# American Institute of Mathematical Sciences

• Previous Article
Existence of solutions for first-order Hamiltonian random impulsive differential equations with Dirichlet boundary conditions
• DCDS-B Home
• This Issue
• Next Article
Vortex Condensation in General U(1)×U(1) Abelian Chern-Simons Model on a flat torus
August  2022, 27(8): 4429-4453. doi: 10.3934/dcdsb.2021235

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

 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

Received  March 2021 Revised  July 2021 Published  August 2022 Early access  September 2021

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.

Citation: Zongmin Yue, Fauzi Mohamed Yusof. A mathematical model for biodiversity diluting transmission of zika virus through competition mechanics. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4429-4453. doi: 10.3934/dcdsb.2021235
##### References:

show all references

##### References:
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
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
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
Taking $a = 0.6, \varepsilon = 2, q = 3, \Lambda_M = 0.5, \Lambda_H = 0.02$, $E_0$ is stable globally
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
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
Co-dimension 2 bifurcation diagrams show the distribution of equilibrium points of system (3) and their stability, where (A)$\varepsilon = 2,q = 3$(B)$\varepsilon = 0.5,q = 0.5$
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
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
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 $q$
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
 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
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 $\begin{array}{l} N_{M1}^* = N_{M1}^{}\\N_{M2}^* = N_{M2}^{} \end{array}$ $N_{M2}^* = N_{M2}^{}$ $\begin{array}{c} N_{M0}^* = \frac{{K{\mu _M} + \kappa aq}}{{2(\varepsilon q - 1)}} \end{array}$ $\begin{array}{c} N_{M0}^* =\frac{{K{\Lambda _M}}}{{K{\mu _M} + \kappa aq}} \end{array}$ $N_{M0}^* = N_{M2}^{}$
 $\varepsilon q > 1$ $\varepsilon q = 1$ $\varepsilon q < 1$ $\Delta > 0$ $\Delta = 0$ —— $\Delta < 0$ $a > \frac{{\varepsilon N_{M1}^*}}{\kappa }$ $\frac{{\varepsilon N_{M2}^*}}{\kappa } < a < \frac{{\varepsilon N_{M1}^*}}{\kappa }$ $a > \frac{{\varepsilon N_{M0}^*}}{\kappa }$ $a > \frac{{\varepsilon N_{M0}^*}}{\kappa }$ $a > \frac{{\varepsilon N_{M0}^*}}{\kappa }$ Two equilibrium One equilibrium one equilibrium One equilibrium One equilibrium $\begin{array}{l} N_{M1}^* = N_{M1}^{}\\N_{M2}^* = N_{M2}^{} \end{array}$ $N_{M2}^* = N_{M2}^{}$ $\begin{array}{c} N_{M0}^* = \frac{{K{\mu _M} + \kappa aq}}{{2(\varepsilon q - 1)}} \end{array}$ $\begin{array}{c} N_{M0}^* =\frac{{K{\Lambda _M}}}{{K{\mu _M} + \kappa aq}} \end{array}$ $N_{M0}^* = N_{M2}^{}$
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
 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
 [1] Yue Qiu, Sara Grundel, Martin Stoll, Peter Benner. Efficient numerical methods for gas network modeling and simulation. Networks and Heterogeneous Media, 2020, 15 (4) : 653-679. doi: 10.3934/nhm.2020018 [2] Jean-Paul Chehab. Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2693-2728. doi: 10.3934/dcdss.2021002 [3] Cristina Anton, Jian Deng, Yau Shu Wong, Yile Zhang, Weiping Zhang, Stephan Gabos, Dorothy Yu Huang, Can Jin. Modeling and simulation for toxicity assessment. Mathematical Biosciences & Engineering, 2017, 14 (3) : 581-606. doi: 10.3934/mbe.2017034 [4] Hui li, Manjun Ma. Global dynamics of a virus infection model with repulsive effect. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4783-4797. doi: 10.3934/dcdsb.2019030 [5] Tsanou Berge, Samuel Bowong, Jean Lubuma, Martin Luther Mann Manyombe. Modeling Ebola Virus Disease transmissions with reservoir in a complex virus life ecology. Mathematical Biosciences & Engineering, 2018, 15 (1) : 21-56. doi: 10.3934/mbe.2018002 [6] Gong Chen, Peter J. Olver. Numerical simulation of nonlinear dispersive quantization. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 991-1008. doi: 10.3934/dcds.2014.34.991 [7] Nicolas Vauchelet. Numerical simulation of a kinetic model for chemotaxis. Kinetic and Related Models, 2010, 3 (3) : 501-528. doi: 10.3934/krm.2010.3.501 [8] Petr Bauer, Michal Beneš, Radek Fučík, Hung Hoang Dieu, Vladimír Klement, Radek Máca, Jan Mach, Tomáš Oberhuber, Pavel Strachota, Vítězslav Žabka, Vladimír Havlena. Numerical simulation of flow in fluidized beds. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 833-846. doi: 10.3934/dcdss.2015.8.833 [9] Paolo Podio-Guidugli. On the modeling of transport phenomena in continuum and statistical mechanics. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1393-1411. doi: 10.3934/dcdss.2017074 [10] H. T. Banks, Robert Baraldi, Karissa Cross, Kevin Flores, Christina McChesney, Laura Poag, Emma Thorpe. Uncertainty quantification in modeling HIV viral mechanics. Mathematical Biosciences & Engineering, 2015, 12 (5) : 937-964. doi: 10.3934/mbe.2015.12.937 [11] Jinhae Park, Feng Chen, Jie Shen. Modeling and simulation of switchings in ferroelectric liquid crystals. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1419-1440. doi: 10.3934/dcds.2010.26.1419 [12] Xiaoli Yang, Jin Liang, Bei Hu. Minimization of carbon abatement cost: Modeling, analysis and simulation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2939-2969. doi: 10.3934/dcdsb.2017158 [13] Michael Herty. Modeling, simulation and optimization of gas networks with compressors. Networks and Heterogeneous Media, 2007, 2 (1) : 81-97. doi: 10.3934/nhm.2007.2.81 [14] Ana I. Muñoz, José Ignacio Tello. Mathematical analysis and numerical simulation of a model of morphogenesis. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1035-1059. doi: 10.3934/mbe.2011.8.1035 [15] Andriy Sokolov, Robert Strehl, Stefan Turek. Numerical simulation of chemotaxis models on stationary surfaces. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2689-2704. doi: 10.3934/dcdsb.2013.18.2689 [16] Sergio Amat, Pablo Pedregal. On a variational approach for the analysis and numerical simulation of ODEs. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1275-1291. doi: 10.3934/dcds.2013.33.1275 [17] Michal Beneš, Pavel Eichler, Jakub Klinkovský, Miroslav Kolář, Jakub Solovský, Pavel Strachota, Alexandr Žák. Numerical simulation of fluidization for application in oxyfuel combustion. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 769-783. doi: 10.3934/dcdss.2020232 [18] Rongsong Liu, Jiangping Shuai, Jianhong Wu, Huaiping Zhu. Modeling spatial spread of west nile virus and impact of directional dispersal of birds. Mathematical Biosciences & Engineering, 2006, 3 (1) : 145-160. doi: 10.3934/mbe.2006.3.145 [19] Jing Chen, Jicai Huang, John C. Beier, Robert Stephen Cantrell, Chris Cosner, Douglas O. Fuller, Guoyan Zhang, Shigui Ruan. Modeling and control of local outbreaks of West Nile virus in the United States. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2423-2449. doi: 10.3934/dcdsb.2016054 [20] Xiulan Lai, Xingfu Zou. A reaction diffusion system modeling virus dynamics and CTLs response with chemotaxis. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2567-2585. doi: 10.3934/dcdsb.2016061

2021 Impact Factor: 1.497