Point: 10 Scientific Computing

by Joa on epischisto.org

• Scientific computing (or computational science) is the field of study concerned to the construction of mathematical models and techniques of numerical solutions using computers to analyze and solve scientific and engineering problems.

• Typically, such models require a large amount of calculation, and usually run on computers with great power scalability (parallel and distributed machines)

• Scientific computing is currently regarded as a third way for science complementing experimentation (observation) and theory.

http://www.springer.com/mathematics/computational+science+%26+engineering/journal/10915

an investigation for 2014-2017…

• Computational Epidemiology of Malaria by Cellular Automata and Stochastic Integer Linear programming

Why malaria again?

OMS Milenium goals

P. falciparum P. vivax

Predominance -----

the know-how...

www.pmt.es www.upc.edu

epischisto.org

THE staff www.ufrpe.br

www.cesar.org.br

www.fiocruz.br

www.ines.org.br

Fundamentals

• Epidemiology is the study of the distribution and determinants of health-related states or events (including disease), and the application of this study to the control of diseases and other health problems.

http://jech.bmj.com/

• The Tipping Point, Epidemics are a function of

the people who transmit infectious agents, the infectious agent itself, and the environment in which the infectious agent is operating. And when an epidemic tips, when it is jolted out of equilibrium, it tips because something has happened, some change has occurred in one (or two or three) of those areas.

Fundamentals

An inflection point is a point on a curve at which the sign of

the curvature (i.e., the concavity) changes. Inflection points may

be stationary points, but are not local maxima or local minima.

The first derivative test can sometimes distinguish inflection points

from extrema for differentiable functions

The second derivative test is also useful. A necessary condition

for to be an inflection point is

A sufficient condition requires and to have opposite signs in

the neighborhood of (Bronshtein and Semendyayev 2004, p. 231).

Fundamentals – Mathematical Epidemiology

Fundamentals – how to solve these Differential Equation systems? Some are unsolved in analytic form, but in numerical one…

Adams-Bashforth-Moulton Method

Adams' Method

Collocation Method

Courant-Friedrichs-Lewy Condition

Euler Backward Method

Euler Forward Method

Galerkin Method

Gauss-Jackson Method

Gill's Method

Isocline

Kaps-Rentrop Methods

Milne's Method

Predictor-Corrector Methods

Relaxation Methods

RK2

RK4

Rosenbrock Methods

Runge-Kutta Method

and for neglected diseases?

with sparse

data

Aitken Interpolation Chebyshev Approximatio... Moving Median

B-Spline Cubic Spline Muller's Method

Berlekamp-Massey Algor... Gauss's Interpolation... Neville's Algorithm

Bernstein-Bézier Curve Hermite's Interpolatin... Newton's Divided Diffe...

Bézier Curve Internal Knot NURBS Curve

Bézier Spline Interpolation NURBS Surface

Bezigon Lagrange Interpolating... Richardson Extrapolation

Bicubic Spline Lagrange Interpolation Spline

Bulirsch-Stoer Algorithm Lagrangian Coefficient Thiele's Interpolation...

C-Determinant Lebesgue Constants Thin Plate Spline

Cardinal Function Moving Average

Solving sparse systems

Interpolation?

and some hidden scenarios…

a cellular automaton Cellular automaton A is a 4-upla A = <G, Z, N, f>,

where • G – set of cells • Z – set of possible cells states • N – set, which describes cells neighborhood • f – transition function, rules of the automaton:

– Z|N|+1Z (for automaton, which has cells “with memory”)

– Z|N|Z (for automaton, which has “memoryless” cells)

Statistical mechanics of cellular automata Rev. Mod. Phys. 55, 601 – Published 1 July 1983

Simple initial conditions: Homogeneous states or Self-similar patterns Random initial conditions:

Self-organization phenomena

Moore Neighbourhood (in grey) of the cell marked with a dot in a 2D square grid

Disease cycle: http://www.wellcome.ac.uk/en/labnotes5/animation_popups/schisto.html (animation 5´24´´)

http://wwwnc.cdc.gov/travel/images/map3-14-distribution-schistosomiasis.jpg

Research? for example, with schistosomiasis... we would like to provide an almost real-time and future risk map for it... by monitoring the self-organization endemics states...

no more deaths... Mar, 27th. 2009

a REALLY neglected disease in Brazil...

No data No case reports No statistical series No reliable data Only poor comunities Fiocruz (Schistosomiasis Laboratory) works to discover, to control and to report Fiocruz starts a new study in 2006...

http://200.17.137.109:8081/xiscanoe/infra-estrutura

2006 starts a new monitoring

Praia Carne de Vaca

Praia Enseada dos Golfinhos Praia do Forte

Praia Pau Amarelo

Praia do Janga

Lagoa do Náutico

Praia Porto de Galinhas

BRAZIL

why Carne de Vaca? Tourism interest Isolated population Identified cases Not analysed yet FIOCRUZ starts a new study Near from UFRPE Local support: politicians, population

The village comprises around 1600 people in 1041 households distributed in 70 blocks and covering approximately 4 km2.

2006 – 2007, data collect in-loco

2006 – 2007, data collect in-loco

http://200.17.137.109:8081/xiscanoe/infra-estrutura/expedicoes

Figure 1.

Adjusted Prelavence

0to 10 (3)10to

20 (32)

20to 30 (11)30to 50 (3)

Stream

Prevalence per 100 hab

0 to 1 (15)1 20 (17)

20 60 (14)60 80 (2)80 100 (1)

Breeding sites

to

to

to

to

water-collecting tank

Riacho Doce

1a. Prevalence 1b. Adjusted Prevalence

Male Female Total

Age group Pop1 Posit

2 Prev

3 Pop Posit Prev Pop Posit Prev

up to 9 99 7 7.1 100 3 3.0 199 10 5.0

10 to 19 109 26 23.9 99 24 24.2 208 50 24.0

20 to 29 76 31 40.8 90 21 23.3 166 52 31.3

30 to 39 88 18 20.5 103 23 22.3 191 41 21.5

>= 40* 141 14 9.9 168 18 10.7 310 32 10.3

unreported 16 3 18.8 10 2 20.0 26 5 19.2

Total 529 99 18.71 570 91 15.96 1100 190 17.3

* No information on sex for one individual. 1 population. 2 Number of positives. 3 Prevalence

per 100 inhabitants.

Spatial pattern, water use and risk levels associated with the transmission of schistosomiasis on the north coast of Pernambuco, Brazil. Cad. Saúde Pública vol.26 no.5 Rio de Janeiro May 2010.

http://dx.doi.org/10.1590/S0102-311X2010000500023

2008 – 2009, data analysis and reports... Parasitological exams on 1100 residents

2008 and 2009 data analysis and reports... Summary data for molluscs collected...

Ecological aspects and malacological survey to identification of transmission risk' sites for schistosomiasis in Pernambuco North Coast, Brazil. Iheringia, Sér. Zool. 2010, vol.100, n.1, pp. 19-24.

http://dx.doi.org/10.1590/S0073-47212010000100003

Collecting

Sites

Alive Dead Positive to

S. mansoni

% de

infection

I 0 0

II 1707 129 4 0,23

III 297 198 0 0

IV 0 0

V 0 0

VI 0 0

VII 2355 322 37 1,57

VIII 76 125 3 3,95

IX 0 0

Total 4435 774 44 0,99

2009-2010, modelling with 15 real parameters (?)

Paremeter Ranges (avg) How were obtained?

Susceptible human population 0-23 social inquires (Paredes et al, 2010)

Infected human population 0-23 croposcological inquires (Paredes et al, 2010)

Recovered population of humans 0-23 social inquires (Paredes et al, 2010)

Rate of mobility of humans 0-26% social inquires (Paredes et al, 2010)

Rate of mobility of molluscs 0-2% malacological research (Souza et al, 2010)

Population of healthy molluscs 0-1302 malacological research (Souza et al, 2010)

Population of infected molluscs 0-11 malacological research (Souza et al, 2010)

Area susceptible to flooding 0-45%

LAMEPE - Meteorological Laboratory of Pernambuco (lamepe, 2008)

and environmental inquires (Souza et al, 2010)

Connection to other cells 0-100%

LAMEPE - Meteorological Laboratory of Pernambuco (lamepe, 2008)

and environmental inquires (Souza et al, 2010)

Rate of human infection 0-100% croposcological inquires and social inquires (Paredes et al, 2010)

Rate of human re-infection 0-100% croposcological inquires and social inquires (Paredes et al, 2010)

Recovery rate 0-100% croposcological inquires and social inquires (Paredes et al, 2010)

Mollusc infection rate 0-100% malacological research (Souza et al, 2010)

Rate of sanitation 0-93% social and environmental inquires (Souza et al, 2010)

Rainfall of the area 39-389mm LAMEPE - Meteorological Laboratory of Pernambuco (Lamepe, 2008)

From one year (population 1 snapshot, molluscs 12 snapshots) without previous historical...

Mechanistic epidemic models

Two alternative approaches

Top-down Population-based Models (PbMs)

Bottom-up Agent-based Models (AbMs)

PbM AbM

one proposal: a top-down approach using a cellular automaton

a b

1 km

a ba b

1 km

simulation space, a 10x10 square grid

the dynamics

Mollusk population dynamics a growth model for the number of individuals (N) that

considers the intrinsic growth rate (r) and the maximum

sustainable yield or carrying capacity (C) defined at each

site (Verhulst, 1838):

)1(C

NrN

dt

dN

Human infection dynamics (SIR - SI)

This model splits the human population into three compartments: S (for susceptible), I (for infectious) and R (for recovered and not susceptible to infection) and the snail population into

two compartments: MS (for susceptible mollusk) and MI (for infectious mollusk).

Socioeconomic and environmental factors

environmental quality of the nine collection sites in Carne de Vaca, according to the criteria of Callisto et al (Souza et al, 2010).

rteN

NC

CtN

0

01

)(

the model calculates the local increase of population using equation 1 and calculating N(t+1) out from N(t). The values for r and C are set at each site and each time step, using monthly meteorological inputs and considering the ecological quality of the habitat

(1)

αRχI=dt

dR

χI·S·Mp=dt

dI

αR+p·S·M=dt

dS

IH

I

ISMI

SSMS

rM·I·Mp=dt

dM

rM·I·Mp=dt

dM

(3a)

(3b)

Cells and infection forces

states black: rate of human infection = 100%; red: 80% ≤ rate of human infection < 100%; light red: 60% ≤ rate of human infection < 80%; yellow: 40% ≤ rate of human infection < 60%; light yellow: 20% ≤ rate of human infection < 40%; cyan: 0% ≤ rate of human infection < 20%.

Infection forces Human

S -> I (infected molluscs contact, pH)

I -> R (if treated (1-α), χ) Molluscs

S -> I (infected human contact, pM)

the algorithm

1. Choose a cell in the world;

2. For each human in the cell perform a random walk weighted by the “probability of movement" defined

at each site.

Repeat these steps for every cell in the world. Then update data.

3. Choose a cell in the world;

4. Call the “Events” process;

5. Return the individual to his original cell after the infection phase;

6. Choose a cell in the world;

7. For the mollusk population in that cell, perform a diffusion process weighted by the “rate of movement"

defined at each site;

Repeat these steps for every cell in the world. Then update data.

1. Increase the population of mollusks using the growth model described in Section 3.1;

2. Compute the transition between population compartments of humans using the set of equations (3b)

defined in Section 3.2;

3. Compute the transition between population compartments of humans using the set of equations (3a)

defined in Section 3.2;

Update local data of the spatial cell.

Events process

Main

sumulations

Mathematica 7.0 (Mathematica, 2011) with a processor Intel i5 3GHz, 4MB Cache, 8GB RAM.

Computational costs of a complete simulation when assuming a fixed world size (10x10 cells) and extent (365 time steps) and an increasing number of parameters being swept for rejection sampling (from 1 to 15)

Computational vs Statistical models Day 26 Day 43 Day 88

Day 106 Day 132 Day 365Color Legend

I = 100%80% ≤ I < 100%

60% ≤ I < 80%

40% ≤ I < 60%

20% ≤ I < 40%

0% ≤ I < 20%

(I = percentage of

infected humans)

Temporal

evolution

Day 26Day 26 Day 43Day 43 Day 88Day 88

Day 106Day 106 Day 132Day 132 Day 365Day 365Color Legend

I = 100%80% ≤ I < 100%

60% ≤ I < 80%

40% ≤ I < 60%

20% ≤ I < 40%

0% ≤ I < 20%

(I = percentage of

infected humans)

Temporal

evolution “according to the risk indicator, in the scattering diagram of Moran represented in the Box Map (Figure 2), indicated 18 areas of highest risk for the schistosomiasis, all located in the central sector of the village. Areas with lower risk and areas of intermediate risk for occurrence of the disease were located in the north and central portions with some irregularity in the distribution”

Predictive scenarios

2012 2017 2022 2027Color legend

I = 100%

80% ≤ I < 100%60% ≤ I < 80%

40% ≤ I < 60%

20% ≤ I < 40%

0% ≤ I < 20%

Predictive scenarios generated with the parameter calibration of the year 2007 that show endemic schistosomiasis. I stands for the average percentage of infected humans per spatial cell predicted by the model

STATE OF ART – some of our production on it

• http://www.systems-journal.eu/article/view/172

• http://dl.acm.org/citation.cfm?id=2488022&dl=ACM&coll=DL#

• http://www.ij-healthgeographics.com/content/11/1/51

• http://dx.doi.org/10.1590/S0102-311X2013000200022

Consequences…

INNOVATION on collecting DATA: automatic proposal for diagnosis of schistosomiasis, malaria... (patent in progress)

SEE PROJECTS http://200.17.137.109:8081/xiscanoe/projeto/graduate-projects

some world initiatives on automatic diagnosis… • http://www.fastcoexist.com/3026100/fund-this/a-handheld-device-

that-can-diagnose-diseases-and-drug-resistance-in-15-minutes • http://www.jove.com/blog/2012/05/04/crowd-sourcing-for-

malaria-game-on • http://g1.globo.com/jornal-nacional/noticia/2013/01/empresa-de-

israel-cria-celular-que-faz-exames-medicos.html • http://lifelensproject.com/blog/

but no one on mobile simulation of cellular automata for disease spreading…

INNOVATION on collecting DATA: an integrated plataform www.ankos.com.br

http://ankos.sourceforge.net/

INNOVATION on collecting DATA: an integrated plataform SchistoTrack (patent in progress)

We are Health Map in PE-Brazil! http://healthmap.org/

INNOVATION on mathematical and computational simulation using Cellular Automata

What are we doing now? Running

simulations on Mobile platforms and these

simulations will guide the data collect

the codes of the humans, 2013

by Conway, Cellular Automata are “not just a game”, 1970

by epischisto.org , Schistosomiasis by mobile phones and social machines and simulators based on Cellular Automata, 2011

So, what is the problem for now? an investigation for 2014-2017…

• Computational Epidemiology of Malaria by Cellular Automata and Stochastic Integer Linear programming

Fundamentals

• To find the possible scenarios that match inflection points as optimal conditions for epidemic trends…

• a NP-Complete Problem and polinomial reductions to SILP is possible and… how to solve it?

Stochastic Integer Linear Programming

Sparse points captured by stochastic scenarios and inflexion points by statistical noises

@work…

• a PhD Thesis on this direction (feb, 2014): Statistical confidence of Cellular Automata rules for Schistosomiasis by Genetic Algorithms (PPGBIO-UFRPE)

• IFORS 2014 for solving the SILP by an old approximation…

• Interior-point nethods for solve it? A giant deterministic one by relaxation… maybe…

• What else? – Contact us www.epischisto.org

Thanks! We have several other projects by other researchers... www.epischisto.org