Optimal control of epidemics in metapopulations

Robert E. Rowthorn 2 Ramanan Laxminarayan 1 Christopher A. Gilligan 0 0 Epidemiology and Modelling Group, Department of Plant Sciences, University of Cambridge , Downing Street, Cambridge CB2 3EA , UK 1 Resources for the Future , 1616 P Street NW, Washington, DC 20036 , USA 2 Department of Economics, University of Cambridge , Sidgwick Avenue, Cambridge CB3 9DD , UK Little is known about how best to deploy scarce resources for disease control when epidemics occur in different but interconnected regions. We use a combination of optimal control methods and epidemiological theory for metapopulations to address this problem. We consider what strategy should be used if the objective is to minimize the discounted number of infected individuals during the course of an epidemic. We show, for a system with two interconnected regions and an epidemic in which infected individuals recover and can be reinfected, that equalizing infection in the two regions is the worst possible strategy in minimizing the total level of infection. Treatment should instead be preferentially directed at the region with the lower level of infection, treating the other subpopulation only when there is resource left over. The same strategy holds with preferential treatments of regions with lower levels of infection when quarantine is introduced. 1. INTRODUCTION Many epidemics outstrip the resources available to treat all infected individuals (Lipsitch et al. 2000), especially when disease occurs simultaneously in different but interconnected regions ( Ferguson et al. 2001; Keeling et al. 2001; Dye & Gay 2003). Seeking to control in more than one region poses a dilemma for epidemiologists and health administrators of how best to deploy limited resources among different regions: should preference be given to treating infected individuals in regions with high or with low levels of infection, or to equalizing levels of infection in different regions as fast as possible? Choosing between these options requires a combination of epidemiological and economic insights that hitherto have tended to remain separate: epidemiological models take little account of economic constraints ( Forster & Gilligan 2007; Klein et al. 2007), while economic models mostly ignore the spatial and temporal dynamics of disease (Gilligan 2003), with some recent exceptions (Goldman & Lightwood 2002; Rowthorn & Brown 2003; Smith et al. 2005; Barrett & Hoel 2007; Forster & Gilligan 2007), of which Smith et al. (2005) and Forster & Gilligan (2007) explicitly consider infection space. The influence of the spatial structure of susceptible populations on the invasion and persistence of human, animal and plant pathogens is now well established ( Ferguson et al. 2001; Keeling et al. 2001; Dye & Gay 2003; Stacey et al. 2004). Most contemporary epidemiological theories are focused on the dynamics of disease in so-called structured metapopulations (Gyllenberg et al. 1997; Grenfell & Bolker 1998; Keeling & Gilligan 2000a; Hanski & Ovaskainen 2002) following on from early models that addressed spatial heterogeneity in disease transmission (Lajmanovich & Yorke 1976; Murray & Cliff 1977; Nold 1980) in which epidemics occur in loosely coupled subpopulations. These subpopulations correspond with natural aggregations of susceptibles, such as hospitals, towns, cities or countries. Infecteds and susceptibles mix more or less freely within subpopulations, with a smaller movement of infecteds or inoculum among subpopulations. The system of loose coupling leads to spatially distributed epidemics with local fade-out but global persistence ( Keeling & Gilligan 2000b), as infection is transmitted between infected and healthy subpopulations. It follows that local deployment of control in one region may benefit other regions by reducing the number of infecteds capable of transmitting infection between subpopulations, but the regional benefits of control may also be countermanded by reinvasion from neighbouring populations. Using a combination of optimization methods from the economic theory of disease control (Sethi 1978; Goldman & Lightwood 2002; Rowthorn & Brown 2003; Forster & Gilligan 2007) with a metapopulation model from epidemiological theory ( Hanski 1998; Swinton et al. 1998; Park et al. 2003; Keeling et al. 2004), we show, however, that it is possible to optimize the deployment of control. By formalizing the problem as one of control of a dynamic, spatially structured system subject to economic constraints, it becomes apparent that one plausible intuition to give preference to the most highly infected regions when resources are limited may be the worst possible strategy in limiting the amount of infection suffered by the entire population. 2. METHODS 2.1. The model We consider two coupled subpopulations of susceptible individuals, in which an epidemic is described by a simple susceptibleinfectedsusceptible (SIS ) compartmental model. An SIS model is characteristic of a sexually transmitted disease, such as gonorrhoea, in which infecteds (I ) recover naturally or after treatment ( Lajmanovich & Yorke 1976; Hethcote 1980; Anderson & May 1991). Infected individuals do not gain immunity to the disease, rejoining the susceptible class (S ) and so may be reinfected. This relatively simple model of an epidemic allows a rigorous analysis of strategies for optimal control of disease. Here, we consider a simple control strategy in which a certain drug is administered to some or all of the infected individuals in two regions, each with populations of the same size N. The model is inspired by the analysis of Goldman & Lightwood (2002) for optimal drug use in a single region. We envisage regions as encompassing local districts, counties, provinces or countries. The dynamics of infection for the SIS model in the two regions Ii are given by in which b and g are the transmission rates within and between subpopulations, respectively; mK1 is the infectious period; and a is a measure of the rate at which infecteds are cured by the drug. The number of infecteds receiving treatment in region i is equal to Fi. We assume that the drug is not used as a prophylactic so that only infected individuals receive it, hence Fi%Ii. An alternative scenario, in which treatment is effected through changes in the transmission rate (b), is discussed briefly in the electronic supplementary material. 2.2. Optimal control under a budget constraint Suppose that expenditure on drugs is subject to a budget constraint c(F1CF2)%M. We assume that finance is not transferable through time, so that money which is not spent immediately cannot be saved for the future purchase of drugs. If there are sufficient resources, every infected individual will be treated. Otherwise, drugs are allocated so as to minimize the discounted sum of total infection in the two regions. Hence, we choose F1 and F2 so as to minimize the following integral: N V Z 0 The objective function in equation (2.3) is concerne (...truncated)