Dynamics of a periodic West Nile virus model with mosquito demographics

Zhenguo Bai; Zhiwen Zhang

doi:10.3934/cpaa.2022121

Article Contents

2022, Volume 21, Issue 11: 3755-3775. Doi: 10.3934/cpaa.2022121

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Dynamics of a periodic West Nile virus model with mosquito demographics

Zhenguo Bai^, and
Zhiwen Zhang

School of Mathematics and Statistics, Xidian University, Xi'an, 710126, China

^* Corresponding author: Zhenguo Bai
^* Corresponding author: Zhenguo Bai

Received: August 2022

Early access: August 2022

Published: November 2022

This research was supported by the NSF of China (No. 11971369) and the Fundamental Research Funds for the Central Universities (No. JB210711).

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Abstract

In this paper, we propose a time-delayed model of West Nile virus with periodic extrinsic incubation period (EIP) and mosquito demographics including stage-structure, pair formation and intraspecific competition. We define two quantities $ \mathcal{R}_{\rm min} $ and $ \mathcal{R}_{\rm max} $ for mosquito population and the basic reproduction number $ \mathcal{R}_0 $ for our model. It is shown that the threshold dynamics are determined by these three parameters: (ⅰ) if $ \mathcal{R}_{\rm max}\leq 1 $, the mosquito population will not survive; (ⅱ) if $ \mathcal{R}_{\rm min}>1 $ and $ \mathcal{R}_0<1 $, then WNv disease will go extinct; (ⅲ) if $ \mathcal{R}_{\rm min}>1 $ and $ \mathcal{R}_0>1 $, then the disease will persist. Numerically, we simulate the long-term behaviors of solutions and reveal the influences of key model parameters on the disease transmission. A new finding is that $ \mathcal{R}_0 $ is non-monotone with respect to the fraction of the aquatic mosquitoes maturing into adult male mosquitoes, which can help us implement more effective control strategies. Besides we observe that using the time-averaged EIP has the possibility of underestimating the infection risk.

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Mathematics Subject Classification: Primary: 92D30, 37N25; Secondary: 34K13.

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Figure 1. Flowchart of the WNv model

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Figure 2. Long-term behaviors of infected mosquitoes and birds with $ \mathcal{R}_{\rm min} = 2.3226>1 $ and $ \mathcal{R}_0 = 1.1714>1 $. Initial conditions are $ A_M(\theta) = 8000, \; M_M(\theta) = 4000, \; S_M(\theta) = 2300, \; E_M(\theta) = 1000, \; I_M(\theta) = 700, \; B_S(\theta) = 900, \; B_I(\theta) = 100 $ for $ \theta\in[-\bar{\tau}, 0] $

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Figure 3. Long-term behaviors of infected mosquitoes and birds with $ \mathcal{R}_{\rm min} = 1.3936>1 $ and $ \mathcal{R}_0 = 0.6248<1 $. Initial conditions are the same as in Fig. 2

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Figure 4. The effects of $ q $ and $ \epsilon_M $ on $ \mathcal{R}_0 $. Left panel: $ \mathcal{R}_0 $ vs $ q $ on the range (0, 1). Right panel: $ \mathcal{R}_0 $ vs $ \epsilon_M $ on the range (0.1, 0.74)

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Figure 5. Sensitivity analysis of $ \mathcal{R}_0 $ with respect to parameters

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Figure 6. Comparison of the effects of $ a_0 $ and $ b_0 $ on $ \mathcal{R}_0 $ for different incubation period. Solid blue line represents $ \mathcal{R}_0 $ for time-varying incubation period $ \tau(t) $, and solid red line stands for $ \mathcal{R}_0 $ for time-averaged incubation period $ [\tau] $. Left panel: $ \mathcal{R}_0 $ vs $ a_0 $ for fixed $ b_0 = 0.51 $. Right panel: $ \mathcal{R}_0 $ vs $ b_0 $ for fixed $ a_0 = 1.076 $

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Table 1. Parameter values in simulation

Parameter	Value (range)	Dimension	Resource
$ \gamma_{M} $	0.093 (0.051–0.093) $ \times $ 30.4	month$ ^{-1} $	[25]
$ \Lambda_{B} $	18000/12	birds $ \times $ month$ ^{-1} $	[11]
$ \beta_{MB} $	0.8 (0.8–1)	dimensionless	[25]
$ \beta_{BM} $	0.2 (0.02–0.24)	dimensionless	[25]
$ \mu_{A} $	0.4 (0.213–16.9) $ \times $ 30.4	month$ ^{-1} $	[25]
$ \mu_{M} $	0.025 (0.016–0.07) $ \times $ 30.4	month$ ^{-1} $	[25]
$ \mu_{F} $	0.025 (0.016–0.07) $ \times $ 30.4	month$ ^{-1} $	[25]
$ \mu_{B} $	30.4/1000	month$ ^{-1} $	[22]
$ d_{B} $	0.01 (0–0.5) $ \times $ 30.4	month$ ^{-1} $	[16]

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