Two-sex mosquito model for the persistence of Wolbachia

Figures & data

Table 1. State variables and parameters for the model (1(1a) $\begin{array}{r}\frac{\mathrm{d}{A}_u}{\mathrm{d}t}=B_{uu}+v_u(B_{wu}+B_{ww})-{\mu }_a{A}_u-\psi A_u,\end{array}$(1a) ).

$M_u,F_u,A_u$:	Number of uninfected male, female, and aquatic mosquitoes
$M_w,F_w,A_w$:	Number of male, female, and aquatic mosquitoes infected with Wolbachia
$N_A$:	Total number of aquatic stage of mosquitoes
$K_a$:	Carrying capacity of aquatic stage of mosquitoes
$b_f$:	Fraction of births that are female mosquitoes.
$c_i$:	Fraction of unviable offspring due to cytoplasmic incompatibility.
$b_m$:	Fraction of births that are male mosquitoes $=1-b_f$.
$m_w$:	Fraction of the male mosquitoes that are infected $=M_w/(M_w+M_u)$.
$m_u$:	Fraction of the male mosquitoes that are uninfected $=1-m_w$.
$v_w$:	Fraction of infected mosquito eggs produced by infected female mosquitoes.
$v_u$:	Fraction of uninfected mosquito eggs produced by infected female mosquitoes $=1-v_w$.
${\phi }_u$:	Per capita egg laying rate by Wolbachia-free mosquito eggs. Number of eggs/time
${\phi }_w$:	Per capita egg laying rate by Wolbachia-infected mosquito eggs. Number of eggs/time
ψ:	Per capita development rate of mosquito eggs. Time^−1
${\mu }_a$:	Per capita death rate of aquatic stage of mosquitoes. Time^−1
${\mu }_{fu}$:	Per capita death rate of uninfected female mosquitoes. Time^−1
${\mu }_{fw}$:	Per capita death rate of infected female mosquitoes. Time^−1
${\mu }_{mu}$:	Per capita death rate of uninfected male mosquitoes. Time^−1
${\mu }_{mw}$:	Per capita death rate of infected male mosquitoes. Time^−1

Figure 1. The birthing rates (2(1b) $\begin{array}{r}\frac{\mathrm{d}{A}_w}{\mathrm{d}t}=v_w(B_{wu}+B_{ww})-{\mu }_a{A}_w-\psi A_w,\end{array}$(1b) ) capture that when the uninfected males mate with uninfected females, they produce uninfected offspring. When infected males mate with uninfected females, then CI causes the embryos to die before hatching. Uninfected males mating with infected females produce a fraction, denoted by $v_w$, of infected offspring by vertical transmission. Cross of infected males with infected females produces a fraction of infected offspring.

Table

Male	Female	Offspring
Uninfected	Uninfected	Uninfected
Uninfected	Infected	Some offspring are infected
Infected	Uninfected	Unviable
Infected	Infected	Some offspring are infected

Figure 2. Bifurcation diagrams for Wolbachia vertical transmission. ${\phi }_u$ and ${\mu }_{fu}$ are varying, and other parameter values are the same as those baseline values in Table 2. Denote the intersection of two endemic equilibria, that is, the intersection of the black dashed line and x-axis, as $R_0^{\ast }=\sqrt{4v_w{v}_u}$. When $R_0<1$ and $v_w<0.5$, no endemic equilibria exist. When $R_0<1$ and $v_w>0.5$, as the vertical transmission rate increases so does $R_0^{\ast }$ and the LAS equilibrium approaches a constant. If we increase the number of infected females, then the EE may become stable EE or CIE If we decrease the number of infected females at EE, then the EE may become DFE. WIF denotes Wolbachia-infected female mosquitoes.

Table 2. Baseline values for parameters in Model (1(1a) $\begin{array}{r}\frac{\mathrm{d}{A}_u}{\mathrm{d}t}=B_{uu}+v_u(B_{wu}+B_{ww})-{\mu }_a{A}_u-\psi A_u,\end{array}$(1a) ).

Parameter	Baseline Value	Ranges	References
$c_i$	1		Assume
$b_f$	0.5	0.34–0.6	[7,29,33]
${\phi }_u$	$50/$day	0–75	[7]
${\phi }_w$	51/day	0–75	[7]
ψ	$0.01/$day		Assume
${\mu }_a$	$0.02/$day		Assume
$v_w$	0.9	0–1	Assume
${\mu }_{fu}$	$0.061/$day	1/55–1/11	[7,31,50]
${\mu }_{fw}$	$0.068/$day	1/55–1/11	[7,31,50]
${\mu }_{mu}$	$0.068/$day	1/31–/7	[7,31,50]
${\mu }_{mw}$	$0.068/$day	1/31–/7	[7,31,50]
$K_a$	100,000		Assume

Table 3. Threshold condition for existence of DFE, EE, and CIE and their stability.

Transmission	DFE	EE	CIE
Complete	$R_0<1$ and $R_{0u}>1$, LAS	$R_0<1$ and $R_{0w}>1$, unstable	$R_{0w}>1$, LAS
Incomplete	$R_0<1$ and $R_{0u}>1$, LAS	$0.5<v_w<1$ and $4v_w(v_w-1)<R_0<1$, when $k=k_1$, LAS, when $k=k_2$, unstable	does not exist

Notes: $R_{0u}$ is the threshold for growth of the Wolbachia-free mosquito population, and $R_{0w}$ is the threshold for growth of the Wolbachia-infected mosquito population. The Wolbachia-free population can grow only when $R_{0u}>1$, and the Wolbachia-infected population can grow only when $R_{0w}>1$.

Table 4. Initial condition thresholds for epidemic to occur with different vertical transmission rates.

Initial condition	$v_w=0.9$	$v_w=0.95$	$v_w=1$
$A_{u0}$	$\le 60\mathrm{\%}{A}_u^0$	$\le 66\mathrm{\%}{A}_u^0$	$\le 72\mathrm{\%}{A}_u^0$
$A_{w0}$	$\ge 40\mathrm{\%}{A}_u^0$	$\ge 34\mathrm{\%}{A}_u^0$	$\ge 28\mathrm{\%}{A}_u^0$
$F_{u0}$	$\le 60\mathrm{\%}{F}_u^0$	$\le 66\mathrm{\%}{F}_u^0$	$\le 72\mathrm{\%}{F}_u^0$
$F_{w0}$	$\ge 40\mathrm{\%}{F}_u^0$	$\ge 34\mathrm{\%}{F}_u^0$	$\ge 28\mathrm{\%}{F}_u^0$
$M_{u0}$	$\le 60\mathrm{\%}{M}_u^0$	$\le 66\mathrm{\%}{M}_u^0$	$\le 72\mathrm{\%}{M}_u^0$
$M_{w0}$	$\ge 40\mathrm{\%}{M}_u^0$	$\ge 34\mathrm{\%}{M}_u^0$	$\ge 28\mathrm{\%}{M}_u^0$

Notes: When the number of Wolbachia-carrying mosquitoes is above the threshold, they can petsit, otherwise, they are wiped out by Wolbachia-free mosquitoes.

Figure 3. Thresholds for fraction of infected individuals vary with reproduction number. $A_{u0}+A_{w0}=A_u^0$, $F_{u0}+F_{w0}=F_u^0$, $M_{u0}+M_{w0}=M_u^0$. When $R_0<1$, the smaller $R_0$ is, the larger number of infected female mosquitoes are needed to be released for Wolbachia to be endemic. The Wolbachia infection is only sustained if the fraction of WIF mosquitoes is above the red dotted line.

Table 5. Different population suppression strategies applied before release of Wolbachia infected female mosquitoes can reduce the minimum number of Wolbachia-infected mosquitoes that can lead to persistence of Wolbachia.

Scenario	Minimum release ratio when $R_0=0.85$	Minimum release ratio when $R_0=0.9$	Minimum release ratio when $R_0=0.95$
DFE	1.007	0.571	0.259
KHA	0.549	0.317	0.151
KHM	0.478	0.244	0.107
KHM2	0.329	0.165	0.072
KHMA	0.210	0.112	0.051

Notes: The top row indicates that you have to release 1.007 times as many wolbachia-infected female mosquitoes as there are wild mosquitoes to establish an infection in the wild that starts at the DFE when $R_0=0.85$. The number of infected mosquitoes that need to be released to establish an infection decreases with the strategies: KHA denotes killing half of the aquatic stage mosquitoes, KHM denotes killing half of adult wild mosquitoes, KHM2 denotes killing half adult of wild mosquitoes once, then kill half adult wild mosquitoes again after two weeks, and finally KHMA denotes killing half adult wild mosquitoes and half aquatic stage of mosquitoes.

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