Resource Allocation for Epidemic Control in Metapopulations

Citation: Ndeffo Mbah ML, Gilligan CA ( Resource Allocation for Epidemic Control in Metapopulations Martial L. Ndeffo Mbah 0 Christopher A. Gilligan 0 Michael George Roberts, Massey University, New Zealand 0 1 Yale School of Public Health, Yale University , New Haven , Connecticut, United States of America, 2 Department of Plant Sciences, University of Cambridge , Cambridge , United Kingdom Deployment of limited resources is an issue of major importance for decision-making in crisis events. This is especially true for large-scale outbreaks of infectious diseases. Little is known when it comes to identifying the most efficient way of deploying scarce resources for control when disease outbreaks occur in different but interconnected regions. The policy maker is frequently faced with the challenge of optimizing efficiency (e.g. minimizing the burden of infection) while accounting for social equity (e.g. equal opportunity for infected individuals to access treatment). For a large range of diseases described by a simple SIRS model, we consider strategies that should be used to minimize the discounted number of infected individuals during the course of an epidemic. We show that when faced with the dilemma of choosing between socially equitable and purely efficient strategies, the choice of the control strategy should be informed by key measurable epidemiological factors such as the basic reproductive number and the efficiency of the treatment measure. Our model provides new insights for policy makers in the optimal deployment of limited resources for control in the event of epidemic outbreaks at the landscape scale. - Funding: This work was supported by a Gates Cambridge Trust Scholarship (MLNM) and a BBSRC (Biotechnology and Biological Research Council) Professorial Fellowship (CAG) which we gratefully acknowledge. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. The management of diseases involves the expenditure of limited resources, which more often than not are outstripped by the demand for controlling all infected individuals [13]. This is often the case when disease occurs simultaneously in different but interconnected regions [2,4,5]. Treatment of infection in one region such as a state, city, or hospital may affect the potential for spread to another region when there is movement of individuals between the regions. Seeking to control disease outbreaks in more than one region, poses a dilemma for epidemiologists and health administrators of how best to deploy limited resources, such as drugs or trained personnel, amongst the different regions [611]. One common objective is to minimise the numbers of infected individuals and hence to minimize the burden of infection during the course of an epidemic [4,12]. For epidemics of the SIS (Susceptible-Infected-Susceptible) form, in which individuals can be re-infected, Rowthorn et al. [10] showed that rather than targeting the region with most infecteds, as might have been intuitively expected, it is instead optimal to give preference to treating the region with the lower levels of infecteds: the remaining regions are treated as residual claimants, receiving treatment only when there is resource left over. The epidemiological intuition underpinning the optimal strategy is understood by noting that since there are only two types of host (susceptible or infected), preferential treatment in a region with low level of infection is equivalent to giving preference to the region with the highest level of susceptibles available for infection. Since, on average an infected individual infects more than one susceptible, removing infecteds where susceptibles are plentiful reduces the force of infection of the epidemic and so is likely to bring the epidemic under control. But what happens when there are more than two epidemiological classes? For many diseases, reinfection is often preceded by a period of temporary immunity, yielding a third class of removed individuals in the population that complicates the identification of an optimal strategy for control. In this paper, we focus on this much broader class of epidemics described by an SIRS model. We consider an SIRS-type epidemic in which infected individuals cease to be infectious and move into a temporary immune (R) class, after which they become susceptible once again. This is characteristic of many diseases, such as malaria [13,14], tuberculosis [15] and syphilis [16], in which infecteds (I) recover naturally or after treatment. Infected individuals gain a temporal immunity to the pathogen, after which they rejoin the susceptible class (S) and can be reinfected. We assume that treatment is not used as a prophylactic so that only infected individuals receive treatment. Hence, the proportion of treated individuals is given as fI I (0f 1): To address the problem of resource allocation for disease management in multiple regions, we use a combination of optimization methods from economic theory of disease control [17,18] with a metapopulation model from epidemiological theory [19,20]. This enables us to formalize the problem and to derive criteria for optimality so as to minimize the total number of infections over time. Not infrequently, strict criteria for optimization identify strategies that may be logistically impractical, for example by requiring a change in pattern of control at a switching time that may be difficult to monitor [17]. Strictly optimal strategies may also be challenged on grounds of social equity, whereby every infected individual does not have an equal chance of being treated [21,22]. Accordingly we assess the tractability of optimal control strategies and consider also how adaptations may be made to balance, optimality, tractability and social equity. For the sake of simplicity, the analysis is initially carried out for two interconnected regions (e.g.cities, towns or states) and the robustness of the results to spatial structure are later tested for two other simple and realistic spatial configurations. Model We consider two coupled sub-populations (regions) of susceptible individuals each with a fixed size N, in which an epidemic is described by a simple SIRS compartmental model: with i=j and i,j~1,2: Each sub-population is composed of susceptible (S), infectious (I) and recovered (R) individuals, and are scaled here as proportions. The transmission rate for each subpopulation is given by b. The coupling strength between subpopulations is given by 0v v1. The infectious period is given by m{1; n is the rate of loss of immunity, and s is rate of birth/death. g is a measure of the incremental increase in the recovery rate of treated individuals, and fi is the proportion of infected individuals in sub-population i that receive treatment. When all infected individuals receive treatment (f1~f2~1), th (...truncated)