Modelling the individual vaccination decisions: a …turinici/images/...Modelling the individual...

Modelling the individual vaccination decisions: astructural explanation

Gabriel Turinici, in collaboration with Laetitia Laguzetfinancial support from ANR EMAQS

CEREMADE,Universite Paris Dauphine

ECMTB 2014

Gabriel Turinici, Laetitia Laguzet (CEREMADE) Modelling vaccination decisions 15-19 June 2014 1 / 34

Disclaimer:

What follows is a theoretical epidemiological investigation. It is not meantto be used directly for health-related decisions; if in need to take such adecision please seek professional medical advice.


Outline

1 Motivations

2 Modelization of the problem

3 Technical details: boundary condition

4 Results concerning the value function

5 Taking into account the individual decisionsPrevious works

6 Individual cost function

7 Perspectives


Vaccine scares:

Influenza A (H1N1) (flu) (2009-10)

• At 15/06/2010 flu (H1N1): 18.156 deads in 213 countries (WHO)• France: 1334 severe forms (out of 7.7M-14.7M people infected)

Vaccination in France

• Adjuvant suspected of some neurological undesired effects; massvaccination uncertainty (few previous studies for this size)• Very costly campaign (500M EUR),• Low efficiency (8% to 10% in France with respect to e.g., 24% US or74% Canada).


Vaccine scares:

Example : Influenza A in 2009 - 2010.Vaccination Coverage expected and realized in different countries onpercentage of population: (See schedule 3 of the french parliamentaryreport number 2698)

Countries Target coverage Effective rate of vaccination

Germany 100 % 10%Belgium 100 % 6 %

Spain 40 % < 4%France 70 - 75 % 8.5 %Italy 40 % 1.4 %


Vaccine scares

Previous vaccine scares (some have been disproved):• France: hepatitis B vaccines cause multiple sclerosis• US: mercury additives are responsible for the rise in autism• UK: the whooping cough (1970s), the measles-mumps-rubella (MMR)

(1990s).

Vaccine Scares : ”as cases of a disease decrease, people becomecomplacent about their risk, and the threat of vaccines (imagined or real)seems greater than the threat of disease” (C. Bauch)

Question: individual decisions sum up to give a global response.How to model this ?


Outline

1 Motivations






7 Perspectives


The SIR-V model

Susceptible Infected Recovered

Vaccinated

−dV

−βX1X2dt −γX2dt

Figure: Graphical illustration of the SIR-V model. In this model all individuals areidentical.

dX1 = −βX1X2dt − dV (t)

dX2 = (βX1X2 − γX2)dt

dX3 = γX2dt

dV = dV .

β : probability of contamination,γ : recovery rates,dV (t) : measure of vaccination

Since X1(t) + X2(t) + X3(t) + V (t) = cst = 1 ∀t > 0, the variable X3 isdependent on the others, we denote X = (X1,X2)T .


The vaccination model

Several possibilities to model vaccination:

? dV (t) = λ(t)X1(t)dt : probability of individual vaccination in[t, t + ∆t] is λ∆t; λ(t) ∈ [0, λmax ] or can also take value λ→∞ (forcertain vaccination)

? dV (t) = u(t)dt, a speed of vaccination model, u(t) ∈ [0, umax ]; thencorresponding individual vaccination probability λ(t) cannot be largerthan u(t)/X1(t) (when this makes sense)

? General case : dV (t) is a (positive) measure on [0,∞]; in particularV can include (a countable set of) Dirac masses if vaccination of anon-negligible proportion of population can be instantaneous.

Number (proportion) of individuals vaccinated up to time ’t’ is∫ t

0 dV (s);it is increasing which implies that V is BV. This allows to give a meaningto the ODE (e.g., BV 2 functions would not be ok).


Vaccination cost functional

Cost for an infected person : rI .Cost of the vaccine (including side-effects) : rV .

Global cost for the society (from the initial state X0 = (X1(0),X2(0))T ) :

J(X0,V ) = rI

∫ ∞0

βX1X2dt + rV

∫ ∞0

dV (t) (1)

It is an optimal control problem. The value function :

V(X ) = infw∈Ω

J(X ,w)

Here, Ω is some set of admissible functions; e.g., measurable functionsw : [0,∞[→ [0, umax ] and such as 0 ≤ Xi (t) ≤ 1,∀t ≥ 0, i ∈ 1, 2.


Problem details

The value function V must satisfy the HJB equation :

−H(X ,∇V) = 0

Let X = (x1; x2)T and f (X ,w) = (−βx1x2 − w ;βx1x2 − γx2)

H(X , p) = minw∈[0,umax ]

[f (X ,w) · p + rIβx1x2 + rVw ]

= −umax(p1 − rv )+ + βx1x2(rI + p2 − p1)− γx2p2.

But there is no a priori certainty that the solutions are C1 (possiblediscontinuity introduced by V ).


Existing literature

There has been a lot of work on this subject in the literature :

Abakuks, Andris, 1974 ”Optimal immunisation policies for epidemics”,

Behncke, Horst, 2000, ”Optimal control of deterministic epidemics”,

Piunovskiy, Alexei B. and Clancy, Damian, 2008, ”An explicit optimal intervention policyfor a deterministic epidemic model”

Funk, Sebastian and Salathe, Marcel and Jansen, Vincent A. A, ”Modelling the influenceof human behaviour on the spread of infectious diseases: a review”

C. T. Bauch, 2005, ”Imitation dynamics predict vaccinating behaviour”

Hethcote, Herbert W. and Waltman, Paul, 1973, ”Optimal vaccination schedules in adeterministic epidemic model”

Sethi, Suresh P. and Staats, Preston W., 1978, ”Optimal control of some simpledeterministic epidemic models”,

Morton, R. and Wickwire, K. H., 1974, ”On the optimal control of a deterministicepidemic”Ledzewicz, Urszula and Schattler, Heinz, 2011, ”On optimal singular controls for a generalSIR-model with vaccination and treatment”

Andris Abakuks, 1972, ”Some optimal isolation and immunisation policies for epidemics”

but none shows any regularity of the value function


Regularity issues

Previous works show that optimal strategy is of the form : maximumintensity vaccination on [0, θ(X )]. Thus V(X ) = F(X , θ(X )) with Fregular.The question is : Is θ(X ) regular ? Let us plot θ(X ).


Regularity issues

0.40.6

0.81

0

5 · 10−2

0.1

0

2

4

·10−3

X1X2

θ(X)

X11

1X2

O

Left image : (zoom) vaccination time θ(X ) for a particular choice of parameters.

Right image : multiple solutions exist on the frontier where θ(X ) is not regular. The solidregion is the vaccination region (θ(X ) ≥ 0). The figure illustrates two strategies that areequivalent : the solid path corresponds to a no vaccination strategy, the dashed path to a partialvaccination strategy; both have same cost, i.e., non-uniqueness of the optimal strategy.

Generic results suggest that the solution will not be C1. The derivation under C1 hypothesis hasto be checked under weaker assumptions (or hypothesis to be proved).

We use the concept of viscosity solution introduced by Pierre-Louis Lions and Michael Crandall(1992, 1997). Widely used for the optimal control problem.


Outline

1 Motivations






7 Perspectives


Technical details: boundary condition

X11γβ

X2

1

O

On red boundary there is no natural boundary condition to use. If thesystem starts e.g., with X2 = 0 on the red boundary it will remain withX2 = 0 at all times but this behavior is unstable. As soon as X2(0) > 0(even very very small) and X1(0) > γ/β the value function takes largevalues (larger than min(rI , rV )(X1(0)− γ/β)) and do not converge to zerowhen X2(0) tends to 0.The black arrow indicates the typical evolution of the SIR system; alltrajectories converge to points on the green boundary.


Other technicalities

• The cost function J(X0,V ) =∫∞

0 βX1X2rIdt +∫∞

0 rV dV (t) has nodumping term, need to work in infinite horizon. This is a problemwhen trying to obtain Lipschitz regularity for the value function.

• In general, a convenient hypothesis (cf. also Crandall, Ishii, Lions[1992]) for the uniqueness of F (x ,V ,DV ) = 0 is that F be strictlymonotone in the second argument. Here H does not depend on thisargument.


Other technicalities : solutions

• The boundary problem : prove uniqueness on increasing domains Dαconverging to full domain :

X11A = ( γβ , 0)

X2

1

O

Bα = (1− α, α)Dα

••

• No dumping term : note that the vaccination time is less thanTmax = 1/umax and vaccination occurs at the beginning. Thus on[0,Tmax ] obtain Lipschitz estimates for the value function and thenuse properties of J(X , 0) (because control is null after Tmax);

• Non-monotonicity of F : change of variables (Kruzkov) V 7→ 1− e−V .


Outline

1 Motivations






7 Perspectives


Results

Results obtained:

? the HJB equation admits a unique solution; furthermore the solutionis C1 for rV < r critv , otherwise it is Lipschitz.

? explicit construction for the solution and the optimal vaccinationstrategy;

? rigorous justification of the limit umax →∞;

Global optimal vaccination strategy : decomposition in vaccination andnon-vaccination region. The non vaccination is attractive.


Results for γ/β = 0.5

Optimal strategy for rV = 0.5rI .

X11

X2

1

O

Optimal strategy for rV = 1.1rI

X11

X2

1

O

This calls for asking the nature of optimality: how can be optimal tovaccinate someone when rV > rI ?


Outline

1 Motivations






7 Perspectives


Taking into account the individual decisions: previousliterature

Bauch & Earn (PNAS 2004) consider the SIR-V model with births anddeaths :

dS = µ(1− p)− βSIdt − µSdI = (βSI − γI )dt − µIdR = γIdt − µRdV = µp.

β : probability of contamination,γ : recovery rates,dV (t) : measure of vaccinationµ : the birth / death ratep the probability to vaccinate at birth

? the vaccination is at birth only (probability p) ; at equilibriumvaccination occurs when the probability to be infected ≥ rV /rI .

? no time dependence; individual strategies are rather simple.? equilibrium ok if everybody does the same? no eradication possible through voluntary vaccination, endemic state

(coherent with literature)? “vaccine scare behavior” : let other vaccinate (when perceived risk

differs from that of majority)Gabriel Turinici, Laetitia Laguzet (CEREMADE) Modelling vaccination decisions 15-19 June 2014 23 / 34


Galvani & Reluga & Champan (PNAS 2006) consider a double SIR-Vperiodic model of flu with two age groups (break at 65yrs). Vaccination isseparated from dynamics, once at the beginning of each season.

Results : show that actual vaccine coverage is consistent with individualoptimum; explain impact of age-targetted campaigns (children).



Bauch (2005) :

• time dependent vaccination rate p(t);

• probability to be infected is approximated by a “rule of thumb”(proportional to I (t));

• the corresponding dynamics is a phenomenological proposal (m aparameter) :

dp(t)

dt= kp(1− p)(mrI I (t)− rV )


Taking into account the individual decisions

Remarks on the individual vaccination strategy:

• the simplest one: vaccinate now or never; simple but is it alwaysoptimal ? stable equilibrium ?Equilibrium vision : all individuals are the same if one vaccinateseverybody does (100% coverage !!!) ... free ride effect.

• a naive one: “vaccinate with certainty at some given instant in thefuture, instant that can be computed now”: naive because theindividual level equations are RANDOM processes. Suppose optimalvaccination time is t, one is not sure to not be infected by time t.

• more realistic: mixed probabilistic strategies: the individual has aprobability λI (t)δt to vaccinate in the time interval [t, t + δt]


Outline

1 Motivations






7 Perspectives


Individual cost functional

dX1 = −βX1X2dt − uG (t)dt

dX2 = (βX1X2 − γX2)dt

dX3 = γX2dt

dV = uG (t)dt.

vaccination at the society level with rateuG (t).

Individual decision: the strategy λI (truncated at umaxX1(t) cf. above).

dU(t) = (−βX2 − λI )U(t)dt. U(t) = survival rate of the susceptibles(the ”probability” to not be infected nei-ther vaccinate up to time t, i.e., percent-age still in class X1).

JI (λI ) =∫∞

0 U(t) [rIβX2 + rVλI ] dt. JI = the cost functional for the individ-ual

Equilibrium condition for the global strategy to arise from individualdecisions: uG (t) = λI (t)X1(t). (when this operation makes sense) (MeanField Games, cf. Lasry, Lions!).


Theoretical results

Theoretical results:

? the individual vaccination is at the beginningλI = umax (t)

X1(t) · 1[0,θI (X )](t);

? to optimize for equilibrium t 7→ JI (X ,umax (·)X1(·) · 1[0,t](·)); non-smooth

(not C 1) ;

? technical remark: the overall infection risk∫∞t βX1(τ)X2(τ)dτ

X1(t)

(depending on uG ) is decreasing with respect to time (convexity fort 7→ JI (...)).

? a stable equilibrium exists, no free-ride effect

? at equilibrium the optimal switch point is also∫∞t βX1(τ)X2(τ)dτ

X1(t) = rVrI

? the frontier of the vaccination region is the surface ζ(X1,X2)X1

= rVrI

.Here ζ(X ) is the size of a non-controlled epidemic starting from X .


Theoretical results

? Remark: vaccination region independent of umax (!), can pass easilyto the limit umax →∞

? Mean cost for an individual in the stable individual-global equilibriumis larger than the cost for the, non stable, societal (non-individual)equilibrium. The stable strategy will be preferred even if it is morecostly for everyone.

X11

X21

O

Red: vaccination in the first modeland in the individual model, green :vaccination for the first model butnot for the individual, blue: novaccination in both models.


Application to Influenza A 2009/2010 vaccination

The SIR model was fit to the observed vaccination coverage (J.P. Guthman et al. BEH 2010);the other parameters were chosen consistent with ranges from the literature (large CIs !).Time axis = weeks starting from W19Y2009; peak = W49Y2009 (30), vaccination peak =W51Y2009 (31-32), vaccination end W05Y2010 (40).

30 40 50 600

0.005

10 20

0.01

0.015

0.02

10 20 30 40 50 600

0.02

0.04

0.06

0.08

0.1

Real

Model

Real

Model

Comparison of cumulative numbers of cases

Comparison of numbers of cases

30 40 50 600

0.01

10 20

0.02

0.03

0.04

0.05

0.06Real

Modeled

Comparison of cumulative numbers of vaccination


Application to Influenza A 2009/2010 vaccination

0

-0.2

1

50

0.8

0.6

0.4

0.2

40302010

Model free, data-driven criterion

Simpler criterion, model based

Model based, optimal risk criterion0.3

0.25

0.2

0.15

0.1

0.05

05040302010

Vaccination intensity

0.35Infection risk

60

60

Time axis = weeks starting from W19Y2009; peak = W49Y2009 (30), vaccination peak =W51Y2009 (31-32), vaccination end W05Y2010 (40).

The results indicate that the first ones to stop vaccination were people in a group with

rV /rI ' 10%− 15% (week 32). The last one correspond to rV /rI < 1% (week 40; model cannot

be more precise with available data).Gabriel Turinici, Laetitia Laguzet (CEREMADE) Modelling vaccination decisions 15-19 June 2014 32 / 34

Outline

1 Motivations






7 Perspectives


Work in progress and perspectives

• More general strategies,

• Birth / deaths taken into account,

• Structured population models: age, geographical location,

• Stochastic dynamics,

• impact of the societal penalty for non-vaccination; do we obtain the(societal, non individual) solution even for rV ≥ rI ?

• ...


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