The impact of predators of mosquito larvae on Wolbachia spreading dynamics

Zhongcai Zhua Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou, People's Republic of China;b College of Mathematics and Information Sciences, Guangzhou University, Guangzhou, People's Republic of ChinaView further author information

Yuanxian Huic School of Mathematics and Statistics, Huanghuai University, Zhumadian, People's Republic of ChinaView further author information

Linchao Hua Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou, People's Republic of China;b College of Mathematics and Information Sciences, Guangzhou University, Guangzhou, People's Republic of ChinaCorrespondence[email protected]
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Table 1. Model parameters and their interpretations.

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Figure 1. Let the parameters $α, d_{0}, μ$ and $d_{1}$ be specified as in (Equation10(10) $α = 0.129, d_{0} = 0.05, μ = 0.06 a n d d_{1} = 0.035.$ (10) ). Then we get $β^{*} \approx 0.1665$ . By selecting $β = 0.1 \in (0, β^{*})$ , and plotting the graphs of $J_{U} (t), J_{I} (t), A_{U} (t)$ and $A_{I} (t)$ in panels $(A), (B), (C)$ and (D), respectively, we obtain the above figure, which illustrates the global attractivity of ${\bar{E}}_{0}$ .

Figure 1. Let the parameters α,d0,μ and d1 be specified as in (Equation10(10) α=0.129,d0=0.05,μ=0.06andd1=0.035.(10) ). Then we get β∗≈0.1665. By selecting β=0.1∈(0,β∗), and plotting the graphs of JU(t),JI(t),AU(t) and AI(t) in panels (A),(B),(C) and (D), respectively, we obtain the above figure, which illustrates the global attractivity of E¯0.

Figure 2. Suppose that the parameters $α, d_{0}, μ$ and $d_{1}$ in system (Equation1(1) ${\begin{cases} \frac{d J_{U}}{d t} = \frac{1}{2} β A_{U} \frac{A_{U}}{A_{U} + A_{I}} - [d_{0} + d_{1} (J_{U} + J_{I})] J_{U} - α J_{U} := {\bar{f}}_{1} (J_{U}, J_{I}, A_{U}, A_{I}), \\ \frac{d J_{I}}{d t} = \frac{1}{2} β A_{I} - [d_{0} + d_{1} (J_{U} + J_{I})] J_{I} - α J_{I} := {\bar{f}}_{2} (J_{U}, J_{I}, A_{U}, A_{I}), \\ \frac{d A_{U}}{d t} = α J_{U} - μ A_{U} := {\bar{f}}_{3} (J_{U}, J_{I}, A_{U}, A_{I}), \\ \frac{d A_{I}}{d t} = α J_{I} - μ A_{I} := {\bar{f}}_{4} (J_{U}, J_{I}, A_{U}, A_{I}), \end{cases}$ (1) ) are in line with (Equation10(10) $α = 0.129, d_{0} = 0.05, μ = 0.06 a n d d_{1} = 0.035.$ (10) ). Then $β^{*} \approx 0.1665$ . By choosing $β = 0.25 > β^{*}$ , we get the graphs of $J_{U} (t), J_{I} (t), A_{U} (t)$ and $A_{I} (t)$ in panels $(A), (B), (C)$ and (D), respectively.

Figure 2. Suppose that the parameters α,d0,μ and d1 in system (Equation1(1) {dJUdt=12βAUAUAU+AI−[d0+d1(JU+JI)]JU−αJU:=f¯1(JU,JI,AU,AI),dJIdt=12βAI−[d0+d1(JU+JI)]JI−αJI:=f¯2(JU,JI,AU,AI),dAUdt=αJU−μAU:=f¯3(JU,JI,AU,AI),dAIdt=αJI−μAI:=f¯4(JU,JI,AU,AI),(1) ) are in line with (Equation10(10) α=0.129,d0=0.05,μ=0.06andd1=0.035.(10) ). Then β∗≈0.1665. By choosing β=0.25>β∗, we get the graphs of JU(t),JI(t),AU(t) and AI(t) in panels (A),(B),(C) and (D), respectively.

Figure 3. Let the relevant parameters in (Equation2(2) ${\begin{cases} \frac{d J_{U}}{d t} = \frac{1}{2} β A_{U} \frac{A_{U}}{A_{U} + A_{I}} - [d_{0} + d_{1} (J_{U} + J_{I})] J_{U} - α J_{U} \\ - c \frac{m J_{U}}{J_{U} + a} P := {\tilde{f}}_{1} (J_{U}, J_{I}, A_{U}, A_{I}, P), \\ \frac{d J_{I}}{d t} = \frac{1}{2} β A_{I} - [d_{0} + d_{1} (J_{U} + J_{I})] J_{I} - α J_{I} - c \frac{m J_{I}}{J_{I} + a} P := {\tilde{f}}_{2} (J_{U}, J_{I}, A_{U}, A_{I}, P), \\ \frac{d A_{U}}{d t} = α J_{U} - μ A_{U} := {\tilde{f}}_{3} (J_{U}, J_{I}, A_{U}, A_{I}, P), \\ \frac{d A_{I}}{d t} = α J_{I} - μ A_{I} := {\tilde{f}}_{4} (J_{U}, J_{I}, A_{U}, A_{I}, P), \\ \frac{d P}{d t} = \frac{m J_{U}}{J_{U} + a} P + \frac{m J_{I}}{J_{I} + a} P - δ P := {\tilde{f}}_{5} (J_{U}, J_{I}, A_{U}, A_{I}, P), \end{cases}$ (2) ) are specified as in Example 3.3. Then the graph of F against x is shown in the above figure, where the left and right red dashed straight lines in the thumbnail represent $x = \frac{a δ}{2 m - δ}$ and $x = \frac{a δ}{m - δ}$ , respectively, and the intersection point of F and x-axis, marked with asterisk, corresponds to the first component of the unique equilibrium of (Equation2(2) ${\begin{cases} \frac{d J_{U}}{d t} = \frac{1}{2} β A_{U} \frac{A_{U}}{A_{U} + A_{I}} - [d_{0} + d_{1} (J_{U} + J_{I})] J_{U} - α J_{U} \\ - c \frac{m J_{U}}{J_{U} + a} P := {\tilde{f}}_{1} (J_{U}, J_{I}, A_{U}, A_{I}, P), \\ \frac{d J_{I}}{d t} = \frac{1}{2} β A_{I} - [d_{0} + d_{1} (J_{U} + J_{I})] J_{I} - α J_{I} - c \frac{m J_{I}}{J_{I} + a} P := {\tilde{f}}_{2} (J_{U}, J_{I}, A_{U}, A_{I}, P), \\ \frac{d A_{U}}{d t} = α J_{U} - μ A_{U} := {\tilde{f}}_{3} (J_{U}, J_{I}, A_{U}, A_{I}, P), \\ \frac{d A_{I}}{d t} = α J_{I} - μ A_{I} := {\tilde{f}}_{4} (J_{U}, J_{I}, A_{U}, A_{I}, P), \\ \frac{d P}{d t} = \frac{m J_{U}}{J_{U} + a} P + \frac{m J_{I}}{J_{I} + a} P - δ P := {\tilde{f}}_{5} (J_{U}, J_{I}, A_{U}, A_{I}, P), \end{cases}$ (2) ) in Example 3.3.

Figure 3. Let the relevant parameters in (Equation2(2) {dJUdt=12βAUAUAU+AI−[d0+d1(JU+JI)]JU−αJU−cmJUJU+aP:=f~1(JU,JI,AU,AI,P),dJIdt=12βAI−[d0+d1(JU+JI)]JI−αJI−cmJIJI+aP:=f~2(JU,JI,AU,AI,P),dAUdt=αJU−μAU:=f~3(JU,JI,AU,AI,P),dAIdt=αJI−μAI:=f~4(JU,JI,AU,AI,P),dPdt=mJUJU+aP+mJIJI+aP−δP:=f~5(JU,JI,AU,AI,P),(2) ) are specified as in Example 3.3. Then the graph of F against x is shown in the above figure, where the left and right red dashed straight lines in the thumbnail represent x=aδ2m−δ and x=aδm−δ, respectively, and the intersection point of F and x-axis, marked with asterisk, corresponds to the first component of the unique equilibrium of (Equation2(2) {dJUdt=12βAUAUAU+AI−[d0+d1(JU+JI)]JU−αJU−cmJUJU+aP:=f~1(JU,JI,AU,AI,P),dJIdt=12βAI−[d0+d1(JU+JI)]JI−αJI−cmJIJI+aP:=f~2(JU,JI,AU,AI,P),dAUdt=αJU−μAU:=f~3(JU,JI,AU,AI,P),dAIdt=αJI−μAI:=f~4(JU,JI,AU,AI,P),dPdt=mJUJU+aP+mJIJI+aP−δP:=f~5(JU,JI,AU,AI,P),(2) ) in Example 3.3.

Figure 4. Assume that the parameters in system (Equation2(2) ${\begin{cases} \frac{d J_{U}}{d t} = \frac{1}{2} β A_{U} \frac{A_{U}}{A_{U} + A_{I}} - [d_{0} + d_{1} (J_{U} + J_{I})] J_{U} - α J_{U} \\ - c \frac{m J_{U}}{J_{U} + a} P := {\tilde{f}}_{1} (J_{U}, J_{I}, A_{U}, A_{I}, P), \\ \frac{d J_{I}}{d t} = \frac{1}{2} β A_{I} - [d_{0} + d_{1} (J_{U} + J_{I})] J_{I} - α J_{I} - c \frac{m J_{I}}{J_{I} + a} P := {\tilde{f}}_{2} (J_{U}, J_{I}, A_{U}, A_{I}, P), \\ \frac{d A_{U}}{d t} = α J_{U} - μ A_{U} := {\tilde{f}}_{3} (J_{U}, J_{I}, A_{U}, A_{I}, P), \\ \frac{d A_{I}}{d t} = α J_{I} - μ A_{I} := {\tilde{f}}_{4} (J_{U}, J_{I}, A_{U}, A_{I}, P), \\ \frac{d P}{d t} = \frac{m J_{U}}{J_{U} + a} P + \frac{m J_{I}}{J_{I} + a} P - δ P := {\tilde{f}}_{5} (J_{U}, J_{I}, A_{U}, A_{I}, P), \end{cases}$ (2) ) are defined the same as in Example 3.3. Then the conditions for Theorem 3.4 are satisfied, and there exists six equilibria: ${\tilde{E}}_{0}, {\tilde{E}}_{1}, {\tilde{E}}_{2}, {\tilde{E}}_{3}, {\tilde{E}}_{4}$ and ${\tilde{E}}^{*}$ . Moreover, equilibria ${\tilde{E}}_{0}, {\tilde{E}}_{1}, {\tilde{E}}_{2}$ and ${\tilde{E}}^{*}$ are unstable, equilibria ${\tilde{E}}_{3}$ and ${\tilde{E}}_{4}$ are asymptotically stable.

Figure 4. Assume that the parameters in system (Equation2(2) {dJUdt=12βAUAUAU+AI−[d0+d1(JU+JI)]JU−αJU−cmJUJU+aP:=f~1(JU,JI,AU,AI,P),dJIdt=12βAI−[d0+d1(JU+JI)]JI−αJI−cmJIJI+aP:=f~2(JU,JI,AU,AI,P),dAUdt=αJU−μAU:=f~3(JU,JI,AU,AI,P),dAIdt=αJI−μAI:=f~4(JU,JI,AU,AI,P),dPdt=mJUJU+aP+mJIJI+aP−δP:=f~5(JU,JI,AU,AI,P),(2) ) are defined the same as in Example 3.3. Then the conditions for Theorem 3.4 are satisfied, and there exists six equilibria: E~0,E~1,E~2,E~3,E~4 and E~∗. Moreover, equilibria E~0,E~1,E~2 and E~∗ are unstable, equilibria E~3 and E~4 are asymptotically stable.

Figure 5. We choose the parameters in Figures C and C and set $A_{U} (0) = 25$ . Then obtain the numbers of uninfected adult mosquitoes with and without predators.

Figure 5. We choose the parameters in Figures 2C and 4C and set AU(0)=25. Then obtain the numbers of uninfected adult mosquitoes with and without predators.

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The impact of predators of mosquito larvae on Wolbachia spreading dynamics

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The impact of predators of mosquito larvae on Wolbachia spreading dynamics

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