Optimal control approaches for combining medicines and mosquito control in tackling dengue

Dengue is a debilitating and devastating viral infection spread by mosquito vectors, and over half the world’s population currently live at risk of dengue (and other flavivirus) infections. Here, we use an integrated epidemiological and vector ecology framework to predict optimal approaches for tackling dengue. Our aim is to investigate how vector control and/or vaccination strategies can be best combined and implemented for dengue disease control on small networks, and whether these optimal strategies differ under different circumstances. We show that a combination of vaccination programmes and the release of genetically modified self-limiting mosquitoes (comparable to sterile insect approaches) is always considered the most beneficial strategy for reducing the number of infected individuals, owing to both methods having differing impacts on the underlying disease dynamics. Additionally, depending on the impact of human movement on the disease dynamics, the optimal way to combat the spread of dengue is to focus prevention efforts on large population centres. Using mathematical frameworks, such as optimal control, are essential in developing predictive management and mitigation strategies for dengue disease control.

be externally controlled, for example in drug-interaction simulations the amount of drug administered is directly controlled. We call these control variables, and those that cannot be influenced are referred to as state variables. The goal of optimal control is then to determine, mathematically, the best choice of values for these control variables over time to achieve a certain aim. In the pharmacological example, one may wish to estimate the optimal amount of a drug administered to minimise tumour size while also minimising adverse side-effects [1].
Unsurprisingly, disease dynamics are readily amenable to optimal control problems; often the key epidemiological question is -what is the optimal choice of disease intervention measures (vaccines, vector-control methods) to minimise the number of infected individuals while also trying to minimise the economic cost of such measures. Traditional SIR networks are well formulated for optimal control approaches, and are commonly used in identifying optimal vaccination programmes [2], often in parallel with other methods and/or treatments [3]. One simple example is presented by Yusuf & Benyah (2012) [4].
To begin we define a system of ODEs to describe the behaviour of these state variables x(t) that depends on the control variables u(t): for some function g. We can also express, mathematically, what is wished to be minimised or maximised.
This will be some expression of our state variables and control variables, as we will want to minimise a state (for example, the number of infected hosts), while also not letting the amount of control applied balloon beyond what is feasibly manageable. We express this desire as wishing to minimise (or perhaps maximise), for some function f . Here t 0 and t 1 represent the start and finish of the time period we consider. Again we seek a particular optimal control variable, u * (t), and its corresponding optimal state, x * (t), that will minimise or maximise J.
The key result on which all of optimal control theory is built on is known as Pontryagin's Maximum We consider the Hamiltonian (H) as,

H(t, x(t), u(t), λ(t)) = f (t, x(t), u(t)) + λ(t)g(t, x(t), u(t)),
where f and g are as defined above and λ(t) is a piecewise differentiable function known as the adjoint variable. Pontryagin's Maximum Principle then states that, if u * (t) and x * (t) are optimal such that they maximise (or minimise) J(u), then there exists a particular adjoint variable λ(t) such that, for all control variables u for all times t, if λ(t) is such that, Unsurprisingly this critical point is the optimal maximum. Note that the mathematical explanation follows the exact same procedure for minimisation problems, resulting only in an opposite inequality. As it is an optimum point, it is also the case that ∂H ∂u = 0 at u * . We formally present the three conditions that an optimal problem satisfies, λ(t 1 ) = 0 (Transversality Condition).
For very simple problems this can be solved analytically, however for more realistic, and complex, problem formulations, the solution must be found numerically, via an iterative method. In general we consider some initial states, u 0 , x 0 and λ 0 . Usually we pick the zero function for all three as a starting point, but enforce the initial condition of the system x(0) on x 0 . Then take the following steps.
• Solve the system of equations dx 1 (t) dt = g(t, x 0 (t), u 0 (t)) using the initial condition of the system to obtain a new x 1 .
• Using the optimality condition ∂H ∂u = 0, u can be re-arranged as an expression of x and λ. Use this expression to build a new u 1 from x 1 and λ 1 .
• Repeat these steps until a desired point of convergence.