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doi: 10.3934/dcdsb.2021235
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## 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 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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021235
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##### 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
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