Effects of travel frequency on the persistence of mosquito-borne diseases

Xianyun Chen; Daozhou Gao

doi:10.3934/dcdsb.2020119

Article Contents

2020, Volume 25, Issue 12: 4677-4701. Doi: 10.3934/dcdsb.2020119

This issue Previous Article A delayed differential equation model for mosquito population suppression with sterile mosquitoes Next Article Global mass-preserving solutions for a two-dimensional chemotaxis system with rotational flux components coupled with a full Navier–Stokes equation

Effects of travel frequency on the persistence of mosquito-borne diseases

Xianyun Chen and
Daozhou Gao^,

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

^* Corresponding author: Daozhou Gao (dzgao@shnu.edu.cn)
^* Corresponding author: Daozhou Gao (dzgao@shnu.edu.cn)

Received: September 30, 2019

Early access: March 2020

Published: December 2020

This work was partially supported by National Natural Science Foundation of China grant 11601336, Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (TP2015050), and Shanghai Gaofeng Project for University Academic Development Program

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Abstract

Travel frequency of people varies widely with occupation, age, gender, ethnicity, income, climate and other factors. Meanwhile, the distribution of the numbers of times people in different regions or with different travel behaviors bitten by mosquitoes may be nonuniform. To reflect these two heterogeneities, we develop a multipatch model to study the impact of travel frequency and human biting rate on the spatial spread of mosquito-borne diseases. The human population in each patch is divided into four classes: susceptible unfrequent, infectious unfrequent, susceptible frequent, and infectious frequent. The basic reproduction number $ \mathcal{R}_0 $ is defined. It is shown that the disease-free equilibrium is globally asymptotically stable if $ \mathcal{R}_0\leq 1 $, and there is a unique endemic equilibrium that is globally asymptotically stable if $ \mathcal{R}_0>1 $. A more detailed study is conducted on the single patch model. We use analytical and numerical methods to demonstrate that the model without considering the difference of humans in travel frequency mostly underestimates the risk of infection. Numerical simulations suggest that the greater the difference in travel frequency, the larger the underestimate of the transmission potential. In addition, the basic reproduction number $ \mathcal{R}_0 $ may decreasingly, or increasingly, or nonmonotonically vary when more people travel frequently.

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Mathematics Subject Classification: Primary: 92D30, 91D25, 34C60; Secondary: 34D20, 37N25.

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Figure 1. Flowchart of the mosquito-borne disease model in patch $ i $

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Figure 2. The contour plot of $ \mathcal{R}_0-\hat{ \mathcal{R}}_0 $ versus the relative travel rates $ \tau_{12} $ and $ \tau_{21} $

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Figure 3. The contour plot of $ \mathcal{R}_0 $ versus the relative frequency change rates $ \tau_1 $ and $ \tau_2 $. (a) simultaneously decreases, (b) simultaneously increases, (c) increases in $ \tau_1 $ but decreases $ \tau_2 $, (d) non-monotonically varies with $ \tau_1 $ and decreases in $ \tau_2 $

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Table 1. Descriptions and ranges of parameters (time unit is day)

	Description	Range
$ a_i $	mosquito biting rate	0.1–1
$ b_i^u $	transmission probability from an infectious mosquito	0.01–0.8
	to a susceptible unfrequent traveler per bite
$ b_i^f $	transmission probability from an infectious mosquito	0.01–0.8
	to a susceptible frequent traveler per bite
$ c_i^u $	transmission probability from an infectious unfrequent	0.072–0.64
	traveler to a susceptible mosquito per bite
$ c_i^f $	transmission probability from an infectious frequent	0.072–0.64
	traveler to a susceptible mosquito per bite
$ \gamma_i^u $	recovery rate of infectious unfrequent humans	0.005–0.05
$ \gamma_i^f $	recovery rate of infectious frequent humans	0.005–0.05
$ \mu_i $	mosquito mortality rate	0.05–0.2
$ \sigma_i $	relative HBR of frequent travelers to unfrequent travelers	0.2–5
$ c_{ij}^f $	travel rate of frequent travelers from patches $ j $ to $ i $	0.03–0.1
$ \tau_{ij} $	relative travel rate of unfrequent travelers	0–0.4
	to frequent travelers
$ c_{ij}^u $	travel rate of unfrequent travelers from patches $ j $ to $ i $	$ c_{ij}^u=\tau_{ij}c_{ij}^f $
$ d_{ij} $	travel rate of mosquitoes from patch $ j $ to patch $ i $	0.001–0.03
$ \phi_i^f $	change rate from frequent travelers to	$ 2.7 \times 10^{-4} $
	unfrequent travelers	–$ 9 \times 10^{-4} $
$ \tau_i $	relative change rate of unfrequent travelers	0.1–0.5
	to frequent travelers
$ \phi_i^u $	change rate from unfrequent travelers to	$ \phi_i^u=\tau_i\phi_i^f $
	frequent travelers
$ V/H $	ratio of mosquitoes to humans	1–10

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Table 2. Parameter settings for Figure 3

Symbol	Figure 3a	Figure 3b	Figure 3c	Figure 3d
$ a_1 $	0.438	0.380	0.535	0.161
$ a_2 $	0.402	0.100	0.147	0.051
$ b_1 $	0.548	0.686	0.373	0.656
$ b_2 $	0.642	0.159	0.178	0.716
$ c_1 $	0.398	0.324	0.608	0.586
$ c_2 $	0.636	0.085	0.618	0.403
$ \gamma_1 $	0.049	0.044	0.044	0.009
$ \gamma_2 $	0.026	0.029	0.014	0.020
$ \mu_1 $	0.135	0.066	0.094	0.053
$ \mu_2 $	0.165	0.194	0.174	0.103
$ \sigma_1 $	1	1	1	1
$ \sigma_2 $	1	1	1	1
$ c_{12}^u $	0.021	0.006	0.0128	0.0028
$ c_{21}^u $	0.018	0.002	0.0182	0.0005
$ c_{12}^f $	0.094	0.060	0.0534	0.0550
$ c_{21}^f $	0.092	0.076	0.0943	0.0497
$ d_{12} $	0	0	0	0
$ d_{21} $	0	0	0	0
$ \phi_1^u $	7.73 $ \times 10^{-5} $	2.80$ \times 10^{-4} $	1.40$ \times 10^{-4} $	1.50 $ \times 10^{-4} $
$ \phi_2^u $	5.11$ \times 10^{-5} $	3.46$ \times 10^{-4} $	2.10$ \times 10^{-4} $	1.50$ \times 10^{-4} $
$ \phi_1^f $	7.16 $ \times 10^{-4} $	7.58$ \times 10^{-4} $	5.35$ \times 10^{-4} $	4.82$ \times 10^{-4} $
$ \phi_2^f $	3.24$ \times 10^{-4} $	7.62$ \times 10^{-4} $	4.67$ \times 10^{-4} $	5.55$ \times 10^{-4} $
$ V_1 $	5859	23633	6065	13838
$ V_2 $	18790	29698	17129	19918
$ H $	10000	10000	10000	10000

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