The purpose of this paper is to use analytical method and optimization tools to suggest time-optimal vaccination program for a basic SIR epidemic model with mass action contact rate when supply is limited.
The Lagrange Multiplier Method and Pontryagin’s Maximum Principle are used to explore optimal control strategy and obtain analytical solution for the control system to minimize the total cost of disease with boundary constraint. The numerical simulation is done with Matlab using the sequential linear programming method to illustrate the impact of parameters.
The result highlighted that the optimal control strategy is Bang-Bang control – to vaccinate with maximal effort until either all of the resources are used up or epidemic is over, and the optimal strategies and total cost of vaccination are usually dependent on whether there is any constraint of resource, however, the optimal strategy is independent on the relative cost of vaccination when the supply is limited.
The research indicate a practical view that the enhancement of daily vaccination rate is critical to make effective initiatives to prevent epidemic from out breaking and reduce the costs of control.
The analysis of the time-optimal application of outbreak control is of clear practical value and the introducing of resource constraint in epidemic control is of realistic sense, these are beneficial for epidemiologists and public health officials.
Yang, K., Wang, E., Zhou, Y. and Zhou, K. (2015), "Optimal vaccination policy and cost analysis for epidemic control in resource-limited settings", Kybernetes, Vol. 44 No. 3, pp. 475-486. https://doi.org/10.1108/K-05-2014-0103Download as .RIS
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Mathematical models are generally used to investigate disease spread, and optimal control theory based on mathematical models has gained dramatically wide use for suggesting effective strategies to prevent the widespread and avert the destruction caused by epidemic. One of the most classical models is the susceptible-infectious-recovered (SIR) model proposed by Kermark and McKendrick (1972) and has adopted in many researches to study pandemic flu (Carrat et al., 2006; Mills et al., 2004), seasonal flu (Bridges et al., 2003; Cauchemez et al., 2004), SARS (Lipsitch et al., 2003; Riley et al., 2003), HIV/AIDS (Magombedze et al., 2009; Okosun et al., 2013), smallpox (Elderd et al., 2006; Riley and Ferguson, 2007) and other infectious disease. These studies use SIR models to simulate the disease outbreak and evaluate the effectiveness of selected control measures under various predefined scenarios.
The traditional SIR model could be described by the differential equations:
where S and I are the numbers of susceptible and infected hosts (a dot indicating time derivative), β is the transmission rate, μ is the per capita loss rate of infected individuals through both mortality and recovery, and u is the per capita rates of vaccination.
As vaccine offers one of the best population-level protections against illness, regular and widespread coverage of vaccination is a main concern of epidemiological modeling to implement an optimal vaccine allocation to the population. Some of the earlier works in this area was by Abakuks (1974). Abakuks (1974) investigated the optimal control of a simple deterministic SIR model, and determined the optimal vaccination strategy under the assumption that, at any instant, either all or none of the susceptibles are vaccinated. Shortly after the publication of Abakuks (1974), Morton and Wickwire (1974) studied the same problem with the assumption that some finite rate of isolation or vaccination is possible and the objective is to minimize the total infectious burden over an outbreak. Based on this, Behncke (2000) expanded Wickwire’s results to models with more general contact rates.
In recent works, Tanner et al. (2008) present a stochastic programming framework for finding the optimal vaccination policy for controlling infectious disease epidemics under parameter uncertainty. The problem is initially formulated to find the minimum cost vaccination policy under a chance constraint. Hill and Longini (2003) given a population with m heterogeneous subgroups and developed a method for determining minimal vaccine allocations to prevent an epidemic by setting the reproduction number to 1. Rodrigues et al. (2014) developed vaccination models to simulate a hypothetical vaccine as an extra protection to the population. In a first phase, the vaccination process is studied as a new compartment in the model, and different ways of distributing the vaccines investigated: pediatric and random mass vaccines, with distinct levels of efficacy and durability. In a second step, the vaccination is seen as a control variable in the epidemiological process. In both cases, epidemic and endemic scenarios are included in order to analyze distinct outbreak realities.
Although the analysis of the vaccination scheme in epidemic control has gained attentions, surprising little analysis has been done for a very real practical issue to concern and determine the optimal vaccines distribution for limited resources (Hansen and Day, 2011; Zhou et al., 2014). As in common, the less developed areas where have poor resources such as no advanced medical equipment, no enough amounts of drugs as well as no sufficient funds to combat disease suffer most from the large-scale destruction of infectious disease. Moreover, even in developed areas, the resources are usually limited since the vaccines and other drugs are stockpiled with a fixed amount in preparation for an epidemic outbreak. This raises a critical problem that once the epidemic starts, how to optimally administer these limited resources varies with time so as to achieve some specific goal (e.g. minimizing total cost or outbreak size of disease)? Such time-optimal control strategy is the focus of this paper.
Hansen and Day (2011) investigate the optimal control of epidemics based on the SIR model under the assumption of limited resource to suggest optimal isolation only, vaccination only and combination policy, the objective to minimize the outbreak size followed as:
Based on the research of Hansen and Day (2011), we explore the optimal control strategy of epidemic under the limited resource setting but we only focussed on vaccination strategy, so the inequality constraint of limited vaccine supply follows as:
where the left part represent the population of people being vaccinated and ω represents the total amount of vaccines available over the time interval [t 0, t f ]. By definition, vaccination has a direct effect only on susceptible individuals. From (1), we have:
if the initial states S(t 0) > 0, I(t 0) > 0.
However, sometimes we have to focussed on the cost of control, especially in the less developed areas, so our objective is to design a vaccination program intensity over time such that the total cost (intensity) of disease is minimized (see also in Behncke, 2000; Castilho, 2006; Sethi, 1978; Wickwire, 1975; Zhou et al., 2014), i.e.:
Moreover, we are also interested in the cost changes of vaccination under optimal policy.
Infection imposes a constant per period economic cost of C d per infected per period, which includes the value of lost labor income and leisure, or pain and discomfort. The cost of vaccination is C(u(t)), where this is interpreted as total social cost of vaccination and is assumed to exhibit increasing marginal cost. These costs include materials, facilities and labor used in administering vaccine. It also includes the resources (including time) used by the patient and the value of inconvenience and discomfort in undergoing vaccination. The rationale for increasing marginal costs stems both from short run inflexibilities and the Le Chatelier Principle as well as any inherent diseconomies which may persist in the long run.
In this paper, a function of constant marginal flow cost C(u)=ku was adopted (see also in Behncke, 2000; Castilho, 2006; Sethi, 1978; Wickwire, 1975; Zhou et al., 2014) to in the objective to reveals the intensity of cost of vaccination, where k 0 is a measure of the relative cost of vaccination over a finite time period (k is also regarded as a weight constant) This would be appropriate for the case in which there are no congestion costs – a situation in which the disease was easy to diagnose and evaluate. The use of vaccine would be the case.
Thus, our objective to design vaccination program intensity over time with the goal to minimize the total cost of disease is followed as:
The purpose of this paper is to provide a theoretical framework to investigate an optimal vaccination control problem varies with time under limited resource, the basic SIR model is taken into account to obtain an analytical solution so as a foundation that more complex models can be built upon these rigorous mathematical results, despite the epidemiological model is simple the analysis is not trivial yet. The remaining parts of this paper are organized as follows: In Section 2, we explored the Lemma of our optimal control system to give proof of optimal policy. In Section 3, we gained the analytical results of the optimal control problem. Then sensitive analyses are discussed in Section 4 by simulated results to illustrate the impact of parameters. We draw a conclusion in Section 5.
The primary method applied to solve this problem is the Pontryagine’s Maximum Principle (Pontryagin, 1987) and we introduce Zhou et al. (2014) and Kamien and Schwartz (2012) for more details. We consider the following optimal control problem with isoperimetric constraints:
where, x∈n is the state vector, u∈m is the control vector, , L, L1, L2 and ψ are vector functions of their respective variables, and have continuous partial derivatives with respect to all of their arguments, ω is a constant vector, U is an admissible control region.
Lemma 1. For optimal control problem (5), we can get a Lemma like this: if u(t) is an optimal control with x(t) being the corresponding optimal path, then there exist nontrivial vector functions λ and nontrivial constant vectors λ 1, λ 2 and ν such that the following conditions are satisfied:
where (Equation 10) is Hamiltonian, G(t f , x(t f ))=(t f , x(t f ))+ν t ψ(t f , x(t f )).
Notice that if the optimal system (5) is autonomous, then the Hamiltonian H is constant, i.e.:
3. Analytical results for optimal control
With the preparation of Lemma 1, it is not difficult to see that the optimal control problem (4) admits an optimal solution (Bryson, 1975; Hull, 2003; Zhou et al., 2014). So, we only need to find the necessary conditions.
The corresponding Hamiltonian H(S, I, u, λ S , λ I , λ) for control system (4) is denoted as:
where λ S , λ I are co-state variables, λ is a constant. By the Lemma mentioned before, we have the following necessary conditions:
From (12), u is chosen to minimize H so:
where (Equation 24) represents the marginal cost of vaccination, λ S describes the marginal cost of susceptible. Equation (18) depicts the relationship between the marginal cost of vaccination and the marginal cost of susceptible.
The cost of vaccination over a time period is:
With the assumption of C(u) = ku, we will show that the optimal control is purely Bang-Bang control (that is, has no singular components).
First, in fact, from (12), we then have:
and, by (17):
which implies that:
If, now, k−λ S S+λ S =0 on some sub-interval J⊂[t 0, t f ], then:
i.e., λ S is increasing on J, which implies that there exists a point t c ∈ (t 0, t f ) such that (Equation 32) due to the differentiability of λ S and λ S (t f )=0. This, in turn, will results in:
Thus, from (8), we get:
which contradict (3). Hence, the optimal control must be purely Bang-Bang.
Notice that k−λ S S+λS < 0 cannot occur on some sub-interval [t c , t f ] ⊂ [t 0, t f ] due to λ S (t f )=0, so u = u max on some final sub-interval [t c , t f ] is not optimal.
Next, we will show that u≡0 is not optimal for the problem (4).
Last, we prove that the optimal control u≡0 cannot also begin in initial stages. If not, there exist two switches between 0 and u max , that is, there exist two times (Equation 38) such that:
From (23), (24) and (8), we have:
which contradicts (21).
Therefore, we have the following result:
Theorem 1. For the optimal control problem (4), there exists a τ ∈ [t 0, t f ] such that the optimal vaccination policy is:
It means the practical optimal vaccination policy is to vaccinate with maximal effort until either all of the resources are used up or the epidemic is over. Moreover, the marginal cost of vaccination is:
the marginal cost of susceptible is:
and the cost (intensity) of vaccination over time is:
the total cost of disease is:
4. Sensitivity analysis and discussion
We compute the solution for numerically specified parameter values which are empirically relevant and where we consider one time period as 100 days. The computations are done with Matlab using the sequential linear programming method. In this paper, we set the initial state and assume the basic parameters as S(t 0)=1,000, I(t 0)=10,C d =1, k=1 t 0=0, t f =100, μ=0.03, β=3e−4, u max =0.05, ω=680. The state path S(t)and I(t) and the optimal control u for those basic parameters are shown in Figure 1. It indicates that the optimal policy is to vaccinate with maximal effort with a vaccination rate u max =0.05, the control is consequently effective on the one hand to sharply decrease the population of susceptible individuals and on the other hand although the number of infectious individuals increase at the beginning of the emergence of epidemic it ultimate reduce from the 30th day and finally down to a small scale.
To explore the effect of parameters on optimal vaccination police (established by the variation of switch time τ) as well as on the cost of vaccination C, we carry out a sensitivity analysis by simulations, see Figures 2 and 3 and Table I, which focus on studying the role of varying the total amount of vaccines (ω), the control weight constant (the relative cost of vaccination k) and the control upper bounds (the maximum vaccination rate u max ).
Total amount of vaccines
Figure 2 is plotted with different values of the total amount of vaccines ω and keeping other parameters unchanged. From Figure 2, we can find that for a given u max , as the total amount of vaccines are increased, the switch time τ (which implies the time-length of vaccination) and the cost of vaccination C are also increased until unchanged (ω700) which implies that the optimal system is unconstraint.
We choose three values of the weight constants, k=1, k=3 and k= 5 in two cases without constraint and with constraint, respectively, and keeping the rest of the basic parameters unchanged (similarly hereinafter), a comparison of results is shown in Table I. In the case without constraint such as ω700, the time-length of vaccination is decreased as the weight constants are increased, while the cost of vaccinations is increased with k because we can see from (28) that C = ku max τ since t 0=0.
But, in the case with constraint such as ω680, the time-length of vaccination keeps unchanged, while the cost of vaccination is linearly increasing as the weight constants increased. This can be proved from (15). In fact, with constraint (2), by (15) and (25), we have:
which means that τ is independent on the weight constant k owing to (8) and (9). Therefore, the optimal policy of vaccination is completely independent on the cost of vaccination when supply is limited.
Upper bounds of control
Figure 3 is plotted with different upper bounds umax on controls and two values of the total amount of vaccines, ω=680 and ω=700. From Figure 3 we can find that the variation of switch time τ and the cost of vaccination C can be divided into three stages with the changes of umax.
In the first stage, both τ and C approximately equal to 0 with the extremely small value of umax. This strange phenomenon in actually fit with the fact that a too small vaccination rate gives no help to prevent an outbreak of epidemic no matter how long the vaccination period sustained, so under the objective of minimizing the total cost of disease, the optimal strategy is just do nothing. It worth noting that although the cost of vaccination C is equal to 0, the total cost of disease will hold on a high level because a low vaccination rate lead to a large number of infected people I which makes a rapid increasing in total cost of disease. In the second stage, the switch time τ keeps on a larger value and remained unchanged as 84.5 days, while the cost of vaccination increases linearly with umax, it implies that the resource is sufficient and unconstraint to the corresponding vaccination rates. During the third stage, the switch time τ and the cost of vaccination C sharply decrease with the umax increase, and the tendency sustained despite the speed of reduction gradually slow down. These phenomena indicate a practical view that the enhancement of daily vaccination rate is critical to make effective initiatives to prevent epidemic from out breaking and reduce the costs of control.
The resources are usually limited. It is critical that the resources are administered in a time-optimal fashion. This paper uses analytical method and optimization tools to suggest vaccination program intensity for a basic SIR epidemic model when vaccine supply is limited. We obtain analytical and numerical solution for the control to minimize the total cost of disease. The optimal vaccination strategy is Bang-Bang to vaccinate with maximal effort until either all of the resources are used up or epidemic is over. Specially, the optimal vaccination strategy is independent on the relative cost of vaccination when the supply is limited. Simulated results mean that optimal strategies and total cost of vaccination are usually dependent on whether there is any constraint of resource, and the total cost is direct proportion to the parameter such as the total amount of vaccines, the relative cost of vaccination and the priori maximum vaccination rates.
Dynamic of infection is certainly far more complicated and varied than the one captured by this mathematical model. But, it illustrate the role that mathematical methods can play in formulate treatment strategy.
Professor Kuan Yang can be contacted at: email@example.com
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The authors are grateful to the handling editor and reviewers for their kind comments to our previous manuscript, which were most helpful in its improvement. This research was supported by the Natural Science Foundation of China (No. 71272209), and Humanity and Social Science Foundation of Ministry of Education of China (No. 12YJA630170) and the Natural Science Foundation of Hunan Province (No. 12JJ3081).