Ensuring successful introduction of Wolbachia in natural populations of Aedes aegypti by means of feedback control

Abstract

The control of the spread of dengue fever by introduction of the intracellular parasitic bacterium Wolbachia in populations of the vector Aedes aegypti, is presently one of the most promising tools for eliminating dengue, in the absence of an efficient vaccine. The success of this operation requires locally careful planning to determine the adequate number of individuals carrying the Wolbachia parasite that need to be introduced into the natural population. The introduced mosquitoes are expected to eventually replace the Wolbachia-free population and guarantee permanent protection against the transmission of dengue to human. In this study, we propose and analyze a model describing the fundamental aspects of the competition between mosquitoes carrying Wolbachia and mosquitoes free of the parasite. We then use feedback control techniques to devise an introduction protocol that is proved to guarantee that the population converges to a stable equilibrium where the totality of mosquitoes carry Wolbachia.

Appendices

Appendix A: A sexual version of the infestation model

We here provide a sexual version of model (1). For simplicity, no control input is written. We denote respectively ${\mathbf {m}}_U, {\mathbf {m}}_W, {\mathbf {M}}_U, {\mathbf {M}}_W$, the numbers of uninfected, resp. Wolbachia-infected, males in early and adult phases; and similarly ${\mathbf {f}}_U, {\mathbf {f}}_W, {\mathbf {F}}_U, {\mathbf {F}}_W$ for the females. The model is:

$$\begin{aligned} \dot{{\mathbf {m}}}_U= & {} \lambda _U\alpha _U \frac{{\mathbf {M}}_U}{{\mathbf {M}}_U+{\mathbf {M}}_W} {\mathbf {F}}_U -\nu {\mathbf {m}}_U \nonumber \\&-\,\mu (1+k ({\mathbf {m}}_W+{\mathbf {m}}_U+{\mathbf {f}}_W+{\mathbf {f}}_U)){\mathbf {m}}_U \end{aligned}$$

(39a)

$$\begin{aligned} \dot{{\mathbf {M}}}_U= & {} \nu {\mathbf {m}}_U -\mu _U {\mathbf {M}}_U\end{aligned}$$

(39b)

$$\begin{aligned} \dot{{\mathbf {m}}}_W= & {} \lambda _W\alpha _W {\mathbf {F}}_W -\nu {\mathbf {m}}_W - \mu (1+k ({\mathbf {m}}_W+{\mathbf {m}}_U+{\mathbf {f}}_W+{\mathbf {f}}_U)){\mathbf {m}}_W\end{aligned}$$

(39c)

$$\begin{aligned} \dot{{\mathbf {M}}}_W= & {} \nu {\mathbf {m}}_W -\mu _W {\mathbf {M}}_W\end{aligned}$$

(39d)

$$\begin{aligned} \dot{{\mathbf {f}}}_U= & {} \alpha _U \frac{{\mathbf {M}}_U}{{\mathbf {M}}_U+{\mathbf {M}}_W}{\mathbf {F}}_U -\nu {\mathbf {f}}_U - \mu (1+k ({\mathbf {m}}_W+{\mathbf {m}}_U+{\mathbf {f}}_W+{\mathbf {f}}_U)){\mathbf {f}}_U \end{aligned}$$

(39e)

$$\begin{aligned} \dot{{\mathbf {F}}}_U= & {} \nu {\mathbf {f}}_U -\mu _U {\mathbf {F}}_U\end{aligned}$$

(39f)

$$\begin{aligned} \dot{{\mathbf {f}}}_W= & {} \alpha _W {\mathbf {F}}_W -\nu {\mathbf {f}}_W - \mu (1+k ({\mathbf {m}}_W+{\mathbf {m}}_U+{\mathbf {f}}_W+{\mathbf {f}}_U)){\mathbf {f}}_W\end{aligned}$$

(39g)

$$\begin{aligned} \dot{{\mathbf {F}}}_W= & {} \nu {\mathbf {f}}_W -\mu _W {\mathbf {F}}_W \end{aligned}$$

(39h)

Here $\lambda _U, \lambda _W$ are the sex ratio (ratio of males to females) of the offspring for the uninfected and infected populations. The other parameters have the same meaning than for model (1) (see Table 2). Here they have been chosen identical for both sex, and in such conditions, it is straightforward to see that the variables defined by

$$\begin{aligned} {\mathbf {L}}_U:= {\mathbf {m}}_U+ {\mathbf {f}}_U,\ {\mathbf {L}}_W:= {\mathbf {m}}_W+ {\mathbf {f}}_W,\ {\mathbf {A}}_U:= {\mathbf {M}}_U+ {\mathbf {F}}_U,\ {\mathbf {A}}_W:= {\mathbf {M}}_W+ {\mathbf {F}}_W \end{aligned}$$

(40)

obey the following equations:

$$\begin{aligned} \dot{{\mathbf {L}}}_U= & {} (1+\lambda _U)\alpha _U \frac{{\mathbf {M}}_U}{{\mathbf {M}}_U+{\mathbf {M}}_W}{\mathbf {F}}_U -\nu {\mathbf {L}}_U - \mu (1+k ({\mathbf {L}}_W+{\mathbf {L}}_U)){\mathbf {L}}_U \end{aligned}$$

(41a)

$$\begin{aligned} \dot{{\mathbf {A}}}_U= & {} \nu {\mathbf {L}}_U -\mu _U {\mathbf {A}}_U \end{aligned}$$

(41b)

$$\begin{aligned} \dot{{\mathbf {L}}}_W= & {} (1+\lambda _W) \alpha _W {\mathbf {F}}_W -\nu {\mathbf {L}}_W - \mu (1+k ({\mathbf {L}}_W+{\mathbf {L}}_U)){\mathbf {L}}_W \end{aligned}$$

(41c)

$$\begin{aligned} \dot{{\mathbf {A}}}_W= & {} \nu {\mathbf {L}}_W -\mu _W {\mathbf {A}}_W \end{aligned}$$

(41d)

If moreover $\lambda _U=\lambda _W$ and the sex ratio are initially equal to this common value, that is:

$$\begin{aligned} \frac{{\mathbf {m}}_U(0)}{{\mathbf {f}}_U(0)} = \frac{{\mathbf {M}}_U(0)}{{\mathbf {F}}_U(0)} = \lambda _U,\quad \frac{{\mathbf {m}}_W(0)}{{\mathbf {f}}_W(0)} = \frac{{\mathbf {M}}_W(0)}{{\mathbf {F}}_W(0)} = \lambda _W \end{aligned}$$

(42)

then the same proportions are conserved along the evolution, and it is possible to replace $\frac{{\mathbf {M}}_U}{{\mathbf {M}}_U+{\mathbf {M}}_W}$ by $\frac{{\mathbf {L}}_U}{{\mathbf {L}}_U+{\mathbf {L}}_W}, (1+\lambda _U) {\mathbf {F}}_U$ by ${\mathbf {L}}_U$ and $(1+\lambda _W) {\mathbf {F}}_W$ by ${\mathbf {L}}_W$ in (41a), (41c), showing that (41) boils down to the simpler model (1).

Appendix B: Proof of Theorem 7

1.1 Computation and ordering of the equilibrium points

One here computes the equilibrium points. The latter verify

$$\begin{aligned}&\gamma _U\mathcal{R}_U \frac{ A_U}{ A_U+ A_W} A_U - (1+ L_W+ L_U) L_U = 0 \end{aligned}$$

(43a)

$$\begin{aligned}&\gamma _W \mathcal{R}_W A_W - (1+ L_W+ L_U) L_W =0 \end{aligned}$$

(43b)

$$\begin{aligned}&L_U = \gamma _U A_U,\quad L_W =\gamma _W A_W \end{aligned}$$

(43c)

The point $x_{0,0} := (0,0,0,0)$ is clearly an equilibrium. Let us look for an equilibrium $x_{U,0} := (L_U^*,A_U^*,0,0)$. The quantities $L_U^*,A_U^*$ then have to satisfy

$$\begin{aligned} \gamma _U\mathcal{R}_U A_U^* - \left( 1 + L_U^*\right) L_U^* =0,\quad L_U^* = \gamma _U A_U^*. \end{aligned}$$

(44)

Dividing by $L_U^*\ne 0$ yields $1+ L_U^* = \mathcal{R}_U$. One thus gets the unique solution of this form verifying

$$\begin{aligned} L_U^* = \mathcal{R}_U-1,\quad A_U^* = \frac{\mathcal{R}_U-1}{\gamma _U}, \end{aligned}$$

which is positive due to the sustainability hypothesis (6).

Similarly, one now looks for an equilibrium defined as $x_{0,W} := (0,0,L_W^*,A_W^*)$. The values of $L_W^*,A_W^*$ must verify

$$\begin{aligned} \gamma _W \mathcal{R}_W A_W^* - \left( 1+ L_W^*\right) L_W^* =0,\quad L_W^* = \gamma _W A_W^*. \end{aligned}$$

This is identical to (44), and as for the $x_{U,0}$ case, one gets a unique, positive, solution, namely

$$\begin{aligned} L_W^* = \mathcal{R}_W-1,\quad A_W^* = \frac{\mathcal{R}_W-1}{\gamma _W}. \end{aligned}$$

(45)

We show now that system (7) also admits a unique coexistence equilibrium with positive components $x_{U,W}= (L_U^{**}, A_U^{**}, L_W^{**}, A_W^{**})$. Coming back to (45) and expressing the value of the factor common to the first and second identity leads to

$$\begin{aligned} 1+L_U^{**}+L_W^{**}= & {} \gamma _W\mathcal{R}_W\frac{A_W^{**}}{L_W^{**}} = \mathcal{R}_W\\= & {} \gamma _U\mathcal{R}_U\frac{A_U^{**}}{A_U^{**}+A_W^{**}}\frac{A_U^{**}}{L_U^{**}}\\= & {} \mathcal{R}_U\frac{A_U^{**}}{A_U^{**}+A_W^{**}} \end{aligned}$$

One thus deduces

$$\begin{aligned} \frac{A_U^{**}}{A_U^{**}+A_W^{**}} = \frac{\mathcal{R}_W}{\mathcal{R}_U}, \end{aligned}$$

and one can express all three remaining unknowns in function of $A_W^{**}$:

$$\begin{aligned} L_W^{**} = \gamma _W A_W^{**},\quad A_U^{**} = \frac{\mathcal{R}_W}{\mathcal{R}_U-\mathcal{R}_W} A_W^{**},\quad L_U^{**} = \gamma _U A_U^{**} = \gamma _U\frac{\mathcal{R}_W}{\mathcal{R}_U-\mathcal{R}_W} A_W^{**}. \end{aligned}$$

Using the value of $L_U^{**}$ and $L_W^{**}$ now yields the relation

$$\begin{aligned} \mathcal{R}_W -1 = L_U^{**}+L_W^{**} = \gamma _W\left( 1+ \frac{\gamma _U}{\gamma _W}\frac{\mathcal{R}_W}{\mathcal{R}_U-\mathcal{R}_W} \right) A_W^{**}, \end{aligned}$$

which has a unique, positive, solution when (6) holds. Hence, the fourth equilibrium is given by

$$\begin{aligned} L_U^{**} = \frac{\delta }{1+\delta }(\mathcal{R}_W-1),\quad A_U^{**} = \frac{\delta }{(1+\delta )\gamma _U}(\mathcal{R}_W-1)\\ L_W^{**} = \frac{1}{1+\delta }(\mathcal{R}_W-1),\quad A_W^{**} = \frac{1}{(1+\delta )\gamma _W}(\mathcal{R}_W-1) \end{aligned}$$

where $\delta$ was given in Eq. (14d) in the statement of the theorem.

So far we have found all equilibrium points. Actually, it is easy to see that no equilibria is missing: for $L_U= 0$ we necessarily have $A_U=0,$ and this gives us $x_{0,0}$ and $x_{0,W},$ while for $L_U \ne 0$ we get $A_U\ne 0,$ and this leads us to $x_{U,0}$ and to $x_{U,W}.$

Notice that the last equilibrium can be expressed alternatively by use of the values of the equilibrium $x_{0,W}$:

$$\begin{aligned} L_U^{**}= & {} \frac{\delta }{1+\delta }L_W^*,\quad A_U^{**} = \frac{\delta }{1+\delta }\frac{\gamma _W}{\gamma _U}A_W^* \end{aligned}$$

(46a)

$$\begin{aligned} L_W^{**}= & {} \frac{1}{1+\delta }L_W^*,\quad A_W^{**} = \frac{1}{1+\delta }A_W^* \end{aligned}$$

(46b)

and this provides straightforward comparison result:

$$\begin{aligned} L_U^{**}< L_W^*<L_U^* \quad \text { and } \quad L_W^{**}< L_W^* <L_U^* \end{aligned}$$

(47a)

and thus $A_\eta ^{**} = \gamma _\eta L_\eta ^{**} < \gamma _\eta L_\eta ^* = A_\eta ^*$, for $\eta \in \{U,W\}$ and, therefore,

$$\begin{aligned} A_U^{**}<A_U^*\quad \text { and }\quad A_W^{**} < A_W^*, \end{aligned}$$

(47b)

the second inequality being directly deduced from (46b). The relations (50) allow us to establish the inequalities (15).

1.2 Local stability analysis

The local stability analysis is conducted through analysis of the eigenvalues of the Jacobian matrices. Recall that the Jacobian has been computed in (13).

Stability of $x_{0,0}$. The value of ${ Df}$ is not defined at $x_{0,0}$. To show the instability of the equilibrium $x_{0,0}$, let the function V be defined on $\mathbb {R}_+^4$ by

$$\begin{aligned} V(x):= \rho (L_U+(1+\varepsilon ) A_U) + L_W+(1+\varepsilon )A_W, \end{aligned}$$

for values of $\varepsilon , \rho$ still to be defined. Notice that V is positive definite for any positive $\varepsilon$ and $\rho .$ We will show that there exists $\rho >0$ for which the derivative ${\dot{V}}$ of V along the trajectories is positive definite in a sufficiently small relative neighborhood of $x_{0,0}$ in $\mathbb {R}_+^4.$

One can check that

$$\begin{aligned} \dot{V}(x)= & {} (\varepsilon - L_U-L_W) (\rho L_U+ L_W)\nonumber \\&+\,\rho \gamma _U\left( \mathcal{R}_U\frac{A_U}{A_U+A_W}-1-\varepsilon \right) A_U\nonumber \\&+\,\gamma _W(\mathcal{R}_W-1-\varepsilon )A_W. \end{aligned}$$

(48)

The first term of the last expression is positive for all values of $(L_U,L_W)$ in some relative neighborhood of the origin in $\mathbb {R}_+^2$. Assuming from now on that $\varepsilon \in (0,\mathcal{R}_W-1)$, one verifies easily that the sum of the two remaining terms in the right-hand side of (48) is positive when exactly one of the two numbers $A_U, A_W$ is zero, due to (6). Assume now that e.g. $A_U\ne 0$. Then the sum of the last two terms in (48) is equal to

$$\begin{aligned} \gamma _WA_U\left( b \left( \mathcal{R}_U \frac{1}{1+a}-1-\varepsilon \right) +(\mathcal{R}_W-1-\varepsilon ) a \right) ,\quad a:=\frac{A_W}{A_U},\ b:= \frac{\rho \gamma _U}{\gamma _W}. \end{aligned}$$

We will now prove that there exists $b>0$ (and therefore $\rho >0$) such that the previous expression is positive for any nonnegative a (and therefore for any pair $(A_U,A_W)$ with $A_U>0, A_W\ge 0$).

The map

$$\begin{aligned} a \mapsto b \left( \mathcal{R}_U \frac{1}{1+a}-1-\varepsilon \right) +(\mathcal{R}_W-1-\varepsilon ) a \end{aligned}$$

(49)

is clearly convex. It has a positive value at the origin, where its derivative is equal to $-b\mathcal{R}_U+\mathcal{R}_W-1-\varepsilon$. Taking now $0<b<\frac{\mathcal{R}_W-1-\varepsilon }{\mathcal{R}_U}$, this expression is positive, which ensures that the map (49) takes on positive values on $\mathbb {R}^+$. Set $\rho :=\frac{b\gamma _W}{\gamma _U}.$

Therefore, positive values of $\varepsilon$ and $\rho$ have been exhibited, for which ${\dot{V}}$ is positive definite in a relative neighborhood of $x_{0,0}$ in $\mathbb {R}_+^4.$ This demonstrates the instability of $x_{0.0}.$

Stability of $x_{U,0}$. Using (13) and recalling the value of $x_{U,0}$ given in (16), we see that ${ Df}(x_{U,0})$ is the upper block-triangular matrix

$$\begin{aligned} \begin{pmatrix} 1-2\mathcal{R}_U &{}\quad \gamma _U\mathcal{R}_U &{}\quad 1-\mathcal{R}_U &{}\quad -\gamma _U\mathcal{R}_U\\ 1 &{}\quad -\gamma _U &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad -\mathcal{R}_U &{}\quad \gamma _W\mathcal{R}_W\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad -\gamma _W \end{pmatrix}. \end{aligned}$$

The eigenvalues of this block-triangular matrix have negative real parts if and only if

$$\begin{aligned} \frac{\mathcal{R}_U}{2\mathcal{R}_U-1}<1 \quad \text { and } \quad \mathcal{R}_W<\mathcal{R}_U. \end{aligned}$$

These conditions are satisfied since the sustainability condition (6) holds. In conclusion, the equilibrium $x_{U,0}$ is locally asymptotically stable.

Stability of $x_{0,W}$. From (13) and (16) we get that the Jacobian ${ Df}(x_{0,W})$ of f at $x_{0,W}$ is the lower block-triangular matrix

$$\begin{aligned} \begin{pmatrix} -\mathcal{R}_W &{} 0 &{} 0 &{} 0\\ 1 &{} -\gamma _U &{} 0 &{} 0\\ 1-\mathcal{R}_W &{} 0 &{} 1-2\mathcal{R}_W &{} \gamma _W\mathcal{R}_W\\ 0 &{} 0 &{} 1 &{} -\gamma _W \end{pmatrix}. \end{aligned}$$

(50)

The left-upper $2\times 2$-block is a Hurwitz matrix, while asymptotic stability of the second one is equivalent to the condition

$$\begin{aligned} 2\mathcal{R}_W-1 > \mathcal{R}_W, \end{aligned}$$

that is $\mathcal{R}_W>1$, which holds true, due to hypothesis (6). The equilibrium $x_{0,W}$ is thus locally asymptotically stable.

Stability of $x_{U,W}$. The instability of $x_{U,W}$ can be proved by showing that the determinant of the Jacobian matrix ${ Df}(x_{U,W})$ is negative, which, together with the fact that the state space has even dimension 4, establishes the existence of a positive real root to the characteristic polynomial; and thus that the Jacobian is not a Hurwitz matrix. This argument yields lengthy computations.

It is more appropriate to use here the monotonicity properties of system (7), established in Theorem 5. As a matter of fact, bringing together the inequalities (15) (already proved in the end of the previous section, see (50)), the asymptotical stability of $x_{U,0}$ and $x_{0,W}$ and the strongly order-preserving property of the reference problem, Theorem 2.2 in Smith (1995) shows that the intermediate point $x_{U,W}$ cannot be stable. This finally achieves the stability analysis, as well as the proof of Theorem 7.