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## Wolbachia infection dynamics in mosquito population with the CI effect suffering by uninfected ova produced by infected females

 1 Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China 2 College of Mathematical Sciences, Harbin Engineering University, Harbin, Heilongjiang 150001, China

* Corresponding author: lxue@hrbeu.edu.cn (L. Xue)

Received  November 2019 Revised  December 2019 Published  March 2020

Mosquito-borne diseases pose a great threat to humans' health. Wolbachia is a promising biological weapon to control the mosquito population, and is not harmful to humans' health, environment, and ecology. In this work, we present a stage-structure model to investigate the effective releasing strategies for Wolbachia-infected mosquitoes. Besides some key factors for Wolbachia infection, the CI effect suffering by uninfected ova produced by infected females, which is often neglected, is also incorporated. We analyze the conditions under which Wolbachia infection still can be established even if the basic reproduction number is less than unity. Numerical simulations manifest that the threshold value of infected mosquitoes required to be released at the beginning can be evaluated by the stable manifold of a saddle equilibrium, and low levels of MK effect, fitness costs, as well as high levels of CI effect and maternal inheritance all contribute to the establishment of Wolbachia-infection. Moreover, our results suggest that ignoring the CI effect suffering by uninfected ova produced by infected females may result in the overestimation of the threshold infection level for the Wolbachia invasion.

Citation: Hui Wan, Yunyan Cao, Ling Xue. Wolbachia infection dynamics in mosquito population with the CI effect suffering by uninfected ova produced by infected females. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020123
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##### References:
The distribution of Wolbachia-infected equilibria (WE) in the plane of $(\xi,q)$ when $0<v<1$, $(1-v)(1-f_u)d_{mw} < v(1-\tau)(1-f_w)d_m$ and $R_{0w}>1$. In $D_1$, $R_0>1$, there exists a unique Wolbachia-infected equilibrium $E^*$. There are two coexistence equilibria in $D_2$ where $\hat{R_0}<R_0<1$, and the two equilibria coalesce when the parameters approach the curve segment of $L_3$ between $K_0$ and $K_3$ where $R_0 = \hat{R_0}$. There is a unique coexistence equilibrium on the segment of $L_2$ between $K_3$ and $K_4$, where $R_0 = 1$. In the other regions of the first quadrant, no WE exists
The backward bifurcation occurs at $R_0 = 1$ when $v = 0.9$. Parameter values are taken from Table 4 except that $d_{fw}$ varies
The distribution of Wolbachia-infected equilibria (WE) in the plane of $(\xi,q)$ when $v = 1$ and $R_{0w}>1$. In the region $D_{11}$, where $R_0 < 1-q$, there exists an unstable WE $E_{02}$. In the region $D_{22}$, where $1-q<R_0<1$, There are two WE. One is a coexistence equilibria $E^*$ and the other is the boundary equilibrium, $E_{02}$, which is stable. In the region $D_{12}$, there also exists a unique stable WE $E_{02}$
The distribution of real parts of eigenvalues of Jacobian matrix at $E^*_{1}$
The distribution of product of all six eigenvalues of Jacobian matrix at $E^*_{2}$ and $E_c^*$
Stable manifold of the saddle $E_2^*$ and $E_c^*$. Parameter values used are given in Table 4
The threshold values of Wolbachia-infected male and female mosquitoes. Parameter values are given in Table 4 except for $v$ being varied. Other initial values of state variables are $U_A(0) = 10000$, $U_F(0) = 5335$, $U_M(0) = 5991$, $W_A(0) = 0$
Time series of female mosquito for different releasing strategies. The parameter values are given in Table 4. (Ⅰ) $W_F(0) = W_M(0) = 0$; (Ⅱ) $W_F(0) = 0, W_M(0) = 1,500,000$; (Ⅲ) $W_F(0) = 1, W_M(0) = 150,000$. Other initial values of state variables are $U_A(0) = 4068, U_F(0) = 3335, U_M(0) = 2991, W_A(0) = 0$
Global sensitivity analysis
Bifurcation diagram with respect to $\tau$, $q$, $v$, $\alpha$. The parameter values are given in Table 4
Threshold values of $W_M(0)$ for Wolbachia establishment with respect to $\tau$, $q$, $v$, $\alpha$. Parameter values used are given in Table 4. Other initial values of state variables are: $U_A(0) = 4068,U_F(0) = 3335,U_M(0) = 2991,W_A(0) = 0,W_F(0) = 1$
The impact of the CI effect suffering by uninfected ova produced by infected females when the maternal transmission is imperfect ($v = 0.9$). The parameter values are given in Table 4 except for $\lambda$. Other initial values of state variables are: $U_A(0) = 4068$, $U_F(0) = 3335$, $U_M(0) = 2991$, $W_A(0) = 0$. The dash line corresponds to the case $\lambda = 0$, and the solid line corresponds to the case $\lambda = q$, respectively
State variables in the Model (1) for the transmission dynamics of Wolbachia
 State variables Uninfected Infected Aquatic stage $U_A$ $W_A$ Adult females $U_F$ $W_F$ Adult males $U_M$ $W_M$
 State variables Uninfected Infected Aquatic stage $U_A$ $W_A$ Adult females $U_F$ $W_F$ Adult males $U_M$ $W_M$
Definition of the parameters
 Interpretation Parameter Natural birth rate of mosquitoes $\theta$ Natural death rate of mosquitoes in aquatic stages $d_a$ Natural death rate of adult Wolbachia-uninfected male mosquitoes $d_{m}$ Natural death rate of adult Wolbachia-infected male mosquitoes $d_{mw}$ Natural death rate of adult Wolbachia-uninfected female mosquitoes $d_{f}$ Natural death rate of adult Wolbachia-infected female mosquitoes $d_{fw}$ Density-dependent mortality of immature mosquitoes $\kappa$ Per capita development rate of immature mosquitoes $\varphi$ Fraction of births that are female mosquitoes for Wolbachia-free mosquitoes $f_u$ Fraction of births that are female mosquitoes for Wolbachia-infected mosquitoes $f_w$ Probability of cytoplasmic incompatibility (CI) $q$ The CI parameter of uninfected ova produced by infected females $\lambda$ Fertility cost coefficient of the Wolbachia infection to reproductive output $\alpha$ Maternal transmission rate for Wolbachia infection $v$ Probability of male killing (MK) induced by Wolbachia infection $\tau$
 Interpretation Parameter Natural birth rate of mosquitoes $\theta$ Natural death rate of mosquitoes in aquatic stages $d_a$ Natural death rate of adult Wolbachia-uninfected male mosquitoes $d_{m}$ Natural death rate of adult Wolbachia-infected male mosquitoes $d_{mw}$ Natural death rate of adult Wolbachia-uninfected female mosquitoes $d_{f}$ Natural death rate of adult Wolbachia-infected female mosquitoes $d_{fw}$ Density-dependent mortality of immature mosquitoes $\kappa$ Per capita development rate of immature mosquitoes $\varphi$ Fraction of births that are female mosquitoes for Wolbachia-free mosquitoes $f_u$ Fraction of births that are female mosquitoes for Wolbachia-infected mosquitoes $f_w$ Probability of cytoplasmic incompatibility (CI) $q$ The CI parameter of uninfected ova produced by infected females $\lambda$ Fertility cost coefficient of the Wolbachia infection to reproductive output $\alpha$ Maternal transmission rate for Wolbachia infection $v$ Probability of male killing (MK) induced by Wolbachia infection $\tau$
Existence and stability of equilibria of Model 1 ($\checkmark$, Wolbachia is established; $\times$, Wolbachia is extinct; $?$, Depending on initial values; E, exist; N, does not exist; S, locally asymptotically stable; U, unstable), where $R_{0u} = \frac{f_u\varphi\theta}{d_f(d_a+\varphi)}$, $R_0 = \frac{(1-\alpha)vd_ff_w}{d_{fw}f_u}$, $R_{0w} = \frac{(1-\alpha)\theta vf_w\varphi}{d_{fw}(d_a+\varphi)} = R_{0u}R_0$
 $v$ Threshold condition $E_{01}$ $E_{02}$ $E^{*}$ Case 1 $v=1$ $R_{0w}>1$, $R_{0u}>1$, E, U E, S N $\checkmark$ $R_0>1$ Case 2 $v=1$ $R_{0w}>1$, $R_{0u}\leq1$, N E, S N $\checkmark$ ($R_0>1$) Case 3 $v=1$ $R_{0w}>1$, $R_{0u}>1$, E, S E, S E(one) $?$ $1-q1$, $R_{0u}>1$, E, S E, U N $\times$ $R_0< 1-q$ Case 5 $01$, $R_{0u}>1$, E, U N E(one) $\checkmark$ $R_0>1$ Case 6 $01$, $R_{0u}>1$, E, S N N $\times$ $R_0<1$ $(1-v)(1-f_u)d_{mw} \geq v(1-\tau)(1-f_w)d_m$ Case 7 $01$, $R_{0u}>1$, E, S N E(two) $?$ $\hat{R_0}1$, $R_{0u}>1$, E N E(one) Singularity $R_0=\hat{R_0}$ or $R_0=1$, $(1-v)(1-f_u)d_{mw}< v(1-\tau)(1-f_w)d_m$ Case 9 $01$, $R_{0u}>1$, , E, S N N $\times$ $R_0<\hat{R_0}$ $(1-v)(1-f_u)d_{mw}< v(1-\tau)(1-f_w)d_m$ Case 10 $01$, E, S N N $\times$ ($R_0<1$) Case 11 $0 $ v $Threshold condition$ E_{01}  E_{02}  E^{*} $Case 1$ v=1  R_{0w}>1 $,$ R_{0u}>1 $, E, U E, S N$ \checkmark  R_0>1 $Case 2$ v=1  R_{0w}>1 $,$ R_{0u}\leq1 $, N E, S N$ \checkmark $($ R_0>1 $) Case 3$ v=1  R_{0w}>1 $,$ R_{0u}>1 $, E, S E, S E(one)$ ?  1-q1 $,$ R_{0u}>1 $, E, S E, U N$ \times  R_0< 1-q $Case 5$ 01 $,$ R_{0u}>1 $, E, U N E(one)$ \checkmark  R_0>1 $Case 6$ 01 $,$ R_{0u}>1 $, E, S N N$ \times  R_0<1  (1-v)(1-f_u)d_{mw} \geq v(1-\tau)(1-f_w)d_m $Case 7$ 01 $,$ R_{0u}>1 $, E, S N E(two)$ ?  \hat{R_0}1 $,$ R_{0u}>1 $, E N E(one) Singularity$ R_0=\hat{R_0} $or$ R_0=1 $,$ (1-v)(1-f_u)d_{mw}< v(1-\tau)(1-f_w)d_m $Case 9$ 01 $,$ R_{0u}>1 $, , E, S N N$ \times  R_0<\hat{R_0}  (1-v)(1-f_u)d_{mw}< v(1-\tau)(1-f_w)d_m $Case 10$ 01 $, E, S N N$ \times $($ R_0<1 $) Case 11$ 0
Parameter values
 Parameter Value Range Reference $\theta$ 50 [0, 75] [8] $q$ 0.9 [0.5, 1] [21,50] $v$ 0.9, 0.1 (0.85, 1] [29,42] $d_a$ 0.2 [0.2, 0.75] [11,25] $d_f$ 0.061 [1/55, 1/11] [46] $d_{fw}$ 0.068 [1/55, 1/11] [46] $d_m$ 0.068 [1/31, 1/7] [46] $d_{mw}$ 0.068 [1/31, 1/7] [46] $\varphi$ 0.1 [1/20, 1/7] [28] $\alpha$ 0.1 (0, 1] [49] $f_u$ 0.5 [0.34, 0.6] [46,8] $f_w$ 0.5 [0.34, 0.6] [46,8] $\tau$ 0.01 [0, 1) [13] $\kappa$ 0.01 [0.01, 0.01] Assumed
 Parameter Value Range Reference $\theta$ 50 [0, 75] [8] $q$ 0.9 [0.5, 1] [21,50] $v$ 0.9, 0.1 (0.85, 1] [29,42] $d_a$ 0.2 [0.2, 0.75] [11,25] $d_f$ 0.061 [1/55, 1/11] [46] $d_{fw}$ 0.068 [1/55, 1/11] [46] $d_m$ 0.068 [1/31, 1/7] [46] $d_{mw}$ 0.068 [1/31, 1/7] [46] $\varphi$ 0.1 [1/20, 1/7] [28] $\alpha$ 0.1 (0, 1] [49] $f_u$ 0.5 [0.34, 0.6] [46,8] $f_w$ 0.5 [0.34, 0.6] [46,8] $\tau$ 0.01 [0, 1) [13] $\kappa$ 0.01 [0.01, 0.01] Assumed
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