mathematical models and methods for complex...

Mathematical Models and Methods for Complex

Systems

Nicola [email protected]

Department of Mathematics

Politecnico di Torino

http://calvino.polito.it/fismat/poli/

Mathematical Models and Methods for Complex Systems– p. 1/40

Homepage: http://calvino.polito.it/fismat/poli


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Unraveling Complex (Living) Systems

APRIL 2011 UNRAVELING COMPLEX SYSTEMS

The American Mathematical Society, the American Statistical Association, the

Mathematical Association of America, and the Society for Industrial and Applied

Mathematics announce that the theme of Mathematics Awareness Month 2011 is

”Unraveling Complex Systems.”

From www.mathaware.org

We live in a complex world. Many familiar examples of complexsystems may be found

in very different entities at very different scales: from power grids, to transportation

systems, from financial markets, to the Internet, and even inthe underlying

environment to cells in our bodies. Mathematics and statistics can guide us in

unveiling, defining and understanding these systems,in order to enhance their

reliability and improve their performance.


Lecture 1. - Common Features of Living Systems

Part 1. - Toward a Mathematical Theory of Complex Systems

T.1. Common Features of Living (Complex) Systems

T.2. Foundations of Mathematical Modeling and Scaling

T.3. Representation, Nonlinear Stochastic Games, Pathways, Networks and

Mathematical Tools


Lecture T.1. - Common Features of Living Systems

A Personal Bibliographic Search

• E. Mayr, The philosophical foundation of Darwinism,Proc. American Philosophical

Society, 145 (2001) 488-495.

• A.L. BarabasiLinked. The New Science of Networks, (Perseus Publishing,

Cambridge Massachusetts, 2002.)

• F. Schweitzer,Brownian Agents and Active Particles, (Springer, Berlin, 2003).

• M. Talagrand,Spin Glasses, a Challenge to Mathematicians, Springer, (2003).

• M.A. Nowak and K. Sigmund, Evolutionary dynamics of biological games,Science,

303(2004) 793–799.

• M.A. Nowak,Evolutionary Dynamics, Princeton Univ. Press, (2006).

• F.C. Santos, J.M. Pacheco, and T. Lenaerts, Evolutionary dynamics of social

dilemmas in structured heterogeneous populations, PNAS,103(9), 3490-3494, (2006).

• N.N. Taleb,The Black Swan: The Impact of the Highly Improbable, (2007).

• K. Sigmund,The Calculus of Selfishness, Princeton Univ. Press, (2011).


Lecture T.1. - Common Features of Living Systems

A Personal Quest Through Complexity

• N.Bellomo,Modelling Complex Living Systems. A Kinetic Theory and

Stochastic Game Approach, (Birkhauser-Springer, Boston, 2008).

• N.Bellomo and M. Delitala, From the mathematical kinetic, and stochastic game

theory to modelling mutations, onset, progression and immune competition of cancer

cells,Phys. Life Rev., 5, (2008), 183-206.

• N. Bellomo, A. Bellouquid, J. Nieto J., and J. Soler, Multiscale biological tissue

models and flux-limited chemotaxis from binary mixtures of multicellular growing

systems,Math. Models Methods Appl. Sci., 10 (2010) 1179-1207.

• N. Bellomo and B. Carbonaro, Towards a mathematical theory of living systems

focusing on developmental biology and evolution: a review and perspectives,Physics

of Life Reviews, 8 (2011) 1-18.

• N. Bellomo and C. Dogbe, On The Modelling of traffic and crowds - A survey of

models, speculations, and perspectives,SIAM Review, 53(3)(2011) 409-463.

• N. Bellomo and A. Bellouquid, On the modeling of crowd dynamics looking at the

beautiful shapes of swarms,Networks Heterog. Media, 3, (2011) 383-399.Mathematical Models and Methods for Complex Systems– p. 7/40


• E. Kant 1790, daCritique de la raison pure, Traduction Fracaise, Press Univ. de

France, 1967,

Living Systems: Special structures organized and with the ability to chase a purpose.

Hartwell - Nobel Laureate 2001, Nature 1999

• Biological systems are very different from the physical or chemical systems analyzed

by statistical mechanics or hydrodynamics. Statistical mechanics typically deals with

systems containing many copies of a few interacting components, whereas cells

contain from millions to a few copies of each of thousands of different components,

each with very specific interactions.

• Although living systems obey the laws of physics and chemistry, the notion of

function or purpose differentiate biology from other natural sciences. Organisms exist

to reproduce, whereas, outside religious belief rocks and stars have no purpose.

Selection for function has produced the living cell, with a unique set of properties

which distinguish it from inanimate systems of interactingmolecules. Cells exist far

from thermal equilibrium by harvesting energy from their environment.


Lecture 1. - Common Features of Complex Living Systems

E. Schrodinger, P. Dirac - 1933, What is Life?, I living systems have the ability to

extract entropy to keep their own at low levels.

R. May, Science 2003In the physical sciences, mathematical theory and experimental

investigation have always marched together. Mathematics has been less intrusive in

the life sciences, possibly because they have been until recently descriptive, lacking the

invariance principles and fundamental natural constants of physics.

E.P. Wiegner , Comm. Pure Appl. Math., 1960The miracle of the appropriateness of

the language of mathematics for the formulation of the laws of physics is a wonderful

gift which we neither understand nor deserve.

G. Jona Lasinio , La Matematica come Linguaggio delle Scienze della Natura -

Preprint, La vita e una proprieta emergente della materia?- La vita rappresenta una

fase avanzata di un processo evolutivo e selettivo. Mi pare difficile spiegare il vivente

ignorando la sua dimensione storica. La dinamica delle popolazioni, di cui esiste una

teoria matematica ancora in uno stato abbastanza primitivodovra spiegare l’emergere

per selezione delle dinamiche proprie del singolo vivente.



N.B. H. Berestycki, F. Brezzi, and J.P. Nadal, Mathematics and Complexity in Life

and Human Sciences, Mathematical Models and Methods in Applied Sciences, 2010.

• The study of complex systems, namely systems of many individuals interacting in a

non-linear manner, has received in recent years a remarkable increase of interest

among applied mathematicians, physicists as well as researchers in various other

fields as economy or social sciences.

• Their collective overall behavior is determined by the dynamics of their interactions.

On the other hand, a traditional modeling of individual dynamics does not lead in a

straightforward way to a mathematical description of collective emerging behaviors.

• In particular it is very difficult to understand and model these systems based on the

sole description of the dynamics and interactions of a few individual entities localized

in space and time.



Five Key Questions

1. How far is the state-of-the-art from the development of a biological-mathematical

theory of living systems and how an appropriate understanding of multiscale issues can

contribute to this ambitious objective?Application of methods of the inert matter to

living systems is highly misleading. Moreover multiscale approaches should be

interpreted in the framework of an evolutionary dynamics rather that within a static

framework.

2. Can mathematics contribute to reduce the complexity of living systems by splitting

it into suitable subsystems?Toward a new system biology

3. Can mathematics offer suitable tools to constrain into equations the common

complexity features all living systems?Not at present, but looking ahead to new

mathematical approaches should contribute to this objective



Five Key Questions

4. Should a conceivable mathematical theory show common featuress in all field of

applications?Although a theory should be linked to a specific class of systems, all

theories should have common features.

The last question is also a dilemma

5. Should mathematics attempt to reproduce experiments by equations whose

parameters are identified on the basis of empirical data, or develop new structures,

hopefully a new theory able to capture the complexity of biological phenomena and

subsequently to base experiments on theoretical foundations?This last question

witnesses the presence of adilemma, which occasionally is the object of intellectual

conflicts within the scientific community. However, we are inclined to assert the

second perspective, since we firmly believe that it can also give a contribution to

further substantial developments of mathematical sciences.



Five Common Features and Sources of Complexity

1. Ability to express a strategy: Living entities are capable to develop

specificstrategies andorganization abilitiesthat depend on the state of the

surrounding environment. These can be expressed without the application of any

external organizing principle. In general, they typicallyoperate

out-of-equilibrium. For example, a constant struggle with the environment is

developed to remain in a particular out-of-equilibrium state, namely stay alive.

2. Heterogeneity: The ability to express a strategy is not the same for all

entities:Heterogeneitycharacterizes a great part of living systems, namely, the

characteristics of interacting entities can even differ from an entity to another

belonging to the same structure.In developmental biology, this is due to different

phenotype expressions generated by the same genotype.

3. Learning ability: Living systems receive inputs from their environments and

have the ability to learn from past experience. Therefore their strategic ability and the

characteristics of interactions among living entities evolve in time.Societies can

induce a collective strategy toward individual learningMathematical Models and Methods for Complex Systems– p. 13/40


4. Interactions: Interactions nonlinearly additive and involve immediate

neighbors, but in some cases also distant particles. Indeed, living systems have the

ability to communicate and may possibly choose different observation paths. In some

cases, the topological distribution of a fixed number of neighbors can play a prominent

role in the development of the strategy and interactions. Interactions modify their state

according to the strategy they develop. Living entitiesplay a game at each

interaction with an output that is technically related to their strategyoften related

to surviving and adaptation ability.Individual interactions in swarms can depend on

the number of interacting entities rather that on their distance.

5. Darwinian selection and time as a key variable: All living

systems are evolutionary. For instance birth processes cangenerate individuals more

fitted to the environment, who in turn generate new individuals again more fitted to the

outer environment. Neglecting this aspect means that the time scale of observation and

modeling of the system itself is not long enough to observe evolutionary events. Such

a time scale can be very short for cellular systems and very long for vertebrates.

Micro-Darwinian occurs at small scales, while Darwinian evolution is generally

interpreted at large scales.Mathematical Models and Methods for Complex Systems– p. 14/40


Technical Complexity

• Large number of components: Complexity in living systems is induced by

a large number of variables, which are needed to describe their overall state. Therefore,

the number of equations needed for the modeling approach maybe too large to be

practically treated. Reduction of complexity is the first step of the modeling approach.

• Multiscale aspects: The study of complex living systems always needs a

multiscale approach. For instance, the functions expressed by a cell are determined by

the dynamics at the molecular (genetic) level. Moreover, the structure of macroscopic

tissues depends on such a dynamics.

• Time varying role of the environment: The environment surrounding

a living system evolves in time, in several cases also due to the interaction with the

inner system. Therefore the output of this interaction evolves in time. One of the

several implications is that the number of components of a living system evolves in

time.



MULTISCALE REPRESENTATION OF TUMOUR GROWTH: gene interactions (stochastic

games), cells (kinetic theory), tissues (continuum mechanics), mixed (hybrid models).



EVOLUTION OF CELLULAR PHENOTYPE


Lecture 2. - Modeling and Scaling





Mathematical Tools



Introduction to Modeling and Scaling: This Lecture refers to Chapter 1

of the e-book:

• N. Bellomo, E. De Angelis, M. DelitalaLecture Notes on Mathematical Modelling

From Applied Sciences to Complex SystemsSIMAI e-Lecture Notes, vol. 8, 2010.

Systems belonging to the real world can be observed and phenomenologically studied

in order to reach a knowledge, as deep as possible, of the inner structure and of their

behavior. For such a purpose, experiments often need to be organized, and suitable

experimental devices need to be designed. These may be, in some cases, quite

sophisticated inventions of the human brain. When, throughthe collection and

organization of the experimental data, the systematic observation is developed into the

analysis, interpretation, and prediction of the behavior of the observed system, then a

mathematical model is produced.

In past times, mathematical modeling was essentially related to mathematical physics.

Such a science provided, mainly through the last century, all fundamental models of

the modern mechanical and technological sciences. Mathematical modeling is

nowadays a discipline that plays a crucial role in all applied sciences, from natural to

technological ones.Mathematical Models and Methods for Complex Systems– p. 19/40


Some Definitions

Definition 2.1. Independent variables: That istime t ∈ [0, T ] ⊆ IR + and

space x = x, y, z ∈ D ⊆ IR 3, are theindependent variables. A physical

system can be observed in a time interval[0, T ] ⊆ IR + and in a volumeD ⊆ IR 3. In

order to define the physical volume occupied by the observed system, a fixed system of

orthogonal axesO(xyz) with unit vectorsi, j, k can be used. Thepast is represented

by the negative values of the time,t ∈ IR −; thefuture by positive valuest ∈ IR +. A

system can be observed for positive times, however, the mathematical model can refer

to negative times. In this caset ∈ IR .

Definition 2.2. State variable: Thestate variable is a vector valued variable,

generally a function of the independent variables,u = u(t,x) : [0, T ] ×D 7−→ IR n,

whereu = u1, . . . , ui, . . . , un, such thatu is the set of the variables in charge of

describing (in the mathematical model) the physical state of the observed system. The

state variable is also calleddependent variable. It can take values either

for all values or for discrete values of theindependent variables.



Definition 2.3. Parameters: Theparameters are physical quantities that

characterize the physical system to be modeled. These quantities can be defined either

in a dimensional or in a dimensionless form and can be obtained either by direct

measurements on the system itself or by comparisons betweenthe predictions of the

model and the behavior of the real system. In the latter, the definition of

identification parameters is also used.

Definition 2.4. Mathematical model: A mathematical model is an

equation, or a set of equations, whose solution (that is the solution of the related

mathematical problems) provides the time-space evolutionof thestate variable,

that is the physical behavior, in the framework of the mathematical model, of the

related physical system. The equation that defines the mathematical model can also be

called thestate equation. The structure of the state equation is not, and cannot be

fixeda priori. It depends upon the adopted modeling technique.



The classifications

• Analysis of the structure of the state variable;

• Analysis of the type of the state equation;

• Analysis of the structure of the evolution equation;

• Analysis of the characteristics of the parameters.

Classification by state variableThis type of classification is based upon the analysis

of the structure of the state variable and, in particular, ofits arguments (the

independent variables).

Definition 2.5. Dynamic and static models A mathematical model is

dynamic if the state variableu depends on the time variablet. Otherwise the

mathematical model isstatic.

Definition 2.6. Discrete and continuous models A mathematical model is

discrete if the state variable does not depend on the space variables.Otherwise the

mathematical model iscontinuous.



Definition 2.7. Algebraic model: A mathematical model isalgebraic if the

state equation is an algebraic equation over the state variable and the set of parameters

r:

f(u; t,x, r) = 0.

Definition 2.8. Models and ODE: A mathematical model is expressed in terms

of ordinary differential equations if the state variable satisfies an ordinary differential

equation of the typedu

dt= f(t, u; r).

Definition 2.9. Models and PDE: A mathematical model is expressed in terms

of partial differential equations if the state equation involves partial derivatives of the

state variable with respect to the independent variables.

∂u

∂t= f

(t, x, u,

∂u

∂x,∂2x

∂x2; r

).

Remark If the state variable is a multidimensional vector, then theabove equations are

replaced by systems of equations: One for each componentui of the state variableu.



Classification by structure of the state equation

Definition 2.9. Linear and nonlinear models: A mathematical model is

linear if the state equation can be written in the formLu = 0, whereL is a linear

operator that satisfies the following conditions:

L(λu) = λL(u) ,

whereλ is a scalar and

L(u1 + u2) = Lu1 + Lu2.

A mathematical model isnonlinear if the state equation can be written as

Nu = 0 ,

whereN is a nonlinear operator.



Consistency of Models and Good Position of Problems

• Definition of the consistency properties of a mathematical model.

• Definition of the correct formulation of a mathematical problem.

• Well-posedness of a mathematical problem.

Definition 2.10.Consistency: A mathematical model defined by a system of

PDEs isconsistent if the number of equations is equal to the dimension of the

state-dependent variable.

• Elliptic equations describe systems in the equilibrium or steady state. They can be

seen as equations describing, for instance, the final state reached by a physical system

described by a parabolic equation after the transient term has died out.



• Hyperbolic equations are evolution equations that describe wave-like phenomena.

The solution of a hyperbolic initial value problem cannot besmoother than the initial

data. On the other hand, it can actually develop, as time goesby, singularities even

from smooth initial data, which then characterize the wholeevolution and are

propagated along special curves calledcharacteristics. In particular, if the

solution has initially a compact support, for instance,

u(t = 0, x) = 0 , for |x| > M0 ,

then

∀t , ∃M(t) such that u(t, x) = 0 for |x| > M(t) .

The rate at which the support expands can be interpreted as thespeed of

propagation of the effect modelled by the hyperbolic equation.



• Parabolic equations are evolution equations that describediffusion-like phenomena.

In general, the solution of a parabolic problem is infinitelydifferentiable with respect

to both space and time for anyx andt > 0, even if the initial data are not continuous.

In other words, parabolic systems have asmoothing action and singularities can

neither develop nor be maintained. Moreover, even if the initial data have compact

support, at anyt > 0 the initial condition is felt everywhere, that is

∀t and ∀M , ∃x with |x| > M such that u(t, x) 6= 0 .

This is usually indicated by saying that parabolic systems have aninfinite speed

of propagation.


Lecture 3. - Mathematical Tools





Mathematical Tools


Lecture 3. - Mathematical Tools

Toward a representation of complex systems

Let us consider a system constituted by a large number of interacting living entities,

calledactive particles. Their microscopic state includes, in addition to

geometrical and mechanical variables, also an additional variable, calledactivity,

which is heterogeneously distributed. Moreover, considera system of networks and

nodes such that the role of the space variable is not relevantin the nodes, while

transition from one node to the other has to be properly modeled.

1. Living systems are constituted different types of active particles, which

express several different functions. This source of complexity can be reduced byby

decomposing the system into suitable functional subsystems,

where afunctional subsystem is a collection of active particles, which have the

ability to express cooperatively the sameactivity regarded as a scalar variable.


Lecture 3. - Representation, Nonlinear Stochastic Games,Pathways, and Mathematical Tools

2. The description of the overall state of the system within each node is delivered by a

probability distribution function over such microscopic state, labeled by

the indexi, which denotes the functional subsystem within each node.

fi = fi(t, u) , i = 1, . . . , n,

such thatfi(t, u) du denotes the number of active particles whose state, at timet, is in

the interval[u, u+ du]. If the number of active particles is constant in time, then the

distribution function can be normalized with respect to such a number and

consequently is aprobability density. The physical meaning of the activity

variable differs for each subscript.

3. The overall number of functional subsystems is given by the sum of all of them in

the nodes:fij = fij(t, u) , i = 1, . . . , n, j = 1, . . . ,m, where functional

subsystems differ if are localized in different nodes although they express the same

activity variable.


Lecture 3. - Representation, Nonlinear Stochastic Games,Pathways, Mathematical Tools

Nonlinear Stochastic Games

Active particles interact with a certainencounter rate and play acollective

game at each interaction, which can occur within each node or through the network.

The modeling consists in computing, in probability, the output of the interaction.

Various sources of nonlinearities can be considered. Amongthem:

• Distribution function conditioning the encounter rate also in connection to

hiding and learning dynamics;

• Nonlinearity induced bytopological distribution of the interacting

entities.

• Distribution function conditioning the output of the games;



Nonlinear Stochastic GamesInteraction involves:

• Test particles with microscopic stateu, at the timet: fij = fij(t, u).

• Field particles with microscopic state,u∗ at the timet: fij = fij(t, u∗).

• Candidate particles with microscopic state,u∗ at the timet: fij = fij(t, u∗).

Rule:

i) Thecandidate particle interacts withfield particles and acquires, in probability,

the state of thetest particle.Test particles interact with field particles and lose their

state.

ii) Interactions can: modify the microscopic state of particles; generate proliferation or

destruction of particles in their microscopic state; and can also generate a particle in a

new functional subsystem.

Loss of determinism: Modeling of systems of the inert matter is developed

within the framework of deterministic causality principles, which does not any longer

holds in the case of the living matter.




Interactions within the action domain

C ΩT

F F

F

F

F

Figure 1: – Active particles interact with other particles in their action domain




Interactions with modification of activity and transition Generation of particles into

a new functional subsystem occurs through pathways. Different paths can be chosen

according to the dynamics at the lower scale.

F F F

F C FF

T

T

FF FF FF T

i + 1

i

i − 1

Figure 2: – Active particles during proliferation move fromone functional subsystem

to the other through pathways.



Cooperative/Competitive Games

Cooperative behaviorThe active particle with stateh < k improves its state by

gaining from the particlek , which cooperates loosing part of its state.

Competitive behavior The active particle with stateh < k decreases its state by

contributing to the particlek, which due to competition increases part of its state.


Lecture 3. - Common Features of Complex Living Systems,and Mathematical Tools

Encounter rate ηij : ηij is supposed to decay with the distance between the

interacting active particles, which depends both on their state and on the distance of

their functional subsystems.

ηhk = η0hk ψhk(f) ψhk(f) = e

−αhk(1 + |u∗ − u∗|)(1 + ||fh − fk||) ,

whereη0hk andαhk are positive constants, and

||fh − fk||(t) =

∫

R+

|fh(t, u) − fk(t, u)| du.

Transition probability density: Bhk is supposed to depend on the state of the

interacting pairs and low order moments:

Bhk (u∗ → u|u∗, u∗

,Ep[fk](t)) , E

p(t) =

∫

R+

upfi(t, u) du ,

for p = 1, 2, ....

Proliferation rate The termµihk is supposed to depend on the state of the interacting

pairs and low order moments.



Mathematical Structures

The evolution equation follows from a balance equation for net flow of particles in the

elementary volume of the space of the microscopic state by transport and interactions:

∂tfi(t, u) + Fi(t) ∂ufi(t, u) = Ci[f ](t, u) + Pi[f ](t, u)

where

Ci[f ] =

n∑

j=1

η0ij

∫

Du×Du

ψij(f)Bij (u∗ → u|u∗, u∗

, f)

− fi(t, u)n∑

j=1

η0ij

∫

Du

ψij(f) fj(t, u∗) du∗

.

and

Pi[f ] =

n∑

h=1

n∑

k=1

η0hk

∫

Du

∫

Du

ψhk(f)µihk(u∗, u

∗

, f)fh(t, u∗)fk(t, u∗) du∗

.


Lecture 3. - Common Features of Complex Living Systems,and Mathematical Tools

Models with Space Structure

H.1. Candidate or test particles inx, interact with the field particles inx∗ ∈ Ω located

in the interaction domainΩ. Interactions are weighted by theinteraction rate

ηhk[ρ](x∗) supposed to depend on the local density in the position of thefield

particles.

H.2. The candidate particle modifies its state according to the term defined as follows:

Ahk(v∗ → v, u∗ → u|v∗,v∗, u∗, u

∗, f), which denotes the probability density that a

candidate particles of theh-subsystems with statev∗, u∗ reaches the statev, u after an

interaction with the field particlesk-subsystems with statev∗, u∗.

H.3. Candidate particle, inx, can proliferate, due to encounters with field particles in

x∗, with rateµihk, which denotes the proliferation rate into the functional subsystemi,

due the encounter of particles belonging the functional subsystemsh andk.

Destructive events can occur only within the same functional subsystem.



(∂t + v · ∂x) fi(t,x,v, u) =

[ n∑

j=1

Cij [f ] +

n∑

h=1

n∑

k=1

Sihk[f ]

](t,x,v, u) ,

where

Cij =

∫

Ω×D2u×D2

v

ηij [ρ](x∗)Ahk(v∗ → v, u∗ → u|v∗,v

∗

, u∗, u∗

, f)

× fi(t,x,v∗, u∗)fj(t,x∗

,v∗

, u∗) dv∗ dv

∗

du∗ du∗

dx∗

,

− fi(t,x,v)

∫

Ω×Du×Dv

ηij [ρ](x∗) fj(t,x

∗

,v∗

, u∗) dv∗

du∗

dx∗

Sihk =

∫

Ω×D2u×Dv

ηhk[ρ](x∗)µihk(u∗, u

∗|f)

× fh(t,x,v, u∗)fk(t,x∗

,v∗

, u∗) dv∗

du∗ du∗

dx∗

.



Conjecture 1. Interactions modify the activity variable according to topological

stochastic games, however independently on the distribution of the velocity variable,

while modification of the velocity of the interacting particles depends also on the

activity variable.

Ahk(·) = Bhk(u∗ → u, |u∗, u∗) × Chk(v∗ → v |v∗,v

∗

, u∗, u∗) ,

whereA, B, andC are, for positive definedf , probability densities.

Conjecture 2. Stochastic velocity perturbation

(∂t + v · ∇x

)fi = νi Li[fi] + ηi Ci[f ] + ηi µi Si[f ],

where

Li[fi] =

∫

Dv

(Ti(v

∗ → v)fi(t,x,v∗

, u) − Ti(v → v∗)fi(t,x,v, u)

)dv

∗

,

and whereTi(v∗ → v) is, for theith subsystem, the probability kernel for the new

velocityv ∈ Dv assuming that the previous velocity wasv∗.


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