Relating Eulerian and Lagrangian spatial models for vector-host disease dynamics through a fundamental matrix

Abstract

We explore the relationship between Eulerian and Lagrangian approaches for modeling movement in vector-borne diseases for discrete space. In the Eulerian approach we account for the movement of hosts explicitly through movement rates captured by a graph Laplacian matrix L. In the Lagrangian approach we only account for the proportion of time that individuals spend in foreign patches through a mixing matrix P. We establish a relationship between an Eulerian model and a Lagrangian model for the hosts in terms of the matrices L and P. We say that the two modeling frameworks are consistent if for a given matrix P, the matrix L can be chosen so that the residence times of the matrix P and the matrix L match. We find a sufficient condition for consistency, and examine disease quantities such as the final outbreak size and basic reproduction number in both the consistent and inconsistent cases. In the special case of a two-patch model, we observe how similar values for the basic reproduction number and final outbreak size can occur even in the inconsistent case. However, there are scenarios where the final sizes in both approaches can significantly differ by means of the relationship we propose.

Appendix

1.1 Basic reproduction number

In this section of the Appendix we use the next generation matrix approach (van den Driessche and Watmough 2002) to derive the expressions for the basic reproduction numbers (7) and (12) of systems (3) and (8) respectively.

We first compute the next generation matrix $\left( F^{{\text {lag}}}\right) \left( V^{{\text {lag}}}\right) ^{-1}$ of the Lagrangian system. Let us consider the equations

$$\begin{aligned} \dot{I_i}&= \sum _{j=1}^n \beta _j p_{ji}\frac{S_i}{N_i}I_{v,j} - \left( \gamma _i^{{\text {lag}}} + \mu _i^{{\text {lag}}}\right) I_i \nonumber \\ \dot{I}_{v,i}&= \beta _{v,i} \frac{\sum _{j=1}^n p_{ij}I_j}{\sum _{j=1}^np_{ij}N_j} S_{v,i} \nonumber \\&\qquad + \sum _{j=1}^n m_{ij}^v I_{v,j} - \sum _{j=1}^n m_{ji}^v I_{v,i} - \mu _{v,i} I_{v,i} \end{aligned}$$

(35)

corresponding to the infectious compartments of system (3). From (35) we define the function $\mathcal {F}^{{\text {lag}}}: \mathbb {R}^{2n} \rightarrow \mathbb {R}^{2n}$ by

$$\begin{aligned} \mathcal {F}_i^{{\text {lag}}}(I_1,\ldots ,I_n,I_{v,1},\ldots ,I_{v,n}):= \sum _{j=1}^n \beta _j p_{ji}\frac{S_i}{N_i}I_{v,j}\,, \end{aligned}$$

for $i=1,\ldots ,n$, and

$$\begin{aligned} \mathcal {F}_i^{{\text {lag}}}(I_1,\ldots ,I_n,I_{v,1},\ldots ,I_{v,n}) := \beta _{v,i} \frac{\sum _{j=1}^n p_{ij}I_j}{\sum _{j=1}^np_{ij}N_j} S_{v,i} \,, \end{aligned}$$

for $i=n+1, \ldots ,2n$. The DFE of the Lagrangian systems is determined by $\left( S_i^{{\text {lag}}}\right) ^* = \left( N_i ^{{\text {lag}}}\right) ^*$ and $\left( S_{v,i}^{{\text {lag}}}\right) ^* = \left( N_{v,i}^{{\text {lag}}}\right) ^*$ defined by (4). We then define the Jacobian matrix

$$\begin{aligned}F^{{\text {lag}}} := \left. \partial \mathcal {F}^{{\text {lag}}} / \partial (I_1,\ldots , I_n, I_{v,1}, \ldots , I_{v,n})\right| _{DFE}\,.\end{aligned}$$

We have that $\left. \left( \partial \mathcal {F}_i^{{\text {lag}}} / \partial I_{v,j}\right) \right| _{DFE} = \beta _j p_{ji}$, so the upper-right block of $F^{{\text {lag}}}$ is $P^{{\text {T}}} D_{\beta }$, where $D_{\beta }:= {\text {diag}}\{\beta _i\}$. We also have that $\left. \left( \partial \mathcal {F}_{n+i}^{{\text {lag}}} / \partial I_j\right) \right| _{DFE} = \beta _{v,i} \left( N_{v,i}^*/\hat{N_i}^*\right) p_{ij}$, so the lower-left block of $F^{{\text {lag}}}$ is $D_{\beta _v}^{{\text {lag}}} P$, where $\hat{N_i}^*:= \sum _{j=1}^n p_{ij} \left( N_j^{{\text {lag}}}\right) ^*$ and $D_{\beta _v}^{{\text {lag}}} := {\text {diag}}\left\{ \beta _{v,i} N_{v,i}^*/\hat{N_i}^* \right\}$. Therefore, we have

$$\begin{aligned} F^{{\text {lag}}} = \begin{pmatrix} 0 &{} P^{{\text {T}}} D_{\beta } \\ D_{\beta _v}^{{\text {lag}}}P &{} 0 \end{pmatrix}. \end{aligned}$$

Similarly, we define the function $\mathcal {V}^{{\text {lag}}}: \mathbb {R}^{2n} \rightarrow \mathbb {R}^{2n}$ by

$$\begin{aligned} \mathcal {V}_i^{{\text {lag}}} := \left( \gamma _i^{{\text {lag}}} + \mu _i^{{\text {lag}}}\right) I_i \,, \end{aligned}$$

for $i = 1,\ldots ,n$, and

$$\begin{aligned} \mathcal {V}_i^{{\text {lag}}} := - \sum _{j=1}^n m_{ij}^v I_{v,j} + \sum _{j=1}^n m_{ji}^v I_{v,i} + \mu _{v,i} I_{v,i}\,, \end{aligned}$$

for $i = n+1,\ldots , 2n$. We also define the Jacobian matrix

$$\begin{aligned} V^{{\text {lag}}} := \left. \partial \mathcal {V}^{{\text {lag}}}/ \partial (I_1.\ldots ,I_n, I_{v,1},\ldots , I_{v,n})\right| _{DFE}\,.\end{aligned}$$

If $L_v$ is the graph Laplacian of the vector movement (with adjacency matrix $M^{v} = \left( m_{ij}^v\right) _{i,j\le n}$, see (2)) and $D_\delta ^{{\text {lag}}}:= {\text {diag}}\left\{ \gamma _i^{{\text {lag}}}+\mu _i^{{\text {lag}}}\right\}$, then the upper-left block of $V^{{\text {lag}}}$ is $D_\delta ^{{\text {lag}}}$ and the lower-right block of $V^{{\text {lag}}}$ is $G_v = L_v + D_{\delta _v}$. Therefore,

$$\begin{aligned} V^{{\text {lag}}} = \begin{pmatrix} D_\delta ^{{\text {lag}}} &{} 0 \\ 0 &{} G_v \end{pmatrix}. \end{aligned}$$

Consequently,

$$\begin{aligned} \left( F^{{\text {lag}}}\right) \left( V^{{\text {lag}}}\right) ^{-1} = \begin{pmatrix} 0 &{} P^{{\text {T}}}D_{\beta }G_v^{-1} \\ D_{\beta _v}^{{\text {lag}}}P \left( D_\delta ^{{\text {lag}}}\right) ^{-1} &{} 0 \end{pmatrix} \end{aligned}$$

and

$$\begin{aligned} (\mathcal {R}_0^{{\text {lag}}})^2 = \rho \left( P^{{\text {T}}}D_{\beta }G_v^{-1}D_{\beta _v}^{{\text {lag}}}P \left( D_\delta ^{{\text {lag}}}\right) ^{-1}\right) \,. \end{aligned}$$

We now compute the next generation matrix $\left( F^{{\text {eul}}}\right) \left( V^{{\text {eul}}}\right) ^{-1}$ of system (8). The equations of the infectious compartments of system (8) are

$$\begin{aligned} \dot{I_i}&= \beta _i \frac{S_i}{N_i}I_{v,i} + \sum _{j=1}^n m_{ij}^{{\text {I}}} I_{j} - \sum _{j=1}^n m_{ji}^{{\text {I}}} I_{i} - \left( \gamma _i^{{\text {eul}}} + \mu _i^{{\text {eul}}}\right) I_i \nonumber \\ \dot{I_{v,i}}&= \beta _{v,i} \frac{I_i}{N_i} S_{v,i} + \sum _{j=1}^n m_{ij}^v I_{v,j} - \sum _{j=1}^n m_{ji}^v I_{v,i} - \mu _{v,i} I_{v,i}\, . \end{aligned}$$

(36)

From the equations in (36) we define the function

$$\begin{aligned}\mathcal {F}^{{\text {eul}}}(I_1,\ldots , I_n, I_{v,1},\ldots ,I_{v,n}):= \beta _i \frac{S_i}{N_i}I_{v,i}\,,\end{aligned}$$

for $i=1,\ldots , n$, and

$$\begin{aligned}\mathcal {F}^{{\text {eul}}}(I_1,\ldots , I_n, I_{v,1},\ldots ,I_{v,n}) = \beta _{v,i} \frac{I_i}{N_i} S_{v,i}\,, \end{aligned}$$

for $i=n+1,\ldots , 2n$.

Using the DFE (9) of system (8), we have that $\left. \left( \partial \mathcal {F}_i^{{\text {eul}}} / \partial I_{v,j}\right) \right| _{DFE} = \beta _i$ and $\left. \left( \partial \mathcal {F}_{n+i}^{{\text {eul}}} / \partial \mathcal {F}_{j}^{{\text {eul}}}\right) \right| _{DFE} = \beta _{v,i} N_{v,i}^*/\left( N_i^{{\text {eul}}}\right) ^* \,.$ Therefore, if

$$\begin{aligned} F ^{{\text {eul}}}:= \left. \partial \mathcal {F}^{{\text {eul}}} /\partial (I_1,\ldots , I_n, I_{v,1},\ldots , I_{v,n})\right| _{DFE}\,,\end{aligned}$$

we then have

$$\begin{aligned} F^{{\text {eul}}} = \begin{pmatrix} 0 &{} D_\beta \\ D_{\beta _v}^{{\text {eul}}} &{} 0 \end{pmatrix} \, , \end{aligned}$$

where $D_{\beta }:= {\text {diag}}\{\beta _i\}$ and $D_{\beta _v}^{{\text {eul}}} := {\text {diag}}\left\{ \beta _{v,i} N_{v,i}^*/\left( N_i^{{\text {eul}}}\right) ^* \right\}$. Similarly, we define the function

$$\begin{aligned}\mathcal {V}^{{\text {eul}}}(I_1,\ldots , I_n, I_{v,1},\ldots , I_{v,n}) := - \sum _{j=1}^n m_{ij}^{{\text {I}}} I_{j} + \sum _{j=1}^n m_{ji}^{{\text {I}}} I_{i} + \left( \gamma _i^{{\text {eul}}} + \mu _i^{{\text {eul}}}\right) I_i \,, \end{aligned}$$

for $i = 1,\ldots ,n$, and

$$\begin{aligned} \mathcal {V}^{{\text {eul}}}(I_1,\ldots , I_n, I_{v,1},\ldots , I_{v,n}) := - \sum _{j=1}^n m_{ij}^v I_{v,j} + \sum _{j=1}^n m_{ji}^v I_{v,i} + \mu _{v,i} I_{v,i}\,,\end{aligned}$$

for $i = n+1,\ldots , 2n$. Therefore, if

$$\begin{aligned}V^{{\text {eul}}}:= \left. \partial \mathcal {V}^{{\text {eul}}} / \partial (I_1,\ldots ,I_n,I_{v,1}\ldots , I_{v,n})\right| _{DFE}\,,\end{aligned}$$

we then have

$$\begin{aligned} V^{{\text {eul}}} = \begin{pmatrix} G &{} 0\\ 0 &{} G_v \end{pmatrix}\, , \end{aligned}$$

(37)

where $G := L^{{\text {I}}} + D_{\delta }^{{\text {eul}}}$, $L^{{\text {I}}}$ is the graph Laplacian of the host movement [with adjacency matrix $M^{{\text {I}}} = (m_{ij}^{{\text {I}}})_{i,j\le n}$, see (2)], $D_\delta ^{{\text {eul}}}:= {\text {diag}}\{\gamma _i^{{\text {eul}}}+\mu _i^{{\text {eul}}}\}$, and $G_v = L_v + D_{\delta _v}$. In consequence,

$$\begin{aligned} \left( F^{{\text {eul}}}\right) \left( V^{{\text {eul}}}\right) ^{-1} = \begin{pmatrix} 0 &{} D_\beta G_v^{-1}\\ D_{\beta _v}^{{\text {eul}}}G^{-1} &{}0 \end{pmatrix}\end{aligned}$$

and

$$\begin{aligned} \left( \mathcal {R}_0^{{\text {eul}}}\right) ^2=\rho \left( D_\beta G_v^{-1}D_{\beta _v}^{{\text {eul}}}G^{-1}\right) . \end{aligned}$$

1.2 Comparison of basic reproduction numbers

In this section we prove Proposition 2 of Section 4.2. Let $\beta _{v,1} = \beta _{v,2}$, $N_v^* := N_{v,1}^*=N_{v,2}^*$, $\left( N^{{\text {eul}}}\right) ^* := \left( N_1^{{\text {eul}}}\right) ^* = \left( N_2^{{\text {eul}}}\right) ^*$, and $\beta _v = \beta _{v,1} N_{v,1}^*/\left( N_1^{{\text {eul}}}\right) ^* = \beta _{v,2} N_{v,2}^*/\left( N_2^{{\text {eul}}}\right) ^*$. Define $\left( N_1^{{\text {lag}}}\right) ^*$ and $\left( N_2^{{\text {lag}}}\right) ^*$ such that $\left( N_1^{{\text {eul}}}\right) ^*= p_{11}\left( N_1^{{\text {lag}}}\right) ^* + p_{12}\left( N_2^{{\text {lag}}}\right) ^*$ and $\left( N_2^{{\text {eul}}}\right) ^* = p_{21}\left( N_1^{{\text {lag}}}\right) ^* + p_{22}\left( N_2^{{\text {lag}}}\right) ^*$. Hence A8 holds, and $\delta = \delta _1^{{\text {eul}}} = \delta _2^{{\text {eul}}}$, so we also get $\delta = \delta _1^{{\text {lag}}} = \delta _2^{{\text {lag}}}$ by A9. Let $D_\beta = \beta I$, $D_{\beta _v} = \beta _v I$, $D_\delta ^{{\text {eul}}} = D_\delta ^{{\text {lag}}} = D_\delta := \delta I$, $D_{\delta _v} =\delta _v I$, $L_v = m_v \begin{pmatrix} 1 &{} -1\\ -1 &{} 1\end{pmatrix}, P = \begin{pmatrix} 1-p_{21} &{} p_{12} \\ p_{21} &{} 1-p_{12} \end{pmatrix}$.

We fix all the parameters except $p_{12}$ (we obtain analogous and symmetric results if we fix $p_{21}$). From (7) and (12), we obtain that $\left( \mathcal {R}_{0}^{{\text {lag}}}\right) ^2 = \rho \left( P^{{\text {T}}}D_{\beta }G_v^{-1}D_{\beta _v}P \left( D_\delta \right) ^{-1}\right)$ and $\left( \mathcal {R}_{0}^{{\text {eul}}}\right) ^2=\rho \left( D_\beta G_v^{-1}D_{\beta _v}P(D_\delta \right) ^{-1})$. Therefore, if we define $\mathcal {M}:=D_{\beta }G_v^{-1}D_{\beta _v}P \left( D_\delta ^{{\text {lag}}}\right) ^{-1}$, we get

$$\begin{aligned} \left( \mathcal {R}_{0}^{{\text {lag}}}\right) ^2=\rho (P^{{\text {T}}}\mathcal {M}) \quad \text {and} \quad \left( \mathcal {R}_{0}^{{\text {eul}}}\right) ^2 = \rho (\mathcal {M})\,. \end{aligned}$$

(38)

The following is the statement of the proposition.

Proposition 2

Assume A1–A10 and suppose that $D_\beta = \beta I$, $D_{\beta _v} = \beta _v I$, $D_\delta ^{{\text {eul}}} = D_\delta ^{{\text {lag}}} = D_\delta := \delta I$, $D_{\delta _v} =\delta _v I$ and $m_{12}^v = m_{21}^v = m_v$. Define $\mathcal {M}:=D_{\beta }G_v^{-1}D_{\beta _v}P \left( D_\delta ^{{\text {lag}}}\right) ^{-1}$ and fix $p_{21}$ in the interval (0, 1/2). Then $\rho (P^{{\text {T}}}M)$ is a function of $p_{12}$ where $0<p_{12}<1/2$ and we have that:

• a) $\rho (\mathcal {M}) = (\beta \beta _v)/(\delta \delta _v)$ is constant on $0<p_{12}<1/2$.

• b) $\rho (P^{t}\mathcal {M})$ is decreasing on $(0,p_{21})$, increasing on $(p_{21},1/2)$ and attains one absolute minimum over (0, 1/2) with value $(\beta \beta _v)/(\delta \delta _v)$ at $p_{12}=p_{21}$.

• c) In addition, we have the inequality

  $$\begin{aligned} \rho (P^{{\text {T}}}\mathcal {M})-\rho (\mathcal {M})&\le \frac{\beta \beta _v}{\delta \delta _v (2m_v + \delta _v)}|p_{12}-p_{21}|(1-p_{12}-p_{21})\delta _v\nonumber \\&\le \frac{\beta \beta _v}{\delta \delta _v (2m_v + \delta _v)}\max \left\{ p_{21}(1-p_{21})\delta _v,(1/2-p_{21})^2\delta _v\right\} \nonumber \\&\le \frac{\beta \beta _v}{\delta \delta _v (2m_v + \delta _v)}\frac{\delta _v}{4} \, . \end{aligned}$$

  (39)

Proof

We have that

$$\begin{aligned} \mathcal {M}= & {} D_{\beta }G_v^{-1}D_{\beta _v}P \left( D_\delta ^{{\text {lag}}}\right) ^{-1} \\= & {} \frac{\beta \beta _v}{\delta \delta _v (2m_v + \delta _v)} \begin{pmatrix} m_v + (1-p_{21})\delta _v &{} m_v + p_{12}\delta _v \\ m_v + p_{21}\delta _v &{} m_v + (1-p_{12})\delta _v\end{pmatrix} \, ,\end{aligned}$$

so the eigenvalues of $\mathcal {M}$ in this case are

$$\begin{aligned} \frac{\beta \beta _v}{\delta \delta _v} \quad \text {and}\quad \frac{\beta \beta _v}{\delta \delta _v} \frac{(1-p_{12}-p_{21})\delta _v}{2m_v+\delta _v}.\end{aligned}$$

In consequence, we get $\rho (\mathcal {M}) = (\beta \beta _v)/(\delta \delta _v)$. We can also get $\rho (P^{{\text {T}}}\mathcal {M})$ explicitly by

$$\begin{aligned} \rho (P^{{\text {T}}}\mathcal {M}) = \frac{\beta \beta _v}{\delta \delta _v (2m_v + \delta _v)}\left[ m_v + \delta _v - \left( p_{12}(1-p_{12})+p_{21}(1-p_{21})\right) \delta _v + \right. \nonumber \\ \left. \sqrt{\left[ m_v + \left( p_{12}(1-p_{12})+p_{21}(1-p_{21})\right) \delta _v\right] ^2 + \left[ (p_{12}-p_{21})(1-p_{12}-p_{21})\delta _v\right] ^2} \right] \, . \end{aligned}$$

(40)

From this equation, it follows that when $p_{12}=p_{21}$, we get

$$\begin{aligned} \rho (P^{{\text {T}}}\mathcal {M})= \frac{\beta \beta _v}{\delta \delta _v} = \rho (\mathcal {M})\, . \end{aligned}$$

Moreover, $\partial \rho (P^{{\text {T}}}\mathcal {M})/\partial p_{12} = \kappa \beta \beta _v/\left[ \delta \delta _v (2m_v + \delta _v)\right]$, where

$$\begin{aligned} \kappa = 2p_{12}-1 + \frac{m_v(1-2p_{21}) + \delta _v \left[ 2p_{12}(1-p_{12})(1-2p_{12}) + (p_{12}-p_{21})\left( 2p_{12}(1-p_{12}-p_{21})+p_{12}+p_{21}\right) \right] }{\sqrt{\left[ m_v + \left( p_{12}(1-p_{12})+p_{21}(1-p_{21})\right) \delta _v\right] ^2 + \left[ (p_{12}-p_{21})(1-p_{12}-p_{21})\delta _v\right] ^2}}\,. \end{aligned}$$

(41)

In particular, if $p_{12}=p_{21}$, then

$$\begin{aligned}\frac{\partial \rho (P^{{\text {T}}}\mathcal {M})}{\partial p_{12}}= 0 \, .\end{aligned}$$

Assume that $p_{12} > p_{21}$. Define

$$\begin{aligned} \alpha&:= \sqrt{\left[ m_v + \left( p_{12}(1-p_{12})+p_{21}(1-p_{21})\right) \delta _v\right] ^2 + \left[ (p_{12}-p_{21})(1-p_{12}-p_{21})\delta _v\right] ^2} \\&\le \left[ m_v + \left( p_{12}(1-p_{12})+p_{21}(1-p_{21})\right) \delta _v\right] + \left[ (p_{12}-p_{21})(1-p_{12}-p_{21})\delta _v\right] \, , \end{aligned}$$

and

$$\begin{aligned}\eta:= & {} m_v(1-2p_{21}) + \delta _v \left[ 2p_{12}(1-p_{12})(1-2p_{12}) + (p_{12}-p_{21})\right. \\&\times \left. \left( 2p_{12}(1-p_{12}-p_{21})+p_{12}+p_{21}\right) \right] \,.\end{aligned}$$

We then have that

$$\begin{aligned} \frac{\partial \rho (P^{{\text {T}}}\mathcal {M})}{\partial p_{12}}= \frac{\beta \beta _v}{\delta \delta _v (2m_v + \delta _v)} \left[ 2p_{12}-1 + \frac{\eta }{\alpha }\right] \end{aligned}$$

and

$$\begin{aligned} (2p_{12}-1)\alpha +\eta&\ge (2p_{12}-1)\left[ m_v + \left( p_{12}(1-p_{12})+p_{21}(1-p_{21})\right) \delta _v \right. \\&\quad \left. + (p_{12}-p_{21})(1-p_{12}-p_{21})\delta _v\right] + \eta \\&= 2(p_{12}-p_{21})\left[ m_v + \left( (p_{12}-1)^2 + p_{12}^2\right) \delta _v \right] > 0 \, . \end{aligned}$$

Therefore, $p_{12} > p_{21}$ implies that

$$\begin{aligned}\frac{\partial \rho (P^{{\text {T}}}\mathcal {M})}{\partial p_{12}}= \frac{\beta \beta _v}{\delta \delta _v (2m_v + \delta _v)} \frac{(2p_{12}-1)\alpha + \eta }{\alpha } >0 \,. \end{aligned}$$

Now, let us assume that $p_{12} <p_{21}$. We then have that

$$\begin{aligned} \alpha \ge m_v + \left( p_{12}(1-p_{12})+p_{21}(1-p_{21})\right) \delta _v \end{aligned}$$

and

$$\begin{aligned} (2p_{12}-1)\alpha +\eta&\le (2p_{12}-1)\left[ m_v + \left( p_{12}(1-p_{12})+p_{21}(1-p_{21})\right) \delta _v\right] + \eta \\&=-(p_{21}-p_{12})(m_v + \delta _v ( 1 + (p_{21}-p_{12})(1-2p_{12}) ) )< 0\, . \end{aligned}$$

Therefore, $p_{12} < p_{21}$ implies that

$$\begin{aligned}\frac{\partial \rho (P^{{\text {T}}}\mathcal {M})}{\partial p_{12}}= \frac{\beta \beta _v}{\delta \delta _v (2m_v + \delta _v)} \frac{(2p_{12}-1)\alpha + \eta }{\alpha } <0 \, . \end{aligned}$$

Using that $\alpha \le \left[ m_v + \left( p_{12}(1-p_{12})+p_{21}(1-p_{21})\right) \delta _v\right] + |p_{12}-p_{21}|(1-p_{12}-p_{21})\delta _v$, we get

$$\begin{aligned} \rho (P^{{\text {T}}}\mathcal {M})-\rho (\mathcal {M})&\le \frac{\beta \beta _v}{\delta \delta _v (2m_v + \delta _v)}|p_{12}-p_{21}|(1-p_{12}-p_{21})\delta _v\\&\le \frac{\beta \beta _v}{\delta \delta _v (2m_v + \delta _v)}\max \left\{ p_{21}(1-p_{21})\delta _v,(1/2-p_{21})^2\delta _v\right\} \\&\le \frac{\beta \beta _v}{\delta \delta _v (2m_v + \delta _v)}\frac{\delta _v}{4} \, . \end{aligned}$$

$\square$

Let us try to get some intuition for the inequality ${\mathcal {R}}_0^{lag} \ge {\mathcal {R}}_0^{eul}$ in the previous proposition. Suppose that $p_{21} < p_{12}$ and all the other parameters are assumed to be as in Proposition 2. Assume that the systems (3) and (8) are at the DFE and suppose we introduce the same number of infectious hosts in both patches, say $I_h = I_{h,1} = I_{h,2}$. From the last equation in (3) of the Lagrangian system and A3.1, the rates at which vectors get infected in patches 1 and 2 are

$$\begin{aligned} \beta _{v,1} \frac{p_{11} I_{h,1} + p_{12} I_{h,2}}{ p_{11}\left( N_1^{{\text {lag}}}\right) ^* + p_{12}\left( N_2^{{\text {lag}}}\right) ^*} N_{v,1}^* = \beta _{v} I_h (1+p_{12}-p_{21})\end{aligned}$$

and

$$\begin{aligned} \beta _{v,2} \frac{p_{21} I_{h,1} + p_{22} I_{h,2}}{ p_{21}\left( N_1^{{\text {lag}}}\right) ^* + p_{22}\left( N_2^{{\text {lag}}}\right) ^*} N_{v,2}^* = \beta _{v} I_h(1+p_{21}-p_{12})\end{aligned}$$

respectively. Since $p_{12}>p_{21}$, the vector infection rate in patch 1 is greater than in patch 2 (because $1+p_{12}-p_{21}> 1 > 1+p_{21}-p_{12}$). Therefore, over a period $\Delta t$, the number of infected vectors at patch 1, which is approximately $\Delta I_{v,1}^{{\text {lag}}}:= \beta _{v} I_h (1+p_{12}-p_{21})\Delta t$, would be larger than the amount of infected vectors at patch 2, which is approximately $\Delta I_{v,2}^{{\text {lag}}}:= \beta _{v} I_h (1+p_{21}-p_{12}) \Delta t$. From the second equation in (3), the amount of new host infections in patches 1 and 2 caused by the new infected vectors in DFE would be

$$\begin{aligned} \Delta I_{h,1}^{{\text {lag}}} := \beta _1 p_{11}\left( \Delta I_{v,1}^{{\text {lag}}}\right) + \beta _2 p_{21}\left( \Delta I_{v,2}^{{\text {lag}}}\right) = \beta \left( p_{11}\left( \Delta I_{v,1}^{{\text {lag}}}\right) + p_{21}\left( \Delta I_{v,2}^{{\text {lag}}}\right) \right) \end{aligned}$$

and

$$\begin{aligned} \Delta I_{h,2}^{{\text {lag}}} := \beta _1 p_{12}\left( \Delta I_{v,1}^{{\text {lag}}}\right) + \beta _2 p_{22}\left( \Delta I_{v,2}^{{\text {lag}}}\right) =\beta \left( p_{12}\left( \Delta I_{v,1}^{{\text {lag}}}\right) + p_{22}\left( \Delta I_{v,2}^{{\text {lag}}}\right) \right) \end{aligned}$$

respectively. Therefore, the total amount of new infected hosts would be

$$\begin{aligned} \Delta I_h^{{\text {lag}}}:= \Delta I_{h,1}^{{\text {lag}}}+\Delta I_{h,2}^{{\text {lag}}} = \beta \left( \Delta I_{v,1}^{{\text {lag}}} + \Delta I_{v,2}^{{\text {lag}}} \right) +\beta (p_{12}-p_{21})\left( \Delta I_{v,1}^{{\text {lag}}}-\Delta I_{v,2}^{{\text {lag}}}\right) \,. \end{aligned}$$

(42)

On the other hand, from the last equation in (8) of the Eulerian system, the rates at which vectors get infected in patches 1 and 2 are

$$\begin{aligned} \beta _{v,1} \frac{I_{h,1}}{\left( N_1^{{\text {eul}}}\right) ^*} N_{v,1}^* = \beta _{v} I_h \quad \text {and}\quad \beta _{v,2} \frac{I_{h,2}}{\left( N_2^{{\text {eul}}}\right) ^*} N_{v,2}^* = \beta _{v} I_h\end{aligned}$$

respectively. Therefore, over a period $\Delta t$, the amounts of new infected vectors are $\Delta I_{v,1}^{{\text {eul}}} = \Delta I_{v,2}^{{\text {eul}}} = \beta _v I_h \Delta t$. Notice that

$$\begin{aligned} \Delta I_{v,1}^{{\text {lag}}} + \Delta I_{v,2}^{{\text {lag}}} = 2 \beta _{v} I_h \Delta t = \Delta I_{v,1}^{{\text {eul}}} + \Delta I_{v,2}^{{\text {eul}}}\,. \end{aligned}$$

From the second equation in (8) of the Eulerian system, the total amount of new infected hosts is

$$\begin{aligned} \Delta I_{h}^{{\text {eul}}} = \beta (\Delta I_{v,1}^{{\text {eul}}} + \Delta I_{v,2}^{{\text {eul}}}) = \beta (\Delta I_{v,1}^{{\text {lag}}} + \Delta I_{v,2}^{{\text {lag}}}) \, . \end{aligned}$$

(43)

Since $p_{12}>p_{21}$ and $\Delta I_{v,1}^{{\text {lag}}} > \Delta I_{v,2}^{{\text {lag}}}$, we get that the term $\beta (p_{12}-p_{21})\left( \Delta I_{v,1}^{{\text {lag}}}-\Delta I_{v,2}^{{\text {lag}}}\right)$ in (42) is positive, and then using (43), we have that $\Delta I_h^{{\text {lag}}} > \Delta I_h^{{\text {eul}}}$, which corresponds to ${\mathcal {R}}_0^{{\text {lag}}} \ge {\mathcal {R}}_0^{{\text {eul}}}$. Note that this implicitly relies upon the fact that the removal rates, the rates $\beta _{v,i}$ and the rates $\beta _i$ are the same across patches. As we observed in Figure 5, imposing different removal rates, for instance, can lead to ${\mathcal {R}}_0^{{\text {eul}}} > {\mathcal {R}}_0^{{\text {lag}}}$.