Implicit versus explicit control strategies in models for vector-borne disease epidemiology

Abstract

Throughout the vector-borne disease modeling literature, there exist two general frameworks for incorporating vector management strategies (e.g. area-wide adulticide spraying and larval source reduction campaigns) into vector population models, namely, the "implicit" and "explicit" control frameworks. The more simplistic "implicit" framework facilitates derivation of mathematically rigorous results on disease suppression and optimal control, but the biological connection of these results to realworld "explicit" control actions that could guide specific management actions is vague at best. Here, we formally define the biological and mathematical relationships between implicit and explicit control, and we provide detailed mathematical expressions relating the strength of implicit control to management-relevant properties of explicit control for four common intervention strategies. These expressions allow optimal control and sensitivity analysis results in existing implicit control studies to be interpreted in terms of real world actions. Our work reveals a previously unknown fact: implicit control is a meaningful approximation of explicit control only when resonance-like synergistic effects between multiple controls have a negligible effect on average population reduction. When non-negligible synergy exists, implicit control results, despite their mathematical tidiness, fail to provide accurate predictions regarding vector control and disease spread. The methodology we establish can be applied to study the interaction of phenological effects with control strategies, and we present a new technique for finding impulse control strategies that optimally reduce a vector population in the presence of seasonally oscillating model parameters. Collectively, these elements build an effective bridge between analytically interesting and mathematically tractable implicit control and the challenging, action-oriented explicit control.

Managing disease outbreaks: The importance of vector mobility and spatially heterogeneous control

“…Controls incorporated in this manner are simple but effective models for describing the effects of regularly repeated fixed strategy real-world controls (e.g. an aerial spraying campaign in which airplanes deploy a fixed amount of pesticide repeatedly according to a fixed schedule, or a door-to-door control campaign in which employees visit residences repeatedly according to a fixed schedule) [28]. Under this modeling methodology, the optimal control protocols we find for reducing the basic reproduction number will be naturally phrased in terms of actionable control advice based on real-world control parameters.…”

“…Our model does not contain a larval class, so larvicide controls for both area-wide spraying and door-to-door strategies applied in conjunction with larval source reduction controls are outside the scope of our work (see Ref. [28] for a detailed explanation), and are not considered in this paper. We will therefore always have Λ 0 = Λ N , but we retain the above notation to emphasize the structure of compliant and non-compliant sites.…”

“…In this paper, we apply the control methodology established in Ref. [28]; the modified death and emergence rates in our model describe the time-averaged effects of real-word vector control strategies applied repeatedly at fixed frequencies, where potential resonance-like synergistic effects arising from the interaction and relative timing of controls are ignored. Results obtained from this control modeling methodology are intended to give public health workers a basic intuitive sense of approximately how often different classes of controls need to be applied in order to suppress vector-borne disease outbreaks in a neighborhood setting.…”

“…In contrast, this paper utilizes a hybrid approach to control modeling introduced in Ref. [28]. Specifically, we incorporate control through its overall gross effects on model parameters, where the degrees of control's influence are given as functions of real-world experimentally measurable control parameters, costs, and impulse application frequencies, and we quantify intervention efficacy through the basic reproduction number to determine actionable control advice in the form of cost-optimal impulse application schedules.…”