Transmission dynamics of a general temporal-spatial vector-host epidemic model with an application to the dengue fever in Guangdong, China

Yantao Luo; Zhidong Teng; Xiao-Qiang Zhao

doi:10.3934/dcdsb.2022069

Article Contents

2023, Volume 28, Issue 1: 134-169. Doi: 10.3934/dcdsb.2022069

This issue Previous Article Long, intermediate and short-term well-posedness of high precision shallow-water models with topography variations Next Article Positive periodic solutions for systems of impulsive delay differential equations

Transmission dynamics of a general temporal-spatial vector-host epidemic model with an application to the dengue fever in Guangdong, China

1.
College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China
2.
Department of Mathematics and Statistics, Memorial University of Newfoundland, St.John's, NL A1C5S7, Canada

^*Corresponding author
^*Corresponding author

Received: May 31, 2021

Revised: January 31, 2022

Early access: April 2022

Published: January 2023

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Abstract

Due to the nature of the spread of vector-host epidemic disease, there are many factors affecting its dynamic behaviors. In this paper, a vector-host epidemic model with two seasonal development periods and awareness control of host is proposed to investigate the multi-effects of the spatial heterogeneity, seasonal development periods, temporal periodicity and awareness control. We first address the well-posedness of the model and then derive the basic reproduction number $ R_0 $. In the case where $ R_0<1 $, we establish the global attractivity of the disease-free periodic solution, and in the case where $ R_0>1 $, we show that the disease is uniformly persistent and the system admits at least one positive periodic endemic steady state, and further obtain the global attractivity of the positive endemic constant steady state for the model with constant coefficients. As a case study, we conduct numerical simulations for the dengue fever transmission in Guangdong, China, 2014. We find that the greater heterogeneity of the mosquito distribution and human population may increase the risk of disease transmission, and the stronger awareness control may lower the risk of disease transmission.

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Mathematics Subject Classification: Primary: 92D30, 35K57; Secondary: 35Q92, 37N25.

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Figure 1. The diagram of the vector-host disease transmission

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Figure 2. The temporal-spatial periodic evolution solution of system (12) with with initial value (44) and the Nuemann boundary condition when $ \mathcal{R}_0 = 3.0391>1 $

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Figure 3. The temporal-spatial periodic evolution solution of system (12) with with initial value (44) and the Nuemann boundary condition when $ \mathcal{R}_0 = 0.9701<1 $

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Figure 4. (a) The basic reproduction number $ R_0 $ decreases with $ m $. (b): The basic reproduction number $ R_0 $ increases with $ \eta $;

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Figure 5. The basic reproduction number $ R_0 $ increases with $ \xi $

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Table 1. Parameter values

	Value	Definition	Reference
$ N_h $	603(km$ ^2 $)$ ^{-1} $	Total human	see text
		population density
$ A_{h} $	$ 8.96\times10^{-4} $(km$ ^2 $ month)$ ^{-1} $	Human birth rate	see text
$ \sigma $	$ 3.90\times 10^{-4} $(km$ ^2 $ month)$ ^{-1} $	Human death rate	see text
$ m $	[0–1]	conscious control rate	[46]
		of the susceptible human
$ u $	$ 6.18\times10^{-10} $(km$ ^2 $ month)$ ^{-1} $	Disease-related mortality	[26]
$ \alpha $	$ 30.4/6 $ month $ ^{-1} $	Human constant recovery rate	[50]
$ \tau_p $	$ [3 / 30.4,14 / 30.4] $ month $ ^{-1} $	Development time of	[44]
		dengue virus in human
$ A_w $	to be estimated	Recruitment rate	see text
		of mosquitoes
$ \tau_w $	to be evaluated	Development time	see text
		of dengue virus
		in mosquitoes
$ l $	0.1-0.75	Transmission probability from	[11]
		mosquitoes to hosts per bite
$ p $	0.3-0.75	Transmission probability from	[11]
		hosts to mosquitoes per bite
$ \delta $	to be evaluated	Death rate	see text
		of mosquitoes

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Table 2. Monthly mean temperature Guangdong Province (in $ \left.^{\circ} \mathrm{C}\right) $

Month	Jul	Aug	Sep	Oct	Nov	Dec
Temperature	32.7	32.5	30.4	27.6	23.6	18.4
Month	Jan	Feb	Mar	Apr	May	June
Temperature	16.7	17.2	20.7	25.6	29.4	30.7

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