Optimal control approach for establishing wMelPop Wolbachia infection among wild Aedes aegypti populations

Abstract

Wolbachia-based biocontrol has recently emerged as a potential method for prevention and control of dengue and other vector-borne diseases. Major vector species, such as Aedes aegypti females, when deliberately infected with Wolbachia become less capable of getting viral infections and transmitting the virus to human hosts. In this paper, we propose an explicit sex-structured population model that describes an interaction of uninfected (wild) male and female mosquitoes and those deliberately infected with wMelPop strain of Wolbachia in the same locality. This particular strain of Wolbachia is regarded as the best blocker of dengue and other arboviral infections. However, wMelPop strain of Wolbachia also causes the loss of individual fitness in Aedes aegypti mosquitoes. Our model allows for natural introduction of the decision (or control) variable, and we apply the optimal control approach to simulate wMelPop Wolbachia infestation of wild Aedes aegypti populations. The control action consists in continuous periodic releases of mosquitoes previously infected with wMelPop strain of Wolbachia in laboratory conditions. The ultimate purpose of control is to find a tradeoff between reaching the population replacement in minimum time and with minimum cost of the control effort. This approach also allows us to estimate the number of Wolbachia-carrying mosquitoes to be released in day-by-day control action. The proposed method of biological control is safe to human health, does not contaminate the environment, does not make harm to non-target species, and preserves their interaction with mosquitoes in the ecosystem.

Appendices

Appendix A: Meaning of $\mathcal {R}_0$ in Proposition  1

In order to reveal the meaning of threshold $\mathcal {R}_0$ given by formula (18) in Proposition 1, we propose to follow the next generation operator approach attributed to Diekmann et al. (1990), which was further presented by Castillo-Chávez et al. (2002) in a form of comprehensive tutorial.

In fact, since Wolbachia symbiont is maternally transmitted from a female mosquito to her offspring, the next generation produced by this female will be infected with Wolbachia. Thus, Wolbachia infection can be viewed as a maternally transmitted disease among the mosquito population. The steady state $E_n^{\sharp } = \big ( M_n^{\sharp }, F_n^{\sharp }, 0, 0 \big )$ can be regarded as a disease-free equilibrium in epidemiological terminology. The threshold value $\mathcal {R}_0$ that determines whether the epidemics vanishes or spreads is usually referred to as basic reproductive number.²¹ Diekmann et al. (1990) defined $\mathcal {R}_0$ as the spectral radius of the “next generation operator”, while Castillo-Chávez et al. (2002) provide a method to calculate this spectral radius in case of discrete heterogeneity, i. e. when the population of individuals is subdivided into several groups with fixed characteristics. According to this approach, the ODE system (10)–(11) can be re-written in the following form:

$$\begin{aligned} \frac{d \varvec{Y}}{d t}= & {} \varvec{f}(\varvec{Y},\varvec{Z})=\left( G_1 (\varvec{Y},\varvec{Z}), G_2 (\varvec{Y},\varvec{Z}) \right) ^{\prime },\\ \frac{d \varvec{Z}}{d t}= & {} \varvec{g}(\varvec{Y},\varvec{Z}) = \left( G_3(\varvec{Y},\varvec{Z}), G_4 (\varvec{Y},\varvec{Z}) \right) ^{\prime }, \end{aligned}$$

where $\varvec{Y}= \big ( M_n, F_n \big ) \in \mathbb {R}^2_+$ couples the uninfected individuals, while $\varvec{Z}= \big ( M_w, F_w \big )\in \mathbb {R}^2_+$ engages Wolbachia-infected individuals. Since $E_n^{\sharp } = \big ( M_n^{\sharp }, F_n^{\sharp }, 0, 0 \big )= \big ( \varvec{Y}^{\sharp }, \mathbf{0})$ stands for infection-free equilibrium, we have $\varvec{f}(\varvec{Y}^{\sharp },\mathbf{0}) = \varvec{g}(\varvec{Y}^{\sharp },\mathbf{0}) = \mathbf{0}.$ Let

$$\begin{aligned} A = D_{Z} \left( E_n^{\sharp } \right) = \left[ \begin{array}{ll} \dfrac{\partial G_3}{\partial M_w} &{} \dfrac{\partial G_3}{\partial F_w} \\ \dfrac{\partial G_4}{\partial M_w} &{} \dfrac{\partial G_4}{\partial F_w} \end{array} \right] = \left[ \begin{array}{lll} -\mu _w &{} &{} \dfrac{\epsilon _w \rho _w M^{\sharp }_n e^{-\sigma (M^{\sharp }_n + F^{\sharp }_n)}}{M^{\sharp }_n + F^{\sharp }_n} \\ 0 &{} &{} \dfrac{(1-\epsilon _w)\rho _w M^{\sharp }_n e^{-\sigma (M^{\sharp }_n + F^{\sharp }_n)}}{M^{\sharp }_n + F^{\sharp }_n} -\delta _w \\ \end{array} \right] \end{aligned}$$

and further assume that A can be written in the form $A = \mathcal {M} - D$, with $\mathcal {M} \ge 0$ (that is, $m_{ij} \ge 0$) and $D > 0$, a diagonal matrix.²² Under this assumption

$$\begin{aligned} \mathcal {M} = \left[ \begin{array}{lll} 0 &{} &{} \dfrac{\epsilon _w \rho _w M^{\sharp }_n e^{-\sigma (M^{\sharp }_n + F^{\sharp }_n)}}{M^{\sharp }_n + F^{\sharp }_n } \\ 0 &{} &{} \dfrac{(1-\epsilon _w)\rho _w M^{\sharp }_n e^{-\sigma (M^{\sharp }_n + F^{\sharp }_n )}}{M^{\sharp }_n + F^{\sharp }_n } \\ \end{array} \right] , \quad D = \left[ \begin{array}{ll} \mu _w &{} 0 \\ 0 &{} \delta _w \end{array} \right] . \end{aligned}$$

Then, according to Castillo-Chávez et al. (2002), $\mathcal {R}_0$ can be defined as the spectral radius of

$$\begin{aligned} \mathcal {M} D^{-1}=\left[ \begin{array}{lll} 0 &{} &{} \dfrac{\epsilon _w \rho _w M^{\sharp }_n e^{-\sigma (M^{\sharp }_n + F^{\sharp }_n)}}{\delta _w (M^{\sharp }_n + F^{\sharp }_n)} \\ 0 &{} &{} \dfrac{(1-\epsilon _w) \rho _w M^{\sharp }_n e^{ -\sigma (M^{\sharp }_n + F^{\sharp }_n)}}{\delta _w (M^{\sharp }_n + F^{\sharp }_n)} \\ \end{array} \right] . \end{aligned}$$

Namely,

$$\begin{aligned} \mathcal {R}_0 = \varrho \left( \mathcal {M} D^{-1} \right) = \max \left\{ 0,\frac{(1-\epsilon _w) \rho _w M^{\sharp }_n e^{-\sigma (M^{\sharp }_n + F^{\sharp }_n )}}{ \delta _w (M^{\sharp }_n +F^{\sharp }_n )} \right\} \end{aligned}$$

is the dominant eigenvalue of $\mathcal {M} D^{-1}$. By replacing the values of $M_n^{\sharp }$ and $F_n^{\sharp }$ from (14a), we obtain that

$$\begin{aligned} \mathcal {R}_0 = \frac{(1-\epsilon _w)\rho _w /\delta _w}{(1-\epsilon _n)\rho _n /\delta _n} \end{aligned}$$

only depends on female-related parameters. The latter is quite logical since all Wolbachia-infected females are “spreaders” of the disease. In view of conditions (12), we have $\mathcal {R}_0 < 1$ (cf. (18)), that is, the “birth-death ratio” of Wolbachia-infected females (numerator in the right-hand side of (18)) is less than the “birth-death ratio” of uninfected females (denominator in the right-hand side of (18)). In other words, Wolbachia-infected females exhibit reduced individual fitness in comparison to wild females. If we suppose that both infected and uninfected females have the same chances for successful mating that results in viable offspring (i.e., uninfected males and females have sufficiently high frequencies), then at each next generation of mosquitoes there will be a lesser fraction of Wolbachia-infected individuals and a greater fraction of uninfected individuals. Under this scenario, all Wolbachia-carriers will be eventually driven towards extinction, and the disease-free equilibrium $E_n^{\sharp } = \big ( M_n^{\sharp }, F_n^{\sharp }, 0, 0 \big )$ will be reached.

On the other hand, reduced individual fitness of Wolbachia-infected females is recompensed, at high infection frequencies, by their reproductive advantage derived from CI-phenotype. The latter is explained by Proposition 1.

Hypothetically speaking, if we disregard the conditions (12) and suppose that Wolbachia does not alter neither fecundity nor longevity of the host (that is, $\epsilon _n=\epsilon _w, \rho _n=\rho _w, \mu _n = \mu _w, \delta _n=\delta _w$ so that $\mathcal {R}_0 = 1$), then Wolbachia invasion could be induced by a single release of very small number of Wolbachia-carriers, and this “imaginary” scenario also agrees with our model (10)–(11), as well as with the principle of competitive exclusion.

Appendix B: Technical note on GPOPS-II solver for MATLAB platform

Generally speaking, there are two groups of collocation techniques — the direct collocation and the orthogonal collocation. Under the direct collocation approach, the state and adjoint trajectories of the optimality system are discretized at a set of appropriately chosen grid of nodes (that is, collocation points) in the fixed time interval $[t_0, t_f]$. Then the state and adjoint trajectories are iteratively approximated using the same fixed-degree polynomials (usually, cubic splines) at all subinterval of the grid in order to satisfy the differential constraints (i.e., the discretized ODE system), while the boundary constraints are taken into account at each iteration. Convergence of all methods based on direct collocation is usually achieved by increasing the number of collocation points.

On the other hand, the orthogonal collocation is performed over entire time interval $[t_0, t_f]$ where the collocation points are usually associated with Gaussian quadrature, i.e. they are roots of some orthogonal polynomial, or a linear combination of orthogonal polynomials and its derivatives. Under this approach, the state and adjoint variables of the continuous-time optimality system (as well as their derivatives) are approximated by using Lagrange interpolating polynomials over $[t_0, t_f]$ supported by the collocation points. Convergence of this method can be achieved by increasing the degree of the polynomial approximation.

The GPOPS-II solver implements an adaptive combination of two collocation techniques described above which is also known as Radau pseudospectral method. According to Garg et al. (2009), the Radau pseudospectral method is capable of dealing with free initial $t_0$ or final $t_f$ time since the input time interval $t \in [t_0, t_f]$ (which is in our case is $t \in [0,T]$ with T left free) should be transformed into $\tau \in [-1, 1]$ (with fixed endpoints!) using the affine transformation

$$\begin{aligned} \tau = \frac{2 }{t_f - t_0} \ t - \frac{t_f + t_0}{t_f - t_0}. \end{aligned}$$

In particular, this method uses the Legendre-Gauss-Radau set²³ of orthogonal collocation points what basically explains its name.

Another advantage of Radau pseudospectral method, besides its adaptiveness, is that this method (in contrast to other pseudospectral methods) allows to use the exact formulas for first- and second-order partial derivatives of problem entries and, thus, to solve more accurately the nonlinear programming problems resulting from discretization.

The main limitation of GPOPS-II package is that it requires the continuity of the first- and second-order derivatives of the Hamiltonian and endpoint constraints with respect to all variables. However, the models considered in this paper meet this condition.

Further and more detailed information regarding the GPOPS-II package, as well as some comprehensive examples, can be consulted in the paper by Patterson and Rao (2014).