Global dynamics of a Huanglongbing model with a periodic latent period

Yan Hong; Xiuxiang Liu; Xiao Yu

doi:10.3934/dcdsb.2021302

Article Contents

2022, Volume 27, Issue 10: 5953-5976. Doi: 10.3934/dcdsb.2021302

This issue Previous Article Stability analysis of stochastic delay differential equations with Markovian switching driven by Lévy noise Next Article Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise

Global dynamics of a Huanglongbing model with a periodic latent period

School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

^*Corresponding author: Xiao Yu
^*Corresponding author: Xiao Yu

Received: June 30, 2021

Revised: August 31, 2021

Early access: December 2021

Published: October 2022

The third author is supported in part by the National Natural Science Foundation of China; and Guangdong Basic and Applied Basic Research Foundation Province (12001205, 12026602 and 2019A1515110179)

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Abstract

Huanglongbing (HLB) is a disease of citrus that caused by phloem-restricted bacteria of the Candidatus Liberibacter group. In this paper, we present a HLB transmission model to investigate the effects of temperature-dependent latent periods and seasonality on the spread of HLB. We first establish disease free dynamics in terms of a threshold value $ R^p_0 $, and then introduce the basic reproduction number $ \mathcal{R}_0 $ and show the threshold dynamics of HLB with respect to $ R^p $ and $ \mathcal{R}_0 $. Numerical simulations are further provided to illustrate our analytic results.

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Mathematics Subject Classification: Primary: 92D25, 34D05, 34C60; Secondary: 37C65.

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Figure 1. Tendency of citrus trees when $ \mathcal{R}_0<1 $ and $ \mathcal{R}_0>1 $

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Figure 2. Behaviors of psyllid when $ \mathcal{R}_0<1 $ and $ \mathcal{R}_0>1 $

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Figure 3. infectious psyllid under different $ p $ and $ q $

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Figure 4. $ \mathcal{R}_0 $ vs r

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Table 1. Biological interpretations for parameters in system (1)

Parameter	Description
$ r $	Rate of replanting citrus tree
$ K $	Maximum citrus tree population size
$ \beta_1(t) $	Infection rate of susceptible trees
$ \mu_1 $	Natural death rate of citrus tree population
$ \delta_1 $	Death rate of infected trees
$ \delta_2 $	Rate of removal of infected trees
$ \alpha(t) $	Intrinsic growth rate of psyllid
$ m $	Maximum abundance of psyllid per tree
$ d_v(t) $	Natural death rate of psyllid population
$ \beta_2(t) $	Infection rate of susceptible psyllid
$ \theta(t) $	Killing rate of psyllid with spraying insecticide
$ \tau $	Incubation period in trees
$ \tau_v(t) $	Extrinsic incubation period(EIP) of psyllid

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Table 2. Parameter values in simulation

Parameter	Value	Unit	Reference
$ r $	0.05	month$ ^{-1} $	[17]
$ K $	2000	-	[17]
$ \beta_1(t) $	to be estimated	month$ ^{-1} $	see text
$ \mu_1 $	0.00333	month$ ^{-1} $	[29]
$ \delta_1 $	0.0015	month$ ^{-1} $	[25]
$ \delta_2 $	0.02	month$ ^{-1} $	[25]
$ \alpha(t) $	$ 18.120952 + 14.45466475\cos(2\pi t/12) $	month$ ^{-1} $	[29]
$ m $	$ 1\times 10^{6} $	-	[29]
$ d_v(t) $	to be estimated	month$ ^{-1} $	see text
$ \beta_2(t) $	to be estimated	month$ ^{-1} $	see text
$ \theta(t) $	to be estimated	month$ ^{-1} $	see text
$ \tau $	6	month	[25]
$ \tau_v(t) $	assumed	month	see text

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Table 3. Monthly mean temperature in the South of Jiangxi, 1961-2016($ ^\circ C $)

Month	Jan	Feb	Mar	Apr	May	June
temperature	5.7	7.5	11.6	17.3	22.4	25.0
Month	Jul	Aug	Sep	Oct	Nov	Dec
temperature	28.3	27.6	24.5	19.2	13.2	7.4

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