
Citation: Arindam Mitra, Suman Mukhopadhyay. Biofilm mediated decontamination of pollutants from the environment[J]. AIMS Bioengineering, 2016, 3(1): 44-59. doi: 10.3934/bioeng.2016.1.44
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[9] | Nawei Chen, Shenglong Chen, Xiaoyu Li, Zhiming Li . Modelling and analysis of the HIV/AIDS epidemic with fast and slow asymptomatic infections in China from 2008 to 2021. Mathematical Biosciences and Engineering, 2023, 20(12): 20770-20794. doi: 10.3934/mbe.2023919 |
[10] | B. M. Adams, H. T. Banks, Hee-Dae Kwon, Hien T. Tran . Dynamic Multidrug Therapies for HIV: Optimal and STI Control Approaches. Mathematical Biosciences and Engineering, 2004, 1(2): 223-241. doi: 10.3934/mbe.2004.1.223 |
The SICA (Susceptible-Infected-Chronic-AIDS) compartmental model for HIV/AIDS transmission dynamics was proposed by Silva and Torres in their seminal paper [1]. Since then, the model has been extended to stochastic systems of differential equations [2,3], to fractional-order [4,5] as well as discrete-time dynamics [6], and applied with success to describe very different HIV/AIDS epidemics, like the ones in Cape Verde [7,8] or Morocco [9]. For a survey see [10].
In this work, we consider a SICA epidemic problem of controlling the transmission of the human immunodeficiency viruses (HIV), by considering not only the medical treatment with multiple antiretroviral (ART) drugs, but also the pre-exposure prophylaxis (PrEP), which are medicines taken to prevent getting HIV infection. According to the Centers for Disease Control and Prevention, PrEP is highly effective for preventing HIV, when taken as prescribed, and reduces the risk of getting HIV from sex and from injection drug use by about 99 and 74%, respectively [11].
The main contribution of this work is to propose a model-free based control algorithm that closes the loop between the infected individuals with HIV and PreP medication, in such manner that the medication is driven in 'real-time', according to the number of infected individuals that has to be asymptotically reduced. We highlight that model-free control offers the advantages of a simple Proportional-Integral-Derivative (PID) controller in the framework of model-free design, that is, one whose parameters can be easily tuned without a precise knowledge of the controlled epidemiological model.
The model-free control methodology, originally proposed by Fliess and Join in [12], has been designed to control a priori any "unknown" dynamical system in a "robust" manner, and is referred to as "a self-tuning regulator" in [13]. This control law can be considered as an alternative to PI and PID controllers [14] and the performances are really satisfactory taking into account that the control is calculated based only on the information provided by the controlled input and the measured output signal of the controlled systems. This control law has been extensively and successfully applied to control many nonlinear processes: see, e.g., [12,15,16] and the references therein. In particular, some applications have been dedicated to the control of chemistry and biological processes [15,17,18,19,20], including the development of an artificial pancreas [21]. A derivative-free-based version of this control algorithm has been proposed by the first author in [22], for which some interesting capabilities of online optimization have been highlighted. To the best of our knowledge, the application of model-free control to SICA modeling has never been discussed before.
Here, we compare the solutions obtained by the model-free control method with the corresponding solutions of an optimal control problem for HIV/AIDS transmission from [8], which has a mixed state-control constraint. In [8], the control system is based on a SICAE (Susceptible, HIV-Infected, Chronic HIV-infected under ART, AIDS-symptomatic individuals, E-under PrEP medication) model for the transmission of HIV in a homogeneously mixing population. The control $ u $ represents the fraction of susceptible individuals under PrEP, with $ 0 \leq u(t) \leq 1 $, that is, when $ u(t) = 0 $, no susceptible individual takes PrEP at time $ t $, and when $ u(t) = 1 $ all susceptible individuals are taking PrEP at time $ t $. The mixed state-control constraint refers to the fact that only people who are HIV-negative and at a very high risk of HIV infection should take PrEP, and also to the high costs of PrEP medication. Therefore, the number of susceptible individuals that takes PrEP, at each day, must be bounded by a positive constant. Moreover, the cost functional, which is aimed to be minimized, represents a balance between the number of HIV infected individuals and the costs associated with PrEP implementation.
The paper is structured as follows. In Section 2, we propose a model-free control method and the procedure to minimize the HIV infected cases is described. In Section 3, we present some numerical simulations and provide a comparison of the results obtained using the model-free based approach with the ones in [8] from the Pontryagin maximum principle. Section 4 discusses and compares the results. Finally, some concluding remarks and possible directions for future work are given in Section 5.
In this section, we propose our model-free based control method and apply it to an epidemiological problem of minimizing HIV-infected individuals.
Model-free control was introduced in 2008 and 2009 by Fliess and Join in [23,24]. It is an alternative technique to control complex systems based on elementary continuously updated local modeling via unique knowledge of the input-output behavior. The key feature of this approach lies in the fact that the control system, which might be highly nonlinear and/or time-varying, is taken into account without any modeling procedure [25]. The model-free based control approach has been successfully implemented in concrete applications to diverse fields, ranging from intelligent transportation systems to energy management, etc., see [12] and references cited therein. To the best of our knowledge, no one as yet used this approach in the context of epidemiology.
Consider a nonlinear dynamical system $ f : u \mapsto y $ to control:
$ {˙x=f(x,u),y=g(x), $
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(2.1) |
where $ f $ is the function describing the behavior of a nonlinear system and $ \boldsymbol{x} \in {\mathbb{R}}^n $ is the state vector. The proposed control is an application $ \mathcal{C}_{\pi} : (y, y^*) \mapsto u $, whose purpose is to control the output $ y $ of system (2.1) following an output reference $ y^* $. In simulation, the system (2.1) is controlled in its "original formulation", without any modification or linearization.
For any discrete moment $ t_k, \, k \in \mathbb{N}^* $, one defines the discrete controller $ \mathcal{C}_{\pi} : (y, y^*) \mapsto u $ as an integrator associated to a numerical series $ (\Psi_k)_{k \in \mathbb{N}^*} $, symbolically represented by
$ uk=Cπ(yk,y∗k)=Ψk⋅∫t0Ki(y∗k−yk−1)dτ $
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(2.2) |
with the recursive term
$ Ψk=Ψk−1+Kp(kαe−kβk−yk−1), $
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where $ y^\ast $ is the output (or tracking) reference trajectory; $ K_p $ and $ K_i $ are real positive tuning gains; $ \varepsilon_{k-1} = y^\ast_{k} - y_{k-1} $ is the tracking error; and $ k_\alpha e^{-k_\beta k} $ is an initialization function, where $ k_\alpha $ and $ k_\beta $ are real positive constants. In practice, the integral part is discretized using, for example, Riemann sums.
The set of the $ \mathcal{C}_{\pi} $-parameters of the controller, is defined as the set of the tuning coefficients $ \{K_p, K_i, k_\alpha, k_\beta\} $.
Consider the problem of minimizing the number of infected individuals with HIV, given by the state trajectory $ I $, through the control measure $ u $ associated to PrEP medication, and satisfying a mixed state-control constraint $ S u \leq \gamma_{max} $, where $ \gamma_{max} $ is a positive constant.
The control sequence is divided into two steps, in order to manage, separately, the increasing $ u $ transient and the associated decreasing $ u $ transient that must decrease afterwards both the infected cases and the control input $ u $ to lower values on $ u $ and $ I $. Concerning the control input $ u $, it is expected that $ u(t > t_f) = 0 $, where $ t_f $ is the time from which the medication is stopped.
● A first sequence, associated to the setting up of the medication, aims to progressively increase the medication (and thus start decreasing the infected states) until a certain threshold of infected cases is reached, above which the number of infected cases could be considered as stabilizable around a low value. This control sequence can be managed by a simple $ L $-linear or $ Q $-quadratic slope, such as
$ (slope)u(t)=u0+L⋅t(quadratic)u(t)=u0+Q⋅t2 $
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(2.3) |
driven in open-loop, i.e., independently from any feedback of the infected state, that increases gradually the medication while satisfying the constraint on $ S u \leq \eta $ due to the remaining high level of the susceptible cases. It appears crucial to accelerate the medication at the beginning, in order to reach rapidly $ \gamma_{max} $ and allow a strong decreasing of the infected cases. This point will be discussed later in Section 3.
● Denote $ u_{max} = \max u(t) $, the maximum value of $ u $. The second sequence is associated to the decrease of the medication until the infected state is stabilized around a low value. This sequence is managed by our proposed model-free based control that interacts, in real-time, with the number of infected cases and, consequently, calculates the "optimal" medication $ u $ in order to decrease and stabilize the infected state. According to Eq (2.2), the control reads:
$ uk=Ψk⋅∫t0Ki(I∗k−Ik−1)dτwithΨk=Ψk−1+Kp(kαe−kβk−Ik−1), $
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where $ I^\ast $ denotes the infected cases reference that practically can be chosen as $ I^\ast = \min I(t) $, where the minimum value of $ I $ can be reached online, updating the tracking reference and, therefore, ensuring that the control law is "synchronized" on the lowest value that can be reachable. It is worth to note that, depending on the behavior of the closed-loop, a saturation is added to bound the controlled $ u $: $ 0 \leq u \leq u_{max} \leq 1 $.
The implementation of the control scheme is depicted in Figure 1, where the control sequence starts from the transient slope or the quadratic function and then, once $ u_{max} $ is reached, switches to the proposed model-free based controller. The parameters of the control sequence to be adjusted, comprise the $ \mathcal{C}_{\pi} $-parameters set of the model-free control algorithm and the $ (u_0, L, Q) $ parameters of the first sequence, depending if a linear or a quadratic slope is involved.
Additional constraints. The slope can be adjusted according to a state-control constraint that determines the maximum number of susceptible individuals that take PrEP medication, at each instant of time. This constraint reads as $ S(t) \cdot u(t) < \gamma_{max} $, for all $ t $, where $ \gamma_{max} $ is the corresponding upper bound.
To properly tune each sequence, in order to satisfy both the state-control constraint as well as to minimize the cost criteria, a derivative-free based optimization procedure can be applied [26].
Consider the mathematical model for HIV/AIDS transmission in a homogeneously mixing population proposed in [8] and based in [1,7].
The model subdivides human population into five mutually-exclusive compartments: susceptible individuals ($ S $); HIV-infected individuals with no clinical symptoms of AIDS (the virus is living or developing in the individuals but without producing symptoms or only mild ones) but able to transmit HIV to other individuals ($ I $); HIV-infected individuals under ART treatment (the so called chronic stage) with a viral load remaining low ($ C $); HIV-infected individuals with AIDS clinical symptoms ($ A $); individuals that are under PrEP medication ($ E $). The total population at time $ t $, denoted by $ N(t) $, is given by $ N(t) = S(t) + I(t) + C(t) + A(t) + E(t) $. The model assumptions are the following [1,8]. Effective contact with people infected with HIV is at a rate $ \lambda $, given by
$ λ=βN(I+ηCC+ηAA), $
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where $ \beta $ is the effective contact rate for HIV transmission. The modification parameter $ \eta_A \geq 1 $ accounts for the relative infectiousness of individuals with AIDS symptoms, in comparison to those infected with HIV with no AIDS symptoms. Individuals with AIDS symptoms are more infectious than HIV-infected individuals (pre-AIDS) because they have a higher viral load and there is a positive correlation between viral load and infectiousness [27]. On the other hand, $ \eta_C \leq 1 $ translates the partial restoration of immune function of individuals with HIV infection that use ART correctly [28]. All individuals suffer from natural death, at a constant rate $ \mu $. HIV-infected individuals, with and without AIDS symptoms, have access to ART treatment. HIV-infected individuals with no AIDS symptoms $ I $ progress to the class of individuals with HIV infection under ART treatment $ C $ at a rate $ \phi $, and HIV-infected individuals with AIDS symptoms are treated for HIV, at rate $ \alpha $. HIV-infected individuals with AIDS symptoms $ A $, that start treatment, move to the class of HIV-infected individuals $ I $, moving to the chronic class $ C $ only if the treatment is maintained. HIV-infected individuals with no AIDS symptoms $ I $ that do not take ART treatment progress to the AIDS class $ A $, at rate $ \rho $. Individuals in the class $ C $ that stop ART medication are transferred to the class $ I $, at a rate $ \omega $. Only HIV-infected individuals with AIDS symptoms $ A $ suffer from an AIDS induced death, at a rate $ d $. The proportion of susceptible individuals that takes PrEP is denoted by $ \psi $. It is assume that PrEP is effective, so that all susceptible individuals under PrEP treatment are transferred to class $ E $. The individuals that stop PrEP become susceptible individuals again, at a rate $ \theta $. Susceptible individuals are increased by the recruitment rate $ \Lambda $.
Such model is given by the following system of ordinary differential equations:
$ {˙S(t)=Λ−β(I(t)+ηCC(t)+ηAA(t))N(t)S(t)−μS(t)−ψS(t)+θE(t),˙I(t)=β(I(t)+ηCC(t)+ηAA(t))N(t)S(t)−(ρ+ϕ+μ)I(t)+αA(t)+ωC(t),˙C(t)=ϕI(t)−(ω+μ)C(t),˙A(t)=ρI(t)−(α+μ+d)A(t),˙E(t)=ψS(t)−(μ+θ)E(t). $
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(2.4) |
Recall that PrEP medication should only be administrated to people who are HIV-negative and at very high risk for HIV infection. Moreover, PrEP is highly expensive and it is still not approved in many countries. Therefore, the number of individuals that should take PrEP should be limited at each instant of time for a fixed interval of time $ [0, t_f] $ [8]. The optimal control problem proposed in [8], and considered in this paper for comparison of results, takes into account this health public problem.
The main goal of the optimal control problem is to determine the PrEP strategy $ \psi $ that minimizes the number of individuals with pre-AIDS HIV-infection $ I $ as well as the costs associated with PrEP. Let the fraction of individuals that takes PrEP, at each instant of time, be a control function, that is, $ \psi \equiv u(t) $ with $ t \in [0, t_f] $, and assume that the total population $ N $ is constant: the recruitment rate is proportional to the natural death rate, $ \Lambda = \mu N $, and there are no AIDS-induced deaths ($ d = 0 $). The controlled model is given by
$ {˙S(t)=μN−βN(I(t)+ηCC(t)+ηAA(t))S(t)−μS(t)−S(t)u(t)+θE(t),˙I(t)=βN(I(t)+ηCC(t)+ηAA(t))S(t)−(ρ+ϕ+μ)I(t)+αA(t)+ωC(t),˙C(t)=ϕI(t)−(ω+μ)C(t),˙A(t)=ρI(t)−(α+μ)A(t),˙E(t)=S(t)u(t)−(μ+θ)E(t). $
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(2.5) |
Remark 1. All the parameters of the SICAE model $(2.5)$ are fixed with the exception of the control function $ u(t) $. This system is deterministic and there is no uncertainty. However, the model-free based control proposed in this paper does not use these equations. They are only needed in the classical optimal control approach that is used here for comparison. The sensitivity analysis of the parameters of the SICA model, which is in the basis of the SICAE model $(2.4)$, was studied before in [7].
The classical optimal control problem proposed in [8], and that is considered here in comparison with the model-free control method, considers the cost functional
$ J(u)=∫tf0[w1I(t)+w2u2(t)]dt, $
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(2.6) |
where the constants $ w_1 $ and $ w_2 $ represent the weights associated with the number of HIV infected individuals $ I $ and on the cost associated with the PrEP prevention treatment, respectively. It is assumed that the control function $ u $ takes values between 0 and 1. When $ u(t) = 0 $, no susceptible individual takes PrEP at time $ t $; if $ u(t) = 1 $, then all susceptible individuals are taking PrEP at time $ t $. Let $ \gamma_{max} $ denote the total number of susceptible individuals under PrEP for a fixed time interval $ [0, t_f] $. This constraint is represented by
$ S(t)u(t)≤γmax,γmax≥0,for almost allt∈[0,tf], $
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(2.7) |
which should be satisfied at almost every instant of time during the whole PrEP program.
Let
$ x(t)=(x1(t),…,x5(t))=(S(t),I(t),C(t),A(t),E(t))∈R5. $
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The classical optimal control problem proposed in [8] consists to find the optimal trajectory $ \tilde{x} $, associated with the control $ \tilde{u} $, satisfying the control system (2.5), the initial conditions
$ x(0)=(x10,x20,x30,x40,x50),withx10≥0,x20≥0,x30≥0,x40≥0,x50≥0, $
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the constraint inequality (Eq 2.7), and where the control $ \tilde{u} \in \Omega $,
$ Ω={u(⋅)∈L∞(0,tf)|0≤u(t)≤1}, $
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(2.8) |
minimizes the objective functional Eq (2.6).
We evaluate the accuracy of our model-free proposed approach compared to the classical optimal control one when applied to the SICAE model described in Section 2.3.
Let $ T_e $ denote the final time of the $ u $ treatment such as $ u(t \geq T_e) = 0 $ and $ I(T_e) $ denotes the final value of the state "Infected" at the time $ T_e $. Regarding the "energy" of the control input $ u $ with respect to the behavior of the infected cases and the period of time $ T_e $ for which the medication $ u $ is in effect, i.e., $ u(t \geq T_e) = 0 $, let us consider the following cost criteria:
● Cost criterion for performances over the final time $ T_e $:
$ Ju+I=∫Te0u2+I2dτ, $
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(2.9) |
$ JI=∫Te0I2dτ. $
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(2.10) |
● Time-pondered cost criterion:
$ JTeu+I=∫Te0τ(u2+I2)dτ, $
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(2.11) |
which takes into account the effective period needed to stabilize the infected case, i.e., the period for which $ u > 0 $.
To perform the numerical simulations, we consider the following parameter values, borrowed from [8]: $ N = 10200 $, $ \mu = 1/69.54 $, $ \beta = 0.582 $, $ \eta_C = 0.04 $, $ \eta_A = 1.35 $, $ \theta = 0.001 $, $ \omega = 0.09 $, $ \rho = 0.1 $, $ \phi = 1 $ and $ \alpha = 0.33 $. The weight constants take the values $ w_1 = w_2 = 1 $.
The initial conditions are given by
$ S(0)=10000,I(0)=200,C(0)=0,A(0)=0andE(0)=0, $
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and the mixed state-control constraint is
$ S(t)u(t)≤2000,for almost allt∈[0,tf]. $
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(3.1) |
In Table 1, we evaluate several cases with the cost criteria $ J_{u+I} $, $ J_{I} $ and $ J_{u+I}^{Te} $, according to the final time of medication $ T_e $. We compare the constrained and unconstrained classical optimal control problems and the unconstrained and constrained model-free problems with two types of configurations: slope and quadratic initial transient. The classical optimal control problem corresponds to the one performed in [8]. Figures 2–6 illustrate several scenarios: the constrained case is not satisfied under slope, see Figure 2; the constraint inequality (Eq 3.1) is satisfied under slope, see Figures 3 and 4; the constrained cases are satisfied under quadratic function, see Figures 5 and 6.
Case | $ T_e $ | $ J_{u+I} $ | $ J_{I} $ | $ J_{u+I}^{Te} $ | $ I(T_e) $ | $ {\max\limits_{[0, t_{final}] } S u} $ | $ u_{max} $ |
Unconstrained model-free | 11.3 | $ 5.40 10^4 $ | $ 5.40 10^4 $ | $ 1.15 10^5 $ | 31.12 | 3129 | 0.70 |
Constrained model-free – slope (Ⅰ) | 19.0 | $ 6.45 10^4 $ | $ 6.45 10^4 $ | $ 2.39 10^5 $ | 29.80 | 1990 | 0.80 |
Constrained model-free – slope (Ⅱ) | 22.9 | $ 6.45 10^4 $ | $ 6.45 10^4 $ | $ 2.86 10^5 $ | 28.25 | 2000 | 0.62 |
Constrained model-free – quad. (Ⅰ) | 16.9 | $ 5.83 10^4 $ | $ 5.83 10^4 $ | $ 1.83 10^5 $ | 29.12 | 1989 | 0.70 |
Constrained model-free – quad. (Ⅱ) | 17.2 | $ 6.66 10^4 $ | $ 6.66 10^4 $ | $ 2.39 10^5 $ | 32.29 | 1604 | 0.62 |
Unconstrained classical OC | 25.0 | $ 4.17 10^{4} $ | $ 4.17\, 10^{4} $ | $ 1.69 10^{5} $ | 21.95 | 9750 | 1 |
Constrained classical OC | 25.0 | $ 6.14 10^{4} $ | $ 6.14\, 10^{4} $ | $ 2.72 10^{5} $ | 24.23 | 1989 | 1 |
From Figure 2(a), we see that, although the model-free based control and the classical optimal control approaches propose completely different control functions, the associated number of HIV infected individuals $ I(t) $, $ t \in [0, 25] $, are very similar. Interestingly, the control solution of the model-free based control is active in a much smaller interval of time that the one obtained with the classical optimal control approach: $ T_e = 11.3 $ versus $ T_e = 25 $ (see first line of Table 1). Analogous conclusions are taken from Figures 3(a)–6(a).
It should be noted, however, that in Figure 2(b), the control obtained from the model-free based approach does not satisfy the mixed state-control constraint $ S(t) u(t) \leq 2000 $ for all $ t $. In order to satisfy this constraint, one must increase the time interval where the model-free control is active, see Figures 3(b)–6(b). This depends on the configuration of the initial transient of the control (slope or quadratic), see Figures 3(a)–6(a) and Table 1.
The proposed control sequence can be considered as "quasi-optimal" in the sense that it does not obey to the Pontryagin maximum principle, so it is not an optimal control by definition, but it offers similar properties in terms of cost criteria minimization and reduction of the duration of treatment that is assured by our procedure.
The sequence is fully parametrized thanks to the initial transient coefficients $ (L $ or $ Q) $ associated to $ u_{max} $, including the $ \mathcal{C}_{\pi} $-parameters for the decreasing transient, that must be adjusted according to the evolution of the infected state. In particular, the maximum value on the product $ S u $ depends on the "speed" of the increasing transient as well as the final value $ I(T_e) $, which depends on the "speed" of the increasing transient, the initial value $ u_0 $, and the feedback control that stabilizes $ I $ through the decreasing transient. The transient slope plays a key role in the "accuracy" of the initial decrease of the infected state since a sufficient dose of the medication $ u $ must be injected to the population in order to maintain the infected state to a lower level. Therefore, the maximum value of $ u $ is a trade-off between the constraint $ S u \leq \gamma_{max} $ to be satisfied and the duration of the treatment. Figure 2 illustrates a rather quick treatment, involving thus a fast initial transient but the constraint $ S u \leq 2000 $ is not satisfied; slower medical treatment for which the injection of the medication $ u $ takes more time due to the constraint $ S u \leq 2000 $, can reduce the final asymptotic value $ I(T_e) $ despite not necessarily fully reducing the cost criteria. The model-free based control aims to relax the treatment until $ u = 0 $ is reached. A first tuning has been made according to the gained experience and a more precise tuning can be performed using [26]. The numerical evaluation of the cost criteria shows that our approach is globally better in terms of energy minimization. Moreover, the time-pondered criteria shows that the proposed control procedure is favorable to our approach taking into account the reduction of the duration of the treatment. These results illustrate afterwards that tuning the parameters of the proposed control sequence is a trade-off between considering minimizing the cost criteria, or minimizing the final value $ I(T_e) $, depending of additional constraints.
We remark that the model-free based control could have been also used to drive the initial transient instead of the slope or the quadratic function, but the current algorithm offers slower performances at the very beginning to initiate the increase of the medication that prevents it to deal with, for example, the constraint $ S u \leq 2000 $.
We have considered a SICAE epidemiological model for HIV/AIDS transmission, proposing, for the first time in the literature, a model-free based approach to minimize the number of infected individuals. This approach consists in initializing PrEP medication, using a basic linear or quadratic function, and, after that, creating a direct feedback to control the decrease of infected individuals with respect to the considered measure of infected cases. Globally, the advantages of the proposed approach, when compared with the classical optimal control based on the Pontryagin Maximum Principle, is that it does not need any a priori knowledge of the model and a simple tuning of the proposed control sequence values allows good performances in terms of "energy" minimization as well as minimization of the medical treatment duration. We concluded that our control strategy highlights interesting performances compared with the classical optimal control approach used in [8].
From a biological point of view, our application of the model-free based control approach allows to propose new solutions for the implementation of PrEP in the prevention of HIV transmission, considering the constraints associated to the limitations on the availability of medicines for HIV and on number of individuals that the health systems have capacity to follow up during their treatment. To the best of our knowledge, we were the first to apply the model-free based control approach in the context of epidemiology.
Future work may include replacing the slope or the quadratic initial transient by an optimal control; improvement of the proposed model-free based control; implementation and comparison with the original Fliess-Join version of the model-free control, as it has been done, for example, for the glycemia control [14]. Stability issues regarding the closed loop are very important and a promising LMI framework dedicated to study the stability of optimization algorithms is also of interest [29,30].
This work was partially supported by Portuguese funds through CIDMA, The Center for Research and Development in Mathematics and Applications of University of Aveiro, and the Portuguese Foundation for Science and Technology (FCT–Fundação para a Ciência e a Tecnologia), within project UIDB/04106/2020. Silva is also supported by the FCT Researcher Program CEEC Individual 2018 with reference CEECIND/00564/2018.
The authors are grateful to Reviewers for several constructive suggestions and remarks.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
[1] | Alexander M, Loehr RC (1992) Bioremediation review. Science 258: 874. |
[2] | Prasad MN, Prasad R (2012) Nature's cure for cleanup of contaminated environment—a review of bioremediation strategies. Rev Environ Health 27: 181–189. |
[3] |
Vidali M (2001) Bioremediation. an overview. Pure and Applied Chemistry 73: 1163–1172. doi: 10.1351/pac200173071163
![]() |
[4] |
Gaur N, Flora G, Yadav M, et al. (2014) A review with recent advancements on bioremediation-based abolition of heavy metals. Environ Sci Process Impacts 16: 180–193. doi: 10.1039/C3EM00491K
![]() |
[5] |
Atlas RM, Hazen TC (2011) Oil biodegradation and bioremediation: a tale of the two worst spills in U.S. history. Environ Sci Technol 45: 6709–6715. doi: 10.1021/es2013227
![]() |
[6] |
Bragg JR, Prince RC, Harner EJ, et al. (1994) Effectiveness of bioremediation for the Exxon Valdez oil spill. Nature 368: 413–418. doi: 10.1038/368413a0
![]() |
[7] |
Day SM (1993) US environmental regulations and policies--their impact on the commercial development of bioremediation. Trends Biotechnol 11: 324–328. doi: 10.1016/0167-7799(93)90154-2
![]() |
[8] |
Caplan JA (1993) The worldwide bioremediation industry: prospects for profit. Trends Biotechnol 11: 320–323. doi: 10.1016/0167-7799(93)90153-Z
![]() |
[9] |
Mishra A, Malik A (2014) Novel fungal consortium for bioremediation of metals and dyes from mixed waste stream. Bioresour Technol 171: 217–226. doi: 10.1016/j.biortech.2014.08.047
![]() |
[10] |
Cerniglia CE (1997) Fungal metabolism of polycyclic aromatic hydrocarbons: past, present and future applications in bioremediation. J Ind Microbiol Biotechnol 19: 324–333. doi: 10.1038/sj.jim.2900459
![]() |
[11] | Balaji V, Arulazhagan P, Ebenezer P (2014) Enzymatic bioremediation of polyaromatic hydrocarbons by fungal consortia enriched from petroleum contaminated soil and oil seeds. J Environ Biol 35: 521–529. |
[12] |
Bouwer EJ, Zehnder AJ (1993) Bioremediation of organic compounds--putting microbial metabolism to work. Trends Biotechnol 11: 360–367. doi: 10.1016/0167-7799(93)90159-7
![]() |
[13] |
Bruins MR, Kapil S, Oehme FW (2000) Microbial resistance to metals in the environment. Ecotoxicol Environ Saf 45: 198–207. doi: 10.1006/eesa.1999.1860
![]() |
[14] | Prince RC (2000) Bioremediation. Kirk-Othmer Encyclopedia of Chemical Technology: John Wiley & Sons, Inc. |
[15] | Das S, Dash HR (2014) 1 - Microbial Bioremediation: A Potential Tool for Restoration of Contaminated Areas. In: Das S, editor. Microbial Biodegradation and Bioremediation. Oxford: Elsevier. pp. 1–21. |
[16] | Joutey NT, Sayel H, Bahafid W, et al. (2015) Mechanisms of hexavalent chromium resistance and removal by microorganisms. Rev Environ Contam Toxicol 233: 45–69. |
[17] |
Kumar R, Singh S, Singh OV (2007) Bioremediation of radionuclides: emerging technologies. OMICS 11: 295–304. doi: 10.1089/omi.2007.0013
![]() |
[18] |
Wall JD, Krumholz LR (2006) Uranium reduction. Annu Rev Microbiol 60: 149–166. doi: 10.1146/annurev.micro.59.030804.121357
![]() |
[19] |
Beyenal H, Sani RK, Peyton BM, et al. (2004) Uranium immobilization by sulfate-reducing biofilms. Environ Sci Technol 38: 2067–2074. doi: 10.1021/es0348703
![]() |
[20] | Vogt C, Richnow HH (2014) Bioremediation via in situ microbial degradation of organic pollutants. Adv Biochem Eng Biotechnol 142: 123–146. |
[21] |
Jorgensen KS (2007) In situ bioremediation. Adv Appl Microbiol 61: 285–305. doi: 10.1016/S0065-2164(06)61008-3
![]() |
[22] |
Singh JS, Abhilash PC, Singh HB, et al. (2011) Genetically engineered bacteria: an emerging tool for environmental remediation and future research perspectives. Gene 480: 1–9. doi: 10.1016/j.gene.2011.03.001
![]() |
[23] | Hedlund BP, Staley JT (2001) Vibrio cyclotrophicus sp. nov., a polycyclic aromatic hydrocarbon (PAH)-degrading marine bacterium. Int J Syst Evol Microbiol 51: 61–66. |
[24] |
Nakajima-Kambe T, Ichihashi F, Matsuzoe R, et al. (2009) Degradation of aliphatic–aromatic copolyesters by bacteria that can degrade aliphatic polyesters. Polym Degrad Stab 94: 1901–1905. doi: 10.1016/j.polymdegradstab.2009.08.006
![]() |
[25] |
Costerton JW, Cheng KJ, Geesey GG, et al. (1987) Bacterial biofilms in nature and disease. Annu Rev Microbiol 41: 435–464. doi: 10.1146/annurev.mi.41.100187.002251
![]() |
[26] |
Gieg LM, Fowler SJ, Berdugo-Clavijo C (2014) Syntrophic biodegradation of hydrocarbon contaminants. Curr Opin Biotechnol 27: 21–29. doi: 10.1016/j.copbio.2013.09.002
![]() |
[27] |
Horemans B, Breugelmans P, Hofkens J, et al. (2013) Environmental dissolved organic matter governs biofilm formation and subsequent linuron degradation activity of a linuron-degrading bacterial consortium. Appl Environ Microbiol 79: 4534–4542. doi: 10.1128/AEM.03730-12
![]() |
[28] |
Pratt LA, Kolter R (1999) Genetic analyses of bacterial biofilm formation. Curr Opin Microbiol 2: 598–603. doi: 10.1016/S1369-5274(99)00028-4
![]() |
[29] |
Lacal J, Reyes-Darias JA, García-Fontana C, et al. (2013) Tactic responses to pollutants and their potential to increase biodegradation efficiency. J Appl Microbiol 114: 923–933. doi: 10.1111/jam.12076
![]() |
[30] | Flemming HC, Wingender J (2010) The biofilm matrix. Nat Rev Microbiol 8: 623–633. |
[31] |
More TT, Yadav JSS, Yan S, et al. (2014) Extracellular polymeric substances of bacteria and their potential environmental applications. J Environ Manage 144: 1–25. doi: 10.1016/j.jenvman.2014.05.010
![]() |
[32] | Flemming HC, Wingender J (2001) Relevance of microbial extracellular polymeric substances (EPSs)--Part II: Technical aspects. Water Sci Technol 43: 9–16. |
[33] |
Branda SS, Vik S, Friedman L, et al. (2005) Biofilms: the matrix revisited. Trends Microbiol 13: 20–26. doi: 10.1016/j.tim.2004.11.006
![]() |
[34] |
Jung JH, Choi NY, Lee SY (2013) Biofilm formation and exopolysaccharide (EPS) production by Cronobacter sakazakii depending on environmental conditions. Food Microbiol 34: 70–80. doi: 10.1016/j.fm.2012.11.008
![]() |
[35] | Kreft JU, Wimpenny JW (2001) Effect of EPS on biofilm structure and function as revealed by an individual-based model of biofilm growth. Water Sci Technol 43: 135–141. |
[36] |
Miqueleto AP, Dolosic CC, Pozzi E, et al. (2010) Influence of carbon sources and C/N ratio on EPS production in anaerobic sequencing batch biofilm reactors for wastewater treatment. Bioresour Technol 101: 1324–1330. doi: 10.1016/j.biortech.2009.09.026
![]() |
[37] |
Reysenbach AL, Cady SL (2001) Microbiology of ancient and modern hydrothermal systems. Trends Microbiol 9: 79–86. doi: 10.1016/S0966-842X(00)01921-1
![]() |
[38] |
Edwards KJ, Bond PL, Gihring TM, et al. (2000) An archaeal iron-oxidizing extreme acidophile important in acid mine drainage. Science 287: 1796–1799. doi: 10.1126/science.287.5459.1796
![]() |
[39] |
Matz C, Kjelleberg S (2005) Off the hook--how bacteria survive protozoan grazing. Trends Microbiol 13: 302–307. doi: 10.1016/j.tim.2005.05.009
![]() |
[40] |
Davey ME, O'Toole G A (2000) Microbial biofilms: from ecology to molecular genetics. Microbiol Mol Biol Rev 64: 847–867. doi: 10.1128/MMBR.64.4.847-867.2000
![]() |
[41] |
Mah TF, O'Toole GA (2001) Mechanisms of biofilm resistance to antimicrobial agents. Trends Microbiol 9: 34–39. doi: 10.1016/S0966-842X(00)01913-2
![]() |
[42] |
Sutherland IW (2001) The biofilm matrix--an immobilized but dynamic microbial environment. Trends Microbiol 9: 222–227. doi: 10.1016/S0966-842X(01)02012-1
![]() |
[43] |
Field JA, Stams AJ, Kato M, et al. (1995) Enhanced biodegradation of aromatic pollutants in cocultures of anaerobic and aerobic bacterial consortia. Antonie Van Leeuwenhoek 67: 47–77. doi: 10.1007/BF00872195
![]() |
[44] |
De Philippis R, Colica G, Micheletti E (2011) Exopolysaccharide-producing cyanobacteria in heavy metal removal from water: molecular basis and practical applicability of the biosorption process. Appl Microbiol Biotechnol 92: 697–708. doi: 10.1007/s00253-011-3601-z
![]() |
[45] |
De Philippis R, Paperi R, Sili C (2007) Heavy metal sorption by released polysaccharides and whole cultures of two exopolysaccharide-producing cyanobacteria. Biodegradation 18: 181–187. doi: 10.1007/s10532-006-9053-y
![]() |
[46] |
Micheletti E, Colica G, Viti C, et al. (2008) Selectivity in the heavy metal removal by exopolysaccharide-producing cyanobacteria. J Appl Microbiol 105: 88–94. doi: 10.1111/j.1365-2672.2008.03728.x
![]() |
[47] |
Iwabuchi N, Sunairi M, Urai M, et al. (2002) Extracellular polysaccharides of Rhodococcus rhodochrous S-2 stimulate the degradation of aromatic components in crude oil by indigenous marine bacteria. Appl Environ Microbiol 68: 2337–2343. doi: 10.1128/AEM.68.5.2337-2343.2002
![]() |
[48] |
Li W-W, Yu H-Q (2014) Insight into the roles of microbial extracellular polymer substances in metal biosorption. Bioresource Technology 160: 15–23. doi: 10.1016/j.biortech.2013.11.074
![]() |
[49] |
Pal A, Paul AK (2008) Microbial extracellular polymeric substances: central elements in heavy metal bioremediation. Indian J Microbiol 48: 49–64. doi: 10.1007/s12088-008-0006-5
![]() |
[50] | Ferris FG, Schultze S, Witten TC, et al. (1989) Metal Interactions with Microbial Biofilms in Acidic and Neutral pH Environments. Appl Environ Microbiol 55: 1249–1257. |
[51] |
Zhang HL, Fang W, Wang YP, et al. (2013) Phosphorus removal in an enhanced biological phosphorus removal process: roles of extracellular polymeric substances. Environ Sci Technol 47: 11482–11489. doi: 10.1021/es403227p
![]() |
[52] |
Yuan Z, Pratt S, Batstone DJ (2012) Phosphorus recovery from wastewater through microbial processes. Curr Opin Biotechnol 23: 878–883. doi: 10.1016/j.copbio.2012.08.001
![]() |
[53] |
Burmolle M, Webb JS, Rao D, et al. (2006) Enhanced biofilm formation and increased resistance to antimicrobial agents and bacterial invasion are caused by synergistic interactions in multispecies biofilms. Appl Environ Microbiol 72: 3916–3923. doi: 10.1128/AEM.03022-05
![]() |
[54] |
Logsdon GS, Kohne R, Abel S, et al. (2002) Slow sand filtration for small water systems. J Environ Eng Sci 1: 339–348. doi: 10.1139/s02-025
![]() |
[55] | Kartal B, Kuenen JG, van Loosdrecht MC (2010) Engineering. Sewage treatment with anammox. Science 328: 702–703. |
[56] |
Bengtsson MM, Ovreas L (2010) Planctomycetes dominate biofilms on surfaces of the kelp Laminaria hyperborea. BMC Microbiol 10: 261. doi: 10.1186/1471-2180-10-261
![]() |
[57] | Fuchs S, Haritopoulou T, Schäfer M, et al. (1997) Heavy metals in freshwater ecosystems introduced by urban rainwater runoff—Monitoring of suspended solids, river sediments and biofilms. Water Sci Technol 36: 277–282. |
[58] | Peacock AD, Chang YJ, Istok JD, et al. (2004) Utilization of microbial biofilms as monitors of bioremediation. Microb Ecol 47: 284–292. |
[59] |
Brummer IH, Fehr W, Wagner-Dobler I (2000) Biofilm community structure in polluted rivers: abundance of dominant phylogenetic groups over a complete annual cycle. Appl Environ Microbiol 66: 3078–3082. doi: 10.1128/AEM.66.7.3078-3082.2000
![]() |
[60] |
Arini A, Feurtet–Mazel A, Maury-Brachet R, et al. (2012) Recovery potential of periphytic biofilms translocated in artificial streams after industrial contamination (Cd and Zn). Ecotoxicology 21: 1403–1414. doi: 10.1007/s10646-012-0894-3
![]() |
[61] |
SzabÓ KÉ, Makk J, Kiss KT, et al. (2008) Sequential colonization by river periphyton analysed by microscopy and molecular fingerprinting. Freshwater Biology 53: 1359–1371. doi: 10.1111/j.1365-2427.2008.01967.x
![]() |
[62] |
Dorigo U, Bérard A, Humbert JF (2002) Comparison of Eukaryotic Phytobenthic Community Composition in a Polluted River by Partial 18S rRNA Gene Cloning and Sequencing. Microbial Ecology 44: 372–380. doi: 10.1007/s00248-002-2024-x
![]() |
[63] |
Kostanjsek R, Lapanje A, Drobne D, et al. (2005) Bacterial community structure analyses to assess pollution of water and sediments in the Lake Shkodra/Skadar, Balkan Peninsula. Environ Sci Pollut Res Int 12: 361–368. doi: 10.1065/espr2005.07.271
![]() |
[64] |
Bricheux G, Le Moal G, Hennequin C, et al. (2013) Characterization and evolution of natural aquatic biofilm communities exposed in vitro to herbicides. Ecotoxicol Environ Saf 88: 126–134. doi: 10.1016/j.ecoenv.2012.11.003
![]() |
[65] |
Ancion PY, Lear G, Dopheide A, et al. (2013) Metal concentrations in stream biofilm and sediments and their potential to explain biofilm microbial community structure. Environ Pollut 173: 117–124. doi: 10.1016/j.envpol.2012.10.012
![]() |
[66] |
Navarro E, Guasch H, Sabater S (2002) Use of microbenthic algal communities in ecotoxicological tests for the assessment of water quality: the Ter river case study. J Appl Phycol 14: 41–48. doi: 10.1023/A:1015242301451
![]() |
[67] |
Dorigo U, Bourrain X, Bérard A, et al. (2004) Seasonal changes in the sensitivity of river microalgae to atrazine and isoproturon along a contamination gradient. Sci Total Environ 318: 101–114. doi: 10.1016/S0048-9697(03)00398-X
![]() |
[68] |
Dewez D, Didur O, Vincent-Héroux J, et al. (2008) Validation of photosynthetic-fluorescence parameters as biomarkers for isoproturon toxic effect on alga Scenedesmus obliquus. Environ Pollution 151: 93–100. doi: 10.1016/j.envpol.2007.03.002
![]() |
[69] |
Schmitt-Jansen M, Altenburger R (2008) Community-level microalgal toxicity assessment by multiwavelength-excitation PAM fluorometry. Aquat Toxicol 86: 49–58. doi: 10.1016/j.aquatox.2007.10.001
![]() |
[70] |
Sayler GS, Layton A, Lajoie C, et al. (1995) Molecular site assessment and process monitoring in bioremediation and natural attenuation. off. Appl Biochem Biotechnol 54: 277–290. doi: 10.1007/BF02787926
![]() |
[71] |
Jorgensen KS, Salminen JM, Bjorklof K (2010) Monitored natural attenuation. Methods Mol Biol 599: 217–233. doi: 10.1007/978-1-60761-439-5_14
![]() |
[72] |
Rittmann BE (2004) Definition, objectives, and evaluation of natural attenuation. Biodegradation 15: 349–357. doi: 10.1023/B:BIOD.0000044587.05189.99
![]() |
[73] |
Tyagi M, da Fonseca MM, de Carvalho CC (2011) Bioaugmentation and biostimulation strategies to improve the effectiveness of bioremediation processes. Biodegradation 22: 231–241. doi: 10.1007/s10532-010-9394-4
![]() |
[74] |
Jansson JK, Bjorklof K, Elvang AM, et al. (2000) Biomarkers for monitoring efficacy of bioremediation by microbial inoculants. Environ Pollut 107: 217–223. doi: 10.1016/S0269-7491(99)00140-2
![]() |
[75] |
Gentry T, Rensing C, Pepper IAN (2004) New Approaches for Bioaugmentation as a Remediation Technology. Crit Rev Environ Sci Technol 34: 447–494. doi: 10.1080/10643380490452362
![]() |
[76] |
Morgan P, Watkinson RJ (1989) Hydrocarbon degradation in soils and methods for soil biotreatment. Crit Rev Biotechnol 8: 305–333. doi: 10.3109/07388558909148196
![]() |
[77] |
Grace Liu P-W, Chang TC, Whang L-M, et al. (2011) Bioremediation of petroleum hydrocarbon contaminated soil: Effects of strategies and microbial community shift. Int Biodeterior Biodegradation 65: 1119–1127. doi: 10.1016/j.ibiod.2011.09.002
![]() |
[78] |
Abeysinghe DH, De Silva DG, Stahl DA, et al. (2002) The effectiveness of bioaugmentation in nitrifying systems stressed by a washout condition and cold temperature. Water Environ Res 74: 187–199. doi: 10.2175/106143002X139901
![]() |
[79] |
Boon N, De Gelder L, Lievens H, et al. (2002) Bioaugmenting bioreactors for the continuous removal of 3-chloroaniline by a slow release approach. Environ Sci Technol 36: 4698–4704. doi: 10.1021/es020076q
![]() |
[80] |
Qureshi N, Annous BA, Ezeji TC, et al. (2005) Biofilm reactors for industrial bioconversion processes: employing potential of enhanced reaction rates. Microb Cell Fact 4: 24. doi: 10.1186/1475-2859-4-24
![]() |
[81] |
Bryers JD (1993) Bacterial biofilms. Curr Opin Biotechnol 4: 197–204. doi: 10.1016/0958-1669(93)90125-G
![]() |
[82] |
Rosche B, Li XZ, Hauer B, et al. (2009) Microbial biofilms: a concept for industrial catalysis? Trends Biotechnol 27: 636–643. doi: 10.1016/j.tibtech.2009.08.001
![]() |
[83] |
Singh R, Paul D, Jain RK (2006) Biofilms: implications in bioremediation. Trends Microbiol 14: 389–397. doi: 10.1016/j.tim.2006.07.001
![]() |
[84] |
Wagner-Dobler I (2003) Pilot plant for bioremediation of mercury-containing industrial wastewater. Appl Microbiol Biotechnol 62: 124–133. doi: 10.1007/s00253-003-1322-7
![]() |
[85] | Shieh W, Keenan J (1986) Fluidized bed biofilm reactor for wastewater treatment. Bioproducts: Springer Berlin Heidelberg. pp. 131–169. |
[86] |
Denac M, Dunn IJ (1988) Packed- and fluidized-bed biofilm reactor performance for anaerobic wastewater treatment. Biotechnol Bioeng 32: 159–173. doi: 10.1002/bit.260320206
![]() |
[87] |
Kumar TA, Saravanan S (2009) Treatability studies of textile wastewater on an aerobic fluidized bed biofilm reactor (FABR): a case study. Water Sci Technol 59: 1817–1821. doi: 10.2166/wst.2009.207
![]() |
[88] |
Costley SC, Wallis FM (2001) Bioremediation of heavy metals in a synthetic wastewater using a rotating biological contactor. Water Res 35: 3715–3723. doi: 10.1016/S0043-1354(01)00072-0
![]() |
[89] |
Eker S, Kargi F (2008) Biological treatment of 2,4-dichlorophenol containing synthetic wastewater using a rotating brush biofilm reactor. Bioresour Technol 99: 2319–2325. doi: 10.1016/j.biortech.2007.05.016
![]() |
[90] |
Eker S, Kargi F (2010) COD, para-chlorophenol and toxicity removal from synthetic wastewater using rotating tubes biofilm reactor (RTBR). Bioresour Technol 101: 9020–9024. doi: 10.1016/j.biortech.2010.07.003
![]() |
[91] | Abraham TE, Senan RC, Shaffiqu TS, et al. (2003) Bioremediation of textile azo dyes by an aerobic bacterial consortium using a rotating biological contactor. Biotechnol Prog 19: 1372–1376. |
[92] |
Jeswani H, Mukherji S (2012) Degradation of phenolics, nitrogen-heterocyclics and polynuclear aromatic hydrocarbons in a rotating biological contactor. Bioresour Technol 111: 12–20. doi: 10.1016/j.biortech.2012.01.157
![]() |
[93] |
Sarayu K, Sandhya S (2012) Rotating biological contactor reactor with biofilm promoting mats for treatment of benzene and xylene containing wastewater. Appl Biochem Biotechnol 168: 1928–1937. doi: 10.1007/s12010-012-9908-0
![]() |
[94] | Rittmann BE (2006) The membrane biofilm reactor: the natural partnership of membranes and biofilm. Water Sci Technol 53: 219–225. |
[95] | Nerenberg R, Rittmann BE (2004) Hydrogen-based, hollow-fiber membrane biofilm reactor for reduction of perchlorate and other oxidized contaminants. Water Sci Technol 49: 223–230. |
[96] |
Modin O, Fukushi K, Yamamoto K (2008) Simultaneous removal of nitrate and pesticides from groundwater using a methane-fed membrane biofilm reactor. Water Sci Technol 58: 1273–1279. doi: 10.2166/wst.2008.481
![]() |
[97] | Fathepure BZ, Vogel TM (1991) Complete degradation of polychlorinated hydrocarbons by a two-stage biofilm reactor. Appl Environ Microbiol 57: 3418–3422. |
[98] |
Zhang C, Wang L, Yan N, et al. (2013) Air-lift internal loop biofilm reactor for realized simultaneous nitrification and denitrification. Bioprocess Biosyst Eng 36: 597–602. doi: 10.1007/s00449-012-0814-1
![]() |
[99] |
Zhao Y, Feng C, Wang Q, et al. (2011) Nitrate removal from groundwater by cooperating heterotrophic with autotrophic denitrification in a biofilm–electrode reactor. J Hazard Mater 192: 1033–1039. doi: 10.1016/j.jhazmat.2011.06.008
![]() |
[100] |
White C, Gadd GM (1998) Accumulation and effects of cadmium on sulphate-reducing bacterial biofilms. Microbiology 144: 1407–1415. doi: 10.1099/00221287-144-5-1407
![]() |
[101] |
White C, Gadd GM (2000) Copper accumulation by sulfate-reducing bacterial biofilms. FEMS Microbiology Letters 183: 313–318. doi: 10.1111/j.1574-6968.2000.tb08977.x
![]() |
[102] |
Smith WL, Gadd GM (2000) Reduction and precipitation of chromate by mixed culture sulphate-reducing bacterial biofilms. J Appl Microbiol 88: 983–991. doi: 10.1046/j.1365-2672.2000.01066.x
![]() |
[103] |
Hosseini Koupaie E, Alavi Moghaddam MR, Hashemi SH (2013) Evaluation of integrated anaerobic/aerobic fixed-bed sequencing batch biofilm reactor for decolorization and biodegradation of azo dye acid red 18: comparison of using two types of packing media. Bioresour Technol 127: 415–421. doi: 10.1016/j.biortech.2012.10.003
![]() |
[104] |
Lin YH, Hsien TY (2009) Kinetics of biodegradation of phenolic wastewater in a biofilm reactor. Water Sci Technol 59: 1703–1711. doi: 10.2166/wst.2009.203
![]() |
[105] |
Moreno-Andrade I, Buitron G, Vargas A (2009) Effect of starvation and shock loads on the biodegradation of 4-chlorophenol in a discontinuous moving bed biofilm reactor. Appl Biochem Biotechnol 158: 222–230. doi: 10.1007/s12010-008-8392-z
![]() |
[106] |
Coelhoso I, Boaventura R, Rodrigues A (1992) Biofilm reactors: an experimental and modeling study of wastewater denitrification in fluidized-bed reactors of activated carbon particles. Biotechnol Bioeng 40: 625–633. doi: 10.1002/bit.260400510
![]() |
[107] |
Masic A, Eberl HJ (2014) A modeling and simulation study of the role of suspended microbial populations in nitrification in a biofilm reactor. Bull Math Biol 76: 27–58. doi: 10.1007/s11538-013-9898-2
![]() |
[108] | Martin KJ, Picioreanu C, Nerenberg R (2015) Assessing microbial competition in a hydrogen-based membrane biofilm reactor (MBfR) using multidimensional modeling. Biotechnol Bioeng. |
[109] |
Valls M, de Lorenzo Vc (2002) Exploiting the genetic and biochemical capacities of bacteria for the remediation of heavy metal pollution. FEMS Microbiol Rev 26: 327–338. doi: 10.1111/j.1574-6976.2002.tb00618.x
![]() |
[110] |
Diels L, De Smet M, Hooyberghs L, et al. (1999) Heavy metals bioremediation of soil. Molecular Biotechnology 12: 149–158. doi: 10.1385/MB:12:2:149
![]() |
[111] | Macaskie LE, Yong P, Doyle TC, et al. (1997) Bioremediation of uranium-bearing wastewater: biochemical and chemical factors influencing bioprocess application. Biotechnol Bioeng 53: 100–109. |
[112] | Shukla SK, Mangwani N, Rao TS, et al. (2014) 8 - Biofilm-Mediated Bioremediation of Polycyclic Aromatic Hydrocarbons. In: Das S, editor. Microbial Biodegradation and Bioremediation. Oxford: Elsevier. pp. 203–232. |
[113] | Chen M, Xu P, Zeng G, et al. (2015) Bioremediation of soils contaminated with polycyclic aromatic hydrocarbons, petroleum, pesticides, chlorophenols and heavy metals by composting: Applications, microbes and future research needs. Biotechnol Adv 33: 745–755. |
[114] | Cerniglia C (1993) Biodegradation of polycyclic aromatic hydrocarbons. In: Rosenberg E, editor. Microorganisms to Combat Pollution: Springer Netherlands. pp. 227–244. |
[115] |
Jones KC, de Voogt P (1999) Persistent organic pollutants (POPs): state of the science. Environ Pollut 100: 209–221. doi: 10.1016/S0269-7491(99)00098-6
![]() |
[116] |
Bonefeld-Jorgensen EC, Hjelmborg PS, Reinert TS, et al. (2006) Xenoestrogenic activity in blood of European and Inuit populations. Environ Health 5: 12. doi: 10.1186/1476-069X-5-12
![]() |
[117] | Johnsen AR, Karlson U (2004) Evaluation of bacterial strategies to promote the bioavailability of polycyclic aromatic hydrocarbons. Appl Microbiol Biotechnol 63: 452–459. |
[118] |
Rodriguez S, Bishop P (2008) Enhancing the Biodegradation of Polycyclic Aromatic Hydrocarbons: Effects of Nonionic Surfactant Addition on Biofilm Function and Structure. J Environ Eng 134: 505–512. doi: 10.1061/(ASCE)0733-9372(2008)134:7(505)
![]() |
[119] |
Plosz BG, Vogelsang C, Macrae K, et al. (2010) The BIOZO process--a biofilm system combined with ozonation: occurrence of xenobiotic organic micro-pollutants in and removal of polycyclic aromatic hydrocarbons and nitrogen from landfill leachate. Water Sci Technol 61: 3188–3197. doi: 10.2166/wst.2010.920
![]() |
[120] | Song HG, Bartha R (1990) Effects of jet fuel spills on the microbial community of soil. Appl Environ Microbiol 56: 646–651. |
[121] |
Ron EZ, Rosenberg E (2014) Enhanced bioremediation of oil spills in the sea. Curr Opin Biotechnol 27: 191–194. doi: 10.1016/j.copbio.2014.02.004
![]() |
[122] |
Harayama S, Kasai Y, Hara A (2004) Microbial communities in oil-contaminated seawater. Curr Opin Biotechnol 15: 205–214. doi: 10.1016/j.copbio.2004.04.002
![]() |
[123] | Dasgupta D, Ghosh R, Sengupta TK (2013) Biofilm-mediated enhanced crude oil degradation by newly isolated pseudomonas species. ISRN Biotechnol 2013: 250749. |
[124] |
Mnif I, Mnif S, Sahnoun R, et al. (2015) Biodegradation of diesel oil by a novel microbial consortium: comparison between co-inoculation with biosurfactant-producing strain and exogenously added biosurfactants. Environ Sci Pollut Res Int 22: 14852–14861. doi: 10.1007/s11356-015-4488-5
![]() |
[125] |
Koren O, Knezevic V, Ron EZ, et al. (2003) Petroleum pollution bioremediation using water-insoluble uric acid as the nitrogen source. Appl Environ Microbiol 69: 6337–6339. doi: 10.1128/AEM.69.10.6337-6339.2003
![]() |
[126] | Muyzer G, Stams AJ (2008) The ecology and biotechnology of sulphate-reducing bacteria. Nat Rev Microbiol 6: 441–454. |
[127] | Erable B, Duţeanu NM, Ghangrekar MM, et al. (2009) Application of electro-active biofilms. Biofouling 26: 57–71. |
[128] |
Li Z, Zhang X, Lei L (2008) Electricity production during the treatment of real electroplating wastewater containing Cr6+ using microbial fuel cell. Process Biochemistry 43: 1352–1358. doi: 10.1016/j.procbio.2008.08.005
![]() |
[129] |
Cong Y, Xu Q, Feng H, et al. (2013) Efficient electrochemically active biofilm denitrification and bacteria consortium analysis. Bioresour Technol 132: 24–27. doi: 10.1016/j.biortech.2013.01.004
![]() |
[130] |
Heitzer A, Sayler GS (1993) Monitoring the efficacy of bioremediation. Trends Biotechnol 11: 334–343. doi: 10.1016/0167-7799(93)90156-4
![]() |
[131] |
Perumbakkam S, Hess TF, Crawford RL (2006) A bioremediation approach using natural transformation in pure-culture and mixed-population biofilms. Biodegradation 17: 545–557. doi: 10.1007/s10532-005-9025-7
![]() |
[132] |
Urgun-Demirtas M, Stark B, Pagilla K (2006) Use of Genetically Engineered Microorganisms (GEMs) for the Bioremediation of Contaminants. Crit Rev Biotechnol 26: 145–164. doi: 10.1080/07388550600842794
![]() |
[133] | Cases I, de Lorenzo V (2005) Genetically modified organisms for the environment: stories of success and failure and what we have learned from them. Int Microbiol 8: 213–222. |
[134] | Absalon C, Ymele-Leki P, Watnick PI (2012) The bacterial biofilm matrix as a platform for protein delivery. MBio 3: e00127–00112. |
[135] |
Safa M, Alemzadeh I, Vossoughi M (2014) Biodegradability of oily wastewater using rotating biological contactor combined with an external membrane. J Environ Health Sci Eng 12: 117. doi: 10.1186/s40201-014-0117-3
![]() |
[136] |
Kaindl N (2010) Upgrading of an activated sludge wastewater treatment plant by adding a moving bed biofilm reactor as pre-treatment and ozonation followed by biofiltration for enhanced COD reduction: design and operation experience. Water Sci Technol 62: 2710–2719. doi: 10.2166/wst.2010.938
![]() |
1. | Miled El Hajji, Amer Hassan Albargi, A mathematical investigation of an "SVEIR" epidemic model for the measles transmission, 2022, 19, 1551-0018, 2853, 10.3934/mbe.2022131 | |
2. | Michel Fliess, Cédric Join, Alberto d'Onofrio, Feedback control of social distancing for COVID-19 via elementary formulae, 2022, 55, 24058963, 439, 10.1016/j.ifacol.2022.09.134 |
Case | $ T_e $ | $ J_{u+I} $ | $ J_{I} $ | $ J_{u+I}^{Te} $ | $ I(T_e) $ | $ {\max\limits_{[0, t_{final}] } S u} $ | $ u_{max} $ |
Unconstrained model-free | 11.3 | $ 5.40 10^4 $ | $ 5.40 10^4 $ | $ 1.15 10^5 $ | 31.12 | 3129 | 0.70 |
Constrained model-free – slope (Ⅰ) | 19.0 | $ 6.45 10^4 $ | $ 6.45 10^4 $ | $ 2.39 10^5 $ | 29.80 | 1990 | 0.80 |
Constrained model-free – slope (Ⅱ) | 22.9 | $ 6.45 10^4 $ | $ 6.45 10^4 $ | $ 2.86 10^5 $ | 28.25 | 2000 | 0.62 |
Constrained model-free – quad. (Ⅰ) | 16.9 | $ 5.83 10^4 $ | $ 5.83 10^4 $ | $ 1.83 10^5 $ | 29.12 | 1989 | 0.70 |
Constrained model-free – quad. (Ⅱ) | 17.2 | $ 6.66 10^4 $ | $ 6.66 10^4 $ | $ 2.39 10^5 $ | 32.29 | 1604 | 0.62 |
Unconstrained classical OC | 25.0 | $ 4.17 10^{4} $ | $ 4.17\, 10^{4} $ | $ 1.69 10^{5} $ | 21.95 | 9750 | 1 |
Constrained classical OC | 25.0 | $ 6.14 10^{4} $ | $ 6.14\, 10^{4} $ | $ 2.72 10^{5} $ | 24.23 | 1989 | 1 |
Case | $ T_e $ | $ J_{u+I} $ | $ J_{I} $ | $ J_{u+I}^{Te} $ | $ I(T_e) $ | $ {\max\limits_{[0, t_{final}] } S u} $ | $ u_{max} $ |
Unconstrained model-free | 11.3 | $ 5.40 10^4 $ | $ 5.40 10^4 $ | $ 1.15 10^5 $ | 31.12 | 3129 | 0.70 |
Constrained model-free – slope (Ⅰ) | 19.0 | $ 6.45 10^4 $ | $ 6.45 10^4 $ | $ 2.39 10^5 $ | 29.80 | 1990 | 0.80 |
Constrained model-free – slope (Ⅱ) | 22.9 | $ 6.45 10^4 $ | $ 6.45 10^4 $ | $ 2.86 10^5 $ | 28.25 | 2000 | 0.62 |
Constrained model-free – quad. (Ⅰ) | 16.9 | $ 5.83 10^4 $ | $ 5.83 10^4 $ | $ 1.83 10^5 $ | 29.12 | 1989 | 0.70 |
Constrained model-free – quad. (Ⅱ) | 17.2 | $ 6.66 10^4 $ | $ 6.66 10^4 $ | $ 2.39 10^5 $ | 32.29 | 1604 | 0.62 |
Unconstrained classical OC | 25.0 | $ 4.17 10^{4} $ | $ 4.17\, 10^{4} $ | $ 1.69 10^{5} $ | 21.95 | 9750 | 1 |
Constrained classical OC | 25.0 | $ 6.14 10^{4} $ | $ 6.14\, 10^{4} $ | $ 2.72 10^{5} $ | 24.23 | 1989 | 1 |