Resource Allocation for Epidemic Control Across Multiple Sub-populations

Bulletin of Mathematical Biology https://doi.org/10.1007/s11538-019-00584-2 Resource Allocation for Epidemic Control Across Multiple Sub-populations Ciara E. Dangerfield1 · Martin Vyska1 · Christopher A. Gilligan1 Received: 2 August 2018 / Accepted: 10 February 2019 © The Author(s) 2019 Abstract The number of pathogenic threats to plant, animal and human health is increasing. Controlling the spread of such threats is costly and often resources are limited. A key challenge facing decision makers is how to allocate resources to control the different threats in order to achieve the least amount of damage from the collective impact. In this paper we consider the allocation of limited resources across n independent target populations to treat pathogens whose spread is modelled using the susceptible– infected–susceptible model. Using mathematical analysis of the systems dynamics, we show that for effective disease control, with a limited budget, treatment should be focused on a subset of populations, rather than attempting to treat all populations less intensively. The choice of populations to treat can be approximated by a knapsacktype problem. We show that the knapsack closely approximates the exact optimum and greatly outperforms a number of simpler strategies. A key advantage of the knapsack approximation is that it provides insight into the way in which the economic and epidemiological dynamics affect the optimal allocation of resources. In particular using the knapsack approximation to apportion control takes into account two important aspects of the dynamics: the indirect interaction between the populations due to the shared pool of limited resources and the dependence on the initial conditions. Keywords Epidemiological modelling · Optimal control of epidemics · Metapopulation model Ciara E. Dangerfield and Martin Vyska contributed equally to this work. B Ciara E. Dangerfield 1 Department of Plant Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EA, UK 123 C. E. Dangerfield et al. 1 Introduction The infection burden of many epidemics outstrips the resources available to treat all individuals (Lipsitch et al. 2000; Kiszewski et al. 2007). Furthermore, characteristics of disease spread may differ between different groups of the populations. The challenge facing central decision makers who seek to control an epidemic at the landscape scale is therefore how to allocate limited resources in order to minimise the impacts of disease across the entire population? Optimising the deployment of control requires consideration of both epidemic dynamics and economic factors, including the costs of the epidemic and control as well as budgetary constraints and availability of resources. Previous studies have used control theory to determine the optimal allocation of limited resources to minimise the impacts from an epidemic (Rowthorn et al. 2009; Ndeffo Mbah and Gilligan 2011; Zaric and Brandeau 2001a, b; Brandeau et al. 2003; Hansen and Day 2011; Zhou et al. 2014). For simplicity, many early studies considered the application of a single control within a single target population (Hansen and Day 2011; Zhou et al. 2014). However, heterogeneities in the host population are known to be important in the invasion and persistence of human, animal and plant pathogens (Ferguson et al. 2001; Dye and Gay 2003; Stacey et al. 2004). Within human populations heterogeneities arise, for example through different contact patterns amongst sub-populations (Wallinga et al. 1999). For animal and plant pathogens, it is often the spatial structure that is critical in the invasion and persistence of the pathogen (Ferguson et al. 2001; Stacey et al. 2004; Keeling et al. 2001). Such heterogeneities in the characteristics related to epidemic spread amongst sub-populations of the host population are typically captured using structured metapopulations (Grenfell and Bolker 1998, 2000). Rowthorn et al. (2009) consider the optimal deployment of limited resources across two different but interconnected regions of equal size. Minimising the discounted number of infected individuals over a fixed time horizon within the susceptible–infected–susceptible (SIS) model, Rowthorn et al. (2009) find an arguably counterintuitive result that treatment should be preferentially directed at the sub-population with the lowest number of infected individuals. The inclusion of temporary immunity, essentially extending from an SIS to an SIRS model, alters the optimal strategy whereby it is initially optimal to preferentially treat the more infected sub-population and then switch to treating the less infected sub-population (Ndeffo Mbah and Gilligan 2011). The limitation of the studies by Rowthorn et al. (2009) and Ndeffo Mbah and Gilligan (2011) is that they only consider two sub-populations, but in reality a larger number of sub-populations is often needed to capture the heterogeneities within a target population. A key goal of the current work is to extend the work of Rowthorn et al. (2009); Ndeffo Mbah and Gilligan (2011) to the problem of n ≥ 2 sub-populations. Work on the allocation of resources between two sub-populations uses an optimisation approach based on the Hamiltonian method (Rowthorn et al. 2009) and the Pontryagin maximum principle (Ndeffo Mbah and Gilligan 2011), which provides analytic insight into the form of the optimal allocation strategy. Extending this approach to the general problem of n populations leads to a large number of equations that cannot be solved analytically. Indeed, Zaric and Brandeau (2001b) show that the general problem of the allocation of limited resource across n coupled sub-populations is 123 Resource Allocation for Epidemic Control Across Multiple… intractable. Therefore, numerical techniques are typically used to solve such problems as in Richter et al. (1999), Zaric and Brandeau (2001a), Zaric and Brandeau (2001b) and Brandeau et al. (2003). However, numerical approaches lose the intuitive insight that analytic approaches provide about underlying mechanisms. The loss of intuitive insight limits the use of optimal control methods in determining generalisable rules and simple heuristics that can be used by decision makers. Indeed, Brandeau et al. (2003) identify the need for simple, easy to use guidelines based on the optimal solution in order to make practical use of optimal control theory by decision makers. The challenge therefore is how to generalise the results from Rowthorn et al. (2009) and Ndeffo Mbah and Gilligan (2011) to the case of n ≥ 2 sub-populations. The primary goal of this paper is accordingly to provide insight into the general form of the optimal allocation of treatment across n sub-populations when the resources available for control are limited. In particular, we seek to answer the following questions: – How do epidemiological dynamics and economic constraints impact the optimal allocation of resources across n sub-populations? – How d (...truncated)