
Citation: Luis Ortega-Paz, Giuseppe Giacchi, Salvatore Brugaletta, Manel Sabaté. Percutaneous Transcatheter Aortic Valve Implantation: A Review Focus on Outcomes and Safety[J]. AIMS Medical Science, 2015, 2(3): 200-221. doi: 10.3934/medsci.2015.3.200
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Within the framework of a nonlocal SIR model, we introduce different vaccination policies and then seek the most effective one. Vaccines are dosed at prescribed ages or at prescribed times to prescribed portions of the susceptible population. The careful choice of the ages, times and individuals to be treated may significantly influence the efficiency of a vaccination campaign. It is thus natural to tackle the resulting optimal control problem, seeking what is in some sense the best vaccination policy.
We denote by $ S = S(t, a) $ the density of individuals at time $ t $ of age $ a $ susceptible to the disease. As usual, $ I = I(t, a) $ is the density of infected individuals at time $ t $ and of age $ a $. The density of individuals that can not be infected by the disease is $ R = R(t, a) $, comprising individuals that recovered as well as those that are, whatever the reason, immune. As a reference to the basic properties of SIR models, originated in [1], see [2,Chapter 6], [3,Chapter 19] or [4,§ 1.5.1].
A possible description of the evolution of $ S $ is provided by the renewal equation [4,Chapter 3] with nonlocal source term
$ ∂tS+∂aS=−(dS(t,a)+∫+∞0λ(a,a′)I(t,a′)da′)S $
|
(1.1) |
where $ d_S = d_S(t, a) \ge 0 $ is the susceptibles' death rate and $ \lambda (a, a') \ge 0 $ quantifies the susceptible individuals of age $ a $ that are infected by individuals of age $ a' $. The nonlocal quantity on the right hand side of (1.1), namely $ \int_0^{+\infty} \lambda(a, a') \; I(t, a') \mathinner{\mathrm{d}{a}}' \, S (t, a) $, represents the total number of susceptible individuals of age $ a $ that become infected at time $ t $ per unit time.
The evolution of the infected portion $ I $ of the population is governed by
$ ∂tI+∂aI=−(dI(t,a)+rI(t,a))I+∫+∞0λ(a,a′)I(t,a′)da′S, $
|
(1.2) |
where $ d_I = d_I(t, a) \geq 0 $ is the death rate of the $ I $ portion of the population and $ r_I = r_I(t, a) \geq 0 $ represents the fraction of infected individuals of age $ a $ that recovers at time $ t $ per unit time. Finally, the evolution of the portion $ R $ of the population that recovers or is unaffected from the disease is
$ ∂tR+∂aR=rI(t,a)I−dR(t,a)R, $
|
(1.3) |
where $ d_R = d_R(t, a) \ge 0 $ is the death rate of the $ R $ population. We assume here that members of the $ R $ population are not dosed with vaccine and that individuals may enter the $ R $ population at birth. In the present setting, dealing with choices different from these latter ones only requires very minor changes and keeps falling in the present general analytic framework.
In a first policy, vaccinations are dosed at any time $ t $ at a time dependent portion $ \eta_j (t) $ of the $ S $ population of the prescribed age $ \bar a_j $, with $ 0 < \bar a_1 < \cdots < \bar a_N $. In this way, the fraction $ \eta_j (t) $ of susceptible population $ S (t, \bar a_j) $ becomes immune. In other words, we use as control the fractions $ \eta_j \colon \mathbb{I} \to [0, 1] $ (for $ j \in \{1, \cdots, N\} $ and $ \mathbb{I} $ being the, possibly unbounded, time interval under consideration). We thus have to supplement the evolution (1.1)–(1.2)–(1.3) with the vaccination conditions
$ S(t,ˉaj+)=(1−ηj(t))S(t,ˉaj−)[∀t, S(t,ˉaj) decreases due to vaccination]I(t,ˉaj+)=I(t,ˉaj−)[the infected population is unaltered]R(t,ˉaj+)=R(t,ˉaj−)+ηj(t)S(t,ˉaj−)[vaccinated individuals are immunized] $
|
(1.4) |
for a.e. $ t > 0 $ and for $ j \in \{1, \cdots, N\} $. A reasonable constraint on the vaccination policy $ \eta $ in (1.4) is that its total cost $ \mathcal{N} (\eta) $ on the considered time interval $ \mathbb{I} $ does not exceed a prescribed maximal cost $ M $. Measuring the total cost by the number $ \mathcal{N} $ of dosed individuals, we have
$ N(η)≤M where N(η)=N∑i=1∫Iηi(t)S(t,ˉai−)dt. $
|
(1.5) |
Alternatively, we also consider vaccination campaigns immunizing an age dependent portion of the whole population at given times $ \bar t_1, \ldots, \bar t_N $. This amounts to substitute (1.4) with
$ S(ˉtk+,a)=(1−νk(a)ηj)S(ˉtk−,a)[∀a, S(ˉtk,a) decreases due to vaccination]I(ˉtk+,a)=I(ˉtk−,a)[the infected population is unaltered]R(ˉtk+,a)=R(ˉtk−,a)+νk(a)S(ˉtk−,a)[vaccinated individuals are immunized] $
|
(1.6) |
where $ \nu_k (a) $ is the percentage of susceptible individuals of age $ a $ that are dosed with the vaccine at time $ \bar t_k $, for $ k = 1, \ldots, N $. Now, a reasonable constraint due to the campaign cost is given by a bound on the number $ \mathcal{N} $ of dosed individuals
$ N(ν)≤M whereN(ν)=N∑i=1∫R+νi(a)S(ˉti−,a)da. $
|
(1.7) |
For simplicity, in both cases above, we assume that $ \eta_j $, respectively $ \nu_i $, represents the percentages of individuals that are successfully vaccinated.
In both cases, our aim is to seek the vaccination campaign that minimizes the total number of infected individuals of all ages over the time interval $ \mathbb{I} $, that is
$ J=∫I∫R+φ(a)I(t,a)dadt $
|
(1.8) |
the function $ \varphi $ being a suitable positive age dependent weight. Here, $ \mathcal{J} $ is a function of $ \eta $ in case (1.4) and of $ \nu $ in case (1.6). In general, in the two cases (1.4) and (1.6), reasonable costs are thus
$ J(η)+N(η)=∫I∫R+φ(a)I(t,a)dadt+N∑j=1∫Iηj(t)S(t,ˉaj−)dt $
|
(1.9) |
$ J(ν)+N(ν)=∫I∫R+φ(a)I(t,a)dadt+N∑i=1∫R+νi(a)S(ˉti−,a)da. $
|
(1.10) |
Clearly, in general, suitable weights can be used to modify the relative relevance of the two costs $ \mathcal{N} $ and $ \mathcal{J} $ in the sums above.
The current literature offers a variety of alternative approaches to similar modeling situations. In the context of models based on ordinary differential equations, that is, without an age structure in the population, this problem has been considered, for instance, already in [5,6,7]. In the recent [8], the vaccination control enters an equation of the type (1.1) through a term $ -u \, S $ in the source on the right hand side of (1.1), implicitly suggesting that vaccination takes place uniformly at all ages. For a recent related investigation focused on cholera, see [9].
The next section summarizes the key analytic properties of the present SIR model, providing the necessary basis to tackle optimal control problems. Section 3 shows the qualitative properties of various vaccination strategies through numerical integrations. Section 4 presents numerical optimizations through the descent method. The paper ends with Section 5.
Denote by $ \mathbb{I} $ either the time interval $ [0, T] $, for a positive $ T $, or $ \left[0, +\infty\right[$. Fix throughout a positive integer $ N $ representing the number of vaccination sessions.
Both models introduced above lead to the initial – boundary value problem
$ {∂tS+∂aS=−(dS(t,a)+∫+∞0λ(a,a′)I(t,a′)da′)S∂tI+∂aI=−(dI(t,a)+rI(t,a))I+∫+∞0λ(a,a′)I(t,a′)da′S∂tR+∂aR=rI(t,a)I−dR(t,a)R $
|
(2.1) |
with initial datum and boundary inflow
$ {S(0,a)=So(a),I(0,a)=Io(a),R(0,a)=Ro(a),a∈R+;S(t,0)=Sb(t),I(t,0)=Ib(t),R(t,0)=Rb(t),t∈I. $
|
(2.2) |
The usual case of the birth terms in the boundary data being assigned through suitable integrals of the populations in reproductive age can be recovered by mainly technical adjustments.
We assume below that the following assumptions are satisfied:
($ {\lambda} $) $ \lambda \in \bf{C}^{0}({\mathbb{R}}^+ \times {\mathbb{R}}^+; {\mathbb{R}}) $ is bounded, with total variation in the first argument uniformly bounded with respect to the second (i.e., $ \sup_{a' \in {\mathbb{R}}^+} \mathinner{{\rm{TV}}} (\lambda(\cdot, a'); {\mathbb{R}}^+) < +\infty $) and Lipschitz continuous in the first argument uniformly with respect to the second (i.e., there exists a $ C > 0 $ such that for all $ a, a_1, a_2 \in {\mathbb{R}}^+ $, $ {\left|{\lambda(a_1, a') - \lambda(a_2, a')}\right|} \leq C \; {\left|{a_1 - a_2}\right|} $).
(d-r) The maps $ d_S, d_I, d_R, r_I \colon \mathbb{I} \times {\mathbb{R}}^+ \to {\mathbb{R}} $ are bounded Caratheodory functions, with $ {\bf{L}^1} $ norm and total variation in $ a $ uniformly bounded in $ t $ (i.e., for $ \varphi = d_S, d_I, d_R, r_I $, $ \sup_{t \in \mathbb{I}} {\left\|{ \varphi (t)}\right\|}_{{\bf{L}^1}({\mathbb{R}}^+; {\mathbb{R}})} < +\infty $ and $ \sup_{t \in \mathbb{I}} \mathinner{{\rm{TV}}} (\varphi (t, \cdot); {\mathbb{R}}^+) < +\infty $) and Lipschitz continuous in $ a $ uniformly in $ t $ (i.e., there exists a $ C > 0 $ such that for all $ t \in \mathbb{I} $ and $ a_1, a_2 \in {\mathbb{R}}^+ $, $ {\left|{ \varphi (t, a_2) - \varphi (t, a_1)}\right|} \leq C \, {\left|{a_2 - a_1}\right|} $).
(IB) The initial datum and the boundary inflow in (2.2) satisfy $ S_o, I_o, R_o \in ({\bf{L}^1} \cap \bf{BV}) ({\mathbb{R}}^+; {\mathbb{R}}^+) $ and $ S_b, I_b, R_b \in ({\bf{L}^1} \cap \bf{BV}) (\mathbb{I}; {\mathbb{R}}^+) $.
For the definition of Caratheodory function, of total variation and of the various functional spaces, we refer for instance to [10]. For all other analytic details specific to the present construction, see [11].
First, we provide the basic well posedness and stability result for the model (2.1)–(2.2) in the case of the vaccination policy (1.4), namely
$ {S(t,ˉaj+)=(1−ηj(t))S(t,ˉaj−)I(t,ˉaj+)=I(t,ˉaj−)R(t,ˉaj+)=R(t,ˉaj−)+ηj(t)S(t,ˉaj−)j=1,…,N. $
|
(2.3) |
Theorem 2.1 ([11]). Under hypotheses ($ {\lambda} $) and (d-r), for any initial and boundary data satisfying (IB), for any choice of the positive vaccination ages $ \bar a_1, \ldots, \bar a_N \in {\mathbb{R}}^+ \setminus \{0\} $ and of the control function $ \eta \in \bf{BV}(\mathbb{I}; [0, 1]^N) $, problem (2.1)–(2.2)–(2.3) admits a unique solution defined on the whole interval $ \mathbb{I} $, depending Lipschitz continuously on the initial datum, through the $ {\bf{L}^1} $ norm, and on $ \eta $, through the $ {{\bf{L}}^\infty } $ norm.
An entirely similar result holds for the vaccination policy (1.6), which we rewrite here as
$ {S(ˉti+,a)=(1−νi(a)ηj)S(ˉti−,a)I(ˉti+,a)=I(ˉti−,a)R(ˉti+,a)=R(ˉti−,a)+νi(a)S(ˉti−,a).i=1,…,N. $
|
(2.4) |
Theorem 2.2 ([11]). Under hypotheses ($ {\lambda} $) and (d-r), for any initial and boundary data satisfying (IB), for any choice of the positive vaccination times $ \bar t_1, \ldots, \bar t_N \in \mathbb{I} \setminus \{0\} $ and of the control function $ \nu \in \bf{BV}({\mathbb{R}}^+; [0, 1]^N) $, problem (2.1)–(2.2)–(2.4) admits a unique solution defined on the whole interval $ \mathbb{I} $, depending Lipschitz continuously on the initial datum, through the $ {\bf{L}^1} $ norm, and on $ \nu $, through the $ {{\bf{L}}^\infty } $ norm.
Below, we restrict the search of optimal controls $ \eta_j $ in (1.4) and $ \nu_i $ in (1.6) to functions of the form
$
ηj(t)=∑mℓ=1ηℓjχ[tℓ−1,tℓ[(t)t0=0,tℓ>tℓ−1,tℓ∈I;νi(a)=∑mℓ=1νℓiχ[aℓ−1,aℓ[(a)a0=0,aℓ>aℓ−1,aℓ∈R+.
$
|
The stability results above ensure that the costs (1.5), (1.7) and (1.8) are Lipschitz continuous functions of the various $ \eta^\ell_j $ and $ \nu^\ell_i $. Hence, a straightforward Weierstraß argument proves the existence of optimal controls minimizing the costs (1.9) or (1.10).
We refer to [11] for the detailed proofs as well as for further stability estimates on the dependence of the solutions on the various parameters.
We now compare different instances of both strategies (2.3) and (2.4) by means of numerical integrations.
In all the integrations below, for the convective part we exploit the upwind method [12,§ 4.8] with mesh size $ \Delta a = 2.5\cdot 10^{-3} $ along the age axis and $ \Delta t = 1.25\cdot 10^{-3} $ along the time axis. The integrals in the right hand side in (2.1) are computed via a rectangle rule and a fractional step method [12,§ 17.1] allows to combine the convective evolution with the source term. The time axis $ a = 0 $ belongs both to the physical boundary and to the numerical one, its treatment being straightforward since all characteristics move inward the domain $ \mathbb{I} \times {\mathbb{R}}^+ $, so that the value of the boundary data can be assigned to the solution. On the other side, along the numerical boundary $ a = 10 $, the usual free flow conditions is consistent with all characteristic speeds being positive.
Our aim in the integrations below is to clearly show the qualitative properties of the model (2.1). Indeed, we do not claim that the chosen numerical values are realistic. In particular, the most adequate units for $ t $ and $ a $ might well be different.
With reference to the Cauchy problem (2.1)–(2.2) and to the costs (1.9)–(1.10), we use throughout the following parameters and functions:
$ ds(t,a)=14a1+a,dI(t,a)=12a1+a,dR(t,a)=14a1+a,rI(t,a)=110,λ(a,a′)=2e−|a−a′|/2,φ(a)=1. $
|
(3.1) |
The mortality rates are increasing with age and higher for the infected population. The choice of $ \lambda $ assumes that the infection propagates preferably among individuals of similar age.
We consider two different initial data, first the case without infection, i.e.,
$ So(a)=3(1−a10),Io(a)=0,Ro(a)=(1−a10), $
|
(3.2) |
and then a case with a $ 20\% $ of infected individuals, with respect to the $ S $ population, at time $ t = 0 $, that is
$ So(a)=2.5(1−a10),Io(a)=0.5(1−a10),Ro(a)=(1−a10). $
|
(3.3) |
Both these initial data are linearly decreasing with age. Note that the total initial population is the same in the two cases (3.2) and (3.3).
Natality is chosen so that no jump appears between the initial values at $ a = 0+ $ in (3.2) and the boundary data at $ t = 0+ $:
$ Sb(t)=3,Ib(t)=0,Rb(t)=1, $
|
(3.4) |
where we assume that no individuals are infected already at birth.
We first consider two extreme situations to be later used for comparisons, namely corresponding to no illness (3.2), and to a spreading illness with no vaccination strategy whatsoever.
In other words, we integrate (2.1)–(2.2)–(3.1) first setting $ I_o \equiv 0 $, so that for all times $ t $, $ I (t) \equiv 0 $ and no illness ever appears, and then setting $ I_o $ as in (3.3) but dose no vaccination, so that the epidemic spreads uncontrolled. The results of these two integrations are displayed in Figure 1. The two solutions are, clearly, entirely different. When $ I \equiv 0 $, a sort of dynamic equilibrium is reached, the resulting contour plot being approximately invariant with respect to translation along the vertical $ t $ axis. The presence of the $ I $ population in the initial datum leads to the spread of the disease and eventually to a dramatic population decrease.
We consider now a simple vaccination strategy consisting in a single campaign, dosing vaccination at a single age $ \bar a $ for all times.
As it is to be expected, the earlier vaccinations are dosed, the better it is in terms of the cost (1.8). In Figure 2 the ratio of the number of infected individuals vs. the total population corresponding to the vaccinations ages $ \bar a = 0.1, \; 0.3, \; 0.5, \; 1.5 $ are portrayed. The leftmost contour plot shows the effects of a very early vaccination, clearly far more effective than later doses.
The striking effect of this early age vaccination is confirmed in Figure 3, where we display the sum $ S+R $ as a function of $ t $ and $ a $ in the same cases of the vaccinations ages $ \bar a = 0.1, \; 0.3, \; 0.5, \; 1.5 $. Clearly, due to the form of the vaccination condition (2.3) and to the present smooth data (3.3)–(3.4), the sum $ S+R $ is continuous across $ \bar a $. In the leftmost diagram, the healthy population apparently recovers from the spread of the disease. When vaccinations are dosed at a later age, i.e., in the three diagrams on the right, the relatively large portion of individuals initially vaccinated survive the disease, but juveniles get infected resulting in the solitary waves, consisting essentially of the $ R $ population, shown in Figure 3.
The counterpart of these solitary waves is the propagating "hole" in Figure 2.
In other words, the vaccination campaign at the age $ \bar a = 0.5 $ looks as (un)successful as the campaign at $ \bar a = 1.5 $. On the contrary, dosing at $ \bar a = 0.1 $ is by far more effective.
Indeed, this strategy keeps the density of the susceptible population in (2.1) very low due to (2.3). Hence, also the nonlocal source term, which is proportional to $ S $, is relatively small and infection hardly spreads. When vaccination is dosed at higher ages, the nonlocal source term is able to bring infection at low ages, thus making the vaccination policy far less effective.
Table 1 confirms that, in particular for what concerns the total number of infected individuals, by far, the best strategy is the one dosing vaccines at the earliest considered age $ \bar a = 0.1 $. This choice also results in the highest number of vaccinated individuals, so that also $ \mathcal{N} $ corresponds to the most expensive policy.
$ \bar a $ in (2.3) | 0.1 | 0.2 | 0.3 | 0.5 | 1.0 | 1.5 | none |
$ \mathcal{J} $ in (1.8) | 33.48 | 41.03 | 44.73 | 46.85 | 47.39 | 47.51 | 48.30 |
$ \mathcal{N} $ in (1.5) | 10.53 | 5.05 | 2.30 | 0.76 | 0.41 | 0.36 | 0.00 |
$ \mathcal{J}+\mathcal{N} $ in (1.9) | 44.01 | 46.08 | 47.03 | 47.61 | 47.80 | 47.87 | 48.30 |
We now simulate another single vaccination campaign, consisting in dosing all individuals of all ages at a given time $ \bar t $, see Figure 4.
Similarly to the integration detailed in § 3.2, campaigns at earlier times results in being more effective. The choices $ \bar t = 0.5 $ and $ \bar t = 1.5 $ are quite similar to each other and, in the lower right part of the $ (a, t) $–plane, to the situation with no vaccination campaign whatsoever, as it stems from a comparison between the two rightmost diagram in Figure 4 and the rightmost one in Figure 1, see also Table 2.
$ \bar t $ in (2.4) | 0.1 | 0.2 | 0.3 | 0.5 | 1.0 | 1.5 | none |
$ \mathcal{J} $ in (1.8) | 31.71 | 38.80 | 43.65 | 46.79 | 47.40 | 47.42 | 48.30 |
$ \mathcal{N} $ in (1.7) | 9.37 | 5.46 | 2.74 | 0.86 | 0.40 | 0.39 | 0.00 |
$ \mathcal{J}+\mathcal{N} $ in (1.10) | 41.08 | 44.26 | 46.39 | 47.66 | 47.80 | 47.80 | 48.30 |
Note that the present strategy is far less effective: vaccinations are dosed only at the given time $ \bar t $, after which the disease spreads completely out of control, see Figure 5.
The values of the costs (1.7), (1.8) and (1.10) presented in Table 2 show that vaccinating the whole $ S $ population at $ \bar t = 0.5 $ or later, makes relatively small difference, the most convenient choice being at the earliest time $ \bar t = 0.1 $. Similarly to what happens in § 3.2, vaccinating all susceptible individuals at $ \bar t = 0.1 $ makes the susceptible portion of the population very small wherever $ a > t - \bar t $, thus essentially canceling the nonlocal source term which represents the spreading of the disease.
We note however an evident key difference between the best vaccination age choice (i.e., $ \bar a = 0.1 $) in § 3.2 and the present best vaccination time choice (i.e., $ \bar t = 0.1 $). The latter one yields lower values of the chosen costs. However, as time grows, the former strategy is prone to by far preferable evolutions. Indeed, the leftmost diagram in Figure 2 (corresponding to $ \bar a = 0.1 $) clearly shows a decreasing $ I $ population, while it is increasing in the latter strategy, see Figure 4, left (corresponding to $ \bar t = 0.1 $).
We now consider in more detail the search for an optimal vaccination strategy in a slightly different setting. First, we assume that infection among individuals of different ages is somewhat more difficult, slightly modifying (3.1) to
$ ds(t,a)=14a1+a,dI(t,a)=12a1+a,dR(t,a)=14a1+a,rI(t,a)=110,λ(a,a′)=e−2|a−a′|,φ(t)=1, $
|
(4.1) |
and choose the initial datum
$ So(a)={3.0(1−a10)a∈[0,3]∪[5,10]1.5(1−a10)a∈]3,5[Io(a)={0a∈[0,3]∪[5,10]1.5(1−a10)a∈]3,5[Ro(a)=(1−a10). $
|
(4.2) |
corresponding to an initial infected population that amounts to $ 50\% $ of the susceptible population in the age interval $ \left]3, 5\right[$. As boundary data, we keep the choice (3.4), corresponding to a constant natality rate for the $ S $ and $ R $ populations, and to no one being infected at birth.
As a first reference situation, we consider the case of no vaccination being dosed to anyone. The resulting integration is displayed in Figure 6. In this reference case, the total number of infected individuals is $ 22.77 $.
We only remark that Figure 6 shows a typical effect of the nonlocality of the source terms in (2.1). There is an increase in the $ I $ population for, approximately, $ t \geq 3 $ and $ a \in [1, 3]$ due to the transmission and growth of the infection, a feature which becomes relevant in regions close to (relatively) high values of the susceptible population $ S $.
Below we restrict the values of the percentages $ \eta_j $ and $ \nu_i $ in (2.3) and (2.4) to the interval $ [0, 0.8] $. Indeed, it is reasonable to assume that not every individual can be vaccinated and that, in some cases, vaccination may fail to immunize a small percentage of susceptible individuals. For the sake of simplicity, we introduce only three vaccination ages $ \bar a_1 $, $ \bar a_2 $ and $ \bar a_3 $ or times $ \bar t_1 $, $ \bar t_2 $ and $ \bar t_3 $.
We present both sample values of the costs (1.9)–(1.10) and the results of optimization procedures. The former correspond to dosing vaccines to $ 60\% $ or $ 80\% $ of the $ S $ population at each prescribed age for all times, or at each prescribed time at all ages. The latter are achieved through the usual descent method, see [13,§ 7.2.2], with suitable projections that restrict to values of $ \eta_1, \eta_2, \eta_3 $ and $ \nu_1, \nu_2, \nu_3 $ in $ [0, 0.8] $. A formal justification of the use of this method is in the Lipschitz continuous dependence of the cost (1.9) or (1.10) on the various parameters, see [11].
In the integrations below, we keep $ \Delta a = 2.5 \cdot 10^{-3} $ and $ \Delta t = 1.25 \cdot 10^{-3} $. However, due to computing time limitations, when applying the steepest descent method we passed to coarse meshes, where $ \Delta a = 5 \cdot 10^{-2} $ and $ \Delta t = 2.5 \cdot 10^{-2} $. Correspondingly, in case of the controls (2.3), optimal controls are sought in the class of locally constant functions of the type $ \eta_j (t) = \sum_{\ell = 1}^{20} \eta^\ell_j \, {\chi_{ {[(\ell-1) /4, \, \ell/4[}}} (t) $. An entirely similar procedure is followed in the case of the age dependent controls (2.4). For completeness, we recall that in the present merely Lipschitz continuous setting, the steepest descent (or gradient) method needs not converge and, even if it does, its limit may well be different from a point of minimum. Therefore, in our application of this method, we keep checking that the values of the cost (do indeed) decrease, but we can not ensure that the values obtained are indeed points of minimum.
Now, in the model (2.1)–(2.2) with parameters (4.1), initial datum (4.2) and boundary datum (3.4) we prescribe $ 3 $ vaccination ages and, correspondingly, $ 3 $ constant values of the percentage of vaccinated individuals, so that, with reference to the notation (2.3),
$ ˉa1=2.0,ˉa2=4.0,ˉa3=6.0 and ηj∈{0.6,0.8} for j=1,2,3. $
|
(4.3) |
First, we perform the resulting $ 8 $ numerical integrations and record the resulting costs (1.5), (1.8) and (1.9). Then, these costs are compared among each other and with the costs resulting from the steepest descent method. The numerical values obtained are in Table 3.
$ \eta_1 $ | $ \eta_2 $ | $ \eta_3 $ | $ \mathcal{J} $ as in (1.8) | $ \mathcal{N} $ as in (1.5) | $ \mathcal{J} + \mathcal{N} $ as in (1.9) |
$ 0.6 $ | $ 0.6 $ | $ 0.6 $ | $ 14.54 $ | $ 7.56 $ | $ 22.11 $ |
$ 0.6 $ | $ 0.6 $ | $ 0.8 $ | $ 14.38 $ | $ 7.87 $ | $ 22.25 $ |
$ 0.6 $ | $ 0.8 $ | $ 0.6 $ | $ 14.18 $ | $ 8.02 $ | $ 22.20 $ |
$ 0.6 $ | $ 0.8 $ | $ 0.8 $ | $ 14.02 $ | $ 8.31 $ | $ 22.83 $ |
$ 0.8 $ | $ 0.6 $ | $ 0.6 $ | $ 13.12 $ | $ 9.22 $ | $ 22.34 $ |
$ 0.8 $ | $ 0.6 $ | $ 0.8 $ | $ 12.96 $ | $ 9.52 $ | $ 22.48 $ |
$ 0.8 $ | $ 0.8 $ | $ 0.6 $ | $ 12.80 $ | $ 9.60 $ | $ 22.40 $ |
$ 0.8 $ | $ 0.8 $ | $ 0.8 $ | $ 12.64 $ | $ 9.88 $ | $ 22.53 $ |
see Figure 8 | $ 16.01 $ | $ 4.67 $ | $ 20.69 $ |
As is to be expected, the lowest amount of infected individuals results from the highest (and most expensive) vaccination dosing, detailed in Figure 7. There, the discontinuities in $ S $ across the ages $ \bar a_j $ resulting from the conditions (2.3) are clearly visible. Choosing $ \bar a_1 = 2.0 $ as the lowest vaccination age allows, for about $ t \geq 4 $, that individuals younger that this age are infected and, at later times, the disease may well spread among juveniles.
The steepest descent procedure yields a value of the cost $ \mathcal{J} + \mathcal{N} $ in (1.9) lower than any of the ones obtained with constant $ \eta_j $ as in (4.3), see Table 3. The optimal controls found by means of the steepest descent method are shown in Figure 8 and the corresponding solutions are in Figure 9.
We now follow a procedure, in a sense, analogous to that detailed in the preceding Paragraph 4.1. Choose the times and controls
$ ˉt1=0.5,ˉt2=1.0,ˉt3=1.5 and νi∈{0.6,0.8} for i=1,2,3. $
|
(4.4) |
The costs of the integrations resulting from these choices and from the optimization are collected in Table 4. Again, the lowest number of infected individuals is obtained with the highest number of vaccinations and the corresponding solutions are plotted in Figure 10. Given that infected individuals can not turn back susceptible, it is clear that the first vaccination of $ 80\% $ of all susceptible individuals greatly hinders the spreading of the disease. The later vaccinations, dosed also to the newly born individuals, block further infections, see Figure 10.
$ \nu_1 $ | $ \nu_2 $ | $ \nu_3 $ | $ \mathcal{J} $ as in (1.8) | $ \mathcal{N} $ as in (1.7) | $ \mathcal{J} + \mathcal{N} $ as in (1.10) |
$ 0.6 $ | $ 0.6 $ | $ 0.6 $ | $ 6.41 $ | $ 13.05 $ | $ 19.47 $ |
$ 0.6 $ | $ 0.6 $ | $ 0.8 $ | $ 6.23 $ | $ 13.75 $ | $ 19.98 $ |
$ 0.6 $ | $ 0.8 $ | $ 0.6 $ | $ 6.17 $ | $ 13.60 $ | $ 19.77 $ |
$ 0.6 $ | $ 0.8 $ | $ 0.8 $ | $ 6.06 $ | $ 14.10 $ | $ 20.16 $ |
$ 0.8 $ | $ 0.6 $ | $ 0.6 $ | $ 5.78 $ | $ 13.89 $ | $ 19.66 $ |
$ 0.8 $ | $ 0.6 $ | $ 0.8 $ | $ 5.67 $ | $ 14.44 $ | $ 20.12 $ |
$ 0.8 $ | $ 0.8 $ | $ 0.6 $ | $ 5.66 $ | $ 14.22 $ | $ 19.88 $ |
$ 0.8 $ | $ 0.8 $ | $ 0.8 $ | $ 5.59 $ | $ 14.65 $ | $ 20.24 $ |
see Figure 11 | $ 7.80 $ | $ 8.18 $ | $ 15.98 $ |
The strategy found through the steepest descent method, shown in Figure 11, clearly takes advantage of the particular shape of the initial datum (4.2). Indeed, the two "humps" in $ \nu_2 $ and $ \nu_3 $ roughly correspond to the wave of infected individuals propagating from $ t = 0 $ onward.
As a consequence, note that the optimal control $ \nu_3 $ vanishes for small ages. This is due to the fact that newborn are healthy and that the preceding vaccination campaigns succeeded in avoiding the transmission of the infection to the youngest part of the population, see Figure 12.
With the data chosen in (4.1) and the parameters (4.2), the best vaccination strategies turn out to be the ones of the type (2.4), i.e., those where a percentage $ \nu_i (a) $ of all susceptible individuals of age $ a $ are dosed at time $ \bar t_i $, as it stems out comparing Table 3 with Table 4. Clearly, this outcome is a consequence of the particular choices in (4.1) and (4.2), but the framework based on (2.1) and (2.3) or (2.4), and its amenability to optimization procedure, is independent of these choices.
We addressed the issue of optimizing vaccination strategies. Sample numerical integrations show various features of the solutions to the integro–differential model (2.1)–(2.2) and the effects of different choice of the controls $ \eta_j $ in (2.3) or $ \nu_i $ in (2.4). Standard numerical optimization procedures, such as the steepest descent method, can be applied to single out optimal choices of the controls.
The present framework is amenable to a variety of different applications and extensions, only a minor part of which were explicitly considered above. For instance, natality, which is represented by the boundary datum, can be assigned as a function of the present, or past, population densities, also allowing for newborn to be infected also at birth. The present age structured formulation allows to take into account the possibility that the disease is transmitted differently in different age groups. Introducing in the $ S $, $ I $ or $ R $ populations sexual distinctions only amounts to consider more equations, the basic analytic framework in [11] remaining essentially unaltered.
Part of this work was supported by the PRIN 2015 project Hyperbolic Systems of Conservation Laws and Fluid Dynamics: Analysis and Applications and by the GNAMPA 2018 project Conservation Laws: Hyperbolic Games, Vehicular Traffic and Fluid dynamics.
The IBM Power Systems Academic Initiative substantially contributed to the numerical integrations.
The authors declare there is no conflict of interest.
[1] | Holmes DR, Mack MJ, Kaul S, et al. (2012) ACCF/AATS/SCAI/STS expert consensus document on transcatheter aortic valve replacement. J Am Coll Cardiol 59(13):1200-1254. |
[2] | Nishimura RA, Otto CM, Bonow RO, et al. (2014) AHA/ACC guideline for the management of patients with valvular heart disease: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines. J Am Coll Cardiol 63(22):e57-185. |
[3] | Cribier A, Eltchaninoff H, Bash A, et al. (2002) Percutaneous transcatheter implantation of an aortic valve prosthesis for calcific aortic stenosis: first human case description. Circulation 106(24):3006-3008. |
[4] | Tchetche D, Van Mieghem NM (2014) New-generation TAVI devices: description and specifications. EuroIntervention: journal of EuroPCR in collaboration with the Working Group on Interventional Cardiology of the European Society of Cardiology 10(U):U90-U100. |
[5] | Rodes-Cabau J (2012) Transcatheter aortic valve implantation: current and future approaches. Nature Rev Cardiolo 9(1):15-29. |
[6] | Piazza N, Martucci G, Lachapelle K, et al. (2014) First-in-human experience with the Medtronic CoreValve Evolut R. EuroIntervention : journal of EuroPCR in collaboration with the Working Group on Interventional Cardiology of the European Society of Cardiology 9(11):1260-1263. |
[7] | Binder RK, Rodes-Cabau J, Wood DA, et al. (2012) Edwards SAPIEN 3 valve. EuroIntervention: journal of EuroPCR in collaboration with the Working Group on Interventional Cardiology of the European Society of Cardiology 8 Suppl Q:Q83-87. |
[8] | Ribeiro HB, Urena M, Kuck KH, et al. (2012) Edwards CENTERA valve. EuroIntervention: journal of EuroPCR in collaboration with the Working Group on Interventional Cardiology of the European Society of Cardiology 8 Suppl Q:Q79-82. |
[9] | Meredith Am IT, Walters DL, Dumonteil N, et al. (2014) Transcatheter aortic valve replacement for severe symptomatic aortic stenosis using a repositionable valve system: 30-day primary endpoint results from the REPRISE II study. J Am Coll Cardiol 64 (13):1339-1348. |
[10] | Holzhey D (2013) Thirty day outcomes from the multicentre European pivotal trial for transapical TAVI with a self-expanding prosthesis. EuroPCR May 21st; Paris. |
[11] | The Medtronic CoreValveTM Evolut RTM CE Mark Clinical Study 2015 [cited 2015 February 27]. Available from: https://www.clinicaltrials.gov/ct2/show/NCT01876420?term=NCT01876420&rank=1. |
[12] | Medtronic CoreValve Evolut R U.S. Clinical Study 2015 [cited 2015 February 27]. Available from: https://www.clinicaltrials.gov/ct2/show/NCT02207569?term=NCT02207569&rank=1. |
[13] | The PARTNER II Trial: Placement of AoRTic TraNscathetER Valves [cited 2015 Febrary 2015]. Available from: https://www.clinicaltrials.gov/ct2/show/NCT01314313?term=THE+PARTNER+II+TRIAL&rank=1. |
[14] | Webb J, Gerosa G, Lefevre T, et al. (2014) Multicenter evaluation of a next-generation balloon-expandable transcatheter aortic valve. J Am Coll Cardiol 64(21):2235-2243. |
[15] | Binder RK, Schafer U, Kuck KH, et al. (2013) Transcatheter aortic valve replacement with a new self-expanding transcatheter heart valve and motorized delivery system. JACC Cardiovascular interventions 6(3):301-307. |
[16] | Edwards CENTERA transcatheter heart valve. EuroPCR; May Paris2013. |
[17] | Kempfert J, Holzhey D, Hofmann S, et al. (2014) First registry results from the newly approved ACURATE TA TAVI systemdagger. European journal of cardio-thoracic surgery : official journal of the European Association for Cardio-thoracic Surgery 25. |
[18] | Grube E (2014) ACURATE neoTM & ACURATE TFTM Delivery System Clinical Program & Results. EuroPCR; May 22; Paris. |
[19] | Longterm Safety and Performance of the JenaValve (JUPITER) 2015 [cited 2015 Febrary 27]. Available from: https://www.clinicaltrials.gov/ct2/show/NCT01598844?term=jenavalve&rank=1. |
[20] | Willson AB, Rodes-Cabau J, Wood DA, et al. (2012) Transcatheter aortic valve replacement with the St. Jude Medical Portico valve: first-in-human experience. J Am Coll Cardiol 60(7):581-586. |
[21] | Schofer J (2014) ProspectiveMulticenter Evaluation of the Direct Flow Medical® Transcatheter Aortic Valve: The DISCOVER Trial: 12-month Outcomes. EuroPCR; May 21; Paris. |
[22] | A Registry to Evaluate the Direct Flow Medical Transcatheter Aortic Valve System (DISCOVER) 2015 [cited 2015 Febrary 27]. Available from: https://www.clinicaltrials.gov/ct2/show/NCT01845285?term=direct+flow+medical&rank=1. |
[23] | Leon MB, Piazza N, Nikolsky E, et al. (2011) Standardized endpoint definitions for transcatheter aortic valve implantation clinical trials: a consensus report from the Valve Academic Research Consortium. Euro Heart J 32(2):205-217. |
[24] | Kappetein AP, Head SJ, Genereux P, et al. (2012) Updated standardized endpoint definitions for transcatheter aortic valve implantation: the Valve Academic Research Consortium-2 consensus document. Euro Heart J 33(19):2403-2418. |
[25] | Kapadia SR, Leon MB, Makkar RR, et al. (2015) 5-year outcomes of transcatheter aortic valve replacement compared with standard treatment for patients with inoperable aortic stenosis (PARTNER 1): a randomised controlled trial. Lancet 15. |
[26] | Leon MB, Smith CR, Mack M, et al. (2010) Transcatheter aortic-valve implantation for aortic stenosis in patients who cannot undergo surgery. The New England j med 363(17):1597-1607. |
[27] | Smith CR, Leon MB, Mack MJ, et al. (2011) Transcatheter versus surgical aortic-valve replacement in high-risk patients. The New England j med 364(23):2187-2198. |
[28] | Kodali SK, Williams MR, Smith CR, et al. (2012) Two-year outcomes after transcatheter or surgical aortic-valve replacement. The New England j med 366(18):1686-1695. |
[29] | Mack MJ, Leon MB, Smith CR, et al. (2015) 5-year outcomes of transcatheter aortic valve replacement or surgical aortic valve replacement for high surgical risk patients with aortic stenosis (PARTNER 1): a randomised controlled trial. Lancet 15. |
[30] | Leon MB (2013) A Randomized Evaluation of the SAPIEN XT Transcatheter Valve System in Patients with Aortic Stenosis Who Are Not Candidates for Surgery: PARTNER II, Inoperable Cohort. ACC; March 10; San Francisco. |
[31] | Webb J (2015) 1-Year outcomes from the SAPIEN 3 trial. EuroPCR; May 19th-22nd; Paris, France. |
[32] | Kodali S (2015) Clinical and Echocardiographic Outcomes at 30 Days with the SAPIEN 3 TAVR System in Inoperable, High-Risk and Intermediate-Risk AS Patients. ACC; March 15; San Diego, CA. |
[33] | Amat-Santos IJ, Dahou A, Webb J, et al. (2014) Comparison of hemodynamic performance of the balloon-expandable SAPIEN 3 versus SAPIEN XT transcatheter valve. The Am J Cardio 114(7):1075-1082. |
[34] | Tarantini G, Mojoli M, Purita P, et al. (2014) Unravelling the (arte)fact of increased pacemaker rate with the Edwards SAPIEN 3 valve. EuroIntervention : journal of EuroPCR in collaboration with the Working Group on Interventional Cardiology of the European Society of Cardiology 19. |
[35] | Schymik G, Lefevre T, Bartorelli AL, et al. (2015) European Experience With the Second-Generation Edwards SAPIEN XT Transcatheter Heart Valve in Patients With Severe Aortic Stenosis: 1-Year Outcomes From the SOURCE XT Registry. JACC Cardiovascular interventions 8(5):657-669. |
[36] | Popma JJ, Adams DH, Reardon MJ, et al. (2014) Transcatheter aortic valve replacement using a self-expanding bioprosthesis in patients with severe aortic stenosis at extreme risk for surgery. J Am Coll Cardiol 63(19):1972-1981. |
[37] | Yakubov SJ (2014) Long-Term Outcomes Using a Self-Expanding Bioprosthesis in Patients With Severe Aortic Stenosis Deemed Extreme Risk for Surgery: Two-Year Results From the CoreValve US Pivotal Trial TCT 13 Washington, DC. |
[38] | Adams DH, Popma JJ, Reardon MJ, et al. (2014) Transcatheter aortic-valve replacement with a self-expanding prosthesis. The New Eng J Med 370(19):1790-1798. |
[39] | Reardon MJ (2015) A Randomized Comparison of Self-expanding Transcatheter and Surgical Aortic Valve Replacement in Patients with Severe Aortic Stenosis Deemed at Increased Risk for Surgery 2-Year Outcomes. ACC 15; San Diego, CA. |
[40] | Wendler O (2011) 1-year mortalityresults from combined Cohort I and Cohort II of The SOURCE Registry. EuroPCR; May; Paris, France. |
[41] | Walther T (2014) 2-year outcomes from the SOURCE XT registry: transfemoral versus transapical approach. EuroPCR; May; Paris, France. |
[42] | One Year Outcomes in Real World Patients Treated with Transcatheter Aortic Valve Implantation. EuroPCR; May; Paris, France2013. |
[43] | Piazza N (2015) Three Year Outcomes in Real World Patients Treated with Transcatheter Aortic Valve Implantation. TVT June 5; Chicago, USA. |
[44] | Holmes DR, Brennan JM, Rumsfeld JS, et al. (2015) Clinical outcomes at 1 year following transcatheter aortic valve replacement. Jama 313(10):1019-1028. |
[45] | Mohr FW, Holzhey D, Mollmann H, et al. (2014) The German Aortic Valve Registry: 1-year results from 13,680 patients with aortic valve disease. European journal of cardio-thoracic surgery: official journal of the European Association for Cardio-thoracic Surgery 46(5):808-816. |
[46] | Moat NE, Ludman P, de Belder MA, et al. (2011) Long-term outcomes after transcatheter aortic valve implantation in high-risk patients with severe aortic stenosis: the U.K. TAVI (United Kingdom Transcatheter Aortic Valve Implantation) Registry. J Am Coll Cardiol 58(20):2130-2138 |
[47] | Ludman PF, Moat N, de Belder MA, et al. (2015) Transcatheter Aortic Valve Implantation in the UK: Temporal Trends, Predictors of Outcome and 6 Year Follow Up: A Report from the UK TAVI Registry 2007 to 2012. Circulation 30. |
[48] | Gilard M, Eltchaninoff H, Iung B, et al. (2012) Registry of transcatheter aortic-valve implantation in high-risk patients. The New Eng J Med 366(18):1705-1715. |
[49] | Bourantas CV, Van Mieghem NM, Soliman O, et al. (2013) Transcatheter aortic valve update 2013. EuroIntervention: journal of EuroPCR in collaboration with the Working Group on Interventional Cardiology of the European Society of Cardiology 9 Suppl:S84-90. |
[50] | Engager Align Study 2015 [cited 2015 Febrary 27]. Available from: https://www.clinicaltrials.gov/ct2/show/NCT02149654?term=engager+aortic+valve&rank=2. |
[51] | Meredith IT (2015) 6-Month Outcomes Following Transcatheter Aortic Valve Implantation With a Novel Repositionable Self-Expanding Bioprosthesis. EuroPCR 5: 19-22. |
[52] | Safety and Efficacy Study of the Medtronic CoreValve® System in the Treatment of Severe, Symptomatic Aortic Stenosis in Intermediate Risk Subjects Who Need Aortic Valve Replacement (SURTAVI). 2015 [cited 2015 Febrary 2015]. Available from: https://www.clinicaltrials.gov/ct2/show/NCT01586910?term=surtavi&rank=1. |
[53] | Abdel-Wahab M, Mehilli J, Frerker C, et al. (2014) Comparison of balloon-expandable vs self-expandable valves in patients undergoing transcatheter aortic valve replacement: the CHOICE randomized clinical trial. Jama 311(15):1503-1514. |
[54] | Abdel-Wahab M, Richardt G (2014) Selection of TAVI prostheses: do we really have the CHOICE? EuroIntervention: journal of EuroPCR in collaboration with the Working Group on Interventional Cardiology of the European Society of Cardiology 10(U):U28-U34. |
[55] | Abdel-Wahab M (2015) One-year outcomes after TAVI with balloon-expandable vs. self-expandable valves. EuroPCR 20; Paris. |
[56] | AM ITM (2014) Repositionable Percutaneous Aortic Valve Replacement: 30-Day Outcomes in 250 High Surgical Risk Patients in the REPRISE II Extended Trial Cohort. PCR London Valve; London. |
[57] | Schofer J, Colombo A, Klugmann S, et al. (2014) Prospective multicenter evaluation of the direct flow medical transcatheter aortic valve. J Am Coll Cardiol 63(8):763-768. |
[58] | Colombo A (2015) 2-year data from the DISCOVER CE Mark Trial, which studied the Direct Flow Medical transcatheter aortic valve replacement system. EuroPCR; Paris, France. |
[59] | TranScatheter Aortic Valve RepLacement System a US Pivotal Trial (SALUS) 2015 [cited 2015 Febrary 27]. Available from: https://www.clinicaltrials.gov/ct2/show/NCT02163850?term=direct+flow+medical&rank=3. |
[60] | Primary endpoint data presented on performance and safety of the JenaValve TAVI system. 2014 [cited 2015 Febrary 27]. Available from: http://www.pcronline.com/News/Press-releases/Primary-endpoint-data-presented-on-performance-and-safety-of-the-jenavalve-tavi-system. |
[61] | Thoracic S, Vahanian A, Alfieri O, et al. (2012) Joint Task Force on the Management of Valvular Heart Disease of the European Society of C, European Association for Cardio. Guidelines on the management of valvular heart disease (version 2012). European Heart J 33(19):2451-2496. |
[62] | Nishimura RA, Otto CM, Bonow RO, et al. (2014) AHA/ACC guideline for the management of patients with valvular heart disease: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines. The J Thoracic Cardio Surg 148(1):e1-e132. |
[63] | Bax JJ, Delgado V, Bapat V, et al. (2014) Open issues in transcatheter aortic valve implantation. Part 1: patient selection and treatment strategy for transcatheter aortic valve implantation. European Heart J 35(38):2627-2638. |
[64] | Watanabe Y, Hayashida K, Lefevre T, et al. (2013) Is EuroSCORE II better than EuroSCORE in predicting mortality after transcatheter aortic valve implantation? Catheterization and cardiovascular interventions: official journal of the Society for Cardiac Angiography & Interventions 81(6):1053-1060. |
[65] | Stahli BE, Tasnady H, Luscher TF, et al. (2013) Early and late mortality in patients undergoing transcatheter aortic valve implantation: comparison of the novel EuroScore II with established risk scores. Cardiolo 126(1):15-23. |
[66] | Iung B, Laouenan C, Himbert D, et al. (2014) Predictive factors of early mortality after transcatheter aortic valve implantation: individual risk assessment using a simple score. Heart 100(13):1016-1023. |
[67] | Mack MJ, Brennan JM, Brindis R, et al. Outcomes following transcatheter aortic valve replacement in the United States. Jama 310(19):2069-2077. |
[68] | Piazza N, Kalesan B, van Mieghem N, et al. (2013) A 3-center comparison of 1-year mortality outcomes between transcatheter aortic valve implantation and surgical aortic valve replacement on the basis of propensity score matching among intermediate-risk surgical patients. JACC Cardiovascular interventions 6(5):443-451. |
[69] | Lange R, Bleiziffer S, Mazzitelli D, et al. (2012) Improvements in transcatheter aortic valve implantation outcomes in lower surgical risk patients: a glimpse into the future. J Am Coll Cardiol59(3):280-287. |
[70] | Wenaweser P, Stortecky S, Schwander S, et al. (2013) Clinical outcomes of patients with estimated low or intermediate surgical risk undergoing transcatheter aortic valve implantation. European Heart J 34(25):1894-1905. |
[71] | Schymik G, Schrofel H, Schymik JS, et al. (2012) Acute and late outcomes of Transcatheter Aortic Valve Implantation (TAVI) for the treatment of severe symptomatic aortic stenosis in patients at high- and low-surgical risk. J Interventional Cardiolo 25(4):364-374. |
[72] | D'Errigo P, Barbanti M, Ranucci M, et al. (2013) Transcatheter aortic valve implantation versus surgical aortic valve replacement for severe aortic stenosis: results from an intermediate risk propensity-matched population of the Italian OBSERVANT study. Inter J Cardiolo 167(5):1945-1952. |
[73] | Thyregod HG, Steinbruchel DA, Ihlemann N, et al. (2015) Transcatheter Versus Surgical Aortic Valve Replacement in Patients With Severe Aortic Valve Stenosis: 1-Year Results From the All-Comers NOTION Randomized Clinical Trial. J Am Coll Cardiol 65(20):2184-2194. |
[74] | Athappan G, Patvardhan E, Tuzcu EM, et al. (2013) Incidence, predictors, and outcomes of aortic regurgitation after transcatheter aortic valve replacement: meta-analysis and systematic review of literature. J Am Coll Cardiol 61(15):1585-1595. |
[75] | Leon MB (2014) Perspectives on the 5 “Hottest” Topics in TAVR for 2014. Transcatherer Valve Therapies; Vancouver, Canada. |
[76] | Daneault B, Koss E, Hahn RT, et al. (2013) Efficacy and safety of postdilatation to reduce paravalvular regurgitation during balloon-expandable transcatheter aortic valve replacement. Circulation Cardiovascular interventions 6(1):85-91. |
[77] | Erkapic D, De Rosa S, Kelava A, et al. (2012) Risk for permanent pacemaker after transcatheter aortic valve implantation: a comprehensive analysis of the literature. J cardiovascular electrophysiolo 23(4):391-397. |
[78] | Piazza N, Onuma Y, Jesserun E, et al. (2008) Early and persistent intraventricular conduction abnormalities and requirements for pacemaking after percutaneous replacement of the aortic valve. JACC Cardiovascular interventions 1(3):310-316. |
[79] | Marcel Weber J-MS, Christoph Hammerstingl, Nikos Werner, et al. (2015) Georg nickenig. permanent pacemaker implantation after tavr—predictors and impact on outcomes. Int Cardiolo Review 10(2):98–102. |
[80] | Eggebrecht H, Schmermund A, Voigtlander T, et al. (2012) Risk of stroke after transcatheter aortic valve implantation (TAVI): a meta-analysis of 10,037 published patients. EuroIntervention: journal of EuroPCR in collaboration with the Working Group on Interventional Cardiology of the European Soc Cardiolo 8(1):129-138. |
[81] | Athappan G, Gajulapalli RD, Sengodan P, et al. (2014) Influence of transcatheter aortic valve replacement strategy and valve design on stroke after transcatheter aortic valve replacement: a meta-analysis and systematic review of literature. J Am Coll Cardiol 63(20):2101-2110. |
[82] | Lansky AJ, Schofer J, Tchetche D, et al. (2015) A prospective randomized evaluation of the TriGuard HDH embolic DEFLECTion device during transcatheter aortic valve implantation: results from the DEFLECT III trial. European Heart J 19. |
[83] | Colombo A, Michev I, Latib A (2014) Is it time to simplify the TAVI procedure? ""Make it simple but not too simple"". EuroIntervention : journal of EuroPCR in collaboration with the Working Group on Interventional Cardiology of the European Society of Cardiology 10 Suppl U:U22-27. |
[84] | Wood DA, Poulter R, Cook R, et al. (2014) TCT-701 A Multidisciplinary, Multimodality, but Minimalist (3M) approach to transfemoral transcatheter aortic valve replacement facilitates safe next day discharge home in high risk patients: 1 year follow up. J Am College Cardiolo 64(11_S). |
[85] | Islas F, Almeria C, Garcia-Fernandez E, et al. (2015) Usefulness of echocardiographic criteria for transcatheter aortic valve implantation without balloon predilation: a single-center experience. Journal of the American Society of Echocardiography: official publication of the American Society of Echocardiography 28(4):423-429. |
[86] |
Bramlage P, Strauch J, Schrofel H (2014) Balloon expandable transcatheter aortic valve implantation with or without pre-dilation of the aortic valve—rationale and design of a multicenter registry (EASE-IT). BMC cardiovascular disorders 14:160. doi: 10.1186/1471-2261-14-160
![]() |
[87] | Blumenstein J, Kempfert J, Van Linden A, et al. (2013) First-in-man evaluation of the transapical APICA ASC access and closure device: the initial 10 patients. European journal of cardio-thoracic surgery: official journal of the European Association for Cardio-thoracic Surgery 44(6):1057-1062 |
[88] | Khawaja MZ, Wang D, Pocock S, et al. (2014) The percutaneous coronary intervention prior to transcatheter aortic valve implantation (ACTIVATION) trial: study protocol for a randomized controlled trial. Trials 15:300. |
[89] | Aspirin Versus Aspirin + ClopidogRel Following Transcatheter Aortic Valve Implantation: the ARTE Trial 2015 [cited 2015 Febrary 27]. Available from: https://www.clinicaltrials.gov/ct2/show/NCT01559298?term=arte+trial&rank=1. |
[90] | Tuzcu EM (2014) Transcatheter Aortic Valve in Valve Replacement for Degenerative Aortic Bioprosthesis: Initial Results from the STS/ACC Transcatheter Valve Therapy Registry. ACC; Washington, DC. |
1. | Rinaldo M. Colombo, Mauro Garavello, Well Posedness and Control in a NonLocal SIR Model, 2020, 0095-4616, 10.1007/s00245-020-09660-9 | |
2. | Rinaldo M. Colombo, Mauro Garavello, Francesca Marcellini, Elena Rossi, An age and space structured SIR model describing the Covid-19 pandemic, 2020, 10, 2190-5983, 10.1186/s13362-020-00090-4 | |
3. | Hiroya Kikuchi, Hiroaki Mukaidani, Ramasamy Saravanakumar, Weihua Zhuang, 2020, Robust Vaccination Strategy based on Dynamic Game for Uncertain SIR Time-Delay Model, 978-1-7281-8526-2, 3427, 10.1109/SMC42975.2020.9283389 | |
4. | Giacomo Albi, Lorenzo Pareschi, Mattia Zanella, Modelling lockdown measures in epidemic outbreaks using selective socio-economic containment with uncertainty, 2021, 18, 1551-0018, 7161, 10.3934/mbe.2021355 | |
5. | Giacomo Albi, Lorenzo Pareschi, Mattia Zanella, Control with uncertain data of socially structured compartmental epidemic models, 2021, 82, 0303-6812, 10.1007/s00285-021-01617-y | |
6. | Christian John Hurry, Alexander Mozeika, Alessia Annibale, Vaccination with partial transmission and social distancing on contact networks, 2022, 2022, 1742-5468, 033302, 10.1088/1742-5468/ac50ae | |
7. | Martin Kröger, Mustafa Turkyilmazoglu, Reinhard Schlickeiser, Explicit formulae for the peak time of an epidemic from the SIR model. Which approximant to use?, 2021, 425, 01672789, 132981, 10.1016/j.physd.2021.132981 | |
8. | Emanuele Bernardi, Lorenzo Pareschi, Giuseppe Toscani, Mattia Zanella, Effects of Vaccination Efficacy on Wealth Distribution in Kinetic Epidemic Models, 2022, 24, 1099-4300, 216, 10.3390/e24020216 | |
9. | Reinhard Schlickeiser, Martin Kröger, Analytical Modeling of the Temporal Evolution of Epidemics Outbreaks Accounting for Vaccinations, 2021, 3, 2624-8174, 386, 10.3390/physics3020028 | |
10. | Erivelton G. Nepomuceno, Márcia L. C. Peixoto, Márcio J. Lacerda, Andriana S. L. O. Campanharo, Ricardo H. C. Takahashi, Luis A. Aguirre, Application of Optimal Control of Infectious Diseases in a Model-Free Scenario, 2021, 2, 2662-995X, 10.1007/s42979-021-00794-3 | |
11. | Alejandro Carballosa, José Balsa-Barreiro, Pablo Boullosa, Adrián Garea, Jorge Mira, Ángel Miramontes, Alberto P. Muñuzuri, Assessing the risk of pandemic outbreaks across municipalities with mathematical descriptors based on age and mobility restrictions, 2022, 160, 09600779, 112156, 10.1016/j.chaos.2022.112156 | |
12. | Candy Sonveaux, Joseph J. Winkin, State feedback control law design for an age-dependent SIR model, 2023, 158, 00051098, 111297, 10.1016/j.automatica.2023.111297 | |
13. | Fernando Córdova-Lepe, Juan Pablo Gutiérrez-Jara, A Dynamic Reaction-restore-type Transmission-rate Model for COVID-19, 2024, 21, 2224-2902, 118, 10.37394/23208.2024.21.12 |
$ \bar a $ in (2.3) | 0.1 | 0.2 | 0.3 | 0.5 | 1.0 | 1.5 | none |
$ \mathcal{J} $ in (1.8) | 33.48 | 41.03 | 44.73 | 46.85 | 47.39 | 47.51 | 48.30 |
$ \mathcal{N} $ in (1.5) | 10.53 | 5.05 | 2.30 | 0.76 | 0.41 | 0.36 | 0.00 |
$ \mathcal{J}+\mathcal{N} $ in (1.9) | 44.01 | 46.08 | 47.03 | 47.61 | 47.80 | 47.87 | 48.30 |
$ \bar t $ in (2.4) | 0.1 | 0.2 | 0.3 | 0.5 | 1.0 | 1.5 | none |
$ \mathcal{J} $ in (1.8) | 31.71 | 38.80 | 43.65 | 46.79 | 47.40 | 47.42 | 48.30 |
$ \mathcal{N} $ in (1.7) | 9.37 | 5.46 | 2.74 | 0.86 | 0.40 | 0.39 | 0.00 |
$ \mathcal{J}+\mathcal{N} $ in (1.10) | 41.08 | 44.26 | 46.39 | 47.66 | 47.80 | 47.80 | 48.30 |
$ \eta_1 $ | $ \eta_2 $ | $ \eta_3 $ | $ \mathcal{J} $ as in (1.8) | $ \mathcal{N} $ as in (1.5) | $ \mathcal{J} + \mathcal{N} $ as in (1.9) |
$ 0.6 $ | $ 0.6 $ | $ 0.6 $ | $ 14.54 $ | $ 7.56 $ | $ 22.11 $ |
$ 0.6 $ | $ 0.6 $ | $ 0.8 $ | $ 14.38 $ | $ 7.87 $ | $ 22.25 $ |
$ 0.6 $ | $ 0.8 $ | $ 0.6 $ | $ 14.18 $ | $ 8.02 $ | $ 22.20 $ |
$ 0.6 $ | $ 0.8 $ | $ 0.8 $ | $ 14.02 $ | $ 8.31 $ | $ 22.83 $ |
$ 0.8 $ | $ 0.6 $ | $ 0.6 $ | $ 13.12 $ | $ 9.22 $ | $ 22.34 $ |
$ 0.8 $ | $ 0.6 $ | $ 0.8 $ | $ 12.96 $ | $ 9.52 $ | $ 22.48 $ |
$ 0.8 $ | $ 0.8 $ | $ 0.6 $ | $ 12.80 $ | $ 9.60 $ | $ 22.40 $ |
$ 0.8 $ | $ 0.8 $ | $ 0.8 $ | $ 12.64 $ | $ 9.88 $ | $ 22.53 $ |
see Figure 8 | $ 16.01 $ | $ 4.67 $ | $ 20.69 $ |
$ \nu_1 $ | $ \nu_2 $ | $ \nu_3 $ | $ \mathcal{J} $ as in (1.8) | $ \mathcal{N} $ as in (1.7) | $ \mathcal{J} + \mathcal{N} $ as in (1.10) |
$ 0.6 $ | $ 0.6 $ | $ 0.6 $ | $ 6.41 $ | $ 13.05 $ | $ 19.47 $ |
$ 0.6 $ | $ 0.6 $ | $ 0.8 $ | $ 6.23 $ | $ 13.75 $ | $ 19.98 $ |
$ 0.6 $ | $ 0.8 $ | $ 0.6 $ | $ 6.17 $ | $ 13.60 $ | $ 19.77 $ |
$ 0.6 $ | $ 0.8 $ | $ 0.8 $ | $ 6.06 $ | $ 14.10 $ | $ 20.16 $ |
$ 0.8 $ | $ 0.6 $ | $ 0.6 $ | $ 5.78 $ | $ 13.89 $ | $ 19.66 $ |
$ 0.8 $ | $ 0.6 $ | $ 0.8 $ | $ 5.67 $ | $ 14.44 $ | $ 20.12 $ |
$ 0.8 $ | $ 0.8 $ | $ 0.6 $ | $ 5.66 $ | $ 14.22 $ | $ 19.88 $ |
$ 0.8 $ | $ 0.8 $ | $ 0.8 $ | $ 5.59 $ | $ 14.65 $ | $ 20.24 $ |
see Figure 11 | $ 7.80 $ | $ 8.18 $ | $ 15.98 $ |
$ \bar a $ in (2.3) | 0.1 | 0.2 | 0.3 | 0.5 | 1.0 | 1.5 | none |
$ \mathcal{J} $ in (1.8) | 33.48 | 41.03 | 44.73 | 46.85 | 47.39 | 47.51 | 48.30 |
$ \mathcal{N} $ in (1.5) | 10.53 | 5.05 | 2.30 | 0.76 | 0.41 | 0.36 | 0.00 |
$ \mathcal{J}+\mathcal{N} $ in (1.9) | 44.01 | 46.08 | 47.03 | 47.61 | 47.80 | 47.87 | 48.30 |
$ \bar t $ in (2.4) | 0.1 | 0.2 | 0.3 | 0.5 | 1.0 | 1.5 | none |
$ \mathcal{J} $ in (1.8) | 31.71 | 38.80 | 43.65 | 46.79 | 47.40 | 47.42 | 48.30 |
$ \mathcal{N} $ in (1.7) | 9.37 | 5.46 | 2.74 | 0.86 | 0.40 | 0.39 | 0.00 |
$ \mathcal{J}+\mathcal{N} $ in (1.10) | 41.08 | 44.26 | 46.39 | 47.66 | 47.80 | 47.80 | 48.30 |
$ \eta_1 $ | $ \eta_2 $ | $ \eta_3 $ | $ \mathcal{J} $ as in (1.8) | $ \mathcal{N} $ as in (1.5) | $ \mathcal{J} + \mathcal{N} $ as in (1.9) |
$ 0.6 $ | $ 0.6 $ | $ 0.6 $ | $ 14.54 $ | $ 7.56 $ | $ 22.11 $ |
$ 0.6 $ | $ 0.6 $ | $ 0.8 $ | $ 14.38 $ | $ 7.87 $ | $ 22.25 $ |
$ 0.6 $ | $ 0.8 $ | $ 0.6 $ | $ 14.18 $ | $ 8.02 $ | $ 22.20 $ |
$ 0.6 $ | $ 0.8 $ | $ 0.8 $ | $ 14.02 $ | $ 8.31 $ | $ 22.83 $ |
$ 0.8 $ | $ 0.6 $ | $ 0.6 $ | $ 13.12 $ | $ 9.22 $ | $ 22.34 $ |
$ 0.8 $ | $ 0.6 $ | $ 0.8 $ | $ 12.96 $ | $ 9.52 $ | $ 22.48 $ |
$ 0.8 $ | $ 0.8 $ | $ 0.6 $ | $ 12.80 $ | $ 9.60 $ | $ 22.40 $ |
$ 0.8 $ | $ 0.8 $ | $ 0.8 $ | $ 12.64 $ | $ 9.88 $ | $ 22.53 $ |
see Figure 8 | $ 16.01 $ | $ 4.67 $ | $ 20.69 $ |
$ \nu_1 $ | $ \nu_2 $ | $ \nu_3 $ | $ \mathcal{J} $ as in (1.8) | $ \mathcal{N} $ as in (1.7) | $ \mathcal{J} + \mathcal{N} $ as in (1.10) |
$ 0.6 $ | $ 0.6 $ | $ 0.6 $ | $ 6.41 $ | $ 13.05 $ | $ 19.47 $ |
$ 0.6 $ | $ 0.6 $ | $ 0.8 $ | $ 6.23 $ | $ 13.75 $ | $ 19.98 $ |
$ 0.6 $ | $ 0.8 $ | $ 0.6 $ | $ 6.17 $ | $ 13.60 $ | $ 19.77 $ |
$ 0.6 $ | $ 0.8 $ | $ 0.8 $ | $ 6.06 $ | $ 14.10 $ | $ 20.16 $ |
$ 0.8 $ | $ 0.6 $ | $ 0.6 $ | $ 5.78 $ | $ 13.89 $ | $ 19.66 $ |
$ 0.8 $ | $ 0.6 $ | $ 0.8 $ | $ 5.67 $ | $ 14.44 $ | $ 20.12 $ |
$ 0.8 $ | $ 0.8 $ | $ 0.6 $ | $ 5.66 $ | $ 14.22 $ | $ 19.88 $ |
$ 0.8 $ | $ 0.8 $ | $ 0.8 $ | $ 5.59 $ | $ 14.65 $ | $ 20.24 $ |
see Figure 11 | $ 7.80 $ | $ 8.18 $ | $ 15.98 $ |