A hybrid Lagrangian-Eulerian model for vector-borne diseases

Abstract

In this paper, a multi-patch and multi-group vector-borne disease model is proposed to study the effects of host commuting (Lagrangian approach) and/or vector migration (Eulerian approach) on disease spread. We first define the basic reproduction number of the model, $R_0$ , which completely determines the global dynamics of the model system. Namely, if $R_0\le 1$ , then the disease-free equilibrium is globally asymptotically stable, and if $R_0>1$ , then there exists a unique endemic equilibrium which is globally asymptotically stable. Then, we show that the basic reproduction number has lower and upper bounds which are independent of the host residence times matrix and the vector migration matrix. In particular, nonhomogeneous mixing of hosts and vectors in a homogeneous environment generally increases disease persistence and the basic reproduction number of the model attains its minimum when the distributions of hosts and vectors are proportional. Moreover, $R_0$ can also be estimated by the basic reproduction numbers of disconnected patches if the environment is homogeneous. The optimal vector control strategy is obtained for a special scenario. In the two-patch and two-group case, we numerically analyze the dependence of the basic reproduction number and the total number of infected people on the host residence times matrix and illustrate the optimal vector control strategy in homogeneous and heterogeneous environments.

Keywords: Basic reproduction number; Eulerian approach; Lagrangian approach; Optimal vector control; Population movement; Vector–borne disease.

Fig. 1

Flow chart of the Lagrangian–Eulerian vector-borne disease model

Fig. 2

Contour plots of the basic reproduction number ${\mathcal{R}}_0$ of model (5.1) in terms of the residence time proportions $p_{11}$ and $p_{22}$ under two homogeneous environments. The black dashed lines are $L_1$ and $L_2$ on which ${\mathcal{R}}_0(p_{11},p_{22})={\mathcal{R}}_0(1/1)$

Fig. 3

Contour plots of the basic reproduction number ${\mathcal{R}}_0$ of model (5.1) in terms of the residence time proportions $p_{11}$ and $p_{22}$ under four heterogeneous environments. The black dashed curves represent ${\mathcal{R}}_0(p_{11},p_{22})=\zeta ={\mathcal{R}}_0(p_{11},1-p_{11})$

Fig. 4

Contour plots of the total number of infected hosts ${I_1^h}^{\ast }+{I_2^h}^{\ast }$ versus the residence time proportions $p_{11}$ and $p_{22}$ in a homogeneous and b heterogeneous environments

Fig. 5

The control reproduction number ${\mathcal{R}}_c$ of model (4.5) versus the number of vectors being culled from the first patch, X, when the total number of vectors being culled is a $\eta =5000$, b $\eta =3000$

Fig. 6

The optimal number of vectors being culled in patch 1, $X^{\ast }$, versus the total number of vectors being culled, $\eta$, in a homogeneous and b heterogeneous environment