}_1)\} }$ , $\mathcal{E}^0$ the mosquito-free steady state and $G[\mathcal{E}^0] = \mathcal{G}[\mathcal{E}^0] + \mathcal{V}[\mathcal{E}^0]$ the linear operator associated to the linearization of System (3.2) around $\mathcal{E}^0$ . From where item 1 of Theorem4.1 directly follows.

For item 2, let $\boldsymbol{w}_0\in X_+$ be an initial condition, and $\Phi (t,\boldsymbol{w}_{0}) = (E_0,E_1,A_0,A_1)$ the associated semiflow. Since $H(E(t))\le 1$ , for all $t\ge 0$ , it comes

\begin{equation*} \begin{cases} \partial _t E_{0}(t,x) \le \int _{\Omega } \int _0^\infty m_{0}(x,y) r_{0}(a,y) A_{0}(t,a,y) {\textrm {d}} a {\textrm {d}} y - (\mu _{0}(x) + \gamma _{0}(x)) E_{0}(t,x), \\[2pt] \partial _t E_{1}(t,x) \le \int _{\Omega } \int _0^\infty m_{1}(x,y) r_{1}(a,y) A_{1}(t,a,y) {\textrm {d}} a {\textrm {d}} y- (\mu _{1}(x) + \gamma _{1}(x))E_{1}(t,x), \\[2pt] \left (\partial _t+\partial _a\right ) A_{0}(t,a,x) = - d_{0}(a,x) A_{0}(t,a,x), \\ \left (\partial _t+\partial _a\right ) A_{1}(t,a,x) = - d_{1}(a,x) A_{1}(t,a,x). \end{cases} \end{equation*}

Therefore, by the comparison theorem in [Reference Magal, Seydi and Wang30], it follows that for all $t\geq 0$ :

(4.10) \begin{equation} \begin{cases} 0_{L^1}\leq E_0(t,\cdot ) \leq Z(t,\cdot ), \\ 0_{L^1}\leq A_0(t,\cdot ,\cdot ) \leq J(t,\cdot ,\cdot ), \end{cases} \end{equation}

where $t\mapsto (Z(t,\cdot ),J(t,\cdot ,\cdot ))$ is the mild solution of the following system:

(4.11) \begin{equation} \begin{cases} \partial _t Z (t,x) = \int _{\Omega } \int _0^\infty m_{0}(x,y) r_{0}(a,y) J (t,a,y) {\textrm {d}} a {\textrm {d}} y - (\mu _{0}(x) + \gamma _{0}(x)) Z (t,x), \\ J (t,a=0,x) = (1-c)\tau (x) \gamma _{0}(x) Z (t,x) ,\\ \left (\partial _t+\partial _a\right ) J (t,a,x) = - d_{0}(a,x) J (t,a,x), \end{cases} \end{equation}

with the initial condition $Z_0({\cdot})= E_0(0,\cdot )$ , and $J_0(\cdot ,\cdot )=A_0(0,\cdot ,\cdot )$ at time $t=0$ .

Note that $O=(0,0)$ is always a stationary state of System (4.11). By setting $w(t) = (Z(t,\cdot ), 0_{L^1(\Omega ,{\mathbb{R}})}, J(t,\cdot ,\cdot ))$ , System (4.11) rewrites as

\begin{equation*} \frac {{\textrm {d}} w(t)}{{\textrm {d}} t} = \left ( \mathcal{V} + \mathcal{G} \right ) w(t), \end{equation*}

where

\begin{equation*} \mathcal{V} \begin{pmatrix} Z(x) \\ 0_{L^{1}} \\ J(a,x) \end{pmatrix} = \begin{pmatrix} - (\mu _0(x)+\gamma _0(x)) Z(x) \\ - J(0,x) \\ - \partial _a J(a,x) - d_0(a,x)J(a,x) \end{pmatrix}\!, \end{equation*}

and

\begin{equation*} \mathcal{G} \begin{pmatrix} Z(x) \\ 0_{L^{1}} \\ J(a,x) \end{pmatrix} = \begin{pmatrix} \int _{\Omega } \int _0^\infty m_{0}(x,y) r_{0}(a,y) J (t,a,y) {\textrm {d}} a {\textrm {d}} y \\ (1-c)\tau (x) \gamma _{0}(x) Z(x) \\[2pt] 0_{L^1((0,\infty ) \times \Omega , {\mathbb{R}})} \end{pmatrix}\!. \end{equation*}

Therefore, by similar arguments as for the proof of Lemma4.2, we find that

\begin{equation*} \lim _{t\to +\infty } \left (\int _\Omega Z(t,x) {\textrm {d}} x+\int _\Omega \int _0^\infty J(t,a,x) {\textrm {d}} a {\textrm {d}} x\right )=0. \end{equation*}

Item 3 then holds from (4.10).

4.3. Uniform persistence

Assume that $r(\boldsymbol{\mathcal{L}})= \max \{(1-c)\mathsf{r}(\mathcal{L}_0), \mathsf{r}(\mathcal{L}_1)\}\gt 1$ . Let the non-negative and continuous map $\rho \,:\, X_+ \to {\mathbb{R}}_+$ , such that, for all $(\boldsymbol{u}_0, 0_{L^{1}}, \boldsymbol{v}_0) \in X_+$ :

\begin{equation*} \rho (\boldsymbol{u}_0, 0_{L^{1}}, \boldsymbol{v}_0)=\int _\Omega \left ( E_{0}(x)+ E_{1}(x)\right ) {\textrm {d}} x + \int _\Omega \int _0^\infty \left ( A_{0}(a,x) + A_{1}(a,x)\right ){\textrm {d}} a {\textrm {d}} x, \end{equation*}

where $\boldsymbol{u}_0(x)= \left (E_{0}(x), E_{1}(x)\right )^T$ and $\boldsymbol{v}_0(a,x)= \left (A_{0}(a,x), A_{1}(a,x)\right )^T$ .

Next, we set

\begin{equation*} M_0\,:\!=\,\left \lbrace \boldsymbol{w}_0= (\boldsymbol{u}_0, 0_{L^{1}}, \boldsymbol{v}_0) \in X_+ \,:\, \rho (\boldsymbol{w}_0)\gt 0 \right \rbrace \!, \end{equation*}

and

\begin{equation*} \partial M_0\,:\!=\,\left \lbrace \boldsymbol{w}_0= (\boldsymbol{u}_0, 0_{L^{1}}, \boldsymbol{v}_0) \in X_+ \,:\, \rho (\boldsymbol{w}_0)=0 \right \rbrace\! . \end{equation*}

Note that $X_+= M_0 \cup \partial M_0$ .

Lemma 4.3. Let Assumptions 2.1 and 2.2 be satisfied. Further assume that $ \max \{(1-c)\mathsf{r}(\mathcal{L}_0),$ $\mathsf{r}(\mathcal{L}_1)\}\gt 1$ . We have

\begin{equation*} \forall t\geq 0, \quad \Phi (t,\partial M_0) \subset \partial M_0, \quad \text{and} \quad \Phi (t, M_0) \subset M_0. \end{equation*}

where $\left \lbrace \Phi (t,\boldsymbol{w}_{0}) = (\boldsymbol{u}(t,\cdot ), 0_{L^1}, \boldsymbol{v}(t,\cdot ,\cdot )) \right \rbrace _t$ is the associated semiflow.

Proof of Lemma 4.3. Let $\boldsymbol{w}_0= (\boldsymbol{u}_0, 0_{L^{1}}, \boldsymbol{v}_0) \in X_+$ and $\left \lbrace \Phi (t,\boldsymbol{w}_{0}) = (\boldsymbol{u}(t,\cdot ), 0_{L^1}, \boldsymbol{v}(t,\cdot ,\cdot )) \right \rbrace _t$ the associated semiflow. The Volterra formulation of System (2.5) leads to

(4.12) \begin{equation} \boldsymbol{v}(t,a,x) = \begin{cases} \boldsymbol{\Pi }(a-t,a,x)\,\boldsymbol{v}_0(a-t,x) \quad &\text{if}\ \ t\leq a, \\[3pt] \boldsymbol{\Pi }(0,a,x)\,\mathcal{C}(x)\,\boldsymbol{u}(t-a,x)& \text{if}\ \ t\gt a, \end{cases} \end{equation}

and

(4.13) \begin{align} \partial _t \boldsymbol{u}(t,x) & = \displaystyle h(\boldsymbol{u})(t) \int _0^t \int _{\Omega } \boldsymbol{m}(x,y) \boldsymbol{r}(a,y) \boldsymbol{\Pi }(0,a,y)\,\mathcal{C}(y)\,\boldsymbol{u}(t-a,y) {\textrm {d}} y {\textrm {d}} a \nonumber \\[4pt] & \quad + h(\boldsymbol{u})(t) \int _t^{\infty } \int _{\Omega } \boldsymbol{m}(x,y) \boldsymbol{r}(a,y) \boldsymbol{\Pi }(a-t,a,y)\,\boldsymbol{v}_0(a-t,y) {\textrm {d}} y {\textrm {d}} a - \mathcal{N}(x) \boldsymbol{u}(t,x). \end{align}

Let $\boldsymbol{w}_0= (\boldsymbol{u}_0, 0_{L^{1}}, \boldsymbol{v}_0) \in \partial M_0$ , i.e. $\displaystyle \int _\Omega \boldsymbol{u}_0(x) {\textrm {d}} x=0$ and $\displaystyle \int _\Omega \int _0^\infty \boldsymbol{v}_0(a,x) {\textrm {d}} a {\textrm {d}} x=0$ . Note that

\begin{equation*} \int _t^{\infty } \int _{\Omega } \boldsymbol{m}(x,y) \boldsymbol{r}(a,y) \boldsymbol{\Pi }(a-t,a,y)\,\boldsymbol{v}_0(a-t,y) {\textrm {d}} y {\textrm {d}} a \le \|\boldsymbol{m}\|_\infty \|\boldsymbol{r}\|_\infty \int _\Omega \int _0^\infty \boldsymbol{v}_0(a,x){\textrm {d}} a {\textrm {d}} x =0. \end{equation*}

From where (4.13) becomes

\begin{eqnarray*} \partial _t \boldsymbol{u}(t,x) = h(\boldsymbol{u})(t) \int _0^t \int _{\Omega } \boldsymbol{m}(x,y) \boldsymbol{r}(a,y) \boldsymbol{\Pi }(0,a,y)\,\mathcal{C}(y)\,\boldsymbol{u}(t-a,y) {\textrm {d}} y {\textrm {d}} a - \mathcal{N}(x) \boldsymbol{u}(t,x), \end{eqnarray*}

i.e. $\displaystyle \int _\Omega \boldsymbol{u}(t,x) {\textrm {d}} x=0$ , for all $t\gt 0$ , as soon as $\displaystyle \int _\Omega \boldsymbol{u}_0(x) {\textrm {d}} x=0$ . Moreover, estimate (4.12) then gives $\displaystyle \int _\Omega \int _0^\infty \boldsymbol{v}(t,a,x){\textrm {d}} a {\textrm {d}} x =0$ , for all $t\gt 0$ . Consequently, $\Phi (t,\partial M_0) \subset \partial M_0$ , for all $t\geq 0$ .

For the second part of the lemma, assume that $\boldsymbol{w}_0= (\boldsymbol{u}_0, 0_{L^{1}}, \boldsymbol{v}_0) \in M_0$ , i.e. $\displaystyle \int _\Omega \boldsymbol{u}_0(x) {\textrm {d}} x + \int _\Omega \int _0^\infty \boldsymbol{v}_0(a,x) {\textrm {d}} a {\textrm {d}} x \gt 0$ . If $\displaystyle \int _\Omega \boldsymbol{u}_0(x) {\textrm {d}} x \gt 0$ , then (4.13) gives $\displaystyle \int _\omega \boldsymbol{u}(t,x) {\textrm {d}} x \gt 0$ , for all $t\gt 0$ . From where, $\Phi (t,M_0) \subset M_0$ , for all $t\geq 0$ . If $\displaystyle \int _\Omega \int _0^\infty \boldsymbol{v}_0(a,x) {\textrm {d}} a {\textrm {d}} x \gt 0$ , then (4.12) gives $\displaystyle \int _\Omega \int _0^\infty \boldsymbol{v}(t,a,x) {\textrm {d}} a {\textrm {d}} x \gt 0$ , for all $t\gt 0$ . From where, $\Phi (t,M_0) \subset M_0$ , for all $t\geq 0$ .

Lemma 4.4. Let Assumptions 2.1 and 2.2 be satisfied. Further assume that $ \max \{(1-c)\mathsf{r}(\mathcal{L}_0),$ $\mathsf{r}(\mathcal{L}_1)\}\gt 1$ . Let $\boldsymbol{w}_0\in M_0$ . Then, there exists a constant $\eta \gt 0$ such that

\begin{equation*} \limsup _{t \to \infty } \rho (\Phi (t,\boldsymbol{w}_0)) \gt \eta . \end{equation*}

Proof of Lemma 4.4. Set $\Phi (t,\boldsymbol{w}_{0}) = (\boldsymbol{u}(t,\cdot ), 0_{L^1}, \boldsymbol{v}(t,\cdot ,\cdot ))$ . We argue by contradiction, i.e. let $\zeta \gt 0$ and small enough such that,

(4.14) \begin{equation} \limsup _{t \to \infty } \rho (\Phi (t,\boldsymbol{w}_0))= \limsup _{t \to \infty } \left ( \int _\Omega \boldsymbol{u}(t,x) {\textrm {d}} x+ \int _\Omega \int _0^\infty \boldsymbol{v}(t,a,x) {\textrm {d}} a {\textrm {d}} x \right ) \le \zeta . \end{equation}

Therefore, we can find $t_0\ge 0$ such that, $\rho (\Phi (t,\boldsymbol{w}_0)) \le \zeta$ , for all $t\ge t_0$ . Consequently, for all $t\ge t_0$ ,

\begin{equation*} \int _{\Omega } \langle \mathbf{1}_2, \boldsymbol{u}(t,x)\rangle {\textrm {d}} x+ \int _{\Omega } \int _0^\infty \langle \boldsymbol{\mathcal{T}}, \boldsymbol{v}(t,a,x)\rangle {\textrm {d}} a {\textrm {d}} x \le \max (1,\sup _x \tau ^{-1}) \rho (\Phi (t,\boldsymbol{w}_0))= \zeta \max (1,\sup _x \tau ^{-1}). \end{equation*}

It then comes,

\begin{equation*} \displaystyle h(\boldsymbol{u})(t) \ge ( 1 + \zeta \max (1,\sup _x \tau ^{-1}) )^{- \kappa }, \quad \forall t\ge t_0. \end{equation*}

By this latter inequality, System (2.5) becomes

\begin{equation*} \begin{cases} \displaystyle \partial _t \boldsymbol{u}(t,x) \ge \bar {h}(\zeta ) \int _0^{\infty }\mathcal{B}[\boldsymbol{v}(t,\cdot ,\cdot )](a,x){\textrm {d}} a - \mathcal{N}(x) \boldsymbol{u}(t,x),\\[2pt] \displaystyle \boldsymbol{v}(t,0,x) =\mathcal{C}(x) \boldsymbol{u}(t,x),\\ \displaystyle (\partial _t+\partial _a) \boldsymbol{v}(t,a,x) = - \mathcal{D}(a,x) \boldsymbol{v}(t,a,x), \end{cases} \end{equation*}

where $\bar {h}(\zeta )= \left ( 1 + \zeta \max \left (1,\sup _x \tau ^{-1}\right ) \right )^{- \kappa }$ . By the comparison principles (e.g. [Reference Magal, Seydi and Wang30]), it comes

\begin{equation*} 0\leq \boldsymbol{u}_\zeta (t,\cdot )\le \boldsymbol{u}(t+t_0,\cdot ), \quad \text{and} \quad 0\leq \boldsymbol{v}_\zeta (t,\cdot ,\cdot )\le \boldsymbol{v}(t+t_0,\cdot ,\cdot ), \end{equation*}

for all $t\ge 0$ , and where $\left (\boldsymbol{u}_\zeta (t,\cdot ), \boldsymbol{v}_\zeta (t,\cdot ,\cdot )\right )_t$ is the mild solution of the following system

(4.15) \begin{equation} \begin{cases} \displaystyle \partial _t \boldsymbol{u}_\zeta (t,x) = \bar {h}(\zeta ) \int _0^{\infty }\mathcal{B}[\boldsymbol{v}_\zeta (t,\cdot ,\cdot )](a,x){\textrm {d}} a - \mathcal{N}(x) \boldsymbol{u}_\zeta (t,x),\\[2pt] \displaystyle \boldsymbol{v}_\zeta (t,0,x) =\mathcal{C}(x) \boldsymbol{u}_\zeta (t,x),\\ \displaystyle (\partial _t+\partial _a) \boldsymbol{v}_\zeta (t,a,x) = - \mathcal{D}(a,x) \boldsymbol{v}_\zeta (t,a,x),\\ \boldsymbol{u}_\zeta (0,x)= \boldsymbol{u}(t_0,x), \quad \boldsymbol{v}_\zeta (0,a,x)= \boldsymbol{v}(t_0,a,x). \end{cases} \end{equation}

By setting, $w_\zeta (t) = (\boldsymbol{u}_\zeta (t,\cdot ), 0_{L^1}, \boldsymbol{v}_\zeta (t,\cdot ,\cdot ))$ , System (4.15) writes

(4.16) \begin{equation} \frac {{\textrm {d}} w_\zeta (t)}{{\textrm {d}} t} = (\mathcal{G}_\zeta + \mathcal{V}) w_\zeta (t), \end{equation}

with

\begin{equation*} \mathcal{G}_\zeta \begin{pmatrix} u(x) \\ 0_{L^{1}} \\ v(a,x) \end{pmatrix} = \begin{pmatrix} \bar {h}(\zeta ) \int _0^{\infty }\mathcal{B}[v (t,\cdot ,\cdot )](a,x){\textrm {d}} a \\[2pt] \mathcal{C}(x) u (t,x) \\ 0_{L^1((0,\infty ) \times \Omega , {\mathbb{R}}^2)} \end{pmatrix}\!, \end{equation*}

and

\begin{equation*} \mathcal{V} \begin{pmatrix} u(x) \\ 0_{L^{1}} \\ v(a,x) \end{pmatrix} = \begin{pmatrix} - \mathcal{N}(x) u(x) \\ - v(0,x) \\ - \partial _a v(a,x) - \mathcal{D}(a,x)v(a,x) \end{pmatrix}\!. \end{equation*}

Similarly, as in Lemma4.1, we find that the spectral bound $\mathsf{s}(\mathcal{V})$ of $\mathcal{V}$ satisfies $\mathsf{s}(\mathcal{V})\lt 0$ , and the linear operator $\mathcal{V}$ is resolvent positive. Let us set

\begin{equation*} \mathcal{S}_{\lambda }^\zeta = \mathsf{r}(\mathcal{G}_\zeta (\lambda \textrm {I}_{\textrm {d}}-\mathcal{V})^{-1}),\quad \forall \lambda \gt \mathsf{s}(\mathcal{V}). \end{equation*}

We claim that

Before dealing with the proof of Claim4.1, we first end with the proof of Lemma4.4. Let $w_\zeta ^0 = (\boldsymbol{u}_\zeta (t_0,\cdot ), 0_{L^1}, \boldsymbol{v}_\zeta (t_0,\cdot ,\cdot )) \in X_0$ , then by (4.16), we have $w_\zeta (t) = T_{(\mathcal{G}_\zeta +\mathcal{V})_0}(t)w_\zeta ^0$ for all $t\ge 0$ . Since $\mathcal{S}_0\gt 1$ , item 2 of Claim4.1 leads to

(4.17) \begin{equation} e^{-\lambda _\zeta t} \left \| T_{(\mathcal{G}_\zeta +\mathcal{V})_0}(t)w_\zeta ^0 \right \| \to \left \| \mathcal{P}_\zeta w_\zeta ^0 \right \|, \text{ as } t \to \infty . \end{equation}

Moreover, since $ \left \| w_\zeta ^0 \right \|= \int _\Omega \boldsymbol{u}_\zeta (t_0,x) {\textrm {d}} x+ \int _\Omega \int _0^\infty \boldsymbol{v}_\zeta (t_0,a,x) {\textrm {d}} a {\textrm {d}} x\gt 0$ , by item 2 of Claim4.1, we also have $\left \| \mathcal{P}_\zeta w_\zeta ^0 \right \|\gt 0$ . Therefore, by (4.17), it comes

\begin{equation*} \left \| T_{(\mathcal{G}_\zeta +\mathcal{V})_0}(t)w_\zeta ^0 \right \|= \int _\Omega \boldsymbol{u}(t,x) {\textrm {d}} x+ \int _\Omega \int _0^\infty \boldsymbol{v}(t,a,x) {\textrm {d}} a {\textrm {d}} x \to \infty , \text{ as }, t \to \infty . \end{equation*}

A contradiction holds with (4.14). This ends the proof of Lemma4.4.

5.1. Model parameters

The probability of mutation of unexposed AFM from IR level $y$ to level $x$ is assumed to follow a Gaussian distribution with mean $0$ and variance varJ0 $= 10^{-2}$ . Specifically,

(5.1) \begin{equation} m_{0}(x,y) = \frac {1}{\sqrt {0.02\pi }}e^{-\frac {1}{0.02}(y-x)^2}. \end{equation}

Similarly, the probability of mutation of exposed AFM from IR level $y$ to level $x$ follows a Gaussian distribution with $0$ and variance varJ1 $= 2\times$ varJ0 (due to high selection pressure from insecticide exposure). Thus,

(5.2) \begin{equation} m_{1}(x,y) = \frac {1}{\sqrt {0.04\pi }}e^{-\frac {1}{0.04}(y-x)^2}. \end{equation}

The egg-laying rate for an AFM depends on the age $a$ of the mosquito and its resistance level $x$ . As in the framework introduced by [Reference Djidjou-Demasse, Goudiaby and Seydi14], we assume that

(5.3) \begin{equation} r_0(a,x)\equiv r_1(a,x)= r_m \left [ 1+\left (\frac {r_m-r_0}{r_0}\right ) \left (\frac {r_0}{r_1}\cdot \frac {r_m-r_1}{r_m-r_0} \right )^x \right ]^{-1} \times \mathbf{1}_{a\gt a_{\textrm {L}}}, \end{equation}

where $r_m\,:\!=\, r({-}\infty ,\cdot )\lt \infty$ is a constant due to physiological constraints and $a_{\textrm {L}}$ is the average age at which AFMs start laying eggs. The constants $r_0$ and $r_1$ are egg-laying rates of the reference ‘sensitive’ and ‘resistant’ mosquitoes $x_0=0$ and $x_1=1$ , with $0\lt r_1\lt r_0\lt r_m$ given in Table 1.

The death rate $d_0(a,x)$ of unexposed AFMs aged $a$ with resistant level $x$ is such that

(5.4) \begin{equation} d_0(a,x)= \kappa _0 \mathbf{1}_{a\gt a_{\textrm {LS}}}, \end{equation}

where $a_{\textrm {LS}}$ is the average lifespan of mosquitoes, and $\kappa _0$ is a positive constant. With this formulation, the average lifespan of mosquitoes is approximately $a_{\textrm {LS}}$ days. Indeed, setting $D_0(a,x)= \exp {\left ({-} \int _0^a d_0(s,x) {\textrm {d}} s\right )}$ as the probability that a mosquito remains alive after $a$ days, the average lifespan is given by $\int _0^\infty D_0(a,x) {\textrm {d}} a= a_{\textrm {LS}} + \frac {1}{\kappa _0}.$ Here, we fix, e.g. $\kappa _0=10$ , such that $\int _0^\infty D_0(a,x) {\textrm {d}} a \approx a_{\textrm {LS}}$ . Thus, the exact value of $\kappa _0$ is not critical as long as this approximation holds.

For AFMs exposed to the insecticide, their death rate $d_1(a,x)$ accounts for the insecticide pressure such that

(5.5) \begin{equation} d_1(a,x)= \kappa _0 \mathbf{1}_{a\gt a_{\textrm {LS}}}+ \bar d_0 \left ( \dfrac {\bar d_1}{\bar d_0} \right )^x, \end{equation}

where $\bar d_0$ and $\bar d_1$ represent the insecticide activity experienced by the reference sensitive mosquito $x_0$ and resistant mosquito $x_1$ , respectively, and are given in Table 1. The survival probabilities of unexposed and exposed mosquitoes are illustrated in Figure 2, and all other model parameters are assumed to be constant and are provided in Table 1.

Figure 2.

The survival probability of mosquitoes population as a function of their age $a$ and resistance level $x$ . (A) For mosquitoes unexposed to insecticide. (B) For mosquitoes exposed to insecticide. Here, the probability of surviving insecticide exposure during one day is $p_S=10^{-10}$ .

5.2. Typical simulated dynamics

For all simulated dynamics, we assume that the mosquito population unexposed to insecticide has reached the equilibrium before the insecticide is introduced at time $t=0$ . The insecticide pressure is then maintained on the mosquito population for a period denoted as $n_y$ years. Furthermore, we also assume that unexposed and exposed mosquitoes population have a sufficient growing capability captures by $r\left (\mathcal{L}_0 \right )$ and $r\left (\mathcal{L}_1 \right )$ such that $r\left (\mathcal{L}_0 \right )\gt r\left (\mathcal{L}_1 \right )\gt 1$ . The inequality $r\left (\mathcal{L}_0 \right )\gt r\left (\mathcal{L}_1 \right )$ assumes a global fitness cost within the population of exposed mosquitoes that are evading the insecticide activity.

The proposed model captures evolutionary outcomes, such as the time of resistance emergence ( $\textrm {T}_{\textrm {emg}}$ ), and epidemiological outcomes, such as the relative gain from introducing the insecticide ( $\textrm {r}_{\textrm {gain}}$ ). More specifically, $\textrm {T}_{\textrm {emg}}$ represents the time at which the initially rare population exposed to insecticide reaches 10% of the total AFMs. Note that $\textrm {T}_{\textrm { emg}}= \textrm {T}_{\textrm {emg}}(c)$ depends on the insecticide exposure strategy $c$ . Additionally, $\textrm {r}_{\textrm {gain}}=\textrm {r}_{\textrm { gain}}(c)$ is calculated as the number of AFMs alive during the insecticide usage period T relative to the number of AFMs alive in the absence of insecticide pressure, i.e.

\begin{equation*} \textrm {r}_{\textrm {gain}}(c)= 1-\dfrac { \left . \int _0^{n_y} \int _{\mathbb{R}} \int _0^\infty \left ( A_{0}(t,a,x) + A_{1}(t,a,x)\right ){\textrm {d}} a {\textrm {d}} x {\textrm {d}} t \right |_{c \neq 0}} {\left . \int _0^{n_y} \int _{\mathbb{R}} \int _0^\infty \left ( A_{0}(t,a,x) + A_{1}(t,a,x)\right ){\textrm {d}} a {\textrm {d}} x {\textrm {d}} t \right |_{c \equiv 0}}. \end{equation*}

We defined the difference in performance $D(c_1,c_2)$ between two insecticide exposure strategies, $c_1$ and $c_2$ , as $D(c_1,c_2)= \textrm {r}_{\textrm {gain}}(c_1)- \textrm {r}_{\textrm {gain}}(c_2),$ which corresponds to the percentage difference in effectiveness between the two strategies. For instance, a positive value of 5% indicates that strategy $c_1$ is 5% more beneficial than strategy $c_2$ , while negative values indicate that $c_2$ outperforms $c_1$ .

Consider the scenario where the fitness functions $\Theta _0$ and $\Theta _1$ for unexposed and exposed mosquito populations are illustrated in Figure 3A. In such a scenario, while the optimal phenotype for unexposed mosquitoes corresponds to the one with minimal resistance, insecticide pressure shifts the optimal phenotype of exposed mosquitoes to a higher resistance level (Figure 3A).

Figure 3.

Evolutionary dynamics with a constant and low insecticide exposure rate $c=0.2$ . (A) The fitness functions. (B) The dynamics of AFMs. (C–E) The dynamics of AFMs for exposed and unexposed populations. (F–H) The dynamics of eggs laid by AFMs for exposed and unexposed populations. Here, the probability of surviving insecticide exposure during one day $p_S=10^{-10}$ and other parameters are given by Table 1.

Figure 4.

Effect of the insecticide exposure rate $c$ on the relative gain from introducing the insecticide $\mathrm {r}_{\mathrm {gain}}$ , and the time of resistance emergence $\mathrm {T}_{\mathrm {emg}}$ . Here, $p_S=10^{-10}$ and other parameters are given by Table 1.

When a constant insecticide exposure is maintained within the mosquito population for $n_y=10$ years, this corresponds to approximately 150 mosquito generations, assuming mosquitoes complete 15 generations per year. When insecticide exposure is maintained at a relatively low rate with $c=0.2$ (Figure 3), the durability of insecticide efficacy is about $\textrm {T}_{\textrm {emg}}=3.1$ years. This insecticide durability is associated with a moderate gain from introducing the insecticide, with a relative gain $\textrm {r}_{\textrm {gain}} \approx 23\%$ (Figure 3B). Therefore, compared to taking no action, a constant and relatively low rate of insecticide exposure results in a moderate gain from introducing the insecticide. This is explained by the relatively short durability period of such a strategy. However, increasing the insecticide exposure rate does not provide significant advantages in either the long term, quantified by $\textrm {T}_{\textrm {emg}}$ , or the short term, quantified by $\textrm {r}_{\textrm {gain}}$ (Figure 4).