Optimal Control applied to life sciences: a numerical method based presentation

Optimal Control
applied to
A numerical method based presentation
Talk under the framework of the project “Stochastic models in medicine and life
science: mathematical analysis, model identification, validation and stability
properties” sponsored by CAPES Foundation, Ministry of Education of
Brazil.
Jorge Guerra Pires
Information Engineering and Science
PhD program
University of L’Aquila/
Institute of Systems Analysis and
Computer Science
jorgeguerrapires@yahoo.com.br
Italy, 2014

June/2014 Jorge G Pires 2
Optimal Control applied to
Abstract: Optimal Control Applied to Life Sciences
Life Sciences might be seen as the connections between medicine, biology,
mathematics, physics, and computer sciences. Further, optimal control might be
defined as the extension of static optimization, or even as some comments, the new
face of Variational Calculus.
In this talk we present several examples from life sciences analyzed with the support
of optimal control. We might apply basically three approaches to optimal control
problems, with their own weaknesses and strengthens: the Pontryagin’s Maximum
Principle, Dynamic Programming, or Static Optimization. All the problems treated
here were analyzed by the Pontryagin’s Maximum Principle.
The problems are solved using numerical schemes implemented in a computer.
Topping the bill, we present two cases from a paper on process of publication:
phototherapy for infants affected by neonatal jaundice and Feed-Forward Loop
Network. We leave as future works comparisons with other approaches such as
dynamic programming, or works on constraints on state space. Furthermore, we
have concentrated on continuous-deterministic problems.
Keywords: Life Sciences, applied optimal control, numerical schemes, Runge-Kutta
Method, forward-backward sweep method.

Cases Considered throughout the endeavor
 Toy models;
 Mold and Fungicide;
 Bacteria ;
 Tasmania Devil facial tumour disease;
 Optimal production of Protein;
 Drug Administration in one-compartment model;
 Applied Optimal Control Theory in Phototherapy of Infants;
 Cancer;

Cases Considered throughout the endeavor
 Bioreactors;
 Predator-Prey Model;
 Discrete Time Models;
 Cancer Therapy with Gompertzian Growth and one-compartment
model;
 Fish Harvesting;
 Epidemic Model;
 HIV Treatment;
 Bear Populations: metapopulation;
 Glucose Model;
 Timber Harvesting;

Cases Reported
 Cancer Therapy with Gompertzian Growth and one-compartment
model;
 Fish Harvesting; (Excluded, but interesting)
 Epidemic Model;
 HIV Treatment;
 Bear Populations: metapopulation; (Excluded, but interesting)
 Optimal production of Protein;
 Drug Administration in one-compartment model; (Excluded, but
interesting)
 Applied Optimal Control Theory in Phototherapy of Infants;

1. Introduction
A straightforward definition of life sciences is no longer simple. It might
be said that in the past, this scientific domain comprised of a set of
united field such as medicine and biology that rarely interfered with each
other; nonetheless, in the present it is a “unique” branch comprised of
researches from a variety of field such as mathematics, medicine, and
biology.
The inclusion of mathematics and other fields such as computer science
(and information sciences) came as a “rebirth” of the field.

1. Introduction

1. Introduction
Optimal control might be defined as a sub-area of control system in which
we look for the best policy (control) over a period of time. It was born in
the 50s from concerns on the aerospace industry, instead of optimizing a
finite set of variables, today named static optimization, they wanted to
optimization a dynamical system behavior, dynamic optimization.
In the majority of applications we take as granted important properties of
the system such as controllability.
The similarities are such as the fact that we can show necessary and
sufficient conditions based upon the same mathematical framework.

1. Introduction

1. Introduction
Each box is defined as:
The tradeoff function represents the balance between doing something
and something else, such as letting the system evolves on its own
dynamics or control. In general, this is a integral over the whole period of
optimization and a payoff function for the final state;
The dynamical system represents the real system itself, using state
variables and state equations;
Further details comprise further details such as bounds on control or
even maximum total amount of the same all of the period.
In the scheme it was not represented a arrow between the further details’
box and the tradeoff’s once the demands on the tradeoff are in general
mathematically demanded such as limited tradeoff function on the optimal
policy and pathway.

1. Introduction

2. Ordinary Differential Equation and Systems
The term dynamic refers to phenomena that produce time-changing
patterns, the characteristics of the pattern at one time being interrelated
with those at other times. The term is nearly synonymous with time-
evolution or pattern of change. It refers to the unfolding of events in a
continuing evolutionary process. The term system was originated as a
recognition that meaningful investigation of a particular phenomenon can
often only be achieved by explicitly accounting for its environment [7].
Dynamical system might be defined as a single or set of “systems” that
evolve in time, in the case of more than one, they are in general coupled,
for example in social networks. The most common methodology applied to
model those systems are differential equations. It is not clearly defined, but
other areas such as biology work sometimes with their own methodology for
modeling dynamical system. We might say that one of the challenge of
complex networks theory is to unify those field, mainly using dynamical
systems theory.

2. Ordinary Differential Equation and Systems
Several issues are treated in the theory of dynamical systems, the most
common are bifurcations, stability, and modeling. Bifurcation is the study of
quantitative behavior of a system close to some special values of the
parameters of the model, called bifurcation parameters. Basically several
systems might “change” their dynamical close to several parameterizations.
Stability is a quite important matter, it is related to the fact that a system can
be kept on a particular state, or at least close to it, called equilibrium points.
Modeling is related to the “ability” of a system to represent real plants, this
is extremely important in control theory, once the study is done on the
model, then it must be applied to the real plant, if the model is too far from
the real plant, quite strange behavior will come up.
Predominantly, we have continuous and discrete systems. They differ on
the tools, for continuous systems, we apply differential equations, whereas
for discrete we apply difference equations. Example of discrete systems is
birth and death modeling, and continuous is density modeling.

2. Ordinary Differential Equation and Systems: gene expression
dynamics
Source: Simulations for BRICS-CCI Brazil 2013

dynamics

dynamics
“Type-1 FFL is a sign-sensitive delay
element that can protect against
unwanted responses to fluctuating
inputs. The magnitude of the delay in
the FFL can be tuned over
evolutionary timescales by varying the
biochemical parameters of regulator
protein y [middle gene], such as its
lifetime, maximum expression, and
activation threshold ” Alon (2007,
p.57).

3. Numerical Schemes
Numerical Scheme can be defined as an human-written sequence of orders
(algorithm) with the purpose to solve problems based upon numbers.
Examples go from a simple root finder scheme to a more complicate scheme for
teaching networks (learning machine), called learning paradigms.
A quite famous application of numerical schemes came with Gauss, the Mean
Squared Error (MSE), when he predicted the trajectory of an unknown planet based
upon observations, this is today the base for numerical schemes, from optimal
control to learning machine and regression.
The second famous application came with Feynman and Collaborators, when they
have been surprised by the result of a numerical scheme, this had confirmed the
corpuscular nature of heat, no chaotic.
It is said that Hodgkin–Huxley had studied their famous model on dynamics of
neuron using a hand-like calculator, the model is based on dynamical systems.

3. Numerical Schemes
Nowadays we have an almost unanimous consent regarding the importance of
numerical methods; nonetheless some still “resist” such as an old professor of mine
from Russia.
If we compare the simple method of Guass to the ones used today, we get
somehow surprised, such as the studies of an old professor of mine from Russia,
he was modeling the sun’s dynamics by numerical schemes.
Besides we always make use of computer for numerical schemes, the majority of
them, even the new ones are based upon those, were developed before the
computer could have been even dreamt of. The consequence is that we need to
improvise, see the variations of the famous method of Newton. Maybe this time for
us develop our own schemes!
It is rare a case where we have a single numerical scheme for solving a problem,
the variations go around the aim “cost-simplicity-accuracy”. The choice in the
majority of cases is a matter of schools or preferences.

3. Numerical Schemes: Euler Method or Tangent Line Method

3. Numerical Schemes: The Runge-Kutta Method

3. Numerical Schemes: Error

3. Numerical Schemes: Shooting Method

3. Numerical Schemes: The Newton’s Method

3. Numerical Schemes: The Secant’s Method

4. Numerical Solutions for optimal control: algorithms
All the numerical schemes presented here are result of the application of
the necessary conditions based on the theory of Hamiltonian, as a result
of applying the Pontryagin’s Maximum Principle. The necessary
conditions used here are those presented in [11].
All the derivations will be presented before any coding is discussed
(omitted). In order to try to simplify the presentations, we will organize the
problems into prototypes.
The prototypes suppose to serve as a reference model. We have chosen
to present those in prototypes once they will demand different schemes
for each of them.
The good news is that they all might be done with an extension of single
code for the prototype 1: Forward-Backward Sweep Method, for short
FBSM.

All the models treated follow a common pattern given by the following
picture (next slide).
The drawback of this approach is the red arrow: one must first transform
the optimal control problem into a system of differential equation, this is
hand-calculation, and for complicate system it might be cumbersome.
One potential approach is directly integrating the system. One possible
way of doing that is integrating the Hamiltonian directly, but it might
introduce further error on the approximations of the derivatives for the
optimality condition and the adjoint equation.
Furthermore, it might be complicate once we are demanded to optimize
the Hamiltonian function.

This is the simplest problem we can formulate in optimal control
that could be applied to real problem. As we will see, this simple approach
sometimes might not be appropriate and further details should be added,
this is the next prototype.

4. Numerical Solutions
for optimal control:
The Forward-Backward
Sweep Method

4. Numerical Solutions for optimal control: The Forward-Backward
Sweep Method

Sweep Method

Sweep Method for Prototype 4

Sweep Method

Sweep Method for prototype 5

Sweep Method

5. Optimal Phototherapy Regimen: the neonatal jaundice case

The case study discussed here is the treatment by phototherapy for
newborns affected the syndrome called Neonatal Jaundice. The
neonatal jaundice is caused by the immaturity of the liver, turning
impossible the conjugation of byproduct of the natural process of
hemoglobin breaking down that happens after birth, therefore the
elimination of the undesired wastes, which are toxic, see for example
Pires et al (2009) and references therein.
According to (Pires et al, 2009; Schoof et al 2012), and further
references, jaundice might be seen as the discoloration of skin, mucous,
and sclera due to the accumulation of bilirubin, a waste of the
hemoglobin breaking down, they turn the skin, mucous, and sclera
yellowish.

We know from studies with light properties that frequency is what
matters, what differentiate the colors, the properties of x-ray compared to
gamma-rays. We also know that the visible spectrum is relatively small,
something from 200 nm to 700 nm in wavelength.
The sunlight is the most complete illumination system, containing from
ultraviolet to more rare wavelengths. Nonetheless, we must recollect that
what heats the earth is present in the sunlight, namely, infrared, or even
the so-feared ultraviolet. Therefore, side effects might turn it impossible
to use it depending on the phototherapy intensity demanded.
In the case of the neonatal jaundice, the wavelength corresponding to
the “blue” is the demanded one. Thus, a good system of illumination for
this phototherapy must “possess the blue spectrum in abundance.”

Before going on, some remarks might be useful. In Pires et al
(2009), it is not discussed on the real correlation between the sensitivity
of the bilirubin and the polymeric solution to the radiation, just taken as
granted the correlation, based on theory and independent experiments.
In the case of the bilirubin, we have a circulatory system, that is, just the
portion of the blood in the skin will be exposed to the blue light, whereas
in the polymeric solution, the expose is higher. Nonetheless, this type of
issue had not limited the application of optimal control.
In Lenhart e Workman (2007), this is presented a bear-control policy
designed by optimal control. In Lenhart e Workman (2007), it has been
presented a control of bear over regions sharing boundaries, called
metapopulations. The control of a region affects the other by the dynamic
on the boundaries.

Simulations for the neonatal phototherapy. The green curve depicts a “bad case” of the syndrome, that is, the body of the
newborn eliminate the bilirubin in a low rate compared to the “production” rate; the red curve represents an arbitrary case, but
better than the green curve; the blue curve represents the case in which the degree of importance to diminish the concentration
of bilirubin is increased by 10 compared to the risks of the phototherapy. We simulate an “intermittent phototherapy;” we apply
the therapy for 1.5 units of time and then stop for another 1.5 units of time. We have used 10 units of concentrations as initial
state. The dashed rectangle in red is the therapeutic window. Source: own elaboration.

Simulations for the phototherapy in newborns using a drug X in junction. The blue curve represents the
simulations using just the phototherapy, whereas the red is the simultaneous use of phototherapy and the drug
X. The dashed line represents the goal (superior limit). We apply the treatment for 1.u units of time. Source:
own elaboration.

6. Optimal Protein Production: the feed-forward loop network motif

The topic treated here is quite intriguing or even provocative, and indeed
the treatment herein overlooks several important topics.
For instance, from an engineering viewpoint, one might think of a
bioreactor and we aim to optimize the workings the same; but from a
biological viewpoint, we might want to understand how a certain gene or
even its circuit was selected under evolution.
This topic has been exploited in the literature, see for instance Alon
(2007) for several discussion from a different angle. For the mathematical
model used here, see Cacace et al (2012), or even Pires (2012), for more
details. For discussion on potential application of this topic in engineering,
see for example Pires and Palumbo (2012), or as alternative Pires
(2013).

On this part of the paper, we discuss on the optimal control of protein
production. We suppose that we are lucky enough to reveal the secrets of
god, and we find a function that is always optimized in the protein production
process. See that Alon (2007) comments that one difficulty in optimization
theories, such as the one used here, is that we may not know the fitness
function in the real world. Let’s neglect this issue.
In simple terms, we have three genes. Further, just one produces what we
need, call it Z, but surprisingly, it cannot be controlled directly, a second gene
must be activated for that, call it X. Thus, the more we have activated X, the
more we will have producing, but something starts to go wrong, then we
identify a third gene, call it Y. We find out that Y is activated by X, but it
inhibits Z, our goal. Then we must activate X as much as possible, but keep Y
as silent as possible. It must have an equilibrium point. The communication
between genes is done by a special group of proteins called transcription
factors, something like papers in a company, functionless, but necessary.
What we have described is called a network motif, namely, it is a feed-
forward loop network motif, an incoherent type I.

Controls for different situations, parameterizations: protein degradation rate, mRNA degradation rate, protein
production rate, mRNA production rate, degree of importance to produce protein, degree of importance for
minimizing the control, and threshold for the Hill functions “K” on the mathematical model. The lines on
continuous style are different values for the parameters, on the dashed lines we have increased by 10 the
importance of producing proteins compared to the cost, the cost is applied just on gene X, input gene. The
green function depicts what happens if we add the final value of protein to be optimized (a payoff term). The
initial condition for all genes are ‘0’, but gene X which is given a small amount of mRNA in time ‘0’. Source:
own elaboration.

7. Epidemic: optimal policy

Epidemics can be defined as the breakout of an infection disease.
Related terms are pandemic (world-spread-out) and endemic (steady-
state).
Epidemics is in general a complex network issue.
Source: http://1.bp.blogspot.com/-WSrc1yadP2U/TzFs3dHA-
TI/AAAAAAAABuo/bVbX1lgTnd8/s1600/epidemic_diffusion.jpg

8. HIV and AIDS

8. HIV and AIDS
Acquired immunodeficiency syndrome (AIDS) is medically devastating to
its victims, and wreaks financial and emotional havoc on everyone, infected
or not.
Keywords: Viral Replication; Immunology.
“Viruses are very small biological structures whose reproduction requires a
host cell.” They are not considered living things.
Helper T-lymphocytes play a key role in the process of gaining immunity to
specific pathogens.
HIV is an especially versatile virus. It not only inserts its genetic information
into its host’s chromosomes, but it then causes the host to produce new
HIV.
A virus cannot reproduce outside a host cell, which must provide viral
building materials and energy.

8. HIV and AIDS
The immune system monitors the body for dangerous pathogens.
When it detects pathogens, the immune system computers and
mobilizes the appropriate responses.
The immune system is made of a vast collection of cells that
communicate and interact in myriad ways.
One of the major tools of the immune system is antibodies.
One of the important roles of the immune system is to scan the cells of
the body for antigens – foreign proteins made by pathogens –.
The scanning process is carried out by T-cells.

8. HIV and AIDS

8. HIV and AIDS
Viral nucleic acid enters the host cell and redirects the host cell’s
metabolic apparatus to make new viruses.
“Many RNA viruses do not use DNA in any part of their life cycle.”
Living Systems
and some virus
Retrovirus

8. HIV and AIDS

8. HIV and AIDS
Key-point 1 : To find an optimal chemotherapy strategy in the treatment
of the human immunodeficiency virus (HIV).
Key-point 2: The model used herein describes the interaction of the
immune system with HIV.
Key-point 3: It is assumed the treatment acts to reduce the infectivity of
the virus for a finite time.
Antiretroviral drugs are medications for the treatment of infection by
retroviruses, primarily HIV. Different classes of antiretroviral drugs act on
different stages of the HIV life cycle. Combination of several (typically
three or four) antiretroviral drugs is known as highly active anti-retroviral
therapy (HAART).

8. HIV and AIDS
Each T cell attacks a foreign substance which it identifies with
its receptor. T cells have receptors which are generated by randomly
shuffling gene segments. Each T cell attacks a different antigen. T cells
that attack the body's own proteins are eliminated in the thymus. Thymic
epithelial cells express major proteins from elsewhere in the body. First, T
cells undergo "Positive Selection" whereby the cell comes in contact
with self-MHC expressed by thymic epithelial cells; those with no
interaction are destroyed. Second, the T cell undergoes "Negative
Selection" by interacting with thymic dendritic cell whereby T cells with
high affinity interaction are eliminated through apoptosis (to avoid
autoimmunity), and those with intermediate affinity survive.

8. HIV and AIDS

9. Optimal Tumor Treatment : Cancer Therapy with Gompertzian
Growth and one-compartment model
Tumor is a state of the body where cells divide (mitosis, multiply) on an
uncoordinated way. This is a type of cancer in some cases. Tumors might
be classified as benign, premalignant, or malignant (cancer). Cancer is so
feared for spreading out, invading neighboring tissues, tumors (premalignant
cancer) does not invade neighboring tissues.
On this section we present the simplest model possible to build applying the
theories presented so far.
The model presented herein is relatively simple. We apply the theory of
optimal control for the treatment of cancer. Using the mentioned theory, we
can optimize the behavior of a set of differential equations, on our case,
ordinary differential equations.
The model discussed here was published by Fister e Panetta (2003) and
studied by (Lenhart e Workman 2007; Swan, 1984); we try to make it better
by adding the dynamic of the drug.

See that the first term of the differential equation represents a ‘growth,’ a
natural process, using a dynamics called Gompertz model, and the
second represent our control by means of the treatment.
It should be pointed out that we can use just this equation on the study of
optimal control applied to tumor therapy and this is what is done on
Lenhart e Workman (2007), but we can do better!
What about the dynamics of the drug? This is what we add here. Drugs
might exhibit peculiar behavior and assuming that we can control the
amount exactly of drug that reaches the site might be a mistake.
As Lenhart e Workman (2007) highlights studies of drugs is a rich and
state of the art field, and one of the models lacking are the ones that
studies multiple drugs taken at the same time. That is, models that takes
into account interactions between different drugs; see that the model
presented here is limited in the sense that it is still kinetics, not dynamics,
models for dynamics are much complicate, we skip them.

Just to provoke these studies on the literature, let’s consider an ideal case of two
drugs taken for eliminating this tumor.
We consider a simple case, the second drug just increase the absorption of the
first, the drug one is the one that really can eliminate the tumor.
This can increase the possibility to maintain the plasma concentration within the
therapeutic window; see that the second drug is increasing the absorption, then it
should eventually increase the concentration above the therapeutic window of the
drug we need to monitor, and this exactly the trick of optimal control, the optimal
control policy will just use what is needed given a goal (therapeutic window).
This is known as bioavailability; this is similar when you are suggested to take a
drug together with milk, milk is not a drug, just increase the bioavailability of the
take drug, or even when you are asked to use a drug with empty stomach. This
type of analysis is concern of noncompartmental models.
Noncompartmental models are considered easier to use due to several reasons
such as it does not change from individual to individual so much as compartment
models. See Rosenbaum (2011) for more details.

References
[1] George W Swan, Applications of optimal control theory in biomedicine, Pure and
Applied Mathematics. Marcel Dekker Inc, 1984.
[2] Wendell H Fleming; Raymond W Rishel. Deterministic and Stochastic Optimal
Control. Applications of Mathematics 1, Springer-Verlag: 1975.
[3] Boyce, William E.; Diprima, Richard C. Elementary differential equations and
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[4] Betts, J. T. Practical Methods for optimal Control and Estimation Using Nonlinear
Programming. Second Edition. Advances in Design and Control, Society for Industrial
and Applied Mathematics. 2010.
[5] Wikipedia. The Secant Method Online: http://en.wikipedia.org/wiki/Secant_method.
Accessed on May/2014.
[6] Wikipedia. The Newton’s Method Online: http://en.wikipedia.org/wiki/Newton
%27s_method. Accessed on May/2014.
[7] David G. Luenberger, introduction to dynamic systems: theory, models, and
application. John Wiley & Sons, 1979.
[8] Lynch, Stephen. Dynamical systems with applications using Mathematica®.
Birkhäuser Boston, 2007.
[9] Wikipedia. Logistic Map http://en.wikipedia.org/wiki/Logistic_map. Accessed on
May/2014.

References
[10] Wikipedia. Tent Map http://en.wikipedia.org/wiki/Tent_map. Accessed on May/2014.
[11] Lenhart, S.; Workman, J.T, Optimal Control Applied to biological models, Chapman
& Hall/ CRC, Mathematical and Computational Biology Series, 2007.
[12] Press, William H. ;Teukilsky, Saul A. ;Vetterling, William T. ;Flannerg, Brian P.
Numerical recipes in C: the art of scientific computing. Second edition. Cambridge
University press: 1992.
[13] Ruggiero, Márcia A. Gomes; Lopes, Vera Lúcia da Rocha. Cálculo numérico:
aspectos computacionais. 2° edição. São Paulo: Pearson Markon Books, 1996.
[14] Devries, Paul L. ; Hasbun, Javier E. A first course in computational physics. Second
edition. Jones and Bartlett Publishers: 2011.

References
ALON, U. An Introduction to systems biology: design principles of biological circuits. Chapman & Hall/CRC,
2007.
CACACE, F., GERMANI, A., PALUMBO, P., The state observer as a tool for the estimation of gene
expression, Journal of Mathematical Analysis and Applications, Vol.391, pp.382-396, 2012.
LENHART, S.; WORKMAN, J. T, Optimal Control Applied to biological models, Chapman & Hall/ CRC,
Mathematical and Computational Biology Series, 2007.
PIRES, J. G. Desenvolvimento de programa baseado no problema da mistura. Pesquisa Operacional:
programação matemática. Simpósio de Engenharia de Produção. XVI: 1-12, 2009.
PIRES, J. G.; DUARTE, A. S.; BIANCHI, R. F.; SANTOS, Z. A. DA S.; BIANCHI, A. G. C. Projeto e
desenvolvimento de Produto: proposta e desenvolvimento de dispositivo eletrônico para auxiliar no
tratamento da icterícia. Gestão do Produto: engenharia do produto. Simpósio de Engenharia de Produção.
XVI: 1-12, 2009.
PIRES J. G. On the applicability of Computational Intelligence in Transcription Network Modeling. Thesis of
master of science. Faculty of Applied Physics and Mathematics, Gdansk University of Technology, Poland.
74:1:46. 2012.
PIRES, J. G.; PALUMBO, P. Engenharia de Software: Planejamento e desenvolvimento de programa
baseado em Inteligência Computacional aplicada a Redes de Expressão Genética. Gestão do Produto:
engenharia do produto. Simpósio de Engenharia de Produção. XIX: 1-12, 2012.
PIRES, J. G. Na importância da biologia e em engenharias: biomatemática e bioengenharias. Educação em
Engenharia de Produção: estudo do ensino de engenharia de produção. Simpósio de Engenharia de
Produção. XX: 1-12, 2013.
ROSENBAUM, S. E Basic pharmacokinetics and pharmacodynamics: an integrated textbook and computer
simulations, John Wiley & Sons, 2011.
Visite: http://www.uri.edu/pharmacy/faculty/rosenbaum/basicmodels.
SCHOOF, C. P.: Zschocke, J.: Potocki, L. Human Genetics: from molecules to medicine. Lippincott Williams
& Wilkins: 2012.
WINSTON, W. L. Operations Research: application and algorithms. Third Edition, 1996.

What is next....?
 Dynamics programming;
 Optimization based schemes;
 Discretization based approaches;
 Stochastic counterparts;
 Extension of the models presented herein for stochastic framework;
 Pharmacogenomics;
 Test some in PK/PD;
 Pharmacokinetic/pharmacodynamic interactions;
 Receptor Interactions;
 Report in September;
 Java Library for Optimal Control in Life Sciences (JAR Executable)!? !?

http://ijcai-15.org/index.php/important-dates

Thank you.

Optimal Control applied to life sciences: a numerical method based presentation

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Optimal Control applied to life sciences: a numerical method based presentation