A mathematical model for biodiversity diluting transmission of zika virus through competition mechanics

Zongmin Yue; Fauzi Mohamed Yusof

doi:10.3934/dcdsb.2021235

Article Contents

2022, Volume 27, Issue 8: 4429-4453. Doi: 10.3934/dcdsb.2021235

This issue Previous Article Vortex Condensation in General U(1)×U(1) Abelian Chern-Simons Model on a flat torus Next Article Existence of solutions for first-order Hamiltonian random impulsive differential equations with Dirichlet boundary conditions

A mathematical model for biodiversity diluting transmission of zika virus through competition mechanics

Zongmin Yue^1,2, , and
Fauzi Mohamed Yusof^1,

1.
Faculty of Science and Mathematics, Sultan Idris Education University, Tanjong Malim, Malaysia
2.
School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi'an, China

^* Corresponding author: Zongmin Yue
^* Corresponding author: Zongmin Yue

Received: March 2021

Revised: July 2021

Early access: September 2021

Published: August 2022

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Abstract

Whether increasing biodiversity will lead to a promotion (amplification effect) or inhibition (dilution effect) in the transmission of infectious diseases remains to be discovered. In vector-borne infectious diseases, Lyme Disease (LD) and West Nile Virus (WNV) have become typical examples of the dilution effect of biodiversity. Thus, as a vector-borne disease, biodiversity may also play a positive role in the control of the Zika virus. We developed a Zika virus model affected by biodiversity through a competitive mechanism. Through the qualitative analysis of the model, the stability condition of the disease-free equilibrium point and the control threshold of the disease - the basic reproduction number is given. Not only has the numerical analysis verified the inference results, but also it has shown the regulatory effect of the competition mechanism on Zika virus transmission. As competition limits the size of the vector population, the number of final viral infections also decreases. Besides, we also find that under certain parameter conditions, the dilution effect may disappear because of the different initial values. Finally, we emphasized the impact of human activities on biological diversity, to indirectly dilute the abundance of diversity and make the virus continuously spread.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. The phase portraits of model (3). The parameters are taken as $ \Lambda_M = 0.5, K = 4, \kappa = 4 $. $ a $ and $ q $ are marked under each image. Fig. 1(A) represents $ \overline{E}_{2*}^{{}} = (1.5087,2.9826) $is a nodal sink, $ \overline{E}_{1*}^{{}} = (2.6513,0.69737) $is a saddle point and $ \overline{E}_{0}^{{}} = (3.2861,0) $is a nodal sink. Fig. 1(B) shows is a nodal sink and $ \overline{E}_{0}^{{}} = (3.2861,0) $is a saddle point. A asymptotically stable non-double node $ \overline{E}_{0*}^{{}} = (2.0001,1.7331) $ is shown in Fig. 1(C). In Fig. 1(D), there exist a globally stable node $ \overline{E}_{0*}^{{}} = (2.0767,4.754) $. The dotted curve is the two nullclines

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Figure 2. Take $ a = 0.2,\varepsilon = 2,q = 3,{{\Lambda }_{M}} = 0.5,{{\Lambda }_{H}} = 0.2 $. (A) Initial value is ($ 1, 0.5, 0.6, 0.2, 1.5, 0.05, 0.05, 1 $), a stable positive equilibrium point exists. (B) Change the initial value to ($ 0.02, 0.02, 0.01, 0.01, 0.15, 0.05, 0.05, 0.05 $), then the aliens $ Z(t) $ convert to zero. The system stabilizes to its boundary equilibrium point

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Figure 3. Taking $ a = 0.2,\varepsilon = 0.6, q = 1, \Lambda_M = 0.5,\Lambda _H = 0.2 $, then $ \varepsilon q<1 $. Interior positive equilibrium point$ E_{{}}^{*} $is stable

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Figure 4. Taking $ a = 0.6, \varepsilon = 2, q = 3, \Lambda_M = 0.5, \Lambda_H = 0.02 $, $ E_0 $ is stable globally

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Figure 5. Taking $ a = 0.2,\varepsilon = 2,q = 3,{{\Lambda }_{M}} = 0.5,{{\Lambda }_{H}} = 0.2 $. Initial value is ($ 1, 0.5, 0.6, 0.2, 1.5, 0.05, 0.05, 1 $). The trends of each variable during a same period

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Figure 6. Take $ {{\Lambda }_{M}} = 0.5 $ and $ {{\Lambda }_{H}} = 0.05,{{\Lambda }_{H}} = 0.02,{{\Lambda }_{H}} = 0.002 $, respectively. $ {{R}_{0}} $ increases as $ {{N}_{M}} $ increases

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Figure 7. Co-dimension 2 bifurcation diagrams show the distribution of equilibrium points of system (3) and their stability, where (A)$ \varepsilon = 2,q = 3 $(B)$ \varepsilon = 0.5,q = 0.5 $

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Figure 8. Take $ a = 0.31 $, $ q = 3,{{\Lambda }_{M}} = 0.5,{{\Lambda }_{H}} = 0.02 $. By decreasing $ \varepsilon $ from 3 to 1 unit, mosquitoes become less competitive

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Figure 9. Take $ a = 0.31 $, $ \varepsilon = 2,{{\Lambda }_{M}} = 0.5,{{\Lambda }_{H}} = 0.02 $. With $ q $ decreasing from 3 to 1 units, mosquitoes become more competitive

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Figure 10. Taking $ \varepsilon = 2,a = 0.31 $. By increasing the maximum capacity of the environment for mosquitoes $ K $, the total mosquito size increases with the same $ q $

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Table 1. Description of the parameters used in model (2)

Symbol	Description
$ {\Lambda _H} $	Recruitment rate of susceptible humans
$ {\Lambda _M} $	Recruitment rate of susceptible mosquitoes
$ {\mu _H} $	Natural death rate in humans
$ {\mu _M} $	Natural death rate in mosquitoes
$ {\beta _H} $	Mosquito-to-human transmission rate
$ {\beta _M} $	Human-to-mosquito transmission rate
$ {\alpha _H} $	The rate of exposed humans moving into infectious class
$ {\rho _{}} $	Human factor transmission rate
$ {r_{}} $	Human recovery rate
$ {\delta _M} $	The rate flow from $ E_M $ to $ I_M $
$ K $	The maximum environmental capacity for mosquitoes without alien
$ \kappa $	The maximum environmental capacity for aliens without mosquitoes
$ q $ and $ \varepsilon $	The inhibition between aliens and mosquitoes
$ a $	Alien's natural growth rate

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Table 2. Different expressions of positive equilibrium points of model (3)

$\varepsilon q > 1$			$\varepsilon q = 1$	$\varepsilon q < 1$
$\Delta > 0$		$\Delta = 0$	——	$\Delta < 0$
$a > \frac{{\varepsilon N_{M1}^*}}{\kappa }$	$\frac{{\varepsilon N_{M2}^}}{\kappa } < a < \frac{{\varepsilon N_{M1}^}}{\kappa }$	$a > \frac{{\varepsilon N_{M0}^*}}{\kappa }$	$a > \frac{{\varepsilon N_{M0}^*}}{\kappa }$	$a > \frac{{\varepsilon N_{M0}^*}}{\kappa }$
Two equilibrium	One equilibrium	one equilibrium	One equilibrium	One equilibrium
$\begin{array}{l} N_{M1}^* = N_{M1}^{}\\N_{M2}^* = N_{M2}^{} \end{array}$	$N_{M2}^* = N_{M2}^{}$	$\begin{array}{c} N_{M0}^* = \frac{{K{\mu _M} + \kappa aq}}{{2(\varepsilon q - 1)}} \end{array} $	$\begin{array}{c} N_{M0}^* =\frac{{K{\Lambda _M}}}{{K{\mu _M} + \kappa aq}} \end{array}$	$N_{M0}^* = N_{M2}^{}$

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Table 3. Fixed parameters values in the numerical simulation of the system (2)

Parameter	Description	Value	Ref
$ \beta_H $	Mosquito-to-human transmission rate	0.2 per day	Bonyah et al. [4]
$ {\beta _M} $	Human-to-mosquito Transmission rate	0.09 per day	Bonyah et al. [4]
$ {\alpha _H} $	The rate of exposed humans moving into infectious class	$ \frac{1}{{5.5}} \approx 0.18 $ per day	Ferguson et al. [12]
$ {\rho _{}} $	Human factor transmission rate	0.0029	assumed
$ {\delta _M} $	The rate flow from $ {E_M} $ to $ {I_M} $	$ \frac{1}{{8.2}} \approx 0.12 $ per day	Ferguson et al. [12]
$ {r_{}} $	Human recovery rate	$ \frac{1}{6} \approx 0.17 $ per day	Ferguson et al. [12]
$ {\mu _H} $	Natural death rate in humans	$ \frac{1}{{360 \times 60}} \approx 0.00005 $ per day	Manore et al. [25]
$ {\mu _M} $	Natural death rate in mosquitoes	$ \frac{1}{{14}} \approx 0.07 $ per day	Manore et al. [25]
$ K $	The maximum environmental capacity for mosquitoes without aliens	40	assumed
$ k $	The maximum environmental capacity for aliens without mosquitoes	30	assumed

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