Resource Allocation for Epidemic Control Across Multiple Subpopulations
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 model1 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 subpopulations (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 subpopulations 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 subpopulation 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 subpopulation and then switch to treating the less infected subpopulation (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 subpopulations, but in reality a larger number of subpopulations 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 \(n\ge 2\) subpopulations.
Work on the allocation of resources between two subpopulations 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 subpopulations is 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 \(n\ge 2\) subpopulations.

How do epidemiological dynamics and economic constraints impact the optimal allocation of resources across n subpopulations?

How does the indirect interaction between subpopulations that arises from the limited availability of resources affect the optimal allocation of resources to the different subpopulations?

Can we develop a simple easytouse allocation strategy that is close to the optimal solution?
2 Methods
2.1 Model
Forests often contain a number of different tree species, and in recent years there has been a move towards mixedspecies in order to make forests more resilient to disturbances and stresses such as those posed by climate change (Kerr et al. 2015). To understand how to allocate resources optimally in order to minimise the impacts of multiple threats across different tree species is challenging using traditional optimal control theory approaches. The problem quickly becomes intractable for the general ndimensional problem when n greater than 2.
We consider the optimal control of epidemics in n subpopulations, each of size \(N_{i}\). Each population is considered to be a different tree species within a forest under threat from multiple different pests or pathogens. This is a common problem facing many noncommercial forests which are typically composed of a large number of different tree species under threat from a number of invasive species (Baker et al. 2014). A pest or pathogen is typically specialised to a given tree species, for example the ash dieback fungus only infects ash trees and Dothistroma needle blight only affects pine trees. Therefore, we assume that infection can only be transmitted within a subpopulation (species) and not between subpopulations. This means the subpopulations are independent, which allows us to reduce the optimal control problem to study the dynamics of the individual subpopulations.
We assume that each epidemic can be described by a susceptible–infected–susceptible (SIS) compartmental framework since it is a very general epidemic model applicable to a wide number of different pathogens (Anderson and May 1991). The SIS model assumes individuals return to the susceptible compartment following natural recovery or treatment. It is therefore characteristic of infections, such as gonorrhoea or Dothistroma, where recovered individuals do not gain immunity (Anderson and May 1991). We consider a treatment that increases the rate of recovery of infected hosts by a fixed amount, \(\eta _i\) (Rowthorn et al. 2009; Ndeffo Mbah and Gilligan 2011). For tree diseases, examples of such treatments are application of pesticides or fungicides directly to the tree (Masoa et al. 2014). Such control options are common especially when aerial spraying is banned, as in the UK, and felling is a less popular option with the general public (Sheremet et al. 2017). In human and animal health, examples of such a treatment could be antibiotics. We assume that the rate of recovery due to treatment is different for each subpopulation since the efficacy for a given pesticide/fungicide is likely to vary for across tree species.
We assume resources are allocated to subpopulations at time \(t=0\) and cannot be reallocated later, which is applicable if reallocation is expensive, for example when decisions are taken by central planners (Zaric and Brandeau 2001a, b; Brandeau et al. 2003). We allow the cost of treatment per host (i.e. amount of resource per host treated), \(k_i\), to vary amongst subpopulations. For example, it may be harder to administer treatment to certain subpopulations. We assume there is a limited amount, M units, of the resource available. Even though there is no direct coupling, the assumption of a limited shared resource pool gives rise to an indirect interaction between subpopulations. When more resources are allocated to one of the subpopulations, the disease prevalence in that subpopulation decreases, but there is consequently less resource for allocation into the other subpopulations. The proportions of infected individuals in the subpopulations are therefore anticorrelated.
Table showing the parameters and variables used, together with their descriptions
Parameter/variable  Description 

\(\beta _i\)  Rate of infection in population i 
\(\eta _i\)  Additional recovery rate provided by the treatment 
\(C_T^{(i)}\)  Endemic disease prevalence given full treatment in population i, \(1\frac{1+\eta _i}{\beta _i}\) 
\(C_0^{(i)}\)  Endemic disease prevalence given no treatment in subpopulation i, \(1  \frac{1}{\beta _i}\) 
\(\gamma _c^{(i)}\)  Proportion of hosts in subpopulation i that require simultaneous treatment necessary for saturation 
\(N_i\)  Size of the \(i\text {th}\) subpopulation 
M  The maximum amount of resources available 
\(x_i\)  Proportion of the resource that is allocated to the subpopulation i 
\(k_i\)  The cost of treating one host in subpopulation i 
n  Number of subpopulations 
2.2 Simple Allocation Strategies
 Proportional allocation The amount of resource allocated to the \(i\text {th}\) subpopulation is proportional to the size of the subpopulation, \(N_i\), so the proportion of resource allocated to subpopulation i is$$\begin{aligned} x_i= \frac{N_i}{\sum _i N_{i}}. \end{aligned}$$(6)

Equal allocation The same amount of resource is allocated to each of the subpopulations, so the proportion of resource allocated to subpopulation i is \(x_i = 1/n\).

Allocate to the largest strategy We look at which subpopulation we need to saturate to achieve the greatest decrease in the objective function and then repeat until we cannot saturate anymore, at which point resources are allocated to the remaining subpopulation that would result in the greatest decrease in the objective function. In the case when both the epidemiological and the cost parameters are identical across all subpopulations, this is equivalent to saturating each subpopulation in order of size from largest to smallest, hence we refer to this as the “allocate to the largest” strategy.

Allocate to the smallest strategy This strategy is the opposite of the allocate to the largest strategy, in that we saturate the subpopulation that leads to the smallest decrease in the objective function and then repeat until we cannot saturate any more, at which point we allocate the remaining resource to the subpopulation that would give the smallest decrease in the objective function. This strategy is equivalent to saturating each subpopulation in order of size from smallest to largest when both the epidemiological and cost parameters are identical. Hence we refer to this as the “allocate to the smallest” strategy.
2.3 Model Analysis
In this section we show that the optimal allocation using analysis of the fixed points is as follows: saturate some subset S of the subpopulations such that no further subpopulations can be saturated and then allocate all the resources left over into one of the remaining unsaturated subpopulations. Therefore, the optimal strategy lies on the boundary of possible allocation strategies and no interior solution to the problem exists. Furthermore, we determine a simple heuristic to determine which subpopulations should be saturated. Since the optimal solution involves the saturation of subpopulations, we begin by considering the minimum amount of resource needed to saturate a subpopulation that depends on the longterm dynamics of the system.
2.3.1 Minimal Treatment to Saturate a Subpopulation
We begin by considering the minimum amount of resource, \(\gamma _C^{(i)}\), necessary to ensure that subpopulation i ends up in the full treatment equilibrium, that is \(I^{(i)}\rightarrow C_{T}^{(i)}\) as \(t\rightarrow \infty \). This depends on the dynamics of the system at equilibrium, which are analysed in detail in “Appendix A”. In particular, the dynamics differ in two distinct regions of parameter space, depending on the epidemiological parameters for the rate of infection, \(\beta _i\), and the rate of recovery following treatment \(\eta _i\).
Table giving the parameter regimes, conditions on the initial conditions and the subsequent formulas for the minimal amount of treatment required to saturate subpopulation i
Parameter region  Initial condition criterion  Allocation 

\(\eta _i<(\beta _i1)/2\)  all \(I_0^{(i)}\)  \(\gamma _C^{(i)}=C_T^{(i)}\) 
\(\eta _i>(\beta _i1)/2\)  \(I_0^{(i)}<C_T^{(i)}\)  \(\gamma _C^{(i)}=C_T^{(i)}\) 
\(\eta _i>(\beta _i1)/2\)  \(C_T^{(i)}<I_0^{(i)}<=C_0^{(i)}/2\)  \(\gamma _c^{(i)} = \frac{\beta _i}{\eta _i}I_0^{(i)}(C_0^{(i)}  I_0^{(i)})\) 
\(\eta _i>(\beta _i1)/2\)  \(C_0^{(i)}/2<I_0^{(i)}\)  \(\gamma _c^{(i)} = \frac{\beta _i(C_{0}^{(i)})^2}{4\eta _i}\) 
The analysis shows dependence of the minimum amount of treatment required to saturate a subpopulation on the initial conditions. This dependence on initial conditions arises due to the dynamics of the SIS model in the presence of an economic constraint on available treatment resources. Therefore, even if all subpopulations are identical in terms of the epidemiological and economic parameters, different amounts of resource may be needed to saturate each subpopulation, depending on the initial level of infection within a given subpopulation. Since resources are limited, differences in the amount of resource required to saturate a subpopulation are important in determining the optimal allocation of resources.
2.3.2 The Optimal Strategy is to Saturate a Subset of Subpopulations
We have thus shown that the optimal allocation must lie on the boundary of the surface that defines the potential optimal strategies, that is we need to saturate subpopulations until further saturation is not possible. A key question therefore is how should the remaining resources be distributed amongst those subpopulations that are not saturated?
The question of where to allocate the remaining resources is the same as the problem of where to allocate resources if we cannot saturate any of the subpopulations. Suppose that in the optimal allocation, some subset X of the subpopulations share the resources, that is more than one subpopulation has nonzero amount of resources allocated to it. There is no coupling between the subpopulations and so we can consider the subset X in isolation. Since none of the subpopulations in X are saturated and none have zero resources allocated to it, as far as X is concerned, this allocation is an interior one. We proved above however, that there can be no interior local minimum of any number of subpopulations. Therefore, when no subpopulations can be saturated, all the resources should be allocated to a single subpopulation, that is \(\gamma _m = M/(k_m N_m)\) for some m and \(\gamma _i = 0\)\(\forall i\not = m\).
 1.
How should the subset of subpopulations that are saturated with treatment, S, be chosen?
 2.
Into which subpopulation should the remaining resource be allocated?
2.3.3 Which Subpopulations Should Receive Treatment?
2.3.4 Knapsack Approximation
One of the standard approaches to solving the knapsack problem computationally and the one we use here is the socalled Meet in the middle method (Horowitz and Sartaj 1974). This algorithm is a variation on the brute force approach searching through all the possible subsets S. The Meet in the middle algorithm consists of the following steps:
 1.
Split the n subpopulations into two subsets of approximately equal size in terms of the total value, A and B.
 2.
Find the total weight (\(\sum _i k_i\gamma _C^{(i)}N_i\)) and the total value (\(\sum _i(C_0^{(i)}  C_T^{(i)})N_i\)) of each subset of A and each subset of B.
 3.
For each subset of A, find the subset of B that maximises the value with the combined weight less than the limit M. This can be done efficiently as follows. First sort the subsets of B by weight. Then remove all subsets of B that have higher weight but smaller value than some other subset of B. That is, if for two subsets of B, weight \((S_1)\ge \text {weight}(S_2)\) but \(\text {value}(S_1)\le \text {value}(S_2)\), remove \(S_1\), because it will definitely not be in the optimal selection. After this procedure, the subsets of B are sorted both in weight and value. To find the subset of B that for some given subset of A maximises the value while having combined weight less than M, we can just use binary search.
2.3.5 Allocation of Remaining Resources
2.4 Calculation of True Optimal Strategy
To assess how well the approximate optimal allocation strategy given by the knapsack problem performs, we compare it with the exact optimal solution. Since we have proved that the optimal solution must saturate a subset of subpopulations, we obtain the exact optimal strategy by finding the optimal saturation set S that minimises the objective function (19) numerically. Specifically, we use a brute force approach and scan through all the subsets \(S\subset \lbrace 1,2,\ldots , n\rbrace \) which can be saturated given the resource constraint and select the one that minimises the objective function. This has computational complexity proportional to \(n2^n\), however since the largest n we consider is 11 such an approach remains computationally tractable.
3 Results
3.1 Performance of Knapsack Approximation
The knapsack approximation performed remarkably well for all three different parameter sets (Fig. 2). It only noticeably misses the optimal solution (black line) for a small range of resource limit values under the parameter regime in Fig. 2a, b, with an \(8\%\) error in the knapsack approximation compared with the exact optimum. In comparison all four simple strategies perform significantly worse than the optimal strategy. Indeed the allocate to the largest strategy does worst of all for a wide range of resource limits (Fig. 2a, c). In particular, the allocate to the largest often performs worse than the equal and proportional allocation strategies. This is surprising because the allocate to the largest strategy, like the optimal strategy, saturates some subset of subpopulations while under the equal and proportional allocation strategies it is possible that no subpopulations are saturated. The poor performance of the allocate to the largest strategy therefore suggests that the choice of which subpopulations to saturate is very important, and picking the ‘wrong’ subpopulations could mean that a socially equitable solution is preferable, even if no subpopulations are saturated.
These results suggest that the knapsack approximation performs well when there are many populations. However, due to the computational cost of solving the exact optimum when there are 11 subpopulations, this limits the number of parameter sets for which the knapsack approximation can be tested. Therefore, the knapsack approximation was also tested against the exact optimum when there are just three subpopulations, \(n=3\). In the case of \(n=3\) the exact optimum is relatively fast to compute, allowing us to compare the knapsack and exact optimum for 50, 000 different parameter sets. These 50, 000 parameter sets were generated as follows. The parameter values in the first subpopulation were kept fixed across all 50, 000 parameter sets with \(\beta _1=2\), \(\eta _1=1.2\), \(k_1 = 1\), \(N_1=100\) and the initial condition set to the endemic equilibrium. For the other two subpopulations, 50, 000 different parameter sets are generated randomly. More specifically a given parameter set for population i (\(i=2,3\)) is generated as follows: \(\beta _i\) is uniform on (2,3), \(\eta _i=\beta _i 1 + r\) where r is a uniformly distributed random number on (0,0.5), \(k_i\) is uniform on (1,1.5) and the population size (\(N_i\)) is uniform on (100, 1000). We note that \(\eta _i\) is set to be a function of \(\beta _i\) to ensure that \(\eta _i>(\beta _i1)/2\). In this way we obtain 50, 000 unique parameter sets that describe the population structure, epidemiological dynamics and economic costs for the 3 different subpopulations. Parameter values are chosen at random to ensure that we test the knapsack approximation over a wide range of parameter space.
The computational cost of solving the exact optimum is proportional to \(n2^n\) (n is the number of subpopulations) and so solving the exact optimum when \(n=11\) involves significant computational time. This limits the number of different parameter sets that could be used to test the knapsack approximation for the population structure given in Fig. 1 as the resource limit is varied. Therefore, the knapsack approximation was also tested against the exact optimum across a wide range of different parameter sets when there are just three subpopulations, \(n=3\). The parameter values for the first subpopulation were kept fixed with \(\beta =2\), \(\eta =1.2\), \(k_1 = 1\), \(N_1=100\) and the initial condition endemic. The parameters for the remaining two subpopulations were selected at random in the following manner. \(\beta \) is uniform on (2,3), \(\eta \) is \(\beta  1 + r\) where r is a uniformly distributed random number on (0,0.5) (this is to ensure that \(\eta >(\beta 1)/2\)), k is uniform on (1,1.5) and the population size (\(N_2\) and \(N_3\)) is uniform on 100, 1000. We considered 50,000 different sets of parameter combinations and for each the largest error as the resource limit is varied was computed, hereafter referred to as the worstcase error.
To understand the conditions under which the error in the knapsack is large (greater than \(6\%\)) we consider the relationship between the knapsack values (Eq. 24) of the two subpopulations whose parameter values are varied across the 50,000 different parameter sets tested, Fig. 3. When the worstcase error is large, the knapsack values in the two subpopulations are almost perfectly correlated with each other and in fact lie along one of three lines, \(v_3 = v_2\) and \(v_3 = v_2 \pm v_1\), where \(v_1\) is the value of the subpopulation whose parameter values are kept fixed for the 50,000 different parameter combinations tested (Fig. 3). These results suggest that the error between the knapsack approximation and the exact optimum will be largest when the knapsack values of one of the subpopulations are close to the knapsack value of one of the remaining subpopulations, or close to the sum of knapsack values of a subset of the remaining subpopulations. In these cases, the discrepancy in the total value from saturating different subsets of the subpopulations will be smaller, making it difficult for the knapsack approximation to determine which set of subpopulations it is best to saturate. Unlike the exact optimal solution, the knapsack ignores the interdependency between the set of subpopulations to saturate and where the resources remaining after no more subpopulations can be saturated should be allocated. The results here suggest that when the value of saturating subpopulations is highly correlated with each other, the interdependency between the choice of subpopulations to saturate and the allocation of the remaining resources becomes more important in determining the optimal allocation of resources. However, we note that while in these situations the knapsack approximation seems to perform less well, the maximum error between the knapsack approximation and exact optimum that we found was still only 16.7%. Furthermore, the worstcase error between the knapsack approximation and exact optimum was greater than 6% in only a very small number of cases (about 1.2% of cases tested here) suggesting that the knapsack approximation is very close to the optimal in the vase majority of cases.
We have shown that the knapsack approximation closely approximates the exact optimum. However, unlike the exact optimum, it provides analytic insight into the way to choose which subpopulations to saturate. This is since we have derived an analytic formula for the values and weights in the knapsack approximation in terms of the model parameters. In the next two subsections we use the knapsack approximation to gain insight into the characteristics that are important to capture within an allocation strategy.
3.2 Insight into Choosing Subpopulations to Saturate Using the Knapsack Approximation
The knapsack approximation associates a value and a weight to saturating each subpopulation. The value of the subpopulation represents the gain that is achieved from saturation (see Eq. 24). It depends on the size of the subpopulation \(N_i\), as well as the disease characteristics that are captured in the endemic equilibrium with and without treatment. The knapsack approximation shows that saturating larger subpopulations (greater \(N_i\)) is more advantageous. It also shows that diseases for which treatment leads to a greater reduction in the endemic equilibrium (greater \(C_0^{(i)}C_T^{(i)}\)) also lead to bigger gains in the objective function. However, unlike the simple strategies (as described in Sect. 2.2), the knapsack approximation also takes into account the cost of saturating a subpopulation (Eq. 25), and in particular how this depends on initial conditions when \(\eta _i>(\beta _i1)/2\).
Considering the subpopulations that are, and are not saturated by the different allocation strategies provides insight into the knapsack approximations superior performance. Both the knapsack and the allocate to the largest strategies saturate the largest subpopulation (subpopulation 3; Fig. 4a). This is because the large size of subpopulation 3 means that there are potentially large gains to be made from focusing treatment in this subpopulation. However, the allocate to the largest strategy also saturates the second largest subpopulation (subpopulation 2) while the knapsack approximation instead focuses treatment on the smallest subpopulation (subpopulation 1). While the value of subpopulation 2 is larger than subpopulation 1 (Fig. 4b), the higher initial level of infection in subpopulation 2 means that a greater amount of resources need to be allocated in order to saturate subpopulation 2. That is, the cost of saturating subpopulation 2 is greater than for subpopulation 1 (Fig. 4b). By focusing resources on subpopulation 1 instead, the knapsack allocation is able to saturate two subpopulations, while the allocate to the largest is only able to saturate subpopulation 3 as there are not enough resources remaining to saturate subpopulation 2 as well. On the otherhand, the allocate to the smallest, like the knapsack approximation, is able to saturate two subpopulations, namely subpopulations 1 and 2. However, as with the allocate to the largest strategy, the allocate to the smallest ignores the fact that saturating subpopulation 2 is much more costly, and less valuable, than saturating subpopulation 3 (Fig. 4b) since the initial level of infection is so high in subpopulation 2. This simple example illustrates that the knapsack performs so well because it accounts for both the value and cost of saturating a subpopulation with treatment. In particular, the cost of saturation is important since the more resources that are used treating one subpopulation, the fewer resources there are left to treat the remaining subpopulations. This indirect coupling that arises due to the shared limited resources is captured by the knapsack approximation but is ignored by all the simple strategies.
3.3 The allocation of Resources Across Identical Subpopulations
4 Discussion
In this paper we have investigated the allocation of limited resources across n separate subpopulations to treat infected individuals for SIStype epidemics. The shared resource pool introduces an effective interaction between the subpopulations, because allocating resource to one of them implies there will be less of the resource left for the others. This has the effect of anticorrelating the levels of infection in the different subpopulations. We assume that once allocated for disease control the resource cannot be reallocated later, meaning that the allocation strategy must take into account the longterm behaviour of the epidemic.
Using understanding of the longterm dynamics of the system, we have shown that the optimal allocation strategy involves saturating a subset of subpopulations with treatment, while other subpopulations receive none. This is similar to the findings for the two population case where resources should be focused within the population that is more/less infected depending on the epidemiological model and parameters, (Rowthorn et al. 2009; Ndeffo Mbah and Gilligan 2011). Characterising the full optimal allocation problem for \(n\ge 2\) subpopulations is complicated by the dependence of where the remaining resources are allocated on the subset of subpopulations that are saturated with treatment. Therefore, it is not possible to characterise the full optimal allocation strategy analytically.
By ignoring where the remaining resources are allocated we are able to approximate the full optimisation problem in a way that provides greater insight into how resources should be allocated between the different subpopulations. We term this approximately optimal strategy the knapsack approximation due to its similarity to the knapsack problem from computer science (Kellerer et al. 2004). Formulating the optimisation problem in this way associates a value and a weight to each subpopulation. The value of a subpopulation is the gain expected from saturation while the weight is the cost of doing so. A key advantage of the knapsack approximation is that it provides formulae for the values and weights of each subpopulation in terms of the epidemiological parameters. We showed that the knapsack closely approximates the exact optimum allocation strategy (obtained by brute force method) for a wide range of different parameter sets. Indeed in the worst cases, the knapsack approximation under performs the exact optimum by only \(10\%\). Furthermore, the knapsack is more computationally efficient than the exact optimum, particularly when the number of subpopulations (n) is large.
 1.
The knapsack captures the indirect coupling between subpopulations that arises due to the shared pool of resources. This is because it accounts for both the value gained in saturating a subpopulation, as well as the cost (weight) of saturation and therefore the reduction in resources left over for the remaining subpopulations.
 2.
The knapsack captures the dependence of the optimal solution on the initial levels of infection in each subpopulation. This arises due to the dependence of the minimum amount of resources needed to saturate a subpopulation on the initial conditions.
The superior performance of the knapsack approximation compared with other allocation strategies that involve saturating subpopulations, such as the ‘allocate to the largest’ and ‘allocate to the smallest’ strategies, shows that the choice of which subpopulations to saturate is highly important. Indeed a strategy that saturates the ‘wrong’ subpopulations can actually do worse than a more equitable allocation strategy. This is similar to the findings of Rowthorn et al. (2009) for an SIS epidemic who found that a strategy that focuses resources on the population with the highest prevalence performs worst of all, in the case of two identical connected populations.
A key goal of this work was to extend the results from Rowthorn et al. (2009), who consider the allocation of resources between two interacting subpopulations for an SIStype infection over a finite time horizon, to the general problem of n heterogeneous subpopulations. Due to the challenges of this ndimensional problem we made two simplifications; we assumed no interaction between subpopulations and we consider the allocation of resources over a long time horizon (\(T\rightarrow \infty \)). Therefore, it is difficult to compare our findings directly with those of Rowthorn et al. (2009). However, by considering 3 subpopulations that are identical except for the initial levels of infection, the similarities between the results we obtain from the knapsack approximation and those reported in (Rowthorn et al. 2009) are evident. We find that in this case, the knapsack strategy involves focusing treatment on the subpopulations with the lowest level of initial infection, since these subpopulations have the lowest costs associated with saturation. This is the same as for Rowthorn et al. (2009) who find that in the case of two subpopulations, limited treatment resources should be focused in the least infected region, where there are the most susceptibles. Therefore, the results from the knapsack approximation suggest that the results from (Rowthorn et al. 2009) hold in the general ndimensional problem for subpopulations whose size and epidemiological behaviour is identical. However, when the subpopulations are heterogeneous in terms of their size and/or epidemic dynamics it is a combination of population size, costs of infection, epidemiological parameters and initial levels of infection that determine where scarce resources should be concentrated. That may not necessarily be to the least infected subpopulations.
The objective function in Eq. (2) we use is a special case of a more general, commonly used objective function (Rowthorn et al. 2009; Ndeffo Mbah and Gilligan 2011) defined as the average cost of infection over a longterm horizon \(T\rightarrow \infty \), considering a short time horizon (Zaric and Brandeau 2001a) could significantly change the results. We do not explicitly consider any discounting of the cost of future infections (Forster and Gilligan 2007). Generally, due to how our objective function is defined, an exponential discount factor \(e^{rt}\) would only multiply the objective function by a constant and thus not affect the analysis in any way.
The situations we consider here are intentionally simplified with an SIS model, in order to gain insight into the optimal allocation. Our assumptions are therefore deliberately restrictive in order to make progress. We briefly consider the implications of the assumptions below and the options for dealing with more general and realistic epidemic scenarios. We have considered epidemics of SIStype, and so individuals can be reinfected which is typical of many diseases such as Chlamydia (Turner et al. 2006). However, for many diseases reinfection is preceded by a period of temporary immunity and so an SIRStype (susceptible–infected–recovered–susceptible) is more appropriate. The addition of a removed class can significantly impact the optimal allocation strategy, (Ndeffo Mbah and Gilligan 2011); therefore, it is important to consider the extension of our findings to SIRStype infections. This is not, however, straightforward since the dynamics of the SIRS model with an economic constraint on treatment are complex. In particular the longterm dynamics involve limit cycles (Vyska and Gilligan 2016). This means it is difficult to obtain a formula for the minimum amount of treatment required to saturate the population in the weights for the knapsack problem.
In this paper we assumed that subpopulations do not interact. While this is applicable to systems where each subpopulation represents a distinct pathogen threat to a given species, or when there is negligible contact between the two groups (e.g. this is typically the assumption when considering the spread of bloodborne diseases within injecting drug users and noninjecting drug users) in many systems interactions between populations are important in contributing to the invasion and persistence of a pathogen (Ferguson et al. 2001; Stacey et al. 2004; Dye and Gay 2003). For the case of two identical subpopulations, Rowthorn et al. (2009) show that interactions between populations can lead to nonintuitive allocation strategies. Therefore, an important extension to the work presented here would be to derive a simple heuristic to determine the approximately optimal allocation of limited resources amongst n interconnected subpopulations. Extending the approach taken here to include coupling between subpopulations is, however, not straightforward. Determining the form of the optimal allocation of resources relies on the ability to fully characterise the equilibrium behaviour of the system. Relaxing the assumption of no coupling between subpopulations means that we are no longer able to determine the equilibrium behaviour for the general problem of n subpopulations. Therefore, our approach does not generalise easily to the problem of n interconnected subpopulations, and more computational methods are most likely needed to determine the optimal allocation strategy in this situation. The challenge is that in moving to more computational based methods to determine the optimal strategy we loose the intuitive insight that the analytic approach taken here provides us, with the solution to the problem being more or less a black box.
We assumed that resources are allocated at the beginning and cannot be reallocated later. While this assumption is applicable to situations where reallocation of resources may be very expensive, it ignores the fact that we may need fewer resources to maintain the endemic treatment equilibrium than are needed to reach this state initially. Therefore, surplus resources could more efficiently be used by reallocation to another population. Another natural extension to our current work would thus be to consider how allowing for reallocation of resources alters the optimal strategy. Indeed in the case of two identical populations the optimal allocation strategy for both the SIS model (Rowthorn et al. 2009) and the SIRS model (Ndeffo Mbah and Gilligan 2011) involves the continual reallocation of resources.
In this paper we consider the optimal allocation of a treatment that increases the recovery period of an individual, for example application of a fungicide or administering antibiotics. Other control measures, such as antivirals which reduce viral load or improving condom use in the case of sexually transmitted diseases, reduce the transmission rate of infection. Brandeau et al. (2003) consider such a case and find that the optimal strategy depends on many factors including the size of each subpopulation, the state of the epidemic in each subpopulation before resources are allocated and the effectiveness of the control program. The approach taken here could be extended to obtain an approximately optimal allocation strategy for a control which reduced the transmission rate instead of increasing the recovery rate. Since we include the economic constraint directly into the model this would involve reframing of the problem and redoing the long time analysis. It is therefore beyond the scope of this paper and we leave such analysis for future work.
Optimal control theory provides a powerful tool to combine epidemiological dynamics with economic factors to determine the optimal allocation of resources, which is of particular importance when resources are limited. However, the solution to such problems can often be complex to implement when for example involving multiple switching times. Previous approaches to optimal control theory may provide little intuitive insight into how epidemic spread and economic constraints impact the way in which scarce resources can be most efficiently deployed. The advantage of the approach taken here is that it provides insight into the form of the optimal solution which allows us to obtain an approximate heuristic that is simple to interpret and implement. Such simple heuristics are important in the translation of theoretical results to application by decision makers for current and future pathogen threats.
Notes
Acknowledgements
We would like to thank Elliott Bussell for useful comments and discussions on early drafts of the manuscript and to an anonymous reviewer for useful comments which we feel have improved the manuscript.
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