Citation: Francesco Salvarani, Gabriel Turinici. Optimal individual strategies for influenza vaccines with imperfect efficacy and durability of protection[J]. Mathematical Biosciences and Engineering, 2018, 15(3): 629-652. doi: 10.3934/mbe.2018028
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[9] | Sheng-I Chen, Chia-Yuan Wu . A stochastic programming model of vaccine preparation and administration for seasonal influenza interventions. Mathematical Biosciences and Engineering, 2020, 17(4): 2984-2997. doi: 10.3934/mbe.2020169 |
[10] | Eunha Shim . Optimal strategies of social distancing and vaccination against seasonal influenza. Mathematical Biosciences and Engineering, 2013, 10(5&6): 1615-1634. doi: 10.3934/mbe.2013.10.1615 |
Vaccination is a widely used epidemic control tool which may (and should) be analyzed from several perspectives, such as the design of fabrication techniques, the study of its action mechanisms, the analysis -at the individual level -of the medical issues of the vaccine, including its side effects, and the global impact on the epidemic spread of some carefully designed vaccination protocols.
Obviously, these different viewpoints are strictly interconnected: for example, the action mechanism of a vaccine determines its features and its protection effect against the target illness, and the public health strategies are a consequence of the former two aspects.
When looking at vaccination policies, two approaches are possible.
The first one supposes that a health authority can decide of a vaccination plan, which is then implemented. The plan optimizes the vaccination strategy as a function of the severity of the epidemic, its medical risks and the (economic and medical) costs associated to the vaccine.
This framework, that is suitable for compulsory vaccination or when the individuals fully adhere to the recommendations of the health authority, has been the first one considered in the literature (see [39,1,52,59,23,4,46]).
However, this kind of studies is oriented to the best possible strategy for the population as a whole, and it does not take into account the individual viewpoints. Indeed, when the vaccination is a choice -on a voluntary basis -or when there are debates on the risks or costs of the vaccine, the previous approach is not valid anymore and the situation is better described by models that take into account the individual decision level.
In this second group of models, the agents decide for themselves whether the vaccination is suitable or not, but they cannot individually influence the epidemic propagation, which is given by the collective choice of the population as a whole.
The study of the collective behavior of large populations of non-cooperative interacting individuals -such as the problem considered in the present article -is a complicated problem, but it recently received a firm mathematical ground thanks to the Mean Field Game (MFG) theory, introduced in the literature by the pioneering works of Lasry and Lions [49,48,50] and of Huang, Malhamé and Caines [41,40]. From the point of view of modeling, mean field game theory combines mean field theories, which are widely used in Physics and Mechanics, together with the notion of Nash equilibria in game theory.
One of the main goals of MFG is the study of the existence of equilibria for the whole population, namely a stable collection of individual strategies such that nobody has any incentive to change his own strategy.
Before the development of the MFG theory, some earlier works were already looking into this direction. We quote, for example, [30,12,35] which study the question of disease eradication, market equilibrium and externalities regarding vaccination. More recent contributions (see [7,6,60,31,58,47]) study the question of Nash equilibria for a large number of individuals dealing with an epidemic. They investigated many aspects, such as the impact of the subjective perceptions and individual behaviors on the equilibrium (see, for example, [19,18,57]), the presence of several groups having distinct epidemic characteristics (see [34,20,16]), particular vaccination strategies or specific models about the available information at the individual level (see [13,8,27,25,26,33,69,11,24]).
In this article, we introduce and analyze a model for a non-compulsory vaccination campaign against influenza viruses, with imperfect vaccine efficacy and limited durability of protection (sometimes also called persistence as a shorthand for persistence of antibodies). Our main purpose is the computation of the optimal individual strategy (which allows to deduce the fraction of the population which chooses to be vaccinated in absence of a specific obligation), with the purpose of helping decision makers in designing efficient public health policies. Since compulsory vaccination is the source of some ethical issues on informed consent and individual freedom, knowledge of the optimal individual strategy is an essential step before deciding that a given vaccine is mandatory. On the other hand, advertisement campaigns for non-compulsory vaccination are effective only if their goal is compatible with the optimal individual strategy.
As the features of the target infectious disease heavily influence the dynamics of the epidemic spread, our model cannot be immediately generalized to the vaccination against other diseases. However, the global strategy can be easily modified, mutatis mutandis, for obtaining models adapted to other situations with a similar behavior (i.e. the vaccine is imperfect and the immunity is not permanent).
We focus our attention on countries with temperate climates, which experience a marked seasonal influenza peak during the winter months [66]. Hence our time horizon will be annual.
Since influenza is a contagious disease, the major available tool against the spread of the illness is given by vaccination.
However, vaccination has no permanent effect. Indeed, as pointed out in [21], the protection against a virus, provided by the corresponding vaccine to an individual, persists after some years and it is still effective in case of slight genetic mutations. But, because of the antigenic drift [15], sufficient changes can accumulate in the virus to allow influenza to reinfect the same host. The protection given by a previous vaccine can hence become useless. In order to overcome this phenomenon, the influenza vaccine formula is annually reviewed. Note that vaccine mismatch is not taken into account in our study, because it would lead to introduce different questions, oriented to the modeling of the vaccine itself, rather than to the vaccination policies (which suppose, of course, that the annual release of the vaccine has a good efficacy).
We moreover suppose that the immunity provided by the vaccine is time-depend-break ent. Indeed, as pointed out in several studies, the estimated protection against infection, based on hemagglutination-inhibiting (HAI) antibody titers has a maximum 2-4 weeks after the vaccination, and it subsequently strictly decreases afterwards [71,54]. In particular, [54] estimates that there is a marked decline of the immunity some months after the vaccination. This behavior is taken into account in our analysis because it can be practically observed before reaching the time horizon of the problem.
The aforementioned features of the illness will be considered, in this article, as given data. Two main attributes of the vaccine are considered:
- the durability of protection, that can span from several months up to several years -see [21,17,5] and the literature therein;
- the vaccine efficacy (noted VE, an input in our model), which is the theoretical success rate (to be distinguished from the vaccine efficiency, which is the practical observed success and is the output of the model - see [70] for a presentation of the differences between the two). The VE can range from several percents to almost perfect efficacy -see the meta-analysis in [56] and also [55]; other references include [51] and [67]. The VE can have effects on susceptibility, infectiousness, disease progression, and so on; we only consider here the impact on susceptibility, thus our VE is more specifically, with notations in [38,Section 2.2], of
Hence, our model is suitable for studying imperfect vaccines and takes into account not only the individual decision about the vaccination, but also the best timing of the vaccination if the individual decides to be vaccinated.
Since the choice of the best timing problem of a vaccination campaign is very actual and it is carefully studied by the health authorities, we hope that our model can give a contribution to a better understanding of the vaccination dynamics in order to suggest efficient policies. In particular, our model forecasts that, in the non-cooperative setting, when the protection given by the vaccine is not optimal, the individual behaviors are only partially in agreement with the suggestions of the World Health Organization (WHO), which encourages vaccination as soon as the vaccine of the corresponding seasonal influenza is available [22]: the agents could tend to delay the vaccination in order to arrive at the peak of the epidemic with the best possible protection (see [62,Section "Vaccination Before October"] and also [3,29,44,9,61] for recent references to intra-season waning of the vaccine-induced immunity and its impact on vaccination timing).
Because of the relatively short time horizon of the model, we do not consider any population dynamics, or any reinfection, since we suppose that antigenic drift is not very important on such small time scales [15].
From the mathematical point of view, in this article we work in a discrete setting and our model is described in terms of Markov chains. As far as the time horizon of seasonal influenza has the order of magnitude of one year, this choice allows us to model the coarse graining of the real situation and makes this model more suitable for the applications.
Firstly, we prove that the individual vaccine model proposed here admits an equilibrium. However, up to our knowledge, the equilibrium is not explicitly known.
Far from being a disadvantage, this situation prompted us into proposing a general numerical method to find the equilibrium; this is a second contribution of this work (see also [65] for some alternatives coming from the physics community for general Mean Field Games). The numerical method is adapted from general works in game theory (see Section 3) and is expected to give accurate results in any situation when an individual chooses the right timing to perform some action (here vaccination) with time-dependent costs. This procedure has been extensively tested in our model and performs in a satisfactory way.
The structure of the paper is the following: the model is presented in Section 2 and the theoretical result guaranteeing the existence of an equilibrium in Section 2.3. The numerical algorithm for finding the equilibrium is presented in Section 3 and the numerical results in Section 4.
First of all, the numerical simulations describe a standard situation for seasonal influenza dynamics. Subsequently, we test our model on two extreme cases (the duration of the immunity is of one or six months only, see also [29]), which show some striking behaviors of the population and which may help to understand the strategic policies of the population.
Section 5 collects some considerations on the pertinence and validity of our approach.
The model studies the dynamics of an epidemic in a population. In what follows we will suppose that
- the infection does not cause the death of the patient (as it is well known, the mortality associated to influenza does not induce significant modifications in the population structure [28]); moreover, by considering a time horizon of twelve months, we suppose that births and deaths, as well as age shifts, are non relevant;
- after the disease, the individuals who have been infected acquire permanent immunity (throughout the time horizon of our model): this means that we will suppose the existence of a predominant virus strain, instead of considering a mixing of viruses and therefore reinfection is a rare phenomenon;
- the incubation period is short when compared to the time scale of the model;
- the individuals can be vaccinated. If the vaccine is successful, the protection of the vaccine is maximal (but possibly not total) after a time delay, it remains high during some period and then it decreases (see [54]);
- the vaccine is imperfect and the imperfections can be of two kinds (see [63,38,64] for further discussions): ⅰ) a all-or-none effect, where a fixed fraction
- the evolution of the epidemic can be influenced by seasonal effects, as is it the case of influenza in temperate regions.
In what follows, we describe the model in pure mathematical terms, the quantification of the different parameters will be then discussed in Section 4.
We suppose that the time horizon
- susceptible individuals:
- infected individuals:
- recovered individuals, is the proportion of individuals once they recover from illness (after leaving the class of infected individuals);
- vaccinated individuals:
- failed vaccinated individuals:
The quantities
The upper bound
Similarly,
The equations of the model, which conserves the total number of individuals, have the following form:
Sn+1=(Sn−Un)−βnΔTIn(Sn−Un) | (1) |
I0n+1=βnΔT[Fn+Sn+N−1∑θ=0αθVθn]In | (2) |
Iω+1n+1=(1−γωΔT)Iωnω=0,…,Ω−1 | (3) |
V0n+1=(1−f)(1−βnΔTIn)Un | (4) |
Vθ+1n+1=(1−βnΔTαθIn)Vθn,θ=0,…,Θ−2 | (5) |
VΘn+1=(1−βnΔTαΘ−1In)VΘ−1n+(1−βnΔTIn)VΘn | (6) |
Fn+1=f(1−βnΔTIn)Un+Fn(1−βnΔTIn) | (7) |
with initial conditions
S0=S0−,Iω0=Iω0−,Vθ0=0,∀θ≥0,F0=0, | (8) |
where
-
- the vector
- The function
- The vector
In what follows, we suppose that there exists a maximal time
Let
We work under the meaningful assumption that
The total societal cost associated to the vaccination strategy
J(S0,I0,U)=rIN∑n=0In+rVN−1∑n=0Un, | (9) |
which has to be minimized (see [39,1,52,59,23,46]) within the set of all admissible vaccination strategies
However this is not the strategy followed by individuals. They rather optimize an individual cost function. In order to define it, we have to consider the individual dynamics (see Figure 2 for an illustration). It takes the form of a controlled Markov chain with several states, susceptible (S), failed vaccination (F), recovered (R), infected (indexed by the time counter
The Markov chain of the individual, denoted
P(Mn+1=S|Mn=S)=(1−λn)(1−βnΔTIn)P(Mn+1=I0|Mn=S)=βnΔTInP(Mn+1=V0|Mn=S)=(1−f)λn(1−βnΔTIn)P(Mn+1=F|Mn=S)=fλn(1−βnΔTIn)P(Mn+1=R|Mn=IΩ)=1P(Mn+1=R|Mn=Iω)=γωΔT,ω=0,…,Ω−1P(Mn+1=Iω+1|Mn=Iω)=1−γωΔT,ω=0,…,Ω−1P(Mn+1=I0|Mn=Vθ)=αθβnΔTIn,θ=0,…,Θ−1P(Mn+1=Vθ+1|Mn=Vθ)=1−αθβnΔTIn,θ=0,…,Θ−1P(Mn+1=I0|Mn=VΘ)=βnΔTIn | (10) |
P(Mn+1=VΘ|Mn=VΘ)=1−βnΔTIn,P(Mn+1=I0|Mn=F)=βnΔTIn,P(Mn+1=F|Mn=F)=1−βnΔTIn. |
The conditions
The conditional rates
There is a mapping between
ξ∞=N−1∏n=0(1−λn),ξn=λnn−1∏k=0(1−λk),n≤N−1 | (11) |
∀n≤N−1:λn={ξnξn+…+ξ∞,ifξn+…+ξ∞>00,otherwise. | (12) |
The cost of a vaccination strategy depends on
- the cost
- the cost
- the cost
Note that an individual may incur both costs if he vaccinates and, moreover, if he is infected. For an individual starting at
Jindi(ξ;U)=rVP(∪n<N{Mn+1=V0,Mn≠V0}|M0=S)+rVP(∪n<N{Mn+1=F,Mn≠F}|M0=S)+rVP(∪n<N{Mn+1=I0,Mn≠I0}|M0=S). | (13) |
This form for
- the probability
ψV,In=1−Θ∏k=n(1−βnΔTαk−n−1Ik), | (14) |
where we introduce the coefficient
- the conditional probability of being infected (strictly) before
φIn=P[∪nk=0{Mk=I}|M0=S,Mk≠V0,Mk≠F,k≤n], |
given by the formula:
φIn=1−n∏k=0(1−βnΔTIk),∀n<N−1. | (15) |
Note that the probability of being infected after the time
1−1−φI∞1−φIn=φI∞−φIn1−φIn, |
where
φ∞=1−N−1∏k=0(1−βnΔTIk). |
Then, after elementary computations:
Jindi(ξ;U)rIφI∞ξ∞+N−1∑n=0[rIφIn+(1−φIn)(rV+(1−f)rIψV,In)+rIf(φI∞−φIn)]ξn. | (16) |
The individual cannot change
gUn={rIφIn+(1−φIn)(rV+(1−f)rIψV,In),+rIf(φI∞−φIn),forn≤N−1rIφI∞forn=N. | (17) |
If we denote the Euclidean scalar product between two vectors
⟨X,Y⟩:=N+1∑k=1XkYk, | (18) |
then
Now, for a given individual policy
Un=λnSn, | (19) |
i.e.
A simplified model can be proposed to tackle the possibility of vaccination failure. Note that, for
gUn=rIfφI∞+(1−f)[rIφIn+(1−φIn)(rV/(1−f)+rIψV,In)]. |
Therefore, since the term
Note however that this is a first order approximation as, in practice, the quantities
Consider now the following mapping: for any given probability law
Let
The goal of this subsection is to deduce the existence of an equilibrium of the system, i.e. a common strategy which is a Nash equilibrium when it is used by all agents of the population. The following result holds.
Theorem 2.1. There exists at least one law
Proof. We use Kakutani's fixed point theorem (see [43,page 457]) for the function
∑N+1={(x0,…,xN)∈RRN+1|xk≥0,x0+…+xN=1}. | (20) |
Recall that the assumptions of the theorem are the following:
1. for any
2. the mapping
The only hypothesis to check is the closed graph property of
We denote by
Let
Consider
Although
Since all rates
Remark 1. The theorem reduces the existence of the equilibrium to the study of the mapping
Remark 2. The result does not give any information about the uniqueness of the fixed point. In the Mean Field Game framework, uniqueness results usually from convexity considerations (see e.g., [49,48,41]) and it is not guaranteed, see [45] for a situation where there is no uniqueness. Although this setting is not convex, in all numerical simulations we pursued, a unique solution has always been found.
The result of the Section 2.3 guarantees the existence of at least one equilibrium. But, it does not prescribe a constructive method to find it.
For arbitrary strategy
E(ξ)=⟨ξ,Cξ⟩−minη∈ΣN+1⟨η,Cξ⟩. | (21) |
Note that
A natural idea is then to try to minimize
Another idea is simpler and intuitively more appealing: the equilibrium will be found by successive approximations in a way that mimics a real-life repeated game (see also [32] for additional considerations). Consider a strategy candidate
ξk+1isaminimizeroverΣN+1ofη↦dist(η,ξk)22τ+⟨η,Cξk⟩, | (22) |
where
The term
ddτξ(τ)=Cξ(τ). |
In order to keep the presentation as simple as possible, we used as distance in (22) the standard euclidian distance on
In practice, the algorithm applied is the following:
Step 1. Choose a step
Step 2. Compute
Step 3. If
In practice Step 2 is computed with a quadratic programming routine (quadprog in Matlab/Gnu Octave) that can accommodate linear constraints.
Remark 3. The procedure proposed above can be extended in a straightforward manner to any 'rational individual' vaccination model, by replacing the vector
In order to test the model, we simulated the situation of an epidemic with several sets of parameters, such as long or short durability of protection, as indicated below.
We first tested the procedure for a situation when the analytic result is known (see [31,47]): we used the parameters in [47, Figure 5] and obtained that the optimum individual strategy is a mixed strategy with
The numerical values used in this simulations are the following: total simulation time
We set the vaccine efficacy to
A(t)=1−c1tc2e−c3t, | (23) |
with constants
We considered first the 'ideal' case of instantaneous and non-decaying immunity. The corresponding equilibrium is a policy when people vaccinate at
Then the equilibrium for
Finally, a different situation when immunity falls to
The differences between these three situations, both in the vaccination timing and in the fraction of vaccinated individuals, are a consequence of the optimal criterion of the agents, of the durability of protection of the vaccine and of the nonlinearity of the contagion process. Due to the partial loss of immunity before the end of the season, the agents tend to delay the vaccination in order to have the best possible protection during the peak of epidemics. On the other hand, the reduction of the efficiency of the vaccine motivates more individuals to be vaccinated in order to increase group immunity and hence to reduce the contagion.
We investigate in this subsection the robustness of the numerical results with respect to various choices of parameters, in particular the durability of protection and efficacy, as detailed below. The results of these tests show that the freedom of choice to obtain the best individual result could lead to more expensive individual costs than those obtained in a regulated setting, where public health authorities prescribe individual policies (this phenomenon is the so-called cost of anarchy, see Subsection 4.3.1). On the other hand, we show that imperfect vaccines (i.e., with short durability of protection and limited efficacy) may be also acceptable, for a fraction of the population, as a tool for reducing the contagion process, even if the vaccination is not compulsory.
To define the durability of protection of the vaccine we set
The results are displayed in Figures 5, 6 and 7. A good quality equilibrium is found, that is, the incentive to change the strategy
The solution is a strategy
It should be mentioned that the solution
The parameters are identical as in Subsection 4.3.1, except the durability of protection of the vaccine time
In this Subsection, we test a situation when the vaccine efficacy is only
- the probability of the non-vaccinating strategy is now
- the cost of the optimal strategy is
Therefore the equilibrium shifts towards a bigger fraction of the population that vaccinate (in order to compensate lower vaccine efficacy). However, the overall number of protected people is lower (
We also compared the previous result with the output of the model obtained by setting the cost
We analyze in this subsection the effects of the failed vaccination rate on the overall vaccination policy. The numerical value of the vaccination cost is
Failed vaccination rate |
Vaccination rate |
When the failure rate
When the failure rate is small, individuals tend to vaccinate more to compensate the decrease in efficacy and therefore to contribute to the group protection and to profit from it. However, after a given threshold, the construction of a group protection is too expensive, and consequently the individuals are reluctant to vaccination (if
We analyzed in this work the vaccination equilibrium in a context of rational individual vaccination choices; the situation is modeled as a Nash equilibrium with an infinity of players. In our work, a special attention is given to the presence of imperfect vaccines. We presented a theoretical approach (existence of an equilibrium via the Kakutani fixed point theorem) and a numerical algorithm (similar to a gradient flow). Both approaches have the advantage to use rather weak assumptions on the structure of the model. For this reason, we hope that our study will be useful even in more general situations, as those listed later on in this section, which take into account more complicated individual and collective behaviors.
In the simulations dealing with an influenza epidemic, we remark that the long-term behavior of the vaccine-induced immunity influences the best timing for the individuals to vaccinate. Indeed, when the protection of the vaccine against influenza does not decrease within the time horizon of the problem, the individuals vaccinate as soon as possible (in agreement with the recommendation given by WHO [22]). However, if the vaccine efficacy decreases, the behavior of the population changes and delays the vaccination for optimizing the vaccine protection around the peak of the epidemic.
In addition, the previous simulations show that the imperfections of the vaccine increase the overall cost. But the obtained equilibrium is such that the increased vaccination rate does not compensate for the lower efficacy (or durability of protection) of the vaccine.
When the failure rate is below a given threshold, the cost for building a group protection is advantageous with respect to the infection cost. In this case, a higher vaccination rate can be optimal to compensate for an increase in the failure rate. However, this individual policy is far from the societal level optimal strategy, which would consist in a global optimization of the vaccination policy.
Several assumptions in this work may motivate further studies:
- a general question is whether the individuals choose their vaccination strategies beforehand; for instance, Fine and Clarkson (see [30]) argue that the individuals will rather respond to the prevalence; see also [8] where the vaccination rate is dependent on the number of people infected. However the "learning" of an equilibrium is a topic in itself in game theory (we refer to the monograph [32] for general considerations). In our specific setting, an encouraging factor is that the "game" is played several times (once each season, although with possible different vaccine efficacies), in such a way that a learning mechanism could be recognized. Moreover, individuals have appropriate feedbacks (through general news for instance) on both the history of the epidemic and the vaccination dynamics, as well as -more importantly -projections for the upcoming season (for example, data on the potential severity of the epidemic and the expected dynamics of vaccination). Other factors can also influence the decision, such as the number of reported cases and public health campaigns. But, of course, the setting presented here remains ideal and the interpretability of the results is dependent on our hypotheses. A model that can detect to which extent the individuals adhere to this assumption would be more versatile. -the individuals are supposed perfectly aware of the past, present and future epidemic dynamics: a model with limited information may be more realistic. Such models can be at the mid-way between the MFG and the feedback (also known as information-based) vaccination models, see [26,25,13];
- the individuals are identical. In particular the cost of the illness is exactly the same, irrespective of age: considering several age groups may give interesting results, especially if their strategies are different;
- the geographical heterogeneity in the propagation of the epidemic is neglected: travels and intra/inter-community contacts may be important for the epidemic propagation.
Some of the previous limitations can be overcome. For example, the geographical heterogeneity in the propagation of the epidemic can be taken into account by converting our model to a PDE-based description, and then by coupling it with a population dynamics model. On the other hand, the stratification by age could be handled by writing a more general model with a supplementary age variable. We aim to take into account some of these perspectives in future studies.
G.T. acknowledges support from the Agence Nationale de la Recherche (ANR), projects EMAQS ANR-2011-BS01-017-01 and CINE-PARA. F.S has been supported by the ANR projects Kimega (ANR-14-ACHN-0030-01) and Kibord (ANR-13-BS01-0004).
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Failed vaccination rate |
Vaccination rate |