Disease Spread in Coupled Populations: Minimizing Response Strategies Costs in Discrete Time Models

Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2013, Article ID 681689, 9 pages http://dx.doi.org/10.1155/2013/681689 Research Article Disease Spread in Coupled Populations: Minimizing Response Strategies Costs in Discrete Time Models Geisel Alpízar1 and Luis F. Gordillo2 1 2 Departamento de Matemáticas, Instituto Tecnológico de Costa Rica, Cartago 30101, Costa Rica Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA Correspondence should be addressed to Luis F. Gordillo; Received 6 September 2012; Accepted 19 April 2013 Academic Editor: Francisco Solı́s Lozano Copyright © 2013 G. Alpı́zar and L. F. Gordillo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Social distancing, vaccination, and medical treatments have been extensively studied and widely used to control the spread of infectious diseases. However, it is still a difficult task for health administrators to determine the optimal combination of these strategies when confronting disease outbreaks with limited resources, especially in the case of interconnected populations, where the flow of individuals is usually restricted with the hope of avoiding further contamination. We consider two coupled populations and examine them independently under two variants of well-known discrete time disease models. In both examples we compute approximations for the control levels necessary to minimize costs and quickly contain outbreaks. The main technique used is simulated annealing, a stochastic search optimization tool that, in contrast with traditional analytical methods, allows easy implementation to any number of patches with different kinds of couplings and internal dynamics. 1. Introduction In contrast with the tremendous benefits that modern transportation systems have brought to our society, their potential as contributors to the global and fast spread of infectious agents has become a major concern [1, 2]. This has been evidenced, for instance, with the recent SARS and swine flu outbreaks in 2003 and 2009, respectively. In those occasions, careful monitoring and efficient control of individuals’ flow among cities were implemented, although it has been determined that introducing travel restrictions does not have a drastic impact on a pandemic development, [3, 4]. However, this kind of measures may introduce a delay in the spread of the disease [5], providing additional time to implement nonpharmaceutical interventions. The important role that transportation has on the spread of infectious diseases has been extensively studied in particular cases. Remarkable efforts are currently made to understand the spread of pandemic and seasonal influenza in spatial structured populations, [6–8]. The detailed study made in [9] shows how social distancing policies implanted in Mexico in 2009, combined with control in the transportation of individuals, lead to the effective interruption of the first two waves of the influenza’s emergence. Health administrators face the nontrivial task of controlling disease spread through optimal responses during the first stages of an outbreak, which include the implementation of social distancing, travel restrictions, medical treatments, and vaccination. Coupled populations scenarios may be difficult to assess if confronted with scarce resources. One of our goals is to show that simulated annealing, a wellknown computational optimization technique, may be useful in the search of appropriate responses for some diseases modeled in discrete time. We use variants of two extensively studied models: SIR and SIS in coupled populations, which are introduced in Section 2. The computational results obtained by application of simulated annealing are presented in Section 3. This technique has the advantage of combining easy implementation and simple theoretical principles [10]. Although theoretical models that describe disease dynamics in metapopulations have been studied in detail, the problem of optimal deployment of control combining epidemiological and economic factors has barely been touched. For continuous time, the SIS formulation with two interconnected patches is successfully addressed in [11], and the optimal allocation of resources for SIRS models in metapopulations has been recently studied in [12]. For discrete time 2 epidemic models, optimal control techniques have been employed only for one patch populations [13, 14]. The usual approach in this cases is to use Pontryagin’s maximum principle adapted to discrete systems [14, 15], which establishes necessary conditions for the existence of an optimal condition that can be recovered using a forward-backward algorithm. This theoretical formulation might turn out to be cumbersome if a large number of interconnected patches combined with a high number of model parameters in each patch are involved. In contrast, the alternative of approximating optimal solutions with numerical simulations only requires a very general framework, easily adaptable to any situation. 2. Coupled Populations: Two Examples For the sake of simplicity, we consider the case of only two coupled populations, for example 𝑥 and 𝑦. The disease dynamics in each patch are described with discrete time equations, capturing the processes of infection and migration at each unit in time (day). We assume that the system does not allow the introduction of individuals from outside these patches and that, within each population, individuals are homogeneously mixed. The technique used below to find optimal control parameter values is very general and independent of the type of discrete time model describing the disease dynamics within each patch. Thus the method could be, in principle, adjusted to numberless possible modeling situations. In this paper, we consider separately for analysis two instances of simple but plausible schemes largely employed to describe dynamics of infectious diseases. For both models we assume that the epidemic evolves fast enough to ignore demographic effects, and consequently our attention is focused on the disease spread between populations only for the first few days after an outbreak appears in one of the patches. The first model, referred to here as Model A, is a mechanistic extension of a discrete approximation to the basic Susceptible-InfectedRemoved (SIR), which is valid for short periods of time and examined here due to its simplicity. The motivation for this model comes from the ideas originally explored in [16]. The second model, Model B, is a Susceptible-Infected-Susceptible (SIS) type, derived originally in [17], which might be useful indescribing infectious diseases where almost immediate host reinfection is possible. In addition to each model we define functionals that represent the economic impact caused by the di (...truncated)