
Citation: Cindy E. McKinney, Katherine M. Baumgarner. CRISPR engineering cardiometabolic disease models using human iPSC[J]. AIMS Cell and Tissue Engineering, 2018, 2(3): 185-202. doi: 10.3934/celltissue.2018.3.185
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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 dS=dS(t,a)≥0 is the susceptibles' death rate and λ(a,a′)≥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 ∫+∞0λ(a,a′)I(t,a′)da′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 dI=dI(t,a)≥0 is the death rate of the I portion of the population and rI=rI(t,a)≥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 dR=dR(t,a)≥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 ηj(t) of the S population of the prescribed age ˉaj, with 0<ˉa1<⋯<ˉaN. In this way, the fraction ηj(t) of susceptible population S(t,ˉaj) becomes immune. In other words, we use as control the fractions ηj:I→[0,1] (for j∈{1,⋯,N} and 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∈{1,⋯,N}. A reasonable constraint on the vaccination policy η in (1.4) is that its total cost N(η) on the considered time interval I does not exceed a prescribed maximal cost M. Measuring the total cost by the number 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 ˉt1,…,ˉtN. 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 νk(a) is the percentage of susceptible individuals of age a that are dosed with the vaccine at time ˉtk, for k=1,…,N. Now, a reasonable constraint due to the campaign cost is given by a bound on the number 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 ηj, respectively νi, 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 I, that is
J=∫I∫R+φ(a)I(t,a)dadt | (1.8) |
the function φ being a suitable positive age dependent weight. Here, J is a function of η in case (1.4) and of ν 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 N and 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 −uS 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 I either the time interval [0,T], for a positive T, or [0,+∞[. 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:
(λ) λ∈C0(R+×R+;R) is bounded, with total variation in the first argument uniformly bounded with respect to the second (i.e., supa′∈R+TV(λ(⋅,a′);R+)<+∞) 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,a1,a2∈R+, |λ(a1,a′)−λ(a2,a′)|≤C|a1−a2|).
(d-r) The maps dS,dI,dR,rI:I×R+→R are bounded Caratheodory functions, with L1 norm and total variation in a uniformly bounded in t (i.e., for φ=dS,dI,dR,rI, supt∈I‖φ(t)‖L1(R+;R)<+∞ and supt∈ITV(φ(t,⋅);R+)<+∞) and Lipschitz continuous in a uniformly in t (i.e., there exists a C>0 such that for all t∈I and a1,a2∈R+, |φ(t,a2)−φ(t,a1)|≤C|a2−a1|).
(IB) The initial datum and the boundary inflow in (2.2) satisfy So,Io,Ro∈(L1∩BV)(R+;R+) and Sb,Ib,Rb∈(L1∩BV)(I;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 (λ) and (d-r), for any initial and boundary data satisfying (IB), for any choice of the positive vaccination ages ˉa1,…,ˉaN∈R+∖{0} and of the control function η∈BV(I;[0,1]N), problem (2.1)–(2.2)–(2.3) admits a unique solution defined on the whole interval I, depending Lipschitz continuously on the initial datum, through the L1 norm, and on η, through the L∞ 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 (λ) and (d-r), for any initial and boundary data satisfying (IB), for any choice of the positive vaccination times ˉt1,…,ˉtN∈I∖{0} and of the control function ν∈BV(R+;[0,1]N), problem (2.1)–(2.2)–(2.4) admits a unique solution defined on the whole interval I, depending Lipschitz continuously on the initial datum, through the L1 norm, and on ν, through the L∞ norm.
Below, we restrict the search of optimal controls ηj in (1.4) and νi 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 ηℓj and νℓi. 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 Δa=2.5⋅10−3 along the age axis and Δt=1.25⋅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 I×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 λ 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 Io≡0, so that for all times t, I(t)≡0 and no illness ever appears, and then setting Io 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≡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 ˉa 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 ˉa=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 ˉa=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 ˉa. 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 ˉa=0.5 looks as (un)successful as the campaign at ˉa=1.5. On the contrary, dosing at ˉa=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 ˉa=0.1. This choice also results in the highest number of vaccinated individuals, so that also N corresponds to the most expensive policy.
ˉa in (2.3) | 0.1 | 0.2 | 0.3 | 0.5 | 1.0 | 1.5 | none |
J in (1.8) | 33.48 | 41.03 | 44.73 | 46.85 | 47.39 | 47.51 | 48.30 |
N in (1.5) | 10.53 | 5.05 | 2.30 | 0.76 | 0.41 | 0.36 | 0.00 |
J+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 ˉt, see Figure 4.
Similarly to the integration detailed in § 3.2, campaigns at earlier times results in being more effective. The choices ˉt=0.5 and ˉt=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.
ˉt in (2.4) | 0.1 | 0.2 | 0.3 | 0.5 | 1.0 | 1.5 | none |
J in (1.8) | 31.71 | 38.80 | 43.65 | 46.79 | 47.40 | 47.42 | 48.30 |
N in (1.7) | 9.37 | 5.46 | 2.74 | 0.86 | 0.40 | 0.39 | 0.00 |
J+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 ˉt, 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 ˉt=0.5 or later, makes relatively small difference, the most convenient choice being at the earliest time ˉt=0.1. Similarly to what happens in § 3.2, vaccinating all susceptible individuals at ˉt=0.1 makes the susceptible portion of the population very small wherever a>t−ˉt, 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., ˉa=0.1) in § 3.2 and the present best vaccination time choice (i.e., ˉt=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 ˉa=0.1) clearly shows a decreasing I population, while it is increasing in the latter strategy, see Figure 4, left (corresponding to ˉt=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 ]3,5[. 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≥3 and a∈[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 ηj and νi 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 ˉa1, ˉa2 and ˉa3 or times ˉt1, ˉt2 and ˉt3.
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 η1,η2,η3 and ν1,ν2,ν3 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 Δa=2.5⋅10−3 and Δt=1.25⋅10−3. However, due to computing time limitations, when applying the steepest descent method we passed to coarse meshes, where Δa=5⋅10−2 and Δt=2.5⋅10−2. Correspondingly, in case of the controls (2.3), optimal controls are sought in the class of locally constant functions of the type ηj(t)=∑20ℓ=1ηℓjχ[(ℓ−1)/4,ℓ/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.
η1 | η2 | η3 | J as in (1.8) | N as in (1.5) | J+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 ˉaj resulting from the conditions (2.3) are clearly visible. Choosing ˉa1=2.0 as the lowest vaccination age allows, for about t≥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 J+N in (1.9) lower than any of the ones obtained with constant ηj 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.
ν1 | ν2 | ν3 | J as in (1.8) | N as in (1.7) | J+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 ν2 and ν3 roughly correspond to the wave of infected individuals propagating from t=0 onward.
As a consequence, note that the optimal control ν3 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 νi(a) of all susceptible individuals of age a are dosed at time ˉti, 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 ηj in (2.3) or νi 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.
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1. | M. Ouzahra, Exponential stability of nondissipative linear system in Banach space and application to unbounded bilinear systems, 2017, 109, 01676911, 53, 10.1016/j.sysconle.2017.08.010 | |
2. | Martin Gugat, Jan Giesselmann, G. Buttazzo, E. Casas, L. de Teresa, R. Glowinski, G. Leugering, E. Trélat, X. Zhang, Boundary feedback stabilization of a semilinear model for the flow in star-shaped gas networks, 2021, 27, 1292-8119, 67, 10.1051/cocv/2021061 | |
3. | Martin Gugat, Stephan Gerster, On the limits of stabilizability for networks of strings, 2019, 131, 01676911, 104494, 10.1016/j.sysconle.2019.104494 | |
4. | Charlotte Rodriguez, Networks of geometrically exact beams: Well-posedness and stabilization, 2022, 12, 2156-8472, 49, 10.3934/mcrf.2021002 | |
5. | Martin Gugat, Günter Leugering, Ke Wang, Neumann boundary feedback stabilization for a nonlinear wave equation: A strict H2-lyapunov function, 2017, 7, 2156-8499, 419, 10.3934/mcrf.2017015 | |
6. | Eduardo Cerpa, Emmanuelle Crépeau, Claudia Moreno, On the boundary controllability of the Korteweg–de Vries equation on a star-shaped network, 2019, 0265-0754, 10.1093/imamci/dny047 | |
7. | Enrique Zuazua, 2013, Chapter 9, 978-3-642-32159-7, 463, 10.1007/978-3-642-32160-3_9 | |
8. | Martin Gugat, Boundary feedback stabilization of the telegraph equation: Decay rates for vanishing damping term, 2014, 66, 01676911, 72, 10.1016/j.sysconle.2014.01.007 | |
9. | Zhong-Jie Han, Gen-Qi Xu, Output feedback stabilisation of a tree-shaped network of vibrating strings with non-collocated observation, 2011, 84, 0020-7179, 458, 10.1080/00207179.2011.561441 | |
10. | Farhat Shel, Exponential stability of a network of elastic and thermoelastic materials, 2013, 36, 01704214, 869, 10.1002/mma.2644 | |
11. | Abdessamad El Alami, Imad El Harraki, Ali Boutoulout, Regional Feedback Stabilization for Infinite Semilinear Systems, 2018, 24, 1079-2724, 343, 10.1007/s10883-017-9386-3 | |
12. | Kaïs Ammari, Farhat Shel, 2022, Chapter 2, 978-3-030-86350-0, 15, 10.1007/978-3-030-86351-7_2 | |
13. | Martin Gugat, Markus Dick, Günter Leugering, 2013, Chapter 26, 978-3-642-36061-9, 255, 10.1007/978-3-642-36062-6_26 | |
14. | Pierre-Olivier Lamare, Antoine Girard, Christophe Prieur, Switching Rules for Stabilization of Linear Systems of Conservation Laws, 2015, 53, 0363-0129, 1599, 10.1137/140953952 | |
15. | Abdessamad El Alami, Ali Boutoulout, 2020, Chapter 6, 978-3-030-26148-1, 67, 10.1007/978-3-030-26149-8_6 | |
16. | Yacine Chitour, Guilherme Mazanti, Mario Sigalotti, Stability of non-autonomous difference equations with applications to transport and wave propagation on networks, 2016, 11, 1556-1801, 563, 10.3934/nhm.2016010 | |
17. | Yacine Chitour, Swann Marx, Guilherme Mazanti, G. Buttazzo, E. Casas, L. de Teresa, R. Glowinski, G. Leugering, E. Trélat, X. Zhang, One-dimensional wave equation with set-valued boundary damping: well-posedness, asymptotic stability, and decay rates, 2021, 27, 1292-8119, 84, 10.1051/cocv/2021067 | |
18. | Markus Dick, Martin Gugat, Gunter Leugering, 2012, Feedback stabilization of quasilinear hyperbolic systems with varying delays, 978-1-4673-2124-2, 125, 10.1109/MMAR.2012.6347931 | |
19. | Alaa Hayek, Serge Nicaise, Zaynab Salloum, Ali Wehbe, Existence, Uniqueness and Stabilization of Solutions of a Generalized Telegraph Equation on Star Shaped Networks, 2020, 170, 0167-8019, 823, 10.1007/s10440-020-00360-8 | |
20. | Yacine Chitour, Guilherme Mazanti, Mario Sigalotti, Persistently damped transport on a network of circles, 2016, 369, 0002-9947, 3841, 10.1090/tran/6778 | |
21. | Markus Dick, Martin Gugat, Günter Leugering, A strict H1-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction, 2011, 1, 2155-3297, 225, 10.3934/naco.2011.1.225 | |
22. | Fatiha Alabau-Boussouira, Vincent Perrollaz, Lionel Rosier, Finite-time stabilization of a network of strings, 2015, 5, 2156-8499, 721, 10.3934/mcrf.2015.5.721 | |
23. | Martin Gugat, Sonja Steffensen, Dynamic boundary control games with networks of strings, 2018, 24, 1292-8119, 1789, 10.1051/cocv/2017082 | |
24. | Martin Gugat, Markus Dick, Günter Leugering, Gas Flow in Fan-Shaped Networks: Classical Solutions and Feedback Stabilization, 2011, 49, 0363-0129, 2101, 10.1137/100799824 | |
25. | Mohamed Ouzahra, Global stabilization of semilinear systems using switching controls, 2012, 48, 00051098, 837, 10.1016/j.automatica.2012.02.018 | |
26. | Pierre-Olivier Lamare, Antoine Girard, Christophe Prieur, 2013, Lyapunov techniques for stabilization of switched linear systems of conservation laws, 978-1-4673-5717-3, 448, 10.1109/CDC.2013.6759922 | |
27. | Martin Gugat, Markus Dick, Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction, 2011, 1, 2156-8499, 469, 10.3934/mcrf.2011.1.469 | |
28. | Markus Dick, Martin Gugat, Michael Herty, Günter Leugering, Sonja Steffensen, Ke Wang, 2014, Chapter 31, 978-3-319-05082-9, 487, 10.1007/978-3-319-05083-6_31 | |
29. | Markus Dick, Martin Gugat, Günter Leugering, Classical solutions and feedback stabilization for the gas flow in a sequence of pipes, 2010, 5, 1556-181X, 691, 10.3934/nhm.2010.5.691 | |
30. | Jon Asier Bárcena-Petisco, Márcio Cavalcante, Giuseppe Maria Coclite, Nicola De Nitti, Enrique Zuazua, Control of hyperbolic and parabolic equations on networks and singular limits, 2024, 0, 2156-8472, 0, 10.3934/mcrf.2024015 | |
31. | Martin Gugat, Stabilization of a cyclic network of strings by nodal control, 2025, 25, 1424-3199, 10.1007/s00028-024-01030-0 | |
32. | Yacine Chitour, Paolo Mason, Mario Sigalotti, 2025, Chapter 7, 978-3-031-82981-9, 145, 10.1007/978-3-031-82982-6_7 |
ˉa in (2.3) | 0.1 | 0.2 | 0.3 | 0.5 | 1.0 | 1.5 | none |
J in (1.8) | 33.48 | 41.03 | 44.73 | 46.85 | 47.39 | 47.51 | 48.30 |
N in (1.5) | 10.53 | 5.05 | 2.30 | 0.76 | 0.41 | 0.36 | 0.00 |
J+N in (1.9) | 44.01 | 46.08 | 47.03 | 47.61 | 47.80 | 47.87 | 48.30 |
ˉt in (2.4) | 0.1 | 0.2 | 0.3 | 0.5 | 1.0 | 1.5 | none |
J in (1.8) | 31.71 | 38.80 | 43.65 | 46.79 | 47.40 | 47.42 | 48.30 |
N in (1.7) | 9.37 | 5.46 | 2.74 | 0.86 | 0.40 | 0.39 | 0.00 |
J+N in (1.10) | 41.08 | 44.26 | 46.39 | 47.66 | 47.80 | 47.80 | 48.30 |
η1 | η2 | η3 | J as in (1.8) | N as in (1.5) | J+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 |
ν1 | ν2 | ν3 | J as in (1.8) | N as in (1.7) | J+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 |
ˉa in (2.3) | 0.1 | 0.2 | 0.3 | 0.5 | 1.0 | 1.5 | none |
J in (1.8) | 33.48 | 41.03 | 44.73 | 46.85 | 47.39 | 47.51 | 48.30 |
N in (1.5) | 10.53 | 5.05 | 2.30 | 0.76 | 0.41 | 0.36 | 0.00 |
J+N in (1.9) | 44.01 | 46.08 | 47.03 | 47.61 | 47.80 | 47.87 | 48.30 |
ˉt in (2.4) | 0.1 | 0.2 | 0.3 | 0.5 | 1.0 | 1.5 | none |
J in (1.8) | 31.71 | 38.80 | 43.65 | 46.79 | 47.40 | 47.42 | 48.30 |
N in (1.7) | 9.37 | 5.46 | 2.74 | 0.86 | 0.40 | 0.39 | 0.00 |
J+N in (1.10) | 41.08 | 44.26 | 46.39 | 47.66 | 47.80 | 47.80 | 48.30 |
η1 | η2 | η3 | J as in (1.8) | N as in (1.5) | J+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 |
ν1 | ν2 | ν3 | J as in (1.8) | N as in (1.7) | J+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 |