The role of increased gonotrophic cycles in the establishment of Wolbachia in Anopheles populations

Abstract

Wolbachia, a bacterium that infects insect populations, has been examined extensively in Drosophila populations and, in recent years, has garnered significant attention for its potential to reduce the spread of dengue in the Aedes mosquito population. Similar applications to Anopheles mosquitoes for the reduction of malaria have not been as thoroughly studied, as Anopheles were previously thought to be devoid of Wolbachia infection. The recent discovery, however, of Wolbachia in two separate wild Anopheles populations suggests further study is needed. We develop and analyze an ordinary differential equation model of Wolbachia infection in Anopheles mosquitoes, which demonstrate different reproductive phenotypes than Aedes mosquitoes when infected with Wolbachia. In particular, they do not show the hallmark cytoplasmic incompatibility phenotype—absence of viable offspring when infected males mate with uninfected females—or other standard sex-biasing phenotypes. Instead, evidence of increased speed of gonotrophic cycles by Wolbachia-infected females has been reported. We show that the ability for Wolbachia to invade for a basic reproductive number less than 1 (R_pop < 1), found in other models, is significantly diminished here. However, the invasion threshold below R_pop < 1 can be partially recovered with the increased speed of laying eggs, as incorporated through gonotrophic cycles. Our results highlight the need for further experimental and theoretical work if Wolbachia is to be considered as a form of malaria control.

Appendices

Appendix 1: Analysis of mosquito population lacking Wolbachia infection

The system of equations for mosquito population dynamics where A is the aquatic population, M is the adult males, and G_j are adult females in gonotrophic cycle j is:

$$ \begin{array}{@{}rcl@{}} \frac{dA}{dt} &=& \phi \sum\limits_{i} G_{i} \left( 1- \frac{A}{K_{a}}\right) - \eta A -\mu_{A}A,\\ \frac{dM}{dt} &=& m\eta A -\mu_{M}M,\\ \frac{dG_{1}}{dt} &=& f\eta A - \sigma G_{1} -\mu_{G_{1}}G_{1},\\ \frac{dG_{j}}{dt} &=& \sigma G_{j-1} -\sigma G_{j} - \mu_{G_{j}}G_{j}, \qquad j \in \{2,\cdots,N-1\},\\ \frac{dG_{N}}{dt} &=& \sigma G_{N-1} -\mu_{G_{N}}G_{N}, \end{array} $$

(11)

where ϕ is the per capita egg laying rate of females, K_a is the carrying capacity of aquatic mosquitoes, η is the development rate of aquatic mosquitoes, μ_i is the mortality rate of females in the i th gonotrophic cycle, m (f ) is the fraction of eggs that are male (female), and σ is the transition rate of adult females between gonotrophic cycles.

Epidemiologically and mathematically well-posed. The model for mosquito population dynamics is epidemiologically and mathematically well-posed. To show this, we demonstrate that the region:

$$ \left. \mathcal{D} = \left\{ \left( \begin{array}{cc} A\\ M\\ G_{1} \\ \vdots\\ G_{N} \end{array} \right) \in \mathbb{R}^{N+2} \right| \begin{array}{c} 0 \leq A \leq K_{a},\\ 0 \leq M \leq \frac{m\eta K_{a}}{\mu_{M}},\\ 0 \leq G_{j}, \\ 0 \leq {\sum}_{j} G_{j} \leq \frac{f \eta K_{a}}{\mu_{g}} \end{array} \right\} $$

is invariant under the flow from system (11) with $\mu _{G_{j}}=\mu _{g}$ for all j, assuming that the initial conditions lie in the domain $\mathcal {D}$.

In order to show this, we note that the right-hand sides of all Eqs. 11 have continuous partial derivatives in the specified domain. Along all the edges of the domain, the time derivatives lead the solution back into the invariant domain. We begin by examining when each of the variables is at its minimum:

$$ \begin{array}{@{}rcl@{}} A &=& 0 \implies \frac{dA}{dt}=\phi \sum\limits_{i} G_{i} \geq 0,\\ M &=& 0 \implies \frac{dM}{dt}= m\eta A \geq 0,\\ G_{1} &=& 0 \implies \frac{dG_{1}}{dt} = f \eta A \geq 0,\\ G_{j} &=& 0 \implies \frac{dG_{i}}{dt} = \sigma G_{j-1} \geq 0,\qquad j \in \{2,\cdots, N-1\},\\ G_{N} &=& 0 \implies \frac{dG_{N}}{dt} = \sigma G_{N-1} \geq 0. \end{array} $$

Furthermore, when each variable is at their maximum we see:

$$ \begin{array}{@{}rcl@{}} A = K_{a} &\implies \frac{dA}{dt}= -(\eta+\mu_{A})K_{a} < 0,\\ M = \frac{m \eta K_{a}}{\mu_{M}} &\implies \frac{dM}{dt} = m \eta A - \mu_{M} M \leq m \eta K_{a} - \mu_{M} M = 0,\\ \sum\limits_{i} G_{i} = \frac{f \eta K_{a}}{\mu_{g}}&\implies \sum\limits_{j} \frac{dG_{j}}{dt} = f \eta A - \mu_{g} \sum\limits_{j} G_{j} \\ & \qquad \qquad \qquad \quad \leq f \eta K_{a} - \mu_{g} \sum\limits_{j} G_{j}=0. \end{array} $$

We expect a similar result when relaxing the assumption that all female adult mortality rates are identical with a bit more complicated formula for the maximum of the adult female population.

Equilibria and stability. Assuming mortality is constant in the adult stage, i.e., $\mu _{G_{i}}=\mu _{g}$, the extinction equilibrium given by:

$$ A^{*}=M^{*}=G_{i}^{*}=0 \qquad i \in \{1,\cdots,N\} $$

is locally asymptotically stable when R_pop < 1, with:

$$ R_{\text{pop}} = \frac{f \eta \phi}{\mu_{g}(\eta+\mu_{A})}. $$

The persistent equilibrium given by:

$$ \begin{array}{@{}rcl@{}} A^{*} &=& K_{a} \left( 1-\frac{1}{R_{\text{pop}}}\right),\\ M^{*} &=& \frac{m \eta A^{*}}{\mu_{M}},\\ G_{i}^{*} &=& \frac{f \eta \sigma^{i-1} A^{*}}{(\mu_{g}+\sigma)^{i}},\qquad i \in \{1,\cdots,N-1\}\\ G_{N}^{*} &=& \frac{f \eta \sigma^{N-1} A^{*}}{\mu_{g}(\mu_{g}+\sigma)^{N-1}}. \end{array} $$

is locally asymptotically stable when R_pop > 1.

To show this, we consider the Jacobian matrix of system 11, which is:

$$ J = \left[ \begin{array}{cccccc} -\phi \frac{{\sum}_{i} G_{i}}{K_{a}}-\eta-\mu_{A} & 0 & \phi\left( 1-\frac{A}{K_{a}}\right) &\qquad {\cdots} \qquad & \qquad {\cdots} \qquad &\phi\left( 1-\frac{A}{K_{a}}\right) \\ m\eta & -\mu_{M} & 0 & {\cdots} & {\cdots} &0 \\ f \eta & 0 & -(\sigma+\mu_{g}) & 0 & {\cdots} & 0 \\ 0 & 0 & \sigma & -(\sigma+\mu_{g}) & {\cdots} & 0\\ 0 & 0 & 0 & \sigma & {\cdots} & 0\\ \\ {\vdots} & {\vdots} & {\vdots} & \vdots& \vdots& \vdots\\ \\ 0 & 0 & 0 & {\cdots} & \sigma & \mu_{g} \end{array} \right] $$

with − (σ + μ_g) along the diagonal and σ along the subdiagonal in the part of the matrix that is suppressed. At the extinction equilibrium, this becomes

$$ J = \left[ \begin{array}{cccccc} -\eta-\mu_{A} & 0 & \phi &\qquad {\cdots} \qquad & \qquad {\cdots} \qquad &\phi \\ m\eta & -\mu_{M} & 0 & {\cdots} & {\cdots} &0 \\ f \eta & 0 & -(\sigma+\mu_{g}) & 0 & {\cdots} & 0 \\ 0 & 0 & \sigma & -(\sigma+\mu_{g}) & {\cdots} & 0\\ 0 & 0 & 0 & \sigma & {\cdots} & 0\\ \\ {\vdots} & {\vdots} & {\vdots} & \vdots& \vdots& \vdots\\ \\ 0 & 0 & 0 & {\cdots} & \sigma & \mu_{g} \end{array} \right] $$

with − (σ + μ_g) along the diagonal and σ along the subdiagonal. The eigenvalues of this matrix are:

$$ \begin{array}{@{}rcl@{}} \lambda_{1,\cdots,N-1} &=& -(\sigma+\mu_{g}),\\ \lambda_{N} & =& -\mu_{M},\\ \lambda_{N+1,N+2} &=& \frac{1}{2}\left( -\alpha\pm\sqrt{\alpha^{2}+4\left( f\eta\phi-\mu_{g}(\mu_{A}+\eta)\right)}\right)\\&& \text{ with } \alpha = \eta+\mu_{A}+\mu_{g}. \end{array} $$

When R_pop < 1, then all eigenvalues are negative and the extinction equilibrium is locally asymptotically stable.

The Jacobian matrix at the persistent equilibrium is:

$$ J = \left[ \begin{array}{cccccc} -\phi \frac{{\sum}_{i} G^{*}_{i}}{K_{a}}-\eta-\mu_{A} & 0 & \phi\left( 1-\frac{A^{*}}{K_{a}}\right) &\qquad {\cdots} \qquad & \qquad {\cdots} \qquad &\phi\left( 1-\frac{A^{*}}{K_{a}}\right) \\ m\eta & -\mu_{M} & 0 & {\cdots} & {\cdots} &0 \\ f \eta & 0 & -(\sigma+\mu_{g}) & 0 & {\cdots} & 0 \\ 0 & 0 & \sigma & -(\sigma+\mu_{g}) & {\cdots} & 0\\ 0 & 0 & 0 & \sigma & {\cdots} & 0\\ \\ {\vdots} & {\vdots} & {\vdots} & \vdots& \vdots& \vdots\\ \\ 0 & 0 & 0 & {\cdots} & \sigma & \mu_{g} \end{array} \right] $$

with − (σ + μ_g) along the diagonal and σ along the subdiagonal. The eigenvalues of this matrix are:

$$ \begin{array}{@{}rcl@{}} \lambda_{1,\cdots,N-1} &=& -(\sigma+\mu_{g}),\\ \lambda_{N} & =& -\mu_{M},\\ \lambda_{N+1,N+2} &=& \frac{1}{2\mu_{g}}\left( -\alpha\pm\sqrt{\alpha^{2}-4\mu_{g}\left( f\eta\phi-\mu_{g}(\mu_{A}+\eta)\right)}\right) \text{ with } \alpha ={\mu_{g}^{2}}+f\eta\phi. \end{array} $$

When R_pop > 1, then all eigenvalues are negative and the persistent equilibrium is locally asymptotically stable.

Appendix 2: Next-generation method matrices

In order to reduce the notational complexity, we assume two gonotrophic cycles for each type of adult female, i.e., N_u = 2 and N_w = 2, and that adult female mortality is only dependent on infection status ($\mu _{G_{i}}=\mu _{g}$ and $\mu _{W_{i}}=\mu _{w}$).

Thus, the Jacobian of the matrix of new infections $\mathcal {F}$ is:

$$ \frac{\partial \mathcal{F}}{\partial \mathcal{S}} = \left[ \begin{array}{cccc} 0 & 0 & x & x\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{array} \right] \qquad \text{with} \qquad x = q_{uw}\phi_{w}\left( 1- \frac{A_{u}^{*}}{K_{a}}\right). $$

Here, $A_{u}^{*}$ refers to the uninfected aquatic population at the DFE.

The Jacobian of the matrix of transitions $\mathcal {V}$ is:

$$ \frac{\partial \mathcal{V}}{\partial \mathcal{S}} = \left[ \begin{array}{cccc} \eta_{w}+\mu_{A_{w}} & 0 & 0 & 0\\ -m \eta_{w} & \mu_{M_{w}} & 0 & 0\\ -f \eta_{w} & 0 & \mu_{w}+\sigma_{w} & 0\\ 0 & 0 & \sigma_{w} & \mu_{w} \end{array} \right]. $$

Appendix 3: Comparison with other Wolbachia-infected mosquito models

In similarly formed models of Wolbachia-infected mosquitoes including both male and female adult mosquitoes (Qu et al. 2018; Xue et al. 2017), the authors made simplifying assumptions on the aquatic and male compartments that we do not include in this paper. They assume identical male mortality ($\mu _{M_{u}}=\mu _{M_{w}}$), identical aquatic mortality ($\mu _{A_{u}}=\mu _{A_{w}}$), and identical aquatic maturation (η_u = η_w) between uninfected and infected mosquitoes. In the following, we discuss how our analysis in “Endemic equilibrium” compares with their results.

Under perfect maternal transmission (q = 1), the authors found $\left (\frac {1}{R_{\text {pop}}}-1\right )$ for what we call z, which determines the endemic equilibria. Under their parameter simplifications, then our α = 1, in which case we would also recover $z=\left (\frac {1}{R_{\text {pop}}}-1\right )$.

In the absence of restrictions on maternal transmission 0 ≤ q ≤ 1, the authors found:

$$ \frac{1}{2p}\left( 2q-1 \pm \sqrt{1-\frac{4pq}{R_{\text{pop}}}}\right) $$

for what we call z. Under their parameter simplifications, in addition to our α = 1 and β = 1, then p_wu = 0 as they always work in the setting of full cytoplasmic incompatibility. Using their parameter simplifications, we would recover:

$$ z=\frac{1}{2p}\left( q-p \pm \sqrt{(p+q)^{2}-\frac{4pq}{R_{\text{pop}}}}\right). $$

Noticing that they also assume that q + p = 1, our result would be identical to their solution.

Our formula for the basic reproduction number R_pop is similar to that found by Qu et al. (2018) and Xue et al. (2017), $R_{\text {pop}} = \frac {R_{\text {pop}}^{w}}{R_{\text {pop}}^{u}}$. If we assume the complete cytoplasmic incompatibility, i.e., p_wu = 0 and their other parameter simplifications, then we recoup the same result.

Our model, despite similarity to previous work, retains the flexibility to allow for differences between the uninfected and infected populations in the aquatic and male stages as well as including variable levels of cytoplasmic incompatibility. In addition, we include a variable number of gonotrophic cycles for adult females and variable mortality across adult female stages.