Searching for the most cost-effective strategy for controlling epidemics spreading on regular and small-world networks

Abstract

We present a combined epidemiological and economic model for control of diseases spreading on local and small-world networks. The disease is characterized by a pre-symptomatic infectious stage that makes detection and control of cases more difficult. The effectiveness of local (ring-vaccination or culling) and global control strategies is analysed by comparing the net present values of the combined cost of preventive treatment and illness. The optimal strategy is then selected by minimizing the total cost of the epidemic. We show that three main strategies emerge, with treating a large number of individuals (global strategy, GS), treating a small number of individuals in a well-defined neighbourhood of a detected case (local strategy) and allowing the disease to spread unchecked (null strategy, NS). The choice of the optimal strategy is governed mainly by a relative cost of palliative and preventive treatments. If the disease spreads within the well-defined neighbourhood, the local strategy is optimal unless the cost of a single vaccine is much higher than the cost associated with hospitalization. In the latter case, it is most cost-effective to refrain from prevention. Destruction of local correlations, either by long-range (small-world) links or by inclusion of many initial foci, expands the range of costs for which the NS is most cost-effective. The GS emerges for the case when the cost of prevention is much lower than the cost of treatment and there is a substantial non-local component in the disease spread. We also show that local treatment is only desirable if the disease spreads on a small-world network with sufficiently few long-range links; otherwise it is optimal to treat globally. In the mean-field case, there are only two optimal solutions, to treat all if the cost of the vaccine is low and to treat nobody if it is high. The basic reproduction ratio, R₀, does not depend on the rate of responsive treatment in this case and the disease always invades (but might be stopped afterwards). The details of the local control strategy, and in particular the optimal size of the control neighbourhood, are determined by the epidemiology of the disease. The properties of the pathogen might not be known in advance for emerging diseases, but the broad choice of the strategy can be made based on economic analysis only.

1. Introduction

Epidemiological modelling has long been used to design strategies to control disease outbreaks [1]. The underlying assumption of these strategies is the wide availability and low economic or social cost of treatment, be it in the form of preventive vaccination or therapy [2]. These assumptions are however not true in many cases, particularly for large outbreaks like cholera [3], AIDS [2], severe acute respiratory syndrome (SARS) [4] or foot-and-mouth disease [5]. There is, therefore, a need for a ‘marriage of economics and epidemiology’ [2] in designing effective strategies for control of disease [6]. Key to this approach is the realization that an optimal policy does not necessarily result in curing everybody in the population at any cost; it might instead be acceptable to tolerate some lower level of disease persistence if the costs of eradication are prohibitively high [7]. Several recent papers have combined epidemiological with economic constraints to identify optimal strategies for disease control or management [8–12]. Most of these studies, however, ignore the spatial components of disease spread and control while searching for an optimum strategy (see, however, Rowthorn et al. [13]). The spatial scale at which control is applied in relation to the spatial scale of the pathogen dispersal has been identified for many diseases, notably for plant diseases in which the spatial component of the location of the hosts plays a particular important role [14,15]. The relationship between the epidemic and control scales can however be affected by economic aspects of both disease and treatment. Simple network models, while capturing the essence of the topology of spread and control, offer a unique opportunity to analyse the relationship between the epidemic and control scales when there are cost constraints [6,16–20]. In this paper, we analyse a model for optimal control of disease spreading on regular and ‘small-world’ networks [6,20]. The importance of long-range transmissions in influencing the efficiency of control strategies has been shown for numerous major epidemics of human (e.g. SARS [4] and influenza [21–23]), animal (e.g. foot-and-mouth disease [5,24]) and plant diseases (e.g. citrus canker [25], sudden oak death [26] and rhizomania of sugar beet [14,15]).

There exist two broad strategies in response to a threat of an infectious disease. The authorities can implement control measures before the potential outbreak (e.g. a preventive vaccination [1]) or prepare a set of reactive measures, with a mixture of palliative care and control implemented only after the outbreak. In this paper, we consider the second case and assume that the outbreak has already started. A successful reactive control strategy needs to combine therapy (i.e. treatment of existing cases) with prevention against secondary cases (e.g. vaccination or culling) [2]. Treatment limited to individuals who are displaying symptoms is usually not enough to stop an outbreak, particularly if the disease includes a pre-symptomatic stage [27]. Thus, by the time a symptomatic individual is detected, the disease will have spread well beyond the original focus. Combination of a palliative with a preventive (although applied after the start of the outbreak) treatment allows the control to be more effective, if enough individuals are included in the population to catch all infectious individuals or to remove susceptible ones from the perimeter of the spreading focus [15]. However, such a strategy is also costly—it invariably leads to treating individuals that might never have been infected and become diseased even when no action were taken. If treatment is simple and cheap, this perhaps does not matter. The experience of large outbreaks of foot-and-mouth disease [28,29] and citrus canker [25] shows, however, that treatment cost may be very important. Thus, the process of designing the optimal strategy must involve in the first step the identification of all potential costs (including disease and control costs) and subsequently finding the right balance between them [3].

In this paper, we identify two main sources of costs associated with a disease outbreak and subsequent control [2]. These are the cost of untreated disease cases and the cost of treating individuals located around those cases (including the cost of surveillance needed to identify existing cases). If no preventive measure is taken, infection, and hence disease, spreads and many individuals become ill and either recover or die. This leads to direct costs associated with, for example, hospitalization and drugs that need to be administered and indirect costs associated with the loss of revenue owing to illness, and with death or incapacity of individuals. Such associated costs can be very high if the epidemic is severe and affects all or most of the population. The main objective of the preventive measures is to lower the total cost by investing in treatment or vaccination in the initial stages of the epidemic, with the hope that this will arrest the disease spread [30]. Control might, for example, involve a mass vaccination as early in the outbreak as possible, or continuous preventive vaccination [1,31,32]. Although there is a potentially large cost associated with such a strategy, the investment is seen as worthwhile if it leads to a significantly reduced number of infections owing to removal of susceptible individuals. Vaccination, culling or other forms of preventive treatment can also be targeted, by concentrating on individuals that exhibit disease symptoms or their neighbours, regardless of their status [5,27,33,34]. Such a form of ‘ring vaccination’ has been identified as a cost-effective measure, since it concentrates the effort where it is needed. The drawback of such strategies is that they require a detailed knowledge of the actual location of infected individuals and their contacts [17], and this might also involve costly surveillance schemes [35].

In this paper, we compare spatially targeted control strategies. We show that, depending on the relative cost of treatment and infection, a choice of three strategies arises: treating nobody (null strategy, NS), treating only selected individuals within a well-defined neighbourhood of each detected (symptomatic) individual (local strategy, LS) and treating as many individuals in the whole population as possible (global strategy, GS). We also show that the randomness of disease distribution in the initial phases of the epidemic plays a very important role in deciding which strategy to choose. This can result either from an initial distribution of disease foci or from topology of interactions. The details of the LS depend on the epidemiology but not on the economic parameters—it is the choice of the strategy that does depend on the relative costs. The ‘bang–bang’ strategy of either treating nobody or treating all individuals has been observed in non-spatial systems where control strategy varies over time [7,8,36], but to our knowledge not for a spatial control strategy.

2. Model

The spatial model that underlies this paper is an extension of the susceptible–infected–removed (SIR) model to account for pre-symptomatic spread [6,20]. We first introduce a spatial model in which control is applied locally in response to observed cases. Subsequently, we construct mean-field approximations for the spatial model.

2.1. Spatial model

For simplicity, we assume that individuals are located at nodes of a square lattice that represents the geographical distribution of hosts. On this lattice, we define a local neighbourhood of order z as a von Neumann neighbourhood in which we include z shells and ϕ (z) = 2z(z + 1) individuals (excluding the central one). Thus, z = 1 corresponds to the four nearest neighbours, while z = ∞ corresponds to the whole population in the limit of infinite size of the system.

The epidemiological model is a version of an SIR model [1], modified to include pre-symptomatic and symptomatic stages of the illness and to account for detection and treatment (figure 1). All individuals are initially susceptible (S). The epidemic is initiated by the introduction of a few infected but pre-symptomatic (I) individuals. Each infectious (pre-symptomatic or symptomatic) individual is in contact with a fixed number of other individuals and infection is transmitted along these contact routes with probability f per contact. Upon successful infection, the susceptible individual moves to the pre-symptomatic class. Stochastic simulations are performed with a fixed time step so that each probability is interpreted as a hazard.

Figure 1.

Block diagram illustrating transitions in the model considered in the paper. Thick lines represent transitions performed at each time step, whereas light lines represent transitions triggered by treatment.

We consider two models for transmission: local-spread and small-world models. In the local-spread model, a fixed number of individuals is chosen in the nearest neighbourhood of order z_inf surrounding each susceptible individual. Each infected individual located within the neighbourhood contributes to the total hazard for this particular susceptible individual. We consider z_inf = 1 with ϕ(z_inf) = 4 individuals in the infection neighbourhood, but the results are similar for other choices of z_inf . A small-world model [6,37] is similar to the local-spread model, but an additional number of non-local links is added randomly to the lattice of local interactions. These links can span the whole population and the probability of passing an infection along any of the long-range links is the same as for local links.

With a probability q each pre-symptomatic individual develops symptoms that can be detected (and hence moves to class D). Both pre-symptomatic and symptomatic individuals can infect susceptible individuals. At each time step, each symptomatic individual can move to a removed class (R) with a probability r or, if it does not recover, can trigger a treatment event with probability v. This process models delays in public health actions leading to preventive treatment (vaccination or culling). Each treatment event affects the central symptomatic individual and all susceptible S, pre-symptomatic I and symptomatic D (but not removed R) individuals located within a von Neumann neighbourhood of order z centred on a detected individual, as they move to the treated class, V. This represents a localized ‘ring’ treatment (vaccination or culling). For convenience, we extend the definition of z to include two cases: z =−1 describes a strategy in which no spatial control is applied, and z = 0 corresponds to a strategy in which the detected individual is treated only. Neither R nor V can infect or be re-infected any more. The number of individuals in each class is denoted by S , I , D , R and V , respectively, and N = S + I + D + R + V is the total number.

2.2. Mean-field equations

The model without control can be described by the following set of mean-field equations:

2.1

The parametrization of the infection force by βϕ(z_inf) allows a direct comparison of the simulations with the fully spatial model, although β can only cautiously be interpreted as an equivalent of f. If the control is just applied to the detected individual (z = 0), these individuals are removed at the rate v and the equation for D is modified by including a term − vD,

2.2

When z > 0, an additional number of individuals, ϕ(z), is selected for treatment. In the spatial model, those individuals are located in the neighbourhood of the detected individual, but, in the mean-field approximation, the spatial information is lost. Thus, the corresponding number of individuals is selected at random from the population at each control event. As the control events occur at the rate −vD, the rate at which individuals are treated equals −vϕ(z)D. Out of these, a proportion of S/N individuals are susceptible, I/N individuals are pre-symptomatic and D/N are symptomatic (the control event does not distinguish between the state of the individuals subject to treatment, except for the removed class). Incorporating the relevant terms into equation (2.1) we obtain

2.3

2.3. Cost of treatment

From an economic point of view, the problem of designing an optimal control strategy can be viewed as a special case of a net present value test [38]. In this approach, the value of future benefits (reduction in the number of infection cases) is compared with the value of future and current costs (associated with a particular control strategy). The values are often discounted if the optimization horizon spans a longer period of time. For simplicity, we assume that the duration of an epidemic is short enough (e.g. within 1 year) so that no discounting is necessary. The strategy is decided at the beginning of the epidemic and is not changed over time. The economic outcome, on the other hand, is deferred until the end of the epidemic when costs are compared with gains. We also assume that there are no budget constraints and so the decision maker can spend as much as is necessary on controlling the disease within the prescribed strategy.

In this paper, we aim to minimize the total cost of the outbreak and we allocate costs to two groups. The first term representing the palliative cost is associated with individuals who are never treated and therefore spontaneously move into the removed class. This term is equal to R(∞) multiplied by a unit cost of treatment, c₁. The second term describes costs associated with treatment of susceptible and pre-symptomatic individuals aimed at prevention of further spread. For simplicity, we assume that this term also includes surveillance costs involving searching for and detection of infected (symptomatic) individuals as well as treatment of any symptomatic individuals (including the one that triggered the treatment event; figure 1). Thus, the second term is equal to c₂V(∞), with c₂ being a unit cost of preventive treatment. These assumptions lead to the following general form for the total cost of the epidemic:

2.4

We are normally not interested in the absolute measure of X, but only intend to use it to compare different strategies. Thus, without loss of generality we can put c₁ = 1 and c₂ = c, so that X = R + cV with c measuring the relative cost of treatment to infection [6,17]. The goal of the simulation is to find an optimal control strategy, identified here with a value of z (and denoted z_c) for the spatial model (and its mean-field approximations), that minimizes the total cost, X, with other parameters fixed. We call X the severity index, as it characterizes the combined severity of the epidemic including individuals that have been through the disease but were not treated (R) and individuals that have been treated both in response to their symptoms and preventively to halt the spread of the disease (V).

We consider two prevention strategies exemplifying our approach, preventive vaccination (or spraying) and culling (or destruction), for three complementary diseases, influenza [39–41], foot-and-mouth disease [24,27] and citrus canker [25], although our approach is more general. Attempts to control an influenza outbreak include preventive vaccination or treatment with anti-viral drugs [42]—a similar approach has been suggested for measles [43] and for Ebola [44]. For foot-and-mouth disease, both vaccination and preventive slaughter of animals on contiguous premises [24,45] have been used to control spread. Likewise, citrus canker can be controlled by early spraying with copper compounds on resistant varieties, but immediate and rapid destruction of infected trees is essential for controlling the spread [46]. The two exemplary treatments differ in costs associated with them. Vaccination (for influenza or foot-and-mouth disease) and preventive spraying (for citrus canker) are typically cheaper than loss of an individual owing to disease (foot-and-mouth disease, canker) or costs associated with inability to work or even hospitalization (influenza). Thus, for example, Weycker et al. [40] estimates the costs of influenza vaccine at c₂ = US$6–24, with direct costs of infection at c₃ = US$70 and indirect at US$351, leading to c ranging from 0.017 to 0.341 (see also Meltzer et al. [39]). Similar estimates can be obtained for rotavirus and hepatitis A [47,48], with c = 0.01 − 0.85. On the other hand, the cost of culling an animal or destroying a tree is typically comparable to or more expensive than the disease, as it includes not only the lost revenue associated with the culled animal or destroyed tree but also the labour associated with treatment; this leads to c∼1.

2.4. Simulations

Simulations were performed on a lattice of 200 by 200 individuals with periodic boundary conditions. The size of the lattice was a compromise between numerical efficiency and small-size effects that we wanted to avoid. We performed simulations for other sizes and found no effect for sufficiently large lattices. We have considered a range of initial numbers of infected individuals, but the results are shown for 40 initial foci (0.1% of the total population) and 400 initial foci (1% of the total population). Smaller numbers of initial foci led to too many cases in which disease died out without spreading, which affected the optimization procedure. Except when indicated otherwise, z_inf = 1, v = 0.1 , r = 0.1 and q = 0.5. Each simulation was run until I(t) + D(t) = 0 and X was computed at the end of the run. For the simulation model, the minimization of X is achieved by sweeping through different values of z while performing only a single simulation for each value of z. For such a sample, the actual minimal value of X and the corresponding value of z are found. This procedure is then repeated 100 times to yield average values of z_c and X_c and their standard deviations. Numerical solution of the differential equations was done using R [49].

3. Results

The long-term behaviour of the spatial model in the absence of control (z = −1) is determined by f, the probability that infection is passed to a susceptible node from any of the four neighbours (z_inf = 1). For small values of f, the disease quickly dies out, whereas, for large values of f, the pathogen and hence disease is highly contagious and spreads through the whole population when no treatment is applied, X ≃ R(∞) ≃ N; compare figure 2a,b. The extreme cases of f are separated by a threshold for disease invasion, with an exact critical value of f depending on the spatial structure of the network and presence or absence of long-range links. For the simplest case of z_inf = 1 and no long-range links, the transition occurs at f = 0.04 (figure 3); addition of the small-world links shifts the threshold towards the value of f = 0.02 that can be compared with the mean-field critical value of β = 0.02 associated with R₀ = 1 (for details of mean-field calculations see below and in particular equation (3.1)). When control is applied, z ≥ 0, R/N declines monotonically with the order of the control neighbourhood, z, for both invasive and non-invasive diseases (figure 2a,b). Thus, the increased control effort leads to a reduction in the number of cases. However, the number of individuals treated, V(∞), increases at the same time (figure 2c,d). The increase is monotonic for a non-invading disease (figure 2c), but non-monotonic for an invading disease (figure 2d). R(∞) and cV(∞) are subsequently combined to form X = R(∞) + cV(∞) (figure 2e,f). The special case of a vaccination that does not cost anything, c = 0 (not illustrated in figures), corresponds to X = R(∞) and leads to an optimal strategy of treating all (GS). If c ≠ 0, various types of global minima can be obtained depending on the value of c and the shape of v.

Figure 2.

(a,b) The proportion of spontaneously removed individuals, R(∞); (c,d) treated individuals, V(∞), as a function of the control neighbourhood, z, for different values of f and c. The corresponding values of the severity index, X, are shown in (e,f). Points mark the global minimum values of X, X_c and the corresponding optimal control radius, z_c (treat all, filled circles; treat some, open circles; treat detected only, filled triangles; treat none, crosses). f = 0.001 in (a,c,e), representing a non-invasive disease; solid line and filled circles, c = 0.0001; dashed line and filled triangles, c = 0.005; dashed-dotted line and crosses, c = 1.5. f = 0.5 in (b,d,f), for an invasive disease; solid line and filled circles, c = 0.0001; dashed line and filled circles, c = 5; dashed-dotted line and crosses, c = 10. Initial condition is I(0) = 40 (0.1% of the total population). Other parameters: z_inf = 1, q = 0.5, r = 0.1, v = 0.1. Note that z = −1 corresponds to no control whereas z = 0 represents controlling the detected individual only.

Figure 3.

Critical value for the range of control neighbourhood, z_c, as a function of the probability of spread, f, for different values of the relative treatment cost, c. Other parameters: q = 0.5, v = 0.1, r = 0.1 and z_inf = 1. The initial condition is I(0) = 40 (0.1% of the total population). The figure at the top shows the average final size of epidemics in the absence of any control, R(∞) (light grey corresponds to low values of R(∞) and dark grey to high values of R(∞)). Vertical line at f = 0.04 corresponds approximately to the threshold for invasion. (Online version in colour.)

First, consider a non-invasive disease (figures 2a,c,e and 3). If vaccination is cheap (small but finite c), X is dominated by R(∞) (the cost of an uncontrolled epidemic) and the minimum value of X occurs at z_c = ∞ corresponding to the GS of treating all individuals (GS) (figure 2e: filled circle). As the cost, c, increases, the minimum rapidly shifts to z = 0, corresponding to treating just the detected individual (a subset of the local strategy, LS; figure 2e: filled triangle). For very high values of c (thick line in figure 2e), the strategy shifts further to z =−1 when nobody is treated (the NS). The value of a critical control radius, z_c, depends strongly on c but only weakly on f for small values of f (figures 2e and 3).

As f increases and the epidemic character changes from non-invading to invading, V(∞) becomes a non-monotonic function of z (figure 2d). While for small values of c, the GS is still the best option (figure 2f), a new type of LS appears for moderate values of c, corresponding to the treatment within a well-defined region around each detected case. For a very high value of c, the minimum of X corresponding to a finite value of z disappears and the NS of treating nobody becomes optimal (figure 2f). The switch from GS to LS and subsequently to NS is clearly seen in figure 3, which also shows the relative independence of the choice of the optimal control strategy on f.

Thus, the choice of the optimal strategy is determined by two main factors: the infectiousness of the disease, f, and the relative cost of the treatment, c. The dependence on the rate of disease spread, f, is relatively weak for most values of c (figure 3). The values of the optimal control neighbourhood, z_c, cluster in two regions. For small c (c < 10⁻⁴), z_c is independent of f and corresponds to a GS, z_c ≃ 45. For moderate c (0.01 < c < 1), z_c is below 10 (for the parameters discussed here) and slowly increases as the disease switches from non-invasive to invasive. For high values of c (10 < c < 100), the dependence on f is non-monotonic as z_c first increases and subsequently drops back to 0 (treat only detected individuals). Finally, for very high costs of treatment, z_c =−1 (refrain from treatment) for almost all values of f.

The economic aspects of the control determine three regions for c (figure 4). To illustrate the details of the behaviour, we assume that each untreated case (i.e. the individual in the removed class, R, at the end of the epidemic) costs £100. (We use arbitrary but realistic values here, to illustrate general principles rather than to focus on a particular disease.) We consider two contrasting cases for the cost of each treated individual (i.e. the individual in the treated class, V, at the end of the epidemic), £0.01 and £1000. We also assume that initially there are I(0) = 40 cases in a population of 40 000. Consider first the costs of the NS, under which nobody is treated and so X ≃ R(∞). For the non-invasive disease (small f), the total cost is approximately £100I(0) = 4000, whereas for the invasive disease (high f) the total cost reaches £100N = 4 million. For the GS, we treat all individuals indiscriminately and as quickly as possible and so the cost is £0.01N = 400 for small c and £1000N = 40 million for large c, independent of f. Finally, for the LS, it is not possible to obtain a simple estimate of the cost as it depends on z and the effectiveness of prevention.

Figure 4.

Critical value for the range of control neighbourhood, z_c, as a function of the relative treatment cost, c, for different values of the probability of spread f. (a) Examples of the dependence of the optimal control radius, z_c, on the treatment cost for various values of f (solid line, f = 0.005; dashed line, f = 0.05; dashed-dotted line, f = 0.5). The optimal control radius, z_c, is also shown in (b) as a function of both f and c. White colour represents z_c = −1 (NS), grey colour corresponds to treating detected individuals only (z_c = 0) and a range of colours represent z_c > 0. Other parameters: z_inf = 1, q = 0.5, v = 0.1, r = 0.1. The initial condition is I(0) = 40 (0.1% of the total population). The vertical line is placed at c = 1 (equal costs of disease and treatment). The horizontal line at f = 0.04 separates invasive and non-invasive epidemics (cf. figure 3). (Online version in colour.)

For the very cheap preventive treatment (e.g. costing £0.01, i.e. c = 10⁻⁴), cN < I(0), the cost of treating the whole population (GS, £400) is smaller than the cost associated with the infection of the initial cases (NS, £4000). Thus, for both invasive and non-invasive diseases, it is better to spend £400 and stop the epidemic immediately than to allow even the initial cases to go through the disease process (at a minimum cost of £4000). If the cost of treating the whole population is comparable to or higher than treating the initial cases, c ≥ I(0)/N, the GS is no longer optimal. For high c and low f, if the treatment cost of just the few initial cases (£40 000) is significantly higher than the cost of allowing the epidemic to run to its completion (NS for low f, £4000), we expect the NS to be optimal. Similarly, the cost of the GS is high (£40 million) compared with the NS (£4 million) for large f and the NS is again optimal. The range of c between those two extremes is occupied by the LS with z_c = 0 (treat only detected individuals) for the non-invasive disease and z_c < 10 for the invasive disease (figure 4).

A remarkable feature of the LS is the stability of z_c as a function of c over a wide range of c and f, (figure 4a; see also figure 3). Interestingly, even in cases when the cost of the preventive treatment, c, exceeds the cost of uncontrolled disease (c > 1), the LS is still optimal for some combinations of c and f (even though the NS is optimal for very high values of c. The mechanism for this behaviour is related to spatial correlations in the spread of the disease. Consider a focus originating with a single pre-symptomatic but infectious individual. Infection subsequently spreads to its nearest neighbour and then to their neighbours, but the focus still remains undetected. It is only when the first individual in the group shows symptoms that the authorities might become aware of the infection (this individual is usually the original source of infection, but owing to the stochastic nature of the process it might also be another one). Further delay (represented by the finite value of v) before any responsive treatment (vaccination or culling) is applied leads to further expansion of the focus. Thus, with a high probability, we can expect pre-symptomatic but infectious individuals in the immediate neighbourhood of a detected one. The optimal local control strategy will aim at treatment of all such pre-symptomatic individuals, but without extending the control neighbourhood too far (which will lead to an unnecessary increase in costs). This is a similar mechanism to herd immunity [1], but local application makes it a very effective strategy. Thus, the epidemic can be stopped within a few steps, even though the rest of the population remains susceptible. We also note that the fewer the initial foci, the less effort is required to stop the outbreak in this case, and so we expect that the critical value of c determining the transition between the LS and the NS will increase with a decreasing number of initial foci. However, once the spatial correlation is destroyed, we expect the LS to be no longer efficient for any value of c > 1.

3.1. Destroying spatial structure

The spatial correlations can be destroyed either by introducing non-local spread, for example in the form of long-range links in a small-world model, or by increasing the number of initial foci. There is not much change in the behaviour for small f (cf. figure 4 with figure 5) where 30 per cent long-range links have been added to the model structure. In this case, it is still preferable to treat individuals locally for a broad range of c . However, as the disease becomes more infectious, the probability of it spreading via long-range links increases. In this case, the region of optimality for the GS extends to higher values of c, whereas the range of the NS extends to lower values of c until they merge at c = 1 for high f (cf. figure 4 with figure 5).

Figure 5.

Critical value of the range of control neighbourhood, z_c, as a function of the relative treatment cost, c, and the probability of spread, f, but for 30% long-range links. Other parameters and labels as in figure 4. (Online version in colour.)

The effect of changing the number of non-local links is shown in figure 6, which is analogous to figures 4a and 5a, but for a smaller range of c and for a single value of f = 0.98. For 40 initial foci (0.1% of the total number) and the purely local spread, the switch between the LS and the NS occurs at approximately c = 10 (thin line in figure 6a) while z_c ≃ 6 for c below 10. The addition of 2 per cent links decreases the range of c for which the LS is still optimal but does not increase z_c (dashed line). However, the number of individuals treated preventively, V(∞), increases markedly compared with the purely local case (figure 6b). Addition of 30 per cent long-range links shifts the critical value for c close to c = 1, while increasing the size of the control neighbourhood to z_c ≃ 40 (thick line in figure 6a). In this case, the proportion of the population that needs to be treated, V(∞)/N, is also very high for c < 1 (figure 6b).

Figure 6.

(a) The optimal control neighbourhood, z_c, and (b) the proportion of treated individuals, V(∞)/N, as functions of the treatment cost, c, for local network, small-world network and mean-field models. For the number of initial foci I(0) = 40 the number of non-local links as a proportion of the total number of nodes is marked as follows: 0%, solid line; 2%, dashed line; 8%, thick dashed line; and 30%, thick line. The dashed-dotted line corresponds to the local spread (0% links), but the number of initial foci is increased to I(0) = 400. The thin-dashed line represents the simulation of the mean-field model. The rate of disease spread is f = 0.98; other parameters: q = 0.5, r = 0.1, v = 0.1. (Online version in colour.)

If the number of initial foci is increased without addition of long-range links, the effect on the critical value of c is similar to the addition of links, although there is no noticeable increase in z_c below the critical value (the dashed-dotted line in figure 6a). Thus, in each treatment event, we are still treating a small number of individuals. However, overall, we still need to treat a large proportion of individuals (figure 6b).

The change in z_c and V(∞) can be very rapid as long-range links are added to the system (figure 7a). This is reminiscent of the rapid transition associated with the small-world model in which addition of only a few links can drastically change the behaviour of the system [37]. If the preventive treatment (e.g. vaccination or culling) is even marginally more costly than allowing the disease to run without control, c = 1.25, the addition of 6 per cent of long-range links renders the LS inefficient (figure 7a). In this case it is best to refrain from any preventive treatment (and follow the NS), even if 4 per cent links still leave the LS optimal. The reason for this critical behaviour is clear from figure 7b. Consider the case of c = 1 for which it is optimal to treat locally even for a large number of non-local links. However, in this case, the proportion of treated individuals exceeds 50 per cent of the total population for 5 per cent or more of long-range links (marked by the arrow in figure 7b). This shows how critical it is to reduce the number of non-local links in the population [6,17], if local control strategies are applied.

Figure 7.

(a) The optimal control neighbourhood, z_c, and (b) the proportion of treated individuals, V(∞)/N, for the small-world model with a varying number of long-range links and for two similar values of c = 1 and c = 1.25. Other parameters as in figure 6; I(0) = 40. The arrow marks the point at which V/N = 0.5 (see text).

3.2. Mean-field limit

With the increase in the number of non-local links, we are approaching the mean-field approximation (figure 6a). In this case, there are only two options for treatment. The GS is optimal for c ≤ 1 and the NS is optimal for c > 1. This can be confirmed by the analysis of the mean-field equations (2.3). The responsive treatment in which the treatment rate depends on the current number of detected cases is not capable of controlling the invasion of the disease. When the basic reproduction ratio, R₀, is computed for equation (2.3), the result does not depend on the rate of treatment v,

3.1

In this formula, βϕ(z) is the rate of infection, 1/q is the average time an infected individual spends before detection and 1/r is the average time a detected individual spends before spontaneous removal. As a consequence, the stability of the disease-free equilibrium (I = 0, D = 0) is unaffected by the control since, for low levels of infection (I,D ≪ N), the control term is very small. Although as the number of cases increases, so does the control effort, but the dependence of the control rate on D means that the effort always follows the infection. Simulations show that the final number of treated individuals, V(∞), and the final number of spontaneously removed individuals, R(∞), are closely related in this case so that, if c = 1, X = R(∞) + V(∞) = N independently of v. Thus, if it is cheaper to prevent the disease than to avoid treatment, c ≤ 1, it is best to treat all individuals (GS). By contrast, if it is cheaper to refrain from treatment, c > 1, it is best not to treat anybody (NS). However, regardless of the applied control strategy, the whole population is affected either by the infection or by the control (figure 6b). The results agree with simulations of a stochastic spatial model in which individuals contact ϕ(z_inf) individuals randomly (not shown here).

4. Discussion

The main objective of the ‘optimal’ control strategy is to stop the epidemic not only in the shortest possible time but also at a manageable cost. Faced with a large outbreak, health authorities need to decide quickly whether to build up a coordinated effort to vaccinate or to treat a large proportion of the population, despite often substantial costs involved [3]. In some cases, refraining from treatment might be a more cost-effective choice than to act. Such decisions are often very difficult, as they involve many unknown factors. Mathematical modelling is then used to provide help and guidance by, among others, pointing to factors that do or do not influence the final outcome of the control process. Among the factors that mainly influence the decision are the costs associated with both preventive control and the disease itself. While the first category can be estimated with a certain degree of accuracy, the second factor might be difficult to determine.

In this paper, we provide a systematic study of the choice of the optimal strategy for a range of diseases for which spread is either localized or not. We have identified three basic strategies, the GS (treat all), the local strategy (treat within a well-defined neighbourhood of any detected individual or treat just the detected individual) and the NS (do not preventively treat any individual). In the last case, the individuals can still be treated for disease symptoms, but no prevention is effected on the population.

The details of the LS (when it is applied) surprisingly do not depend strongly on the cost of treatment, although the decision whether to apply the control locally or globally (or not at all) does depend on the cost. Once we decide on application of the local control, it is the epidemiology and social network structure that determine the spatial extent of LS. The results presented here for the LS show that it is important to match the scale of control with the scale of the disease dispersal; see [14] for a practical application in matching scales for control with the inherent scale of spread for a crop disease at the landscape scale. There are, however, also cases when the balance of costs is an over-riding factor and it is necessary to treat all individuals as quickly as possible (GS) or to refrain from treatment (NS).

When the purely local structure of the disease spread is destroyed by an increase in the number of initial foci or by addition of long-range links, local control can still be applied. Dybiec et al. [6] found that, for a small number of links, the local strategy still works, but at a cost of an increased control neighbourhood. This is necessary to catch the pre-symptomatic individuals before they cause new foci to appear via long-range links. Interestingly, the case of c = 1 that was considered by Dybiec et al. [6] corresponds to the minimal impact of a small-world structure in the order of control neighbourhood. If the cost of treatment is only marginally higher than the cost associated with infection (i.e. c > 1), it might be more profitable to withhold treatment completely rather than to use local control strategies.

Our cost function, equation (2.4), is linear in R and V and we assume that the budget is unlimited. For rapidly spreading epidemics, there might be a situation when the number of cases in a certain locality exceeds the maximum capacity of the control system (either health or veterinary care system). This leads to a rapid increase in costs per treated individual when compared with a small-size outbreak [50,51]. We extended our model to include nonlinear (quadratic) terms in either R and V, but there was no qualitative change compared with the linear cost function. In particular, increasing ɛ in X = (R(∞) + ɛR²(∞)) + cV(∞) shifts the curve in figure 4a horizontally towards the higher values of c (results not shown). Thus, the range of c for which the GS is optimal increases, whereas the range for which the NS is optimal decreases. This can be understood in terms of the penalty against outbreaks with a large value of r, leading to more strict criteria for control. The effect of including a nonlinear (quadratic) term in v is opposite, as the areas of the NS and LS shift in the direction of lower values of c. The critical value of z_c at the plateau is unaffected in both cases. Here we are penalizing against outbreaks leading to large spending on prevention and therefore are more likely to let the disease spread unchecked.

The critical control neighbourhood, z_c, and the resulting severity index, X, are very sensitive to the percentage of long-range links (figure 6). However, precise network structure and the actual number of long-range links are unlikely to be exactly known. In this case, the precautionary principle suggests to expect the worst case scenario and to either use the largest possible number of expected links, if known, or use the mean-field approximation corresponding to a large number of such links. In this case, the critical value of c is 1 and therefore the GS is optimal for c ≤1 and the NS for c > 1.

The current work assumes that the time span of the potential epidemic is very short and so no discounting is applied. In addition, we assume that, once the strategy is decided at the start of the epidemic, the authorities continue with the implementation. Each of these assumptions can be relaxed. A general relationship between the cost and the epidemic variables can be written as

4.1

where F₁ is a functional representing responsive costs, F₂ represents surveillance costs and F₃ corresponds to prevention costs while δ is a discounting factor. Under some simple assumptions on the functionals F₁, F₂ and F₃, we recover equation (2.4) if discounting is ignored and if the costs are only counted at the end of the epidemics. In general, however, the costs need to be evaluated as the epidemics unfold. Similarly, the radius of control neighbourhood z can change in time. This approach would require changes to the simulation procedure as it is no longer efficient to scan all possible values of z to search for z_c as done in this paper.

The model describes a single, relatively short outbreak of a disease that either kills the infected individuals or leads to complete immunity and also ignores influx of new susceptibles. Extension of the model to include recovery and/or re-infection (as in an SIS model) is planned for the future, but would require a different approach to cost calculations. We have also assumed that all social, economic and epidemiological parameters are fixed and well known in advance. This is not the case for emerging diseases. There is, therefore, a need to study the sensitivity of various control strategies to uncertainties in f, z_inf and the structure of the network. The long-term goal is to identify a selection of strategies that can be applied at the beginning of an emerging epidemic, even if we do not know the details of the disease, and then modified as the epidemic unfolds. However, the results of this paper suggest that if c can be reliably estimated in advance, we can decide between the overall control strategies (NS, LS or GS) even without knowing exactly what the value of f is for a given emerging disease.